KINETIC ISOTOPE EFFECTS IN
UNIMOLECULARDECOMPOSffiONS
OF METASTABLE IONS
by COLIN EVAN ALLISON, B.Sc. (Hons.)
This thesis is submitted in fulfilment of the requirements for the degree of Doctor of Philosophy
DEPARTMENT OF PHYSICAL CHEMISTRY
SCHOOL OF CHEMISTRY
UNIVERSITY OF NEW SOUTH WALES
AUGUST 1986 (ii)
DECLARATION
I, COLIN EVAN ALLISON, hereby declare that this submission is my own work and that, to the best of my knowledge and belief, it contains no material previously published or written by another person nor material which to a substantial extent has been accepted for the award of any other degree or diploma of a university or other institute of higher learning, except where due acknowledgement is made in the text.
COLIN EVAN ALLISON (iii)
ACKNOWLEDGEMENTS
I would like to express my gratitude to Prof. Peter Derrick for his supervision and guidance throughout the course of this work. I would also like to thank Dr. Gary
Willett for many helpful discussions, and for allowing me access to the molecular orbital programs.
I am grateful to Dr. Steen Hammerum, Prof. John Bowie and Prof. Fred McLafferty for their generous donations of chemicals. I would also like to thank Dr. Steen
Hammerum for many stimulating discussions.
I would like to extend my thanks to Dr. Kevin Donchi and Mr. Peter Cullis for their assistance with the experimentation and for their companionship. I would also like to thank Dr. Anthony Craig and Dr. Bruce Rumpf for their companionship during the course of my studies. (iv)
ABSTRACT
Isotope effects on ion abundances have been measured and used as a probe of reaction mechanisms for unimolecular ion decompositions. Quasi-equilibrium theory (QET) calculations have been performed to obtain rate coefficients k(E) which have been used within various models for reaction to calculate the isotope effect on ion abundances.
For a prototype simple cleavage, elimination of butyl radical from 2H labelled N-methyl dipentylamine molecular ions, the model of the transition state was varied until the calculations reproduced the experimental isotope effects at 10-11 s and 10-5 s. The transition state model indicated relaxation of the pentyl chain accompanies cleavage.
A more complex reaction, the McLafferty rearrangement, was studied in 3-ethyl-
2-pentanone using 2H and 13c labelling, in benzyl ethyl ether using 2H labelling, and in cx.,cx.'-diethoxy p-xylene using 2H and 180 labelling. Experimental ion abundances indicate that multi-step processes are occurring in 3-ethyl-2-pentanone, but the experimental ion abundances for benzyl ethyl ether and cx.,cx.'-diethoxy p-xylene do not indicate whether a stepwise or concerted mechanism is operating. QET calculations were performed, and reaction models constructed, to determine whether the McLafferty rearrangement in benzyl ethyl ether and cx.,cx.'-diethoxy p-xylene might best be described
as a concerted or stepwise process. Calculations indicated that the model used for the
concerted mechanism reproduced accurately the experimental ion abundances, but that
the model used for the stepwise mechanism failed to accommodate the experimental
results. The McLafferty rearrangement in benzyl ethyl ether and in cx.,cx.'-diethoxy
p-xylene is proposed to occur in a concerted rather than a stepwise fashion in apparent
disagreement with other systems for which a stepwise mechanism is appropriate. (v)
TABLE OF CONTENTS
Declaration (ii) Acknowledgements (iii) Abstract (iv) Table of contents (v)
Chapter 1. Introduction. 1
1. 1. Stable ion structures 1 1.1.1. Ion thermochemistry 2 1.1.2. Experimental methods of determining stable ion structures 4 1.2. Unimolecular ion decompositions 4 1.2.1. Mechanisms of unimolecular ion decompositions 4 1.2.2. Theory of mass spectra 7 1.2.2.1. Relationship between k(E) and ion abundances 8 1.2.3. Kinetic isotope effects 11 1. 2. 3. 1. Relationship between kinetic isotope effects and isotope effects on ion abundances 12 1. 2. 3. 2. Significance of kinetic isotope effects 14 1.3. Metastable ions 17 1.3.1. Mass-analysed ion kinetic energy (MIKE) spectrometry 17 1. 3. 2. Energy release in metastable ion decompositions 19 1. 3. 3. Field ionisation kinetics 20
Chapter 2. Calculation of ion abundances. 22
2.1. Calculation of k(E) 22 2.1.1. Evaluation of W(E) 23 2.1.2. Choice of energy levels 25 2. 2. Energy deposition function, P(E) 27
2.3. Observation times, t1 and 12 28 2.4. Collection efficiency, G 29 2.5. Correlation of calculated and measured ion abundances 30 2.5.1. Adj_ustment of the transition state 31 2.6. Computer programs 33 2.6.1. Program QET 33 2.6.2. Program CFT3 34 (vi)
2. 6. 3. Program PEAKS 35 2.6.4. Adjustment of vibrational frequencies 36 2.6.5. Othercomputerprograms 36
Chapter 3. Instrumental. 37
3.1. History of the instrument 37 3.2. Vacuum system 37 3.3. Ion beam defining slits 38 3.4. The ion source 39 3.5. Magnetic sector 40 3.6. Electric sector 41 3.7. Field-free regions 42 3.8. Detector system 43 3.9. Data acquisition 43
Chapter 4. Alpha-cleavage of N-methyl-dipentylamine molecular ion. 45
4.1. Introduction 45 4. 2. Experimental results 45 4.3. Calculation of ion abundances 46 4.3.1. The kinetic model 46 4.3.2. Reactant ion vibrational frequencies 47 4.3.3. Transition state vibrational frequencies 47 4. 3 .4. Critical energy 48 4.3.5. Optimisation of the transition state 49 4.4. Results of the calculations 50 4.5. Discussion of the transition state model 53
Chapter 5. Investigations of the McLafferty rearrangement. 55
5 .1. Introduction 55 5.2. Evidence for the stepwise nature of the McLafferty rearrangement 56 5. 3. 3-Ethyl-2-pentanone 58 5.4. The McLafferty rearrangement in some substituted benzenes 62 5.5. Benzyl ethyl ether and a.,a.'-diethoxy p-xylene 65 (vii)
5.5.1 Benzyl ethyl ether 66 5.5.2 cx,cx'-Diethoxy p-xylene 68 5.6. Energetics 71 5. 7. Concerted elimination of acetaldehyde from cx,cx'-diethoxy p-xylene 72 5. 7. 1. Modelling of the transition state 73 5.7.2. Results 74 5.8. Hydrogen exchange in the.benzyl 1,1-Drethyl ether radical catio 77 5. 8. 1. The kinetic model for hydrogen exchange 79 5.8.2. Ion structures and energetics 81 5.8.3. Vibrational frequencies and critical energies 83 5. 8. 4. Details of the calculations 86 5.8.5. Results obtained using the thermochemical energetics 87 5.8.6. Results obtained using the MOPAC energetics 88 5. 8. 7. Discussion 90 5. 9. Stepwise elimination of acetaldehyde from cx,cx'-diethoxy p-xylene 91 5. 9. 1. The kinetic model 91 5. 9. 2. Ion structures and energetics 93 5.9.3. Vibrational frequencies and critical energies 94 5. 9 .4. Details of the calculations 97 5.9.5. Results of the calculations based on the thermochemical energetics 99 5.9.6. Results of the calculations based on the MOPAC energetics 101 5. 9. 7. Discussion of the stepwise models 103 5 .10. Conclusion 104
REFERENCES 106
Appendix 1 Computer program QET Al.I Appendix 2 Calculation of metastable ion abundances A2.l Appendix 3 Adjustment of the vibrational frequencies for the N-methyl-dipentylamine molecular ion A3.1 Appendix 4 Estimation of heats of formation for neutral species using Bensons' rules A4.1 Appendix 5 Optimised ion geometries A5.l Appendix 6 Solution of kinetic schemes using Laplace transformation A6.1 Appendix 7 Computer programs CALKT and NKT A7.l Appendix 8 Estimation of heats of formation for distonic ions A8.l 1
Chapter 1. Introduction.
The dynamics of unimolecular decompositions constitute one of the most fascinating aspects of mass spectrometry. Reliable structures have been determined for many gaseous ions, and provide starting points for discussions of reaction dynamics.
Mechanisms describing which atoms are involved have been postulated for many reactions. To elucidate further the mechanisms with regard to the sequence and timing of the atomic movements is more challenging. No reaction mechanism can be proven to be correct (1), but a proposed mechanism can be demonstrated to be consistent with all available data. In the work described here, kinetic isotope effects, or more specifically isotope effects on ion abundances, have been studied with a view to exploring their value for distinguishing concerted from non-concerted, or step-wise, processes.
Particular attention has been paid to the McLafferty rearrangement in benzyl ethyl ether.
1.1. Stable ion structures.
The term "ion structure" as used in this thesis refers to the question of which atoms are
directly connected to which others, rather than to the geometrical relationships among
atoms. Geometric data from experiment are becoming available for small polyatomic
cations [2,3), but most geometric data have been obtained from molecular orbital
calculations [4]. The elucidation of ion structures is important as many molecular ions
isomerise prior to reaction. The [C4H8]+. species exemplify the problem [5-13).
The [C4H8]+. ions formed from 1-butene, cis- and trans- 2-butene, methyl propene, methyl cyclopropane and cyclobutane all produce similar electron impact (EI) mass
spectra [5-10). Presumably the initially generated molecular ions are quite distinct.
Decomposition of the [C4H8]+. ions has been proposed to involve common species (or 2
mixture of species). At short times (10-12 s) decomposition is believed to occur from the molecular ions [11-13], but at longer times decomposition is believed to occur from the common species formed by isomerisation of the molecular ions.
The possibility of isomerisation of molecular ions indicates that it is essential that the structures of ions be known, and hence methods of determining stable ion structures are required. At the same time, it is essential that the thermochemistry pertaining to stable ion structures be characterised.
1.1.1. Ion thermochemistry.
The thermochemical data required in the case of an ionic decomposition are heats of formation (Af-If) for all ionic and neutral species involved in the reaction. These data can be obtained from measurements of the ionisation energies (IE's) and appearance energies (AE's) for ions, together with tabulated [14] or calculated [15] values of .1Hfs
for neutral species. Consider the ionisation and subsequent decomposition of a
molecule AB;
AB - e ➔ [AB]+. ➔ [A]+. + B (1.1)
The thermochemical relationships are illustrated in Fig. 1. 1. If the excess energy
Eexcess is zero (vide infra ), then from known heats of formation for the neutral
molecules AB and B, the heats of formation of the species [AB]+. and [A]+. can be
estimated from IE(AB) and AE([A]+·). Heats of formation are characteristic of an ion
structure.
Experimental determinations of heats of formation of ions rely upon experiments using
energy-controlled ionisation sources, such as photo-ionisation (PI) mass spectrometers Fig. 1.1. Thermochemical relationships for the reaction sequence :
AB - e ➔ [AB)+. ➔ [At· + B
AE([Af·)
IE(AB)
I + Eexcess
Fig. 1.2. Reaction profile for the decomposition :
t AE-IE f Eexcess
[AB]+. [A]+• + B 3
[ 16] or mass spectrometers equipped with mono-energetic electron-ionisation sources
[17]. Values of IE's, AE's and heats of formation for many ions are available in compiled tables [ 18, 19]. Heats of formation for odd-electron ions for which no experimental measurements are available may be estimated using a simple relationship based on functional groups and the size of the molecule [20-23].
Heats of formation may also be calculated for neutral and ionic species using molecular orbital (MO) calculations [4]. MO calculations optimise a molecular geometry by minimising the electronic potential energy. Such calculations are sometimes lengthy.
Estimates of heats of formation are not often used to predict ion structures, but are used to assist in the determination of ion structures. It is conceivable for more than one stable ion structure to have a given value for the heat of formation; a given stable ion structure can only have one value for its heat of formation.
Having obtained the AE for a product ion and the IE for the precursor neutral, an energy barrier to reaction can be determined. This energy barrier is the amount of energy required for the reaction to proceed and for the products to be observed, under the experimental conditions appropriate to the determination of the AE. Consider the reaction profile shown in Fig. 1.2. The energy barrier given by AE([A]+·) - IE(AB) is not the minimum energy required for the reaction to occur, designated as E0. The ener_gy E+ is the energy in excess of the minimum energy E0• E+ is known as the non-fixed energy [24]. The excess energy Eexcess comprises the non-fixed energy E+ and the energy required for the reverse reaction to occur, E0r . The energies E0 and E0r are known as the critical energy [25] and the reverse critical energy respectively, and will be discussed further in Section 1.2.
When products are formed, Eexcess is partitioned amongst the various electronic, 4
rotational, vibrational and translational modes. Often only partitioning amongst the vibrational and translational modes need be considered. Partitioning of Eexcess can be distinguished as either partitioning of E+, which appears to be statistical, or partitioning of E0r, which is determined by the nature of the reaction [24]. Excess energy partitioned into translation should be characteristic of the reaction mechanism and thus dependent upon the ion structure. Energy partitioned into the relative translation of products can be measured. Methods of measuring the translational energy release are discussed in Section 1.3.2.
1.1.2. Experimental methods of determining stable ion structures.
A common approach to determination of stable ion structures, and the one used in this
study, is collisionally induced decompostion (CID) [26,27]. Other methods involve
studying the ion-molecule reactivity of an ion with various neutral species [28].
The CID experiment involves the interaction of a mass-selected ion with keV
translational energy and a neutral, usually chemically unreactive species with thermal
energy. The collision between these two species results in the uptake of an amount of
internal energy by the ionic species which leads to decomposition of the "stable" ion.
The decomposition tends to reflect directly the structure of the stable ion. The CID
spectrum should be largely independent of the instrument used, and is sometimes referred to as a "fingerprint" of the ion. Comparison of the fingerprints of known and
unknown ion structures allows ion structures to be determined.
1.2. Unimolecular ion decompositions.
1.2.1. Mechanisms of unimolecular ion decompositions. 5
Assignment of a structure to a decomposing ion and the observation of specific products may permit a reaction mechanism to be postulated, however more information concerning the products and the specific atoms involved is generally required. Isotopic labelling is frequently used to identify specific atoms involved in a reaction [29], and when used in conjunction with molecular orbital calculations to assign structures to various species [30,31], provides a useful probe of the reaction mechanism. Given that the individual atoms involved in the reaction have been identified, and ion structures identified, a reaction mechanism may be postulated with more confidence.
Reactions which involve the rupture of only one bond are usually considered to occur in
a single kinetic step, i.e. to be an elementary reaction [32]. Reactions which involve
more than one bond (multibond reactions) may occur in either one kinetic step or in a
sequence of kinetically distinct steps. In this thesis, multibond reactions which occur in
a single kinetic step will be termed concerted, and reactions which occur by a sequence
of kinetically distinct steps will be termed non-concerted, or step-wise.
Multibond reactions which occur in a concerted fashion can be described as
synchronous if all processes involved have occurred to similar extents in the transition state for the reaction, and as non-synchronous if they have not. Recently, Dewar [33]
has concluded that multibond reactions will not normally occur as synchronous
processes due to an expectation that high energy barriers will normally be present. Dewar suggests that multibond reactions will normally be either non-synchronous
concerted reactions, or step-wise reactions [33].
The distinguishing characteristic between a synchronous concerted reaction and a step-wise reaction, is the presence of one or more stable intermediate species. Such an
intermediate is depicted as a valley, or well, in the potential energy profile which would
be used to characterise the reaction. Fig. 1.3(a) shows the potential energy profile Fig. 1.3. Reaction profiles (potential energy diagrams) representing
(a) a synchronous concerted reaction, and (b) a step-wise reaction.
(a) Transition state
[A]+. + B
(b) Transition state 2 Transition state 1
Intermediate [AB] +. 6
which would be appropriate to characterise a synchronous reaction. A single transition state is kinetically significant. Fig l.3(b) shows the potential energy profile which would be appropriate to characterise a step-wise reaction. A stable intermediate exists, and two kinetically significant transition states are encountered in the reaction.
A non-synchronous concerted reaction could be described by either of the potential energy profiles in Fig 1.3, however, if the potential energy profile of Fig l.3(b) were appropriate, only one of the transition states would be kinetically significant. As reactions are often represented by potential energy profiles similar to those in Fig 1.3, the question of concerted or step-wise behaviour is important. The only conclusion concerning the reaction mechanism which could be obtained from the potential energy profile shown in Fig 1.3(a) is that the reaction is concerted, i.e. a single kinetic step is
involved. The reaction may or may not be synchronous. A potential energy profile
similar to Fig l.3(b) indicates that the reaction is non-synchronous, but the reaction
may be either concerted or step-wise.
The potential energy profiles in Fig 1.3 allow the relationship between the measured
energy barrier to reaction, i.e. AE - IE, and the critical energy for reaction, E0, to be developed. For the purposes of this discussion it will be assumed that there is no
excess energy. If a reaction is described by the potential energy profile in Fig 1.3(a),
the measured energy barrier will equal the critical energy E0. If the reaction is described by the potential energy profile shown in Fig l.3(b), the measured energy barrier gives
the difference in energy between the reactant ion and the highest energy transition state.
In the case of a non-synchronous concerted reaction, the energy barrier will represent
E0, as the reaction is a single kinetic step and only the higher energy transition state will affect the rate of reaction. Note that in the quasi-equilibrium theory of mass spectra,
described in Section 1.2.2, the shape of the potential energy profile is not important.
Only the relative energies of the reactant and transition state matter. 7
In the case of a step-wise reaction, the energy barrier will provide an upper limit to E0 for the first step, but no other information. If the heat of formation of the intermediate is known, the energy barrier can provide limits for the E0's for the first step and the second step. As the intermediate is a stable species, reaction of the intermediate may
lead back to the reactant Thus a limit for E0r for the first step is also obtained from the energy barrier, if the heat of formation of for the intermediate is known.
1.2.2. Theory of mass spectra.
The most widely accepted theory of unimolecular reactions in mass spectrometry is the
quasi-equilibrium theory (QET) developed by Rosenstock et al. (34,35]. The theory is
based upon earlier theories of unirnolecular reactions proposed by Rice and Rarnsperger
[36,37] and Kassel [38-40], which developed in accord with transition-state theory
[41]. The development of QET ran parallel to the development of a similar theory by
Marcus [42-44], which has come to be known as RRKM (Rice, Ramsperger, Kassel
and Marcus) theory. QET and RRKM theory produce identical expressions for the
microcanonical rate coefficient k(E). QET and RRKM theory have been described in
detail in the literature [28,45,46].
QET assumes that reaction depends only upon the internal energy of the reactant ion,
and that reaction is described by an energy-dependent rate coefficient k(E). QET gives
the following expression for k(E) at energy E [45,46]:
cr G*(E- E0) (1.2) k(E) - h N(E)
E0 is the critical energy for the reaction [25], h is Planck's constant, CJ is the symmetry 8
number or reaction degeneracy, G*( E-E0 ) is the number of internal energy states of the transition state in the range Eo to E, and N(E) is the density of internal energy states of the reactant at energy E. These quantities are discussed further in Chapter 2.
QET relies upon three assumptions which can be summarised as follows [35] :
( 1) Ionisation is rapid with respect to decomposition.
(2) Energy flow within the reactant ion is rapid compared to the rate of
decomposition, i.e. energy is randomised.
(3) Reaction may be described as motion along a "reaction coordinate" which is an
internal degree of freedom, separable from all other degrees of freedom, and
which passes through a critical configuration or transition state.
1.2.2.1. Relationship between k(E) and ion abundances.
Rate coefficients k(E) can usually not be obtained from measurements in mass
spectrometry. While techniques such as photoion-photoelectron coincidence (PIPECO)
[47] and charge exchange (CE) [48] may allow the direct evaluation of these rate
coefficients, the majority of mass spectrometric techniques do not. Typically it is ion
abundances which are measured, and, assuming randomisation of internal energy, these
are related to specific rate coefficients as follows [49,50].
If N0 is the number, per time, of molecular ions formed with a discrete amount of
internal energy E1, and there is only one available reaction channel, the rate of reaction is expressed as the rate of disappearance of the molecular ion:
dN -- =k(E1)N dt (1.3) 9
N is the number, per time, of undecomposed molecular ions with internal energy E1 remaining at time t. k(E1) is the rate coefficient for reaction of molecular ions with internal energy E1. The dimension of time alluded to as "per time" takes into account the fact that mass spectrometry is a flow experiment
If the molecular ion can undergo n parallel reactions, the rate of disappearance of the molecular ions becomes:
dN
dt i=l (1.4)
i=l i=l
The rate of appearance of a particular non-decomposing product ion mi is:
n dNm-1 ~(E1) N0 exp( - L ~(E1)t) (1.5) dt i=l
If the product ion mi can undergo decomposition, the kinetic scheme becomes more
complex. Subsequent decomposition of product ions will be neglected for simplicity.
Equation 1.4 holds for single-valued internal energies, however ions formed in a mass
spectrometer are usually formed with a range of internal energies. If the energy
randomisation assumption of QET holds, a normalised internal energy distribution P(E)
can be defined such that:
Emax (1.6) J P(E)dE = N0 0 10
Emax is the maximum internal energy that a molecular ion may possess. The rate of disappearance of molecular ions becomes:
E dN n max n L J ~(E) P(E) exp( - L ~(E)t)dE (1.7) dt i=l 0 i=l
and the rate of appearance of a particular product ion 11\ becomes:
dNm- Emax 1 J ~(E) P(E) exp( - ki(E)t)dE (1.8) dt 0
Measurement of the product ions relates to a finite time interval known as the
observation window .1.t, defined by the minimum and maximum observation times of
the experiment (t1 and½ respectively). The amount, NIDj, of product ion mi formed is obtained by integrating over the observation window:
l2 Emax n Nmi = J J ~(E) P(E) exp( - L ki(E)t)dEdt (1.9) i=l
t1 a~d ½_ are measured after ionisation, i.e. ionisation occurs at t = o.
Equation 1.8 describes the relationship between rate coefficient ~(E) and the number,
per time, of product ions, Nmi, formed within the observation window of the
experiment. All ions will, however, not normally be detected. The collection efficiency
Gmi , is defined as the proportion of the number, per time, of product ions 11\ formed
in the observation window of the experiment, which are detected. Incorporation of the 11
collection efficiency GJ'J¾ into equation 1.8 yields the detected number, per time, of product ions 1'I¾, i.e. the ion abundance lmi.
Im-1 =Gm-Nm- I l l2 Emax n ki(E) P(E) exp( - L ~(E)t)dEdt = Gmi J J (1.10) t 1 0 i=l
The detected number, per time, of undecomposed molecular ions IM is obtained by integrating from t = O to t = tD, where tD is the time taken for a molecular ion to be detected.
Emax (1.11) IM= GM J P(E) exp( - i ki(E)t0 )dE 0 i=l
Collection efficiencies and internal energy distributions are discussed further in Chapter 2.
1.2.3. Kinetic isotope effects.
Isotopic substitution can provide an indication of the degree of participation or kinetic
sign~ficance of a particular atom in an overall reaction through the effect on reaction
rate. Kinetic isotope effects in ionic unimolecular decompositions are essentially similar
to the isotope effects encountered in thermal systems [51].
The kinetic isotope effect is defined as the ratio of the rate coefficients k1(E)/k11(E),
where the numerator k1(E) is the rate coefficient for a specific reaction which
incorporates isotope I, and the denominator kn(E) is the rate coefficient for an identical 12
reaction which contains isotope II rather than isotope I. If isotope I were hydrogen (1H or H), for example, and isotope II were deuterium (2H or D). the kinetic isotope effect would be expressed as kH(E)/k0 (E). For other isotopic atoms, the numerator is by convention the rate coefficient for the reaction containing the light isotope and the denominator is the rate coefficient for the reaction containing the heavy isotope, e.g. k35c1(E)/k37c1CE).
As rate coefficients are not often measured directly in mass spectrometry (vide supra), the effect of isotopic substitution upon the reaction rate is usually determined from the ion abundances I, of product ions. For the case of hydrogen (deuterium) isotope effects mentioned above, the ion abundance measured for the light isotope H is expressed as IH and that for the heavy isotope Dis expressed as 10 . The isotope effect on ion abundances can then be written as IH / I0 , and in this thesis will also be termed the ion abundance ratio, or simply the isotope effect.
Isotope effects on ion abundances can be classified as either intermolecular or
intramolecular according to whether the isotope effect arises from the study of two
reactant ions (intermolecular) or a single reactant ion (intramolecular). Consider two
molecular ions, M1 and M2, where hydrogen atoms in M 1 have been replaced by
deuterium atoms in M2. If M1 undergoes reaction losing H2, and the identical reaction
of M2 results in the loss of D2, comparison of the ion abundances of [M1 - H2]+- and
[M2 - D2]+. gives an intermolecular isotope effect. If a single molecular ion M3 contains both hydrogen and deuterium atoms, and reaction generates either the products
H2 or D2, comparison of the ion abundances of [M3 - H2]+. and [M3 - D2]+. gives an intramolecular isotope effect.
1.2.3.1. Relationship between kinetic isotope effects and isotope effects on ion abundances. 13
Equation 1.9 can be used to develop the relationship between kinetic isotope effects and the isotope effect on ion abundances.
(a) The intermolecular case.
On the basis of equation 1.9, the ratio of ion abundances I1 / In obtained from the identical reaction of isotopomers M1 and Mu is:
½1 .Emax1 n1 l1 G1 f f k1(E) P1(E) exp( - L ~1(E)t)dEdt t11 0 i=l (1.12)
t2n Emaxll nu III Gn f f k11(E) Pu(E) exp( - L kill(E)t)dEdt trn 0 i=l
The collection efficiencies G1 and Gu, the internal energy distributions P1(E) and Pn(E),
the maximum internal energies Emaxl and EmaxU• and the observation times tu, trn, ½I and ½n are different for each of the reactant ions. The isotope effect on ion abundances
11/ In will contain not only the kinetic isotope effect k1(E) / ku(E) but also the different internal energy distributions [52], different collection efficiencies and the change in
observation window for the experiment. Kinetic isotope effects on competing reaction
channels may also contribute to the observed isotope effect on ion abundances. In
general, it is difficult to correlate kinetic isotope effects with intermolecular ion
abundances.
(b) The intramolecularcase.
In contrast to the intermolecular case, kinetic isotope effects are more easily comparable 14
with intramolecular ion abundances. As product ions are generated from identical reactions, the quantities in equation 1.12 which are dependent upon the reactant ion are common to both products. The intramolecular isotope effect on ion abundances is given by:
t2 Emax n l1 G1 J J k1(E) P(E) exp( - L ki(E)t)dEdt t1 0 i=l (1.13)
t2 Emax n III GnJ J ku(E) P(E) exp( - L ki(E)t)dEdt t1 0 i=l
1.2.3.2. Significance of kinetic isotope effects.
Kinetic isotope effects can be related to molecular properties through the QET
expression for the k(E). For the intermolecular kinetic isotope effect,
cr1 Gt(E - E0(1)) Nu(E) (1.14) <>n G1~(E - E0(1I) N 1(E)
The intermolecular kinetic isotope effect is dependent upon properties of both reactant
ions and both transition states.
For the intramolecular kinetic isotope effect, the reactant ion is common to both
reactions and the dependence upon the properties of the reactant ion disappears. The
intrarnolecular kinetic isotope effect is given by
(1.15) 15
The intramolecular kinetic isotope effect is dependent only upon the properties of the respective transition states. If symmetry factors are corrected for in the ion abundance ratio, the intramolecular kinetic isotope effect is expressed simply as the ratio of the number of states functions for the transition states.
Within the Born-Oppenheimer approximation [53], changes in the properties of a molecule upon isotopic substitution will be purely mass effects involving vibrational frequencies and moments of inertia, i.e. the potential energy surface describing the reaction is unaltered. The resultant changes in vibrational frequencies and moments of inertia are constrained by the Teller-Redlich product rule [54], which relates the changes in vibrational frequencies and moments of inertia to the mass changes in the molecule as:
P-1 n (1.16) k=l [:: l
N is the number of atoms in the molecules, P is the number of vibrational degrees of
freedom (3N-5 for a linear molecule and 3N-6 for a non-linear molecule), Mand M' are
the molecular masses of the "normal" and "substituted" molecules, Cl\ and Cl\' are the
individual atomic masses in Mand M', I3Iblc and I3 'Ib'Ic' are the products of the three
mol'!lents of inertia around the principal axes of M and M', and uk and uk' are the
vibrational frequencies of the kth normal mode of Mand M'.
As the Teller-Redlich product rule can be applied to any isotopically substituted
molecule (isotopomer) and the transition state is a molecule like any other except for one
imaginary frequency, the Teller-Redlich product rule can be applied to transition states.
Equation 1.16 can be re-written as: 16
(1.17)
u*(I) and u*(II) are the imaginary vibrational frequencies of the reaction coordinates for the reactions involving isotopes I and II. For decompositions of the same reactant ion, the right-hand side of equation 1.17 is unity. Equation 1.17 becomes:
( 1.18)
If it is assumed that the moments of inertia for M 1' and M2' are equal, equation 1. u; reduces to:
( 1.19)
Constraint of the vibrational frequencies by the Teller-Redlich product rule allows the effect of isotopic substitution upon the critical energy to be evaluated. The difference in critical energies for reactions involving isotopes I and II will be given by the difference in 7:ero-point energies (ZPE's) of the transition states involving isotopes I and II ( Fig.
1.4 ).
h P-1 (1.20) ZPE(I) = - Li ui(I) 2 i=l
and Fig. 1.4. Difference in critical energies for the intramolecular loss of isotopomeric species I and II.
Transition state II Transition state I
ZPE(II)
Products II
Reactant 17
h P-1 ( 1.21) ZPE(II) = - L ui(II) 2 i=l giving
(1.22)
Through equations 1.19 and 1.22 the Teller-Redlich product rule constrains the transition states for the two reactions. Intramolecular kinetic isotope effects obtained by
the application of QET are tightly constrained for a particular reaction mechanism, and
may therefore be used to probe the nature of the transition state for a reaction.
1.3. Metastable ions.
Ions which decompose after leaving the ion source of a mass spectrometer are known
as metastable ions. The analysis of the decomposition products of metastable ions has
been described in detail in the literature [24,55,56]. In this work, a reverse-geometry
double-focusing mass spectrometer (d.f.m.s.) was used for the study of metastable
ions, and the discussion will be centred on the methods used for this particular
geometry.
1.3.1. Mass-analysed ion kinetic energy (MIKE) spectrometry.
In a reverse-geometry d.f.m.s., ions which leave the ion source are passed through a
magnetic field and separated according to the mass-to-charge ratio m/z. The mass
resolved ion beam is then passed through an energy analyser (consider a cylindrical 18
electric sector (e.s.)), which is normally adjusted to allow only those ions with the full ion accelerating kinetic energy, eV 0, to pass. If a metastable ion, m1+, decomposes to m2+ ( and neutral m3 ) after leaving the ion source, the division of kinetic energy will result in the product ion having less kinetic energy than the parent ion. The kinetic energy, eV, of the product m2+, is given as:
m2 eV0 eV=--- ( 1.23)
If the metastable ion decomposes prior to the magnetic field, the product ion will be passed by the magnet at the apparent mass m* (eq. 1.24), but will not be passed by the electric sector.
m2 2 m*=-- (1.24) m1
Specific product ions, formed from metastable ions in the first field-free region, can be
detected by lowering the voltage of the e.s. to detect the energy deficient product ions.
To detect all the product ions, of a particular metastable ion, the magnet and e.s. are
scanned in a linked manner to maintain a constant B/E ratio of the magnetic field (B)
and the electric field (E) [57,58].
If a metastable ion decomposes after being transmitted by the magnet at mass m1, the product ion will not normally be transmitted by the electric sector, but can be
transmitted by lowering the electric sector voltage to allow ions with kinetic energy e V
to pass. If the electric sector is scanned from Oto the voltage required to transmit ions
with the full kinetic energy eV 0, while the magnetic field is held constant, all product
ions from the mass-selected reactant ion will be transmitted by the electric sector and 19
detected (neglecting transmission losses). This technique is known as mass-analysed ion kinetic energy (MIKE) spectrometry [24], and is one of the major methods for the analysis of metastable ions.
The principles of the MIKE experiment apply in the CID technique (vide supra)
[26,27]. Ions studied by CID are not metastable ions, but are generally stable ions for which structural information is required. The same procedure as for the MIKE experiment is followed for the CID experiment, except that collision gas is admitted to a collision cell located in the ion path, usually at the intermediate slit (see Chapter 3). As the MIKE and CID experiments are similar, CID spectra often contain contributions from MIKE processes as well as CID processes. If the metastable ion decompositions are strong, CID spectra of the products of these decompositions may also be observed.
Similarly, CID processes may be observed in MIKE spectra if the collision
cross-section is large [24,26].
1.3.2. Energy release in metastable ion decompositions.
The MIKE spectrum of a metastable ion shows broadening of the individual peaks due
to the energy released as translation during the decomposition. As mentioned earlier
(Section 1.1.1. ), the translational energy released in a decomposition is dependent on
the details of the reaction, in particular the reverse critical energy E0r. Measurement of the energy released as translation is relatively straightforward from the analysis of the
peak corresponding to the product ion, known as the "metastable peak" [56], in the
MIKE spectrum.
The shapes of metastable peaks can vary from simple Gaussian to flat-topped and
dished [24]. Peak shape is affected by the energy release and by a variety of
instrumental parameters [24,59]. The most common method of analysis of the energy 20
release is based on the reaction
(1.25)
for which the following relationship is obtained [24]:
(1.26)
E 1 is the centre of the metastable peak, ~E is the width of the peak at a convenient point, commonly half-height, and eV is the metastable ion kinetic energy. If ~Eis measured at half-height, the calculated energy release, T0_5, is sometimes called the
"average" energy release. T 0_ 5 has been determined as being a reasonably reliable estimate of the true "mean" energy release [59]. Several methods have been described for the detailed analysis of the shape of metastable peaks [59-62].
1.3.3. Field ionisation kinetics.
In a field ionisation source, neutral molecules are ionised by interaction with a strong electric field, which also acts to accelerate the ions away from the ionisation region. If the molecular ion, m 1 +, decomposes to a product ion m2 + in the electric field, the kinetic energy of the product ion, e V, will be less than that of the molecular ion, e V 0:
m2 eV e(V - V 1) + eV 1 (1.27)
VO is the potential at which the molecular ion, m 1+, was formed, and V 1 is the potential at which the product ion was formed. The kinetic energy can then be analysed as for 21
the first field-free region decomposition of metastable ions, either by scanning the magnet and electric sector in a d.f.m.s., or by scanning the accelerating voltage, V 0 [63]. Mathematical operations [64,65] allow the conversion of the ion current, I, vs "kinetic energy" curves obtained to be converted into ion abundance, I, vs time
following ionisation curves. Resolution of the order of 10-12 scan be achieved using
blade emitters [66], allowing reactions to be studied in detail at short times. The
technique is particularly powerful when used in conjunction with isotopic labelling
[67]. 22
Chapter 2. Calculation of ion abundances.
In this chapter, the methods used to calculate ion abundances are described. The principles of the methods are described first, and then the computer programs which implement the methods are described.
2.1. Calculation of k(E).
Calculation of the microcanonical rate coefficient k(E) rests primarily on a knowledge of
"density-of-states" functions of the reactant ion and the transition state. Classically, the
total number of states G(E), at or below energy E, is related to the density of states
function N(E) as follows: E G(E) = J N(E) dE 0
The density of states, N(E), is defined as the number of states per unit energy range at
energy E, and G(E) is the total number of states between zero and E. G(E) is
sometimes referred to as the integrated density of states [45).
A more useful form of the relationship between N(E) and G(E) is E G(E) = L W(E) 0
where
W(E) = N(E) oE.
W(E) is the number of states at energy E and oE is the allowance in E. If oE is equal to
the unit energy range, then
W(E) = N(E). 23
To calculate k(E), the total number of states is required for the transition state and will be represented as G*(E -E0). N(E) is required for the reactant ion.
2.1.1. Evaluation of W(E).
The evaluation of W(E) functions is a statistical problem, i.e. finding the number of
ways in which a certain amount of energy E can be distributed amongst the available
energy levels. Each distribution of energy corresponds to one state. The simplest
method of evaluating a W(E) function is by direct counting. Given a certain value of
the energy E, the number of ways in which this energy can be distributed amongst the
available energy levels is counted. For large energies and many energy levels, direct
counting can be very time consuming. A simple and efficient algorithm for counting
states has been developed by Beyer and Swinehart [68].
The Beyer and Swinehart (BS) algorithm works in a somewhat unorthodox manner.
Rather than counting the number of ways in which a given amount of energy may be
distributed amongst the available energy levels, the BS algorithm counts the number of
ways in which a chosen energy level can accommodate the required energy. Each
energy level is considered individually. Results obtained using the BS algorithm are
identical to results obtained when a "normal" count is performed. Operation of the BS
algorithm may be described as follows [69]:
(1) A suitable energy increment, 8E, is selected. 8E is known as the grain-size of the
calculation, and is discussed further in section 2.1.2.. The maximum energy and
each energy level are reduced to integer multiples, M and Ni respectively, of the
grain-size.
(2) An array W(I) is established for l=l to M with the initial values of 1 in W(l) and 24
zero in all other elements of W.
(3) For each Ni the following sequence of additions is performed,
W(Ni + 1) = W(l) + W(Ni + 1)
W(Ni + 2) = W(2) + W(Ni + 2)
W(I) = W(I - Ni)+ W(I)
W(M)
After all energy levels have been manipulated, the number of states W(E) will be contained in element W(Nm + 1), where
Nm oE=E.
N (E) is obtained simply as N(E) = W(E) oE
G(E) is obtained by summing all array elements up to, and including, the array element
corresponding to the energy of interest, i.e. for 1=2 to M,
W(I) = W(I) + W(I - 1).
This gives
G(E) = W(Nm + 1)
= I W(E).
The total number of states G(E) is now stored in the location which previously held the
number of states, W(E). The same procedure is applied to two arrays, one array giving
W(E), and the other array giving I W*(E - E0), i.e. G*(E - E0). k(E) is then equal to
the ratio of the elements containing I W*(E - E0) and W(E), multiplied by the quantity ooE/h
o is the reaction degeneracy, h is Planck's constant and oE is the grain-size used to
calculate W(E) for the reactant ion. 25
2.1.2. Choice of energy levels.
k(E) is dependent upon the energy levels associated with the reactant ion and the transition state, hence the choice of the degrees-of-freedom which are used to determine these energy levels are important. The types of degrees-of-freedom which need to be represented are rotational energy degrees-of-freedom associated with free rotors, and vibrational degrees of freedom. In order to keep simple the procedure for calculating k(E), harmonic vibrational frequencies were considered. Rotations, which otherwise could have been considered to act as free rotors, were treated as hindered rotations, i.e. torsions. Anharmonicity corrections were neglected.
Ideally the vibrational frequencies obtained from analysis of experimental data would be used, however such data are not readily available. Alternative methods for obtaining
suitable vibrational frequencies were used.
The simplest method of assigning vibrational frequencies to ions is based on the
assumption that there is little change in the force constants of a molecule following
ionisation. The vibrational frequencies of the ion are then considered the same as those
of the neutral molecule. This simplification may be valid if the electron removed by
ionisation is from a non-bonding orbital, however, if the electron is removed from a
bonding or anti-bonding orbital, changes in the force constants of the ion will result
[70].
If the effects of electron removal are neglected, the vibrational frequencies of neutral
molecules may be used to represent those of the ion, provided that such frequencies are
available. If frequencies are not available for a particular molecular structure, estimates
using the frequencies of similar model structures can be made. Compilations of 26
observed vibrational frequencies and their normal mode analyses are available [71-73].
Another method for obtaining vibrational frequencies is to perform a molecular orbital
(MO) calculation [74,75]. These methods are computationally expensive but provide extensive data, such as optimised ion geometries (minimised electronic energy) and
~Hf's, in addition to the molecular force-field required to calculate the harmonic vibrational frequencies. It may be possible under favourable circumstances to calculate the properties of a transition state using these MO methods [76].
Once the force constants used to calculate the vibrational frequencies are calculated, it is
a simple step to calculate the vibrational frequencies associated with an isotopomeric
species. Such calculations obviate the necessity of checking the effect of isotopic
substitution within the constraints of the Teller-Redlich product rule, and allow a more
systematic approach to the calculation. The frequencies obtained as a result of MO
calculations are typically in error by approximately 10% [74,75]. Stretching and
bending frequencies tend to be overestimated, and torsions underestimated. Correction
can be made to the vibrational frequencies to compensate for the error.
Once a set of vibrational frequencies has been obtained, they need to be integerised to
multiples of the grain-size in order to calculate k(E) efficiently using the BS algorithm.
The integerisation of the vibrational frequencies can be obtained simultaneously with
optimising the grain-size 6E for the calculation by minimising
u. - v. nv l l I: u. i=l l
nv is the number of vibrational frequencies, uj are the "true" frequencies and vj are the
frequencies integerised to a multiple of the grain-size, Nj (Section 2.1.1 ), i.e.
Vj=Nj 6E. 27
If 8E is very small, the above expression is minimised, but in order to reduce computer storage and computation time, other constraints are imposed upon 8E (see Section
2.6.1).
2.2. Energy deposition function, P(E).
The energy deposition function may be estimated as the appropriate derivative of the total ionisation with respect to the energy of the ionising agent [77,78]. This is the first derivative for photo-ionisation and the second derivative for electron ionisation. If the energy of the ionising agent is not well defined, as in the case of the normal electron
ionisation source, then it may not be possible to estimate P(E).
A number of approximations to P(E) have been suggested [79,80], but these are not
rigorous, and the effect of P(E) on the calculated mass spectrum is, in fact, minor [45].
The form of P(E) for electron ionisation used in this work was constant, i.e. P(E) = 1.0
for all values of E. The reasoning behind this decision was that if P(E) has little effect
on calculated mass spectra where a large range of internal energies are being sampled, it
should have even less effect if only a small range of internal energies are being
sampled. Further, only intramolecular isotope effects were considered, so P(E) is
common to all decompositions being compared. A number of P(E) distributions were
in fact tested, and similar ion abundance ratios were predicted for all P(E) distributions
considered.
The choice of Emax in the calculation of ion abundances is to a certain extent arbitrary. The maximum value of k(E) which contributes to the ion abundance may not
correspond to k(Emax). If k(E1) is the maximum value of k(E) which can contribute to
the ion abundance in the observation window, then k(Ei) for E1 $ Ei $ Emax will not 28
contribute to the ion abundance for the observation window. If Emax is less than E1, i.e. the maximum internal energy is less than the maximum energy for which k(E) contributes, then decomposition in the observation window of the experiment may be underestimated.
Emax was generally chosen such that the value of k(Emax) was in excess of the maximum value which could contribute to the ion abundance in the observation
window. A typical value for k(E) at the maximum internal energy was 1010 s- 1. If
Emax were raised to values which were much greater than the maximum contributing
energy, the overall ion abundances would differ. The relative ion abundances,
corresponding to the metastable time-window of the experiment, were not affected.
2.3. Observation times, t1 and½·
The observation time of the experiment is determined by the time taken for the reactant
ion to traverse the observation region. In the case of metastable ion decompositions
studied in the second field-free region (FFR2) of the reverse-geometry double-focusing
mass spectrometer, the observation window is determined as ½ - t1, where t1 is the time taken for the decomposing ion to reach the entrance of the FFR2 and 12 is the time taken for the decomposing ions to reach the exit of the FFR2. These times may be calculated from the kinetic energy of the reactant ions, e V, as I tl = X (~) /2 + tR I 2eV
and
Mis the mass of the reactant ion, x1 and x2 are the distances between the end of the ion 29
acceleration region and the entrance and exit of the FFR2 respectively, and tR is the time taken for the ions to leave the ion source after ionisation.
For the instrument used in this work (see Chapter 3), tR is approximately 10-6 s, x1 is
1.97m and x2 is 4.66m. For a singly charged ion of mass 100 daltons, formed at an accelerating voltage of 10 kV, t1 and 12 are 1.52x10·5s and 3.46x10·5s respectively.
2.4. Collection efficiency, G.
The collection efficiency, G, is defined as "that proportion of the actual number, per
time, of ions formed in the observation window of the experiment which reaches the
detector" [50]. In many previous studies, collection efficiencies have been neglected,
i.e. the collection efficiency has been assumed to be the same for all ions.
A method [59] for obtaining collection efficiencies was used which takes account of the
kinetic energy release accompanying unimolecular decomposition. A number of
instrumental parameters such as slit widths and the accelerating voltage are considered
in addition to the kinetic energy release. The method rests upon trajectory calculations for the decomposing ions. The proportion of fragment ions produced by
decomposition which will be detected is calculated. The method has been described in detail elsewhere [59].
Collection efficiencies obtained in this manner suggest that in the hypothetical case in
which a single reactant ion may decompose to either of two product ions of differing masses, but via processes which release identical amounts of kinetic energy as
translation, different numbers of the two product ions will be detected. Thus, if an ion
decomposes into two isotopomeric fragments without an isotope effect on kinetic 30
energy release, and there is no kinetic isotope effect, i.e. k1(E) equals k11(E), the abundance of the fragments will appear to differ because of different collection efficiencies [81].
Detector discrimination is another factor affecting the collection efficiency. If an electron multiplier detector is used, there are possible mass and energy effects, where the differing masses and kinetic energies of impinging ions may produce different numbers of secondary electrons, and chemical effects where impinging ions fragment further upon impact on the first dynode [82]. In the case of isotopomeric fragment ions as in this study, chemical effects can be neglected.
Mass and energy effects combined, have been measured for the elimination of 35Cl and
37Cl from metastable [C2H 435Cl37CI]+. ions. The ratio [M - 35CI]+. / [M - 37Cl]+. was measured to be less than 1.04 [83]. Assuming an energy release of 20 meV, the
ratio of the fractional transmittances for these two ions is 1.032, which gives a
corrected ratio for [M - 35Cl]+. / [M - 37c11+. of less than 1.008. The detector
discrimination was considered to be negligible in comparison to the fractional
transmittances, over one or two mass units, under the experimental conditions
employed.
2.5. Correlation of calculated and measured ion abundances.
If collection efficiencies, G, are estimated, correction of the measured ion abundances
for the discriminatory effects is easier than incorporation of the effects of the collection
efficiency into the calculation of the ion abundances. Ion abundances are corrected for
Gas 31
I (measured) I (corr) = ---- G (calculated)
Where ion abundances are used to calculate an isotope effect, Iifin, the ratio of ion abundances is corrected as follows: II II Gn - (corr) = (measured) - (calculated) In In GI
When II/In(corr) is compared with a calculated ratio, II/In(calc), the comparison gives an indication of the reliability of the transition states used in the calculations and, hence, of the modelled reaction mechanism. As the reaction mechanism is the same for both isotopes, the transition states (for I and II) have the same "structure". By adjusting the transition state for the reaction, a greater degree of agreement can be achieved.
2.5.1. Adjustment of the transition state.
The approach used in this study and recommended generally for the mechanistic
interpretation of isotope effects in metastable ion abundances is as follows. The initial
choice of a trial transition state is based on the reactant structure, but is governed by the
products of the reaction. If the products are formed by a simple bond cleavage, the
transition state is selected so as to resemble closely the reactant structure. One
vibrational frequency is removed to represent the reaction coordinate. If the products are
formed by a more complex mechanism, the transition state is still modelled upon the
reactant structure, but frequencies other than the reaction coordinate may be adjusted to
represent changes in bonding. For transition states of isotopically distinguished
reactions, the Teller-Redlich product rule holds (Section 1.2.3.2.) and constrains the
transition state vibrational frequencies in a similar manner to those of reactant ions.
Once an isotope effect on ion abundances is calculated and compared with a corrected 32
observed isotope effect, the transition state model can be modified. Each atom can be considered to affect three vibrational frequencies, i.e. any changes in the bonding of one atom will be manifest in three vibrational frequencies. (For normal modes of a molecule all vibrational frequencies are affected to some extent by changes in each
atom.) To achieve agreement between calculated and measured ion abundance ratios,
the transition state is modified by adjusting the vibrational frequencies associated with one, or more, isotopically distinguished atoms. The frequencies associated with a
particular atom are adjusted by an amount, e.g. 10%, and the ion abundance ratio
re-calculated and compared with the corrected ratio. If the frequencies of an isotopically
distinguishable atom are adjusted, the adjustment is also made to the unlabelled atoms in
the identical, isotopically normal transition state. Correction of the critical energy for
the isotopically distinguished reaction for the change in vibrational frequencies must
also be made.
If the bonding of the isotopically distinguished atoms in the transition state is
considered to have strengthened, the associated frequencies are increased. If the
bonding is considered to have weakened, the associated frequencies are decreased. If
the calculated isotope effect is greater than the corrected isotope effect, the bonding in
the transition state model is strengthened, which reduces the magnitude of the calculated
isotope effect. If the calculated isotope effect is less than the corrected measured value,
then the bonding in the transition state model is weakened which increases the
magnitude of the calculated isotope effect. Comparison of isotope effects can only be
used to adjust the vibrational frequencies associated with the isotopically distinguished
atoms. No information concerning the other vibrational frequencies is obtained.
Adjustment of the transition state is made until the calculated ion abundance ratio is
equal to the corrected measured isotope effect. The real test of a transition state, and 33
hence of the model of a reaction mechanism, is in the ability to predict results. This can be achieved in two ways. First, a different observation window can be used in both the experiment and the calculation. Second, a different isotopomeric species can be investigated. Both of these approaches have been used in this study (Chapters 4 and 5 respectively).
2.6. Computer programs.
All calculations of the ion abundances, i.e. calculation of k(E), G and I were performed
at the University of New South Wales using a Digital Equipment Corporation
VAX-11/785 computer. All programs were written in the VAX-11 FORTRAN
language.
2.6.1. Program QET.
Program QET calculates the microcanonical rate coefficient k(E). The method
employed is that described above [69]. Input to the program consisted of the number n
of vibrational degrees-of-freedom of the reactant ion, the n vibrational frequencies of
the reactant ion, the n-1 vibrational frequencies of the transition state, and energetic
data. The energetic data include critical energy E0 for the reaction, the maximum internal energy Emax of the reactant ion, the energy increment for the calculation, the
reaction degeneracy cr, and a quantity ERROR, used in the determination of the
grain-size for the number-of-states calculation.
After accepting the input data, the vibrational frequencies of the reactant ion were
integerised by subroutine INIT as described in Section 2.1.2.. The grain-size was
selected by INIT subject to the constraint that it be approximately 2 x ERROR, while 34
satisfying the constraint given in Section 2.1.2.. The integerised frequencies were next used by subroutine DENS to evaluate W(E) for the reactant ion. The vibrational frequencies of the transition state were then integerised by INIT, which would re-calculate the grain-size. The grain-size calculated for the transition state did not necessarily have to be equal to the grain-size calculated for the reactant ion, although often the two were the same. W*(E-Eo) was then calculated for the transition state and
then summed to provide G*(E-E0). k(E) was then calculated as k(E) = a oE LW*(E - E ) hW(E)
c5E is the grain-size calculated for the reactant ion. The k(E) obtained was then
interpolated to provide k(E) at regular increments by subroutine SETUP. Output from
SETUP was used in further calculations.
Program QET is listed in Appendix 1.
2.6.2. Program CFf3.
Program CFf3 was used to calculate the effect of the kinetic energy release on the
collection efficiency G. The program was written by Dr. B.A. Rumpf, and is described
in detail elsewhere [59].
Input to program CFf3 consisted of the reactant ion mass, the product ion masses, the
kinetic energy of the reactant ion, and the kinetic energy releases measured from the
metastable peak corresponding to the product ions. The output consisted of a fractional
transmittance for the product ions at each of the input kinetic energy releases. These·
fractional transmittances could be used individually, i.e. as G1 and G11 for product ions containing isotopes I and II respectively, or could be combined to give an overall G, 35
given by G1/GIJ, which corresponds to the mass effect on collection efficiency.
2.6.3. Program PEAKS.
Program PEAKS calculated the isotope effect on ion abundances for two competing reactions, using as input two sets of k(E) data produced by program QET. The reactant
ion mass and its kinetic energy were input to program PEAKS to calculate the
observation window for the calculation.
A modified form of equation (1.8) was used to evaluate the ion abundance for each of
the product ions. A quantity k(t) was calculated, given by
Emax n k-(t) = f k.(E) P(E) exp( -L ki(E)t)dE I I . I 0 I=
The ion abundance is then given as tz Ii = f ki(t) dt t1
The ion abundances were calculated as the area under the curves obtained when ~(t) is
plotted against t. The isotope effect on ion abundance is given by the ratio of the two areas.
Prog:am PEAKS initially calculates t1 and½, then calculates k1(t1), ku(t1), k1(t2) and
ku(t2). The integration to obtain 11 and In was performed by calculating the area underneath the curve obtained when ln(ki(t)) was plotted against ln(t). The method is
described in detail in Appendix 2, which also contains a listing of program PEAKS.
The results were tested by comparison with those obtained using Romberg integration
[84) to integrate the area under the ki(t) vs. t curves. Almost identical results were
obtained with the Romberg integration routine being considerably slower. 36
2.6.4. Adjustment of vibrational frequencies.
The adjustment of the transition state vibrational frequencies was carried out in the manner described above. To remove the possibility of errors occurring from the manual adjustment of the vibrational frequencies, computer programs were written to perform the adjustment. Specific programs were written for each system modelled. In general, the computer program would generate a complete set of input data for program
QET. Reactant ion frequencies and adjusted transition state frequencies were generated for all isotopomers of interest. The programs would calculate the zero-point energy changes in the transition state and correct the critical energies accordingly.
2.6.5. Other computer programs.
Other computer programs were developed for situations not covered by program
PEAKS. Program TIME (also listed in Appendix 2), for example, was written for the
case of two competing reactions, as was program PEAKS, but calculates k(t) over a
large time range for comparison with FIK data. Programs which model reaction
mechanisms more complex than the simple case of two competing single-step processes are discussed in the relevant sections of Chapter 5. 37
Chapter 3. Instrumental.
The experimental results presented in this thesis were obtained using a large reverse-geometry double-focusing mass spectrometer located in the School of
Chemistry at the University of New South Wales. This instrument has been described in detail elsewhere [85-89], and only a brief description is given below.
3.1. History of the instrument.
The mass spectrometer was a reverse-geometry double-focusing instrument, which was constructed in the Department of Physical Chemistry at LaTrobe University and was transferred to the School of Chemistry at the University of New South Wales in 1981.
The design of the mass spectrometer was in accordance with a particular solution of the
ion optical equations of Hintenberger and Konig [90], which gives second order
double-focusing when fringing fields are neglected. A schematic diagram of the mass
spectrometer is shown in Figure 3.1.
The reverse geometry and large dimensions of the mass spectrometer made it ideally
suitable for the MIKE technique described previously (Chapter 1).
3.2. Vacuum system.
The ion source chamber was 360 mm in diameter and 400 mm in length, with an
approximate volume of 41 dm3. The ion source chamber was evacuated by an Edwards
160/700 water-cooled oil diffusion pump, which was backed by an Edwards E2M40
direct drive rotary pump. A molecular sieve trap was located between the oil diffusion Second field-free region ( FFR2) Electromagnet Electric sector ( e.s. ) ( median path length 0.753 m) ( median path length 1.422 m )
I .. 'I ..._ _, --- :---...... ,, ,,. ,..9.:---.. ., : <;c'-/ :' (}v; l~"\,<> r··---- 0,37 • ---;:I Intermediate-slit and collision cell --L_m : ,(2 ~ t'~ \, \ /{o \ ',/ Electron multiplier detector , , ·--10 · Om-..j ' ~..? ~
\_.-----~~~~/ -----!"".
Figure 3.1. A schematic diagram of the reverse-geometry double-focusing mass spectrometer used in this work. The total median path length of the instrument is 6.5 m. 38
pump and the rotary pump to minimise back-streaming of pump oil from the rotary pump. An Edwards SPEEDIVAC Mercury Vapour Trap was used as a cryotrap above the oil diffusion pump to decrease the pump-down time, and to achieve a low ultimate ion source chamber pressure. The ion source chamber could be isolated from the rest of
the vacuum chamber using a gate-valve.
The rest of the vacuum chamber was evacuated by six Edwards 160/700 water-cooled oil diffusion pumps, which were backed by various Edwards rotary pumps. The oil diffusion pump closest to the ion source chamber was fitted with an Edwards
SPEED IV AC Mercury Vapour Trap, which acted as a cryotrap.
The pressure inside the instrument was monitored at seven points along the length of
the instrument. Edwards IG-50 ionisation gauges (Bayard-Alpert type) were used for
this purpose. The ionisation gauges were controlled by Edwards ION-7 ion gauge
controllers.
The backing pressures for the oil diffusion pumps were monitored by Edwards Pirani
gauges (PR-105), which were controlled by Edwards Pirani II controllers.
The entire vacuum system was controlled by a Safety/Interlock system [88], which
would in the case of a power failure, cooling water failure or rise in the pressure within
the vacuum chamber, turn off the oil diffusion pumps.
3.3. Ion beam defining slits.
The ion beam was defined by the source-slit, which was located 100 mm from the
ionisation region. The ion beam was focused by an electrostatic lens system prior to 39
reaching the source-slit. The width of the source-slit was adjustable from 2 µm to 2 mm, and defined the width of the ion beam. 250 µm was a typical setting for the width of the source-slit. Adjustments to the source-slit width were made using a linear motion micrometer, mounted externally and connected through a port on the top of the ion source chamber.
The electromagnet (Section 3.5) focused the ion beam emerging from the source-slit onto the intermediate-slit. The width of the intermediate-slit was adjustable from 0.1
mm to 12 mm, using an externally mounted linear motion micrometer.
The electric sector (Section 3.6) focused the ion beam originating from the
intermediate-slit onto the collector-slit. The width of the ion beam at the collector-slit
was determined by the source-slit width and by the magnification of the instrument.
Typically, for a 250 µm source-slit width, a collector-slit width of 400 µm would allow
the ion beam to be transmitted without significant attenuation.
The width of the collector-slit was adjustable from 2 µm to 2 mm, and the slit could be
moved in a direction normal to the ion beam over a 12 mm path in the horizontal plane.
Both of these adjustments were made using externally mounted linear motion
micrometers.
3.4. · The ion source.
The results presented in this thesis were obtained using an electron ionisation (EI)
source, shown schematically in Figure 3.2 [91]. The potentials shown in Figure 3.2
are representative of those required to record a mass or MIKE spectrum at an
accelerating potential of 8kV. Focus Electron repeller 8100 V 7870V /ode Source-slit
Sample
:x::s-;x;w;,.:~~"%$.X-;~;-~~;.-;x;:: Ion repeller 8180V
:❖"-;::::-;-;❖::::t;;::,"WS.X:❖'X-;;:;.;~:~::x:;;-;,:.(fx; §~~ Filament 7930V
Grid/ z-Axis deflector 8000V ~Einzel lens 5300V 6600V
Figure 3.2. Schematic diagram of the electron ionisation source used in this study. Potentials shown are typical values required for an accelerating potential of 8 kV. 40
The EI source produced ions from a gaseous sample, or from a liquid or solid sample exerting a suitable vapour pressure. The flow of gas or vapour was controlled by a
Granville-Phillips variable leak valve (model 203). The sample inlet line consisted of glass sample containers, which were connected by Cajon fittings to a stainless steel line and the leak valve. The stainless steel line from the leak valve was connected to a vacuum feed-through in the ion source chamber. The gas line to the EI source within the vacuum chamber was made from Teflon to provide electrical insulation.
Ions were focused by an electrostatic lens system within the source chamber (Figure
3.2). The lens system was a modified Einzel lens, which has been described elsewhere
[97,88].
The ion source was floated at a high voltage to provide the ions' initial acceleration.
This accelerating voltage, typically 8kV, was provided by a Fluke 410B l0kV power
supply. The high voltages required by the electrostatic lens system were provided by a
Brandenburg 827 Ensign Model 30kV power supply and a Spellman RHSR30P60
30kV power supply.
3.5. Magnetic sector.
Mass dispersion was effected by a large, water cooled laminated electromagnet [87].
The included angle of the magnetic sector was 0.96 radians, and the median ion path had a radius of 0.78 m. The pole gap of the magnet was 22 mm.
The magnetic field was controlled by a 0 to 10 volt reference voltage supply, which controlled an Alpha Scientific current regulated power supply. This power supply consisted of a control unit, which was controlled itself by the reference voltage supply, 41
and a power unit, which was a water-cooled power supply producing 80 amps at 60 volts d.c .. When 80 amps were passed through the electromagnet coils, a magnetic field of approximately 1.6 Tesla was generated between the magnet pole pieces [89].
The magnetic field between the pole pieces was monitored by a Siemens FP110L60 magneto-resistor, mounted in an aluminium block maintained at 313K [89]. The output from the magneto-resistor could be monitored directly, or as was more common, the
output could be monitored by a dedicated microcomputer [89]. The microcomputer used
the magneto-resistor output to access a look-up table, from which the mass represented
by the magnetic field at a given accelerating voltage would be obtained and displayed.
3.6. Electric sector.
Energy focusing of the ion beam was effected by the electric sector (e.s.). This was a
1.42 radian sector of concentric aluminium plates, which had been cut from 25 mm
thick aluminium plate. The plates had been rolled into the approximate shape, mounted
on the base plate and machined in place. Specifications after machining were [88]: mean
radius 999.975 mm, included angle 1.42 radians, mean plate separation 33.508 mm
and height of plates 149.5 mm.
To correct the effects of fringing fields, Matsuda plates [92] were mounted 24 mm
above and below the e.s. plates, and Herzog shunts [93] were mounted 5.6 mm in front
of the e.s. entrance aperture and 5.6 mm behind the e.s. exit aperture.
The voltages on the two e.s. plates were controlled by a 0 to 1000 volt power supply,
which developed-both the positive and negative potentials required. A 0 to 10 volt
representation of the e.s. potentials was also output by the power supply for use in data 42
acquisition (see Section 3.9).
3.7. Field-free regions.
There were three regions of the mass spectrometer in which the ion beam would ideally not experience the influence of applied magnetic or electric fields. These regions are
termed field-free regions and are labelled in Figure 3.1.
The first and third field-free regions, i.e. before the electromagnet and after the e.s.,
acted only as drift regions, while the second field-free region (FFR2, 2.7 m in length),
was, in effect, used as a chemical reactor for both the MIKE and CID experiments
(Chapter 1). The intermediate-slit was located 2.16 m along the second field-free
region, and the collision cell used for the CID experiments was located immediately in
front of the intermediate-slit.
The collision cell was a cylindrical cavity 64 mm in diameter and 10 mm in length
(volume 32 cm3). The entrance and exit apertures of the collision cell were 3mm wide
and 7mm high rectangular slits. Gas was admitted to the collision cell through a
Granville-Phillips variable leak valve (model 203). The pressure outside the collision cell was monitored using an Edwards IG-5G ionisation gauge.
Helium was used as the collision gas in the CID experiments performed in this work.
Measurements had shown that the pressure in the collision cell, P 1, was related to the
pressure outside the collision cell, P0, by the expression
P1 = "f Po where y is a constant dependent upon the collision gas and has been measured as 80 ± 5 for helium [88). 43
3.8. Detector system.
The ion beam was monitored using a Balzers secondary electron multiplier (SEV 217).
This was a 17 stage multiplier with Cu/Be electrodes (dynodes), which produced a gain of ~ 106 when a potential difference of 2kV was applied between the first and last dynodes.
The output from the multiplier was input to an operational amplifier, which provided a
selectable gain of 109, 1010 or 1011 in the resultant output signal. A capacitive
smoothing circuit provided a selectable time constant on the output signal of 0.33 ms,
1.0 ms, 3.3 ms, 11 ms, 33 ms, 110 ms, 330 ms, 1.0 s or 3.3 s. This output was used
as the input for data acquisition.
3.9. Data acquisition.
The analog output from the detector system was connected to an Hitachi V-209
oscilloscope for direct monitoring. Normal mass spectra were recorded on a San-ei
Visigraph SL35 ultra-violet chart recorder with the detector system output controlling
the Y-axis deflection. The chart recorder internal timing circuitry controlled the X-axis.
Three input channels were available on the chart recorder. Two channels were used to
record the mass spectra. The third channel was used to record information, produced by
the dedicated microcomputer system (Section 3.5), concerning the mass of the ions.
When scans of the e.s. were required, as in the MIKE experiments, the detector system
output was recorded on a Nuclear (Chicago) 7590C X-Y recorder. The detector system
output signal was recorded as the Y-axis. The X-axis was used to record the voltage 44
supplied by the Oto 10 volt e.s. voltage supply (Section 3.6).
Another method of data acquisition involved a microcomputer system based on a
Synertek SYM-1 microcomputer to which a number of peripheral devices could be connected. To record a MIKE scan, for example, an Analog Devices 574 12-bit analog-to-digital converter (ADC) and an Analog Devices 1136K 16-bit digital-to-analog converter (DAC) were connected to the microcomputer system. The
ADC was used to connect the analog output from the detector system into a form which the microcomputer could manipulate. The DAC converted the microcomputer output into an analog form, which was used to control the e.s. power supply and hence the e.s. potentials.
The spectra recorded by the microcomputer system could be output, either as an X-Y
plot using a Watanabe Digi Plot WX4671 digital plotter, or as a listing of digital counts
each corresponding to a chosen resolution of the microcomputer system.
The software used to control the microcomputer system was written in two sections.
The fundamental section was a machine-code program, written by P.G. Cullis [89],
which controlled the actual data acquisition. A BASIC language program, written by the
author of this thesis, acted as an interface between the machine-code program and the
user, and controlled the output of data. These programs have been described elsewhere
[89,91]. 45
Chapter 4. Alpha-cleavage of N-methyl-dipentylamine molecular ion.
4.1. Introduction.
Hydrogen isotope effects on ion abundances have been reported in which the substitution of deuterium for hydrogen exerts an influence on the loss of an iodine radical from alkyl-halides at up to six bonds away from the point of cleavage [94]. The reported isotope effects are intermolecular. Ingemann and Hammerum [95] have reported intramolecular hydrogen isotope effects for the a-cleavage reaction of
N-methyl-dipentylamine (1) molecular ions (see Scheme 4.1) where the isotopic
substitution was up to three bonds away from the point of cleavage. Five
systematically deuterated compounds were studied. Due to the extensive isotopic
labelling and the apparent simplicity of the a-cleavage reaction, this system was chosen
for study using the QET to calculate ion abundances.
4.2. Experimental results.
The five labelled N-methyl-dipentylamines used in this study were donated by Dr. S.
Hammerum. The labelled compounds, (2) - (6), are identified in Figure 4.1. Note that
coml?ound (2) contains five deuterium atoms to allow the product ions to be
distinguished as both product ions always retain both a- hydrogens.
The MIKE spectra of the five compounds are shown in Figure 4.2. The spectra
establish that a-cleavage is the dominant process in the observation window of the
experiment. A small peak corresponding to the loss of pentene is present at low
intensity. The ion abundances for the a-cleavage products were obtained from the Scheme 4.1 a-cleavage ofN-methyl-dipentylamine.
Figure 4.1 N-methyl-dipentylamine and the five deuterated compounds used in this study. a B y o E CH3CH2CH2CH2CH2'-. /CH2CH2CH2CH2CH3 (1) N I CH3
(2) a, a, E, E, E - D5
(3) B, B - D2 (4) y, y- D 2 (5) o, o- D2 (6) E, E, E - D3 (a) Compound (2).
0.6771
0.6601
0.6203 artefact 0.5739
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(b) Compound (3) 0.6705 (c) Compound (4 ). (d) Compound (5). (e) Compound (6) 0.6705 0.6705 0.6724
0.6590 0.6590 0.6522
0 6590
0.5954 0.5977 0.5954 0.5954 0.5805 0.5838 0.5838 ----'
1 1 1 I I 1 I 1060 o s~ 060 0.70 0.50 0.60 0.70 EtE 0 EtE EtE 0 E/E 0 0
Figure 4.2. MIKE spectra of the molecular ions of the five deuterated amines (2) - (6). A complete MIKE spectrum is shown for (2). Partial spectra are shown for (3) - (6). 46 areas under the metastable peaks. The ratio of the two peak areas for each compound was used as a measure of the isotope effect on ion abundance. These ratios are shown in Table 4.1 along with the previously reported ion abundance ratios [95].
The measured ratios of peak areas (Table 4.1) were corrected for the differing fractional transmittances of product ions of different masses as described in Section 2.4. The average energy release for each decomposition was estimated for each decomposition from the peak width at half-height, i.e. T 05, and found to be 18 meV in all cases. (Expanded MIKE spectra of compounds (2) and (3) are shown in Figure 4.3.) This value was then used to calculate the fractional transmittances GH and G0 , using program CFT3. The ratio G ( = GH / G0 ) was used to correct the measured metastable peak area ratios, and the result was taken to be the ratio of the product ion abundances.
G8 , G0 and G are given in Table 4.2. The product ion abundance ratios corrected for G are shown in Table 4.3.
4.3. Calculation of ion abundances.
4.3.1. The kinetic model.
The a-cleavage reaction was considered to be a one-step, i.e. concerted, process. The
kinetic model chosen to represent the two competing reactions is depicted in Scheme
4.2 for compound (2). The energy dependent microcanonical rate coefficient for
a-cleavage in which unlabelled butyl radical (C4H9·) is lost is denoted by k8 (E), and
that for loss of the labelled butyl radical (C4H 6D 3 ·) is denoted by k0 (E). The
abundances of the immonium ions formed as a result of these losses, 18 and 10 respectively, were_obtained by application of the expressions presented in Section 2.6.3
using program PEAKS. All calculations were performed with a constant energy
deposition function, i.e. P(E) = 1.0, a grain size of 0.20 kJ moi- 1 (16.7 cm- 1) and a Table 4.1. Ratios of metastable peak areas.
Compound Ratio of areas
(2) 1.32 (1.29) (a), (b)
(3) 1.74 (1.66) (a)
(4) 1.23 (1.21) (a)
(5) 1.08 (1.04) (a)
(6) 1.12 (1.12) (a)
(a) The ion abundance ratios reported in [95] were reported per deuterium. The reported values were converted to observed ion abundance ratios by correcting for the number of deuterium atoms present.
(b) Compound (2) exhibits isotope effects from deuterium at both the a and E carbons. The value of the ratio of peak areas reflects both the a and E contributions. 0.00281
0.65 0.66 0.67 0.68 0 59
0.66 0.67
Figure 4.3. Expanded MIKE spectra of the molecular ions of compounds (2) and (3). The widths of the peaks are in units of E/E0. Table 4.2. Fractional transmittances for (2) and (3) at the measured T 0_5 energy release of 18 meV.
Compound GH G (a)
(2) 0.37633 0.36697 1.026
(3) 0.36976 0.36047 1.026
(a) Collection efficiency G is evaluated as GH / G0 .
Table 4.3. Ratios of metastable peak areas, collection efficiencies G and corrected ion abundance ratios.
Compound Ratio of peak areas G Ion abundance ratio
(2) 1.32 1.026 1.28
(3) 1.74 1.026 1.69
(4) 1.23 1.026 1.19
(5) 1.08 1.026 1.05
(6) 1.12 1.026 1.09 Scheme 4.2 ex-cleavage reactions eliminating C4~· and C4H6D3• from the molecular ion of (2).
CH3CH2CH2CH2CH2" / CD2CH2CH2CH2CD~ +.
N I CH3 47
maximum internal energy of 600 kJ moi-1. The observation window of the experiment was calculated for the molecular ions of each compound, using the accelerating potential used of 8 kV. For (2), molecular weight 176, the times defining the observation window, t1 and½_, were 2.20x10-5 sand 5.17xl0-5 s. For compounds (3),(4) and (5), molecular weight 173, these times were 2.16x 1 o-5 s and 5.08x 1 o-5 s, and for compound (6), molecular weight 174, the times were 2.18xlo-5 sand 5.1 lxlo-5 s.
4.3.2. Reactant ion vibrational frequencies.
Vibrational frequencies for (1) were estimated from published normal mode analyses of n-alkanes [72] and methylamine [96]. These vibrational frequencies were considered representative of the vibrational frequencies of the molecular ion. These estimated vibrational frequencies of the molecular ion (M+·) of (1) are given in the first column of Table 4.4.
The vibrational frequencies for the molecular ions of (2) - (6) were obtained by
adjusting the frequencies constructed for (1) within the constraints of the Teller-Redlich product rule. Six frequencies were assumed to be associated with each carbon of the
backbone and considered to be isotopically sensitive. Each of the six frequencies was
assumed equally sensitive to isotopic substitution, thus all frequencies were adjusted by
the same proportion. The adjustment of the vibrational frequencies is described further
in Appendix 3. The vibrational frequencies chosen to represent the molecular ions
(M+-) of (2) - (6) are given in Table 4.4.
4.3.3. Transition state vibrational frequencies.
The vibrational frequencies associated with the transition state for a-cleavage were
considered to have the same form as those of the reactant ion, except that one C-C (1) (2) (3) (4) (5) (0)
M+• M+. TSH TSO M+• TSH TSO M+. TSH TSD M+. TSH TSD M+. TSH TSD
10u0. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1~00. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000 2985. 2985. 2985. 2985. 2985. 2985. 2985. 2985. 2985. 2985. 2985. 2985. 2985. 2985. 2985. 2985 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2962. 2520. 2820. 2820. 2820. 2820. 2820. 2820. 2820. 2820. 2820. 2820. 2820. 2820. 2820. 2820. 2820. ~-•J. 500. SOO. SOO. 500. SOO. 500. 500. soo. 500. SOO. 500. soo. soo. 500. 500. ,..,__1'85. 1'85. 1485. 1'85. 1'85. 1•8s. 1'85. 1•8s. 1•8s. 1•8s. 1'85. 1'85. 1'85. 1'85. 1•8s. 1•8~. 147•. 147•. n1•. 147•. n1•. 1'7'. 1474. UH. n1•. n1•. 101•. 147'. 147•. 1074. 1117;._ llt)O. 1'30. 14}0. 1'30. 1•Jo. 1430. 14}0. 1'30. 1430. 1430. 1430. 1430. 1430. 1430. 1430. 1•30. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 1195. 11 JO. 1130. 1130. 1130. 1130. 1130. 1130. 1130. 1130. 1130. 1130. 1130. 1130. 1130. , 1 Ju. 11 Ju. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. 1000. h.1JO. 1 100. IIOO. 1100. 400. 400. 1100. •oo. 1100. •oo. 1100. 1100. 400. 400. 400. ttoo. 400. 350. 350. 350. 350. 350. }SO. 350. 350. 350. 350. 350. 350. 350. 350. 350. 350. 120. 120. 120. 120. 120. 120. 120. 120. 120. 120. 120. 120. 120. 120. 120. 120. 2928. 2928. 2867. 2928. 2928. 2867. 2928. 2928. 2867. 2928. 2928. 2867. 2928. 2928. 2867. 2928. 2855. 2855. 2795. 2855. 2855. 2795. 2855. 2855. 2795. 2855. 2855. 2795. 2855. 2855. 2795. 2855. 1075. 1•75. 1111111. 1475. 1475. 141111. 1'75. 1'75. 11111111. 1475. 1475. 1111111. 14"/5. 1075, 111411. 1475. 11165. 11165. 14}5. 1'65. 1'65. 14}5. 1465. 1•65. 14}5. 1'65. 1465. 1435. 1465, 1465, 14}5. 1%5. 11146. 111116. 1'&46. 111116. 1416. 111116. 1ll16. 111'16. 1•16. Jlft16 0 111116. 11&16. 111t16. 1lll&6. 11n6. 11&116 1107. 110'(. 1084. 1107. 1107. 108". 1107. 1107. 10811. 1107. 1107. 1084. 1107. 1107. 10811. 1107. 29JJ. 29JJ. 2620. 29Jl. 29H. 2620. 293}. 293}. 2620. 2933. 29H. 2620. 29B. 2933. 2620. 29)3 2853. 2853. 2548. 2853. 2853. 2548. 2853. 2853. 2548. 2853. 2853. 2548. 2853. 2853. 2548, 28~3- 1462. 1462. 1306, 1'62. 1'62. 1306. 1'62. 1462. 1306. 1462. 1462. 1306. 1"62. 1462. 1J06. 1462. I J 17. 1} 17. 1177. 1317. 1317. 1177. 1317. 1317. 1177. 1317. 1317. 1177. 1} 17. I} 17. 1177. 1317, 12'111. 1214. 11}8. 1274. 121•. 11}8. 121•. 127ll. 1138. 1274. 121,. 1138. 1274. 1274. 1138. 12·1•. 1.:09_ 1209. 1080. 1209. 1209. 1080. 1209. 1209, 1080. 1209. 1209. 1080. 1209. 1209. 11)00. 1209. 2928. 2928. 2798. 2928. 2928. 2798. 2928. 2928. 2798. 2928. 2928. 2798. 2928. 2928. 2/9b. 2928. 2055, 2855. 2728. 2855. 2855. 2728. 2855. 2855. 2728. 2855. 2855. 2728. 2855, 2855. 21>.0. 281>';. l'f.46. JIP&6. 1382. 1046. 1'46. 1}82. 1••6. 1'46. 1382. 111116. 1'46. 1J82. 11-1116. 111116. 1382. lllllb, 1i.16. 11116, 115}. 1416. 1416. 1353. 1416. 11116. 135}. 1416. 1416, 1}53. 1416. 11116. lJSJ. 14H, 1 JOS. 1 }05. 1247. 1 JOS. 1}05. 1247. 1305. 1305. 1247. 1305. 1305. 1247. 1305. 1305. 1247. l]U~. 1?211. 1224. 1170. 122•. 122Q. 1170. 122•. 1224. 1170. 12211. 1224. 1170. l22lt. 12211. 1170. 1224. 2923. 2923. 2894. 2923. 2923. 289•. 2923. 2923. 2894. 2923. 2923. 289•. 2923. 2923. 2894. 292]. 2858. 2858. 2830. 2858. 2858. 2830. 2858. 2858. 2830. 2858. 2858. 2830, 2858. 2858. 28}0. 2858. 11.i'j)_ 1•51. 14J9. 1'53. 1•53. 1'39. 1453. 1115]. 1'39. 1'53. 1453. 14}9. 1•53_ 1•53. 1439. 1453. 1·,1i11. 13611. 1]51. 1]64. 1364. 1351. 136•. 1364. 1351. 1]64. 1364. 1351. 1]64. 116•, 1351. 13611. i 11.2. 1122. 1111. 1122. 1122. 1111. 1122. 1122. 1111. 1122. 1122. 1111. 1122. 1122. 1111. 1122. 1d11. 1074. 10611. 10'(4. 1074. 10611. 101•. 10711. 10611. 1074. 1074. 1064. 10711. 10711, 10611. 10711, 2962. 2%2. 2936. 2962. 2962. 2936. 2962. 2962. 29}6. 2962. 2962. 2936. 2962. 2%2. 29J6, 2962. 29i1. 2961. 2935, 2961. 2961. 2935. 2961. 2961. 2935. 2961. 2961. 2935. 2961. 2961. 2935. 2961. 2883. 288J, 2858. 288}. 2883. 2858. 2883. 2883. 2858. 2883. 2883. 2858. 2883. 2883. 2858. 288). 1467. 1467. 1455. 1•61. 1467. 1"55. 1467, 1"67. 1455. 1467. 1"67. 1455. llH)7. 1467. '455. 1067, Pl&O. lltf>U. 1llll8, 1460. 1460. 1448. 1q&o. 1460. 1448. 11160. 11160. 11148. 1%0. 1•60. Ill~!:! ll!t,r,. 105]. 105]. 10411. 1053. 1053. 1044. 1053. 1053. 104ll. 1053. 1053. 10411, 1U53. 1053. 1044. 10~:. :;oo. 500. 500. 500. 500. 500. soo. 500. 500. 500. 500. 500; 500. 500. 500. 501·. 999, 999. 999. 999. 999. 999. 999. 999. 999. 999, 999. 999. 999. 999. 9'J'J. 9r,11_1. 026. 926. 926. 926. 926. 926. 926. 926, 926. 926. 926. 926. 926. 926. (j~t... 12f:. hdl. 881. 881. 881. 881. 881. 881. 881. 881. 881. 881. 881. 881. 881. 881. C.f11. 8"2. 842. 842, 842, 842. 842. 842. 8•2, 8112. 8•2. 842. 842. 842. 842. 842. 011 ;~. "/62, 762. 762. 762. 762. 762. 762. 762. 762. 762. 762. 762, 762, 762. "/62. 76-'. "/25. 725, "/25. 725. 725. 725. 725. 725. 725. 725. 725. 725. 725, 725. 725. 72:,. 491. 491. 491. •91. •91. •91. 491. 491. 491. •91. "91. •91. 491. 491. 491, 49'. 155. 355, ]55, 355. 355. 355. 355. 355. 355. 355. 355. 355. 355. }55. !55. 3'-15. 286. 286. 286, 286. 286. 286. 286. 286. 286. 286. 286. 286. 286. 286. 286. 28~. 100. 100. 100. 100. 100. 100. 100, 100. 100. 100. 100. 100. 100. 100. 100. lOC. 188. 188, 188. 188. 188, 188. 188. 188, 188. 18a. 188. 188, 188. 188. 18e. 18~. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 9':J. 82. 82. 82. 82. 82, 82. 82. 82, 82, 82. 82. 82, 82, 82. 82, 82. 1026, 1026. •Re• 1026. 1026. 1 1026, Re• 1026. •Re• 1026. 1026. •ne• 1026. 1026. 1 RC 1 1026. 2028. 19011. 1904. 1864. 2928. 2928. 2867. 2928. 2928. 2867. 2928. 2928. 2867. 2928. 2928. 286",. . ,i:55, 1856 . 1856. 1817. 2855. 2855. 2795. 2855. 2855. 2795. 2855. 2855. 2"195. 2855. 2855. 2795. 11,75. 959. 14114, 959. 939. 1475. 1"75. 1"75. 1'75. 1qqq_ 1'75. 1'75. 11144. 1'75. 14"/',. 111111:. 1•65. 95]. 953. 933. 1'65. 1465. 1'35. 1465. 1465. 1"35. 1"65. 1465, 1•3s. 1"65, 1'65. 1'1]'). I ll46, 9110. 940, 921. 14ll6. 1446. 1,16. 1446. 14ll6, 1"16. 14116. }11116, 11116. 1446. 1446. 14 ll. I !07. -,iO. ·120. 705, 1107. 1107. 1084, 1107. 1107. 10011. 1107. 1107, 10811. 1107. 1107. 108'.!. dll. 29]J, 293}. 2620. 2081. 2081. 1858. 29J}. 2933. 2620. 2933. 29JJ. 26 ... 0. d33. 293J. 262U. 28">3. 2853. 2853. 2548. 2024, 20211, 1807. 2853. 2853. 2548. 2853. 2853. 2~·•~. 2853. 2H5). 25111\. 1462. lll62, 1462. 1}06. 1037. 1037. 926. 1062. 1062. 1306. 1"62. 1462. 1]06. 1"62. 11162. 1)05, 1 J17. 1 l 17. 1317. 11"/7. 935. 935. 835. 1J 17, 1317. 1177, 1317. 1317. 1177. I 317. 1317. 1177. 12"/11. 12'/li. 1274, 11 J8. 9011. 904. 807. 1274. 12711. 1138. 121•. 1274. 11 38. 12'74. 127li. 113,1. l.?09. 1209. 1209. 1080, 858. 858. 766. 1209. 1209. 1080. 1209. 1209. 1080. 12U9. 1209. 1060, ?928. 2928. 2928. 2798. 2928. 2928. 2798. 2077. 2077. 1984. 2928. 2928. 2"198. 2928. 292U. 2796. C J';:,5, 2855. 2855. 2"/28. 2855. 2855. 2728. 2025. 2025. 1935. 2855. 2855. 2728. 28'>">. 20'>5. 272~. 111!6. Jllll6. 1li46. 1382, 1•46. 1006. 1382. 1026. 1026. 980, 111116. 14116. 1362. 1'11'6. JIJJH,. 138,'. 1 1 H,. 11116. 1416. 1353, 1416. 1416. 1353. 1005. 1005. 960. 11.116. 11116. 1}53. 11116. ll116, 135 l. I }05, 1 JOS. 1305. 1247. 1)05. 1305. 1247, 926. 926. 885. 1305. 1305, 121-17. 1305. 1305. 1214'f. 122li. 11211. 12211. 1170. 1224, 1224. 1170. 869. 869. 8JO. 122ll. 12211. 1170. 12211. 12211, 117•J. 2Y2l. 292 3. 2923. 2694, 2923, 2923. 289li, 2923. 2923. 2894. 2073. 2073. 2053. 2923. 2923. 28911. ?858. 2858. 2858. 2830. 2858. 2858. 2830. 2858. 2858, 2830. 2027. 2027, 2007. 2858. 2858. 283J. l 115J. 1115). 1115]. 1'39. 11153. 1053. 1•39. ••s3. 1"53. 1439. 1031. 10] 1. 1021. 11153. lli5J. 111 j'). 1]611. ljlJII. I ]61• IJ51. 136•. 1J6•. 1351. 136•. 136•. 1351. 968. 966, 958. 1 36•1. 1 J(ir1. 1j5 I, 1121. I Ill. 1122. 1111. 1122, 1122. 1111. 1122. 1122. 1111. 796. 796. "/86, 1122, 1121. 1111. 10"/4. 10.,ti. 107'4. 10611. 10711. 1074. 10611. 1074. 1074. 10611. 762. 762. 755. 10711. 1Ll1't. 10(..11. i'';62. 1926. 1926. 1909. 2962, 2962. 2936. 2962. 2962. 2936. 2962. 2%2. 29}6. 1"/70. 1"/70 1"15•. ~9bl, ins. 1925. 1908, 2961. 2961. 2935, 2961. 2961. 2935. 2961. 2961. 29]5. 1"/69. 1"169. n'>4. 2b8J. 1s·11.1. 18"/4. 1858. 2883. 2883. 2858. 288J. 2883. 2858. 2883. 288]. 2856. 172J. 17.::l. 1707. 111b7, IJ'>ll. 9'.>11. 906. 1467. 1'67. 1"55. 11167, 1467. 1•s5. 1467, 11l67. 1'155. 877, 877. 869. -,,1(:iO. 9'1'L 9119, 9111, 11160, 11160. lll46. 1460, 1ll60. 14118. 11160. 11.160. 1'1118. 8"/J. 87 J. 6b5. 10':.ij. 665. 685. 679. 1053, 1053. 1011'4. ltJ", J. 1053. 101111, 1053. 1U5J. 1044; 629. 629. 6c•. ')00, '.}UO. 'JOO, 500, 500. soo. 500. 500, 500. 500. 500. 500. sou. soo. Silo. 500. 999. 999. 999. 999. 999. 999. 999. 999. 999. 999. 999. 999, 999. 999. 999. ~99. rue,, 926. 926. 926. 926. 926. 926. 926, 926. 926, 926. 926. 926, 926. 92&. W6. 861. 861. 881. 881. 881. 881. 881. 881. 881. 681. 881. 881. 881. 881. 881, 6d1. 6 112. 842. 8'2. 842. 8•2. 842. 8•2. 8•2. 8•2. 842. 8•2. 842, 8•2. 842. 842. e:.2. 762. 762. 762. 762. 762. 762. 762. 762. 762. 762. 762. 762. 762. 762. 762. 762, 7'5. 725, 725. 725. 725. 725. 725. 725. 725. 725. 725. 725. 725. 725. "125. "/25. 11111. ll91, •91. 491. 491. 491. 1191, •91. •91. 1191. 1&91. 491. 491. 491. 4<) l. tic;,. 155. 355, 355. 355, 355. 355. 355. 355. 355. 355. 355. 355. 355. 355. 3':.6. )'JS. 286. 286. 286. 286. 286, 286. 286. 286. 286. 286, 286. 286. 286. 286. 266. 286. 100. IOU. 100, 100. 100. 100. 100. 100. 100, 100, 100. 100. 100. 100. 1110. 100. 108. lbd, 188, 188. 188. 188. 188. 188. 188, 188. 188. 188. 188. 188. 188. 188. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 95. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 82. 1026. 1026, 1026. 82. •11e• 1026. 1026. •11c• 1026. 1026. 1026. 1026. 1026. •11c• 1026, 1026. •Re•
Table 4.4. Vibrational frequencies used in this srudy. The frequencies of the molecular ions (M+·) of compounds (1) - (6) and the frequencies of the ten transition states (two for each compound) are given. The transition states butyl leading 10 loss of C4H9· are labelled TSH and the transition states leading to the eli':'1ination of labelled radical are labelled TSO. The reaction coordinates, labelled as •Re•, were 1026 cm I for all trans111on states. 48 stretch was taken to represent the reaction coordinate. The reaction coordinate, chosen as 1026 cm-1, was considered to be unaffected by isotopic substitution and was kept constant for all reactions.
4.3.4. Critical energy.
The critical energy for the a-cleavage reaction was estimated from measurements made on related compounds by Dr. J.C. Traeger of LaTrobe University using photo-ionisation mass spectrometry [97]. The loss of a butyl radical from the molecular ion of n-pentylamine and the loss of a butyl radical from the molecular ion of
N,N-dimethyl-n-pentylamine were studied. Butyl radical loss from the molecular ion of N-methyl-dipentylamine was not studied. The ionisation energy of n-pentylamine
was measured as 8.64 eV and the appearance energy of the m/z 30 fragment measured
as 9.58 eV. This gives an energy barrier to reaction of 0.94 eV ( 90.7 kJ moi- 1 ),
identical to that given by Bowen and Maccoll for the same reaction [98]. The
ionisation energy of N,N-dimethyl-n-pentylamine was measured as 7.72 eV and the
appearance energy of the m/z 58 fragment measured as 8.62 eV. This gives an energy
barrier to reaction of 0. 90 e V ( 86.8 kJ mol-1 ). As the kinetic model assumes that the
a-cleavage is a concerted process, the energy barrier to reaction is taken to be
equivalent to the critical energy, assuming that the excess energy in the transition state,
E+, is zero. The lower value of 86.8 kJ moi- 1 was chosen to represent the critical
energy for the a-cleavage loss of butyl radical from the molecular ion of (1 ).
For the molecular ions of (2) - (6), the two competing a-cleavages have different
critical energies, due to the zero-point energy (ZPE) difference of the respective
transition states. In all cases, the critical energy for the loss of the labelled butyl radical
was set to 86.8 kJ mol- 1 plus the ZPE difference calculated for the two transition states. 49
4.3.5. Optimisation of the transition state.
As a starting point, the transition state vibrational frequencies were taken to be identical to the corresponding frequencies of the molecular ions. Calculations based on this model produced an ion abundance ratio of 1.0, as they ought since the transition states for both a-cleavages were identical. As the observed ion abundance ratio l8 /10 is greater than 1.0 in each case (Table 4.3), the rate of the labelled butyl radical loss must be less than the rate of the unlabelled butyl radical loss. If this effect arises primarily from a larger critical energy for the labelled reaction, the isotopically sensitive vibrational frequencies on moving from reactant to transition state must decrease, i.e. the transition state must be "looser" than the molecular ion.
For each of the compounds (2) - (6), the vibrational frequencies at the labelled position
were considered to be isotopically sensitive, and were lowered until the calculated ion
abundance ratio was in good agreement with the experimental ion abundance ratio. The
vibrational frequencies of both the labelled and unlabelled transition states were lowered
by the same proportion. Once the appropriate lowering for a particular position had
been determined, this lowering was included in further calculations, whether or not the
compound was isotopically labelled in the position. As the transition state model was
progressively adjusted, minor corrections to the previously determined optimum
frequency lowerings were necessary in order to reproduce accurately the experimental
valu~s.
The final model of the transition state, which reproduced closely the experimental ion
abundance ratios, is described in Table 4.5. There is an apparent general relaxation
along the carbon backbone, which is greatest at the ~-carbon and which decreases as
the distance from the Ca - c13 linkage is increased. Table 4.5. Transition state adjustments, critical energy differences and calculated ion abundance ratios. For all compounds E0 for the loss of C4H9· was 86.8 kJ moi-1 and E0 for the loss of labelled butyl radical was 86.8 + ~ZPE kJ moi-1.
Compound Transition state ~ZPE Ion abundance lowering/% I kJ mol-1 ratio
(2) 2.81 1.oo(a) 1.26
(3) 10.32 2.01 1.68
(4) 5.26 0.81 1.17
(5) 1.65 0.24 1.05
(6) 0.90 0.31 1.11
(a) The critical energy difference ~ZPE is larger for compound (2) than for compound (4), which has a larger transition state lowering, due to the effect of 5 deuterium atoms in (2) as opposed to 2 deuterium atoms in (4). (See Appendix 3.) 50
The transition state adjustments, critical energy differences and calculated ion abundance ratios for the five compounds are shown in Table 4.5. The vibrational frequencies for the ten transition states used (two for each compound) are given in
Table 4.4.
4.4. Results of the calculations.
In order to test the predictive nature of the model, isotope effects on relative rates as a function of time were calculated. A grain-size of 0.20 kJ moi-1 and Emax of 600 kJ mol- 1 were used. The results (Figure 4.4) indicate that normal isotope effects, i.e. kH(t) I k0 (t) > 1.0, will be present for compounds (2), (3), (4) and (6) at short times, and the isotope effect increases with time. No isotope effect, i.e. kH(t) / k0 (t) = 1.0, is predicted for compound (5) at short times, although the isotope effect becomes normal at longer times.
FIK results for compounds (3), (4) and (5) were obtained by Prof. N.M.M. Nibbering
[99], and these results are indicated on the relevant diagrams in Figure 4.4. Also
indicated are the ion abundance ratios following field ionisation obtained by Prof.
Nibbering in the first and second field-free regions of the mass spectrometer used. The
agreement between calculated and measured ratios is good for compounds (3) and (4),
but for compound (5) the FIK results suggest that a small inverse isotope effect, i.e.
kH(t) / k0 (t) < 1.0, is operating. As the rates k(t) are dependent upon rate coefficients
k(E) at large energies E, Emax was increased to 1200 kJ moi- 1 and the isotope effects on
relative rates recalculated for all compounds. The resultant curves for kH(t) / k0 (t) are shown in Figure 4.5. Negligible effect on the calculated ratios at times greater than
10- 1o s is predicted, while the ratios at times less than 10- 10 s decrease for all
compounds. For compounds (4) and (5) the ratio k8 (t) / k0 (t) becomes less than 1.0, i.e. an inverse isotope effect is predicted, and for compound (6) a ratio of 1.0 is Figure 4.4. Calculated kH(t)/k0 (t) vs. t curves for compounds (2) - (5) with Emax set at 600 kJ moi-1. * indicates FIK results from Prof. Nibbering. • indicates first field-free region metastable ratio from Prof. Nibbering.
■ indicates second field-free region metastable ratio from Prof. Nibbering. Results measured in this work are indicated by H. 00
-3
-5.00 -4.00
00
-6
00
s )
-7
00
tin1e/
-8
(
0
00
1091
-9
5
00
-10
a,a,e,e,e-D
:
(2)
Compound
(a)
-12.00 -11.00
4.4.
Figure
-13.00
00
-14
0 C ru
0 -
CD
0 uJ "q' ...--1 0
0
ru
0 CD 0 C 0
I
0
L 0
Q) > Q) (f) 0 Q) L
_'::,(_
+--'
_'::,(_
+--'
+--'
-
+--'
------
------'-._./ ~
---- '-._./
'-._./ 0 0 ru Figure 4.4. (b) Compound (3): P,P-D2
,,..---.,. 0 m ,,--..._ ...... , - '---..../
Q I
0 < 1 ,,--..._...... , ~l- • '---..../ • ~~I (f) - Q) ...... , 0 L ** 0 ru $*~ Q)> - ...... , * 0 Q) 0 L O ....,
0 m 0 -14 00 -13 00 -12.00 -11. 00 -10.00 -9.00 -8.00 -7.00 -6.00 -5.00 -4.00 -3.00
109 10 ( time/s ) 0 0
(T) I
0 0 V I
0 0 tn I
0 0
(D I
,,,,---...._ 0 0 r--- (f) I ~ Q)
0 E 0 +--' CD I ..____,., 0- 0 0, 0 0 OJ I
0 0 0 ..---< I
0 0
..---< ~ I -0 c:: ::,
8. 0 E 0 0 ru u ..---< I
0 0
(T) ..---< I
0 0 V ..---< 00 c' 08 o' 0 0
(Y) I
0 0 '<1" I
0 0 LO •• I
0 0 CD I
,,----...._ 0 • 0 r--- (/) I ~ Q)
0 E 0 --+-' CD I '-----"
a
0 0, 0 0 OJ I
0 0 0 -I
0 0 ,....._ ~ -I -0 * C ::l 0 0.. 0 E 0 0 u ("\J -I
0 0
(Y) -'
0 0 '<1" 00 c 08 T 09 T Qt, T Oc l 00 T 08 o' 0 0
(") I
0 0 'Sr '
0 0 lD I
0 0 co I
~ 0 0 r-- (/) I ~ Q)
0 E 0 -+--' CD I -..____.,,,
0
0 (}) 0 0 m
0 0 0 -'
0 0
§: -' -0 C ::, 0 0. 0 E 0 8 (\J -'
0 0
(") -'
0 0 'Sr
00 c' 08 T 09 T 017 T 0c' T 00 T 08 o' Figure 4.5. Calculated k8 (t)/k:0 (t) vs. t curves for compounds (2) - (5) with Emax set at 1200kJ moi-1. * indicates FIK results from Prof. Nibbering. • indicates first field-free region metastable ratio from Prof. Nibbering.
■ indicates second field-free region metastable ratio from Prof. Nibbering. Results measured in this work are indicated by H. 0 0
(T) I
0 0 ,q- I
0 0 LO I
0 0 CD I
,------. 0 0 r-- (/) I ~ Q) E 0 0 -+--' CX) I ------0- 0 0) 0 0 m I
0 0 0 -I
0 0 ~ ~ -0 I i:: g_;::s
E 0 u0 0 (\J~, I
0 0
(T) -I
0 0 ,q- 00 2 Oc' 1 00. t 08 o' 0 0
(T) I
0 0 "'3" I
0 0 U1 • I
0 0 CD I
,,------,_ 0 - 0 r--- (/) I ~ (I)
0 E 0 +--' CD I ______, 0- 0 CTl 0 0 m !
0 0 0 N -I Cl I *~ C!l. c::i 0 ,..,. **** 0 C -I -0 C: ;:) 0 0.. 0 8 0 0 u (\J e -I V") ..,. 0 ... 0 0 "'3" OCJ c 08 T 09 T ov t Oct 00. t 08 o'- (- (:l) 0 >1/(+t>1 ) Sd+DJ d/\!+DldJ 0 0 (T) I 0 0 "'1" I 0 0 r.n I 0 0 co I ,,,..._ 0 0 r--- (f) I ~ Q) 0 E 0 +--' CD I "------" -D 0 0) 0 0 m I 0 0 0 -I 0 0 ~ "O C: -I g_ e 0 0 u 0 ru -I 0 0 (T) -I I I 0 I 0 I "'1" 00 c 00. l 08 o' 0 0 (T) I 0 0 .... ! 0 0 lD •• I 0 0 CD I 0 • 0 ------r--- (/) I ~ Q) E 0 0 -+--' (I) I '-----" 0 ~ 0 Q) 0 0 Ol I 0 0 0 ..---, I 5: 0 -0 0 c:: ..... 5 I c. * e u0 0 0 ru -I 0 0 (T)..... I 0 0 .... 00 c 00 T os o' 0 0 (Y) I 0 0 V I 0 0 lD I 0 0 CD I ,,----..., 0 0 r--- (f) I ~ Q) E 0 0 +--' co I ...._____., 0 0 CJ) 0 0 0) I 0 0 0 -I 0 0 -I 0 0 ru -I 0 0 (Y) -' 0 0 V 00 c 08 T 09 T Ot:r T Oc' T 00 T 08 o'- 51 predicted. The agreement between the calculated ratios and the FIK results does not change appreciably, however, the presence of the inverse isotope effects is significant. The inverse isotope effect at short times for compounds (4) and (5) arises as a consequence of the small difference between the critical energies of the competing a-cleavage reactions, and the different numbers-of-states for the two transition states. At short times, rate coefficients k(E) at high energies E determine the ratio of the relative rates. In the transition state for C4H7D2 • loss, deuterium substitution lowers the vibrational frequencies such that the number-of-states in that transition state, G0 *(E), is greater than the number-of-states in the transition state for C4H9· loss, GH*(E). For compound (5), the small difference in critical energies (0.24 kJ moi-1) is not sufficient to overcome the number-of-states effect at high energies E, and so loss of C4H7D2 • is faster than loss of C4H9 ·, i.e. k0 (E) > kH(E) at high energies E, and gives rise to the inverse isotope effect at short times. For compound (4) the number-of-states effect displaces the critical energy difference of 0.81 kJ moi-1. The increase in k0 (E) over kH(E) is demonstrated by the log10(k(E)) vs. E curves shown in Figure 4.6. With compounds (2), (3) and (6), kH(E) > k0 (E) at energies up to 1000 kJ moi- 1, although at high energies kH(E):::::: k0 (E). For compounds (4) and (5), however, k8 (E) > k0 (E) at low energies, but at energies above 800 kJ moi- 1 k0 (E) > k8 (E). In compounds (2) and (3) the difference in critical energies displaces this number-of-states effect, while in compounds (4) and (5) the critical energy differences are displaced by the number-of-states effects. In compound (6) the two effects cancel at the maximum energy considered. The variation in the calculated isotope effect on relative rates k8 (t) / k0 (t) over the time range 10- 14 s to 10-3 s as Emax is varied is shown in Table 4.6. The appearance of an inverse isotope effect can be directly traced back through the Figure 4.6. Log10(k(E)) vs. E curves for the five deuterated compounds (2) - (6). Schematic representations of expanded regions of the curves are indicated at low and high energies. oo oo 120 120 100.00 100.00 110.00 (E) (E) (E) (E) 90.00 90.00 ko ko kH kH oo oo so so oo oo 10 10 ) ) oo oo a a 1 1 50 50 ~.:1 ~.:1 mol- oo oo so so kJ kJ oo oo 40 40 5 5 Energy/( Energy/( oo oo 30 30 (E) (E) a,a,E,E,E-D 0 : : k (2) (2) 20.00 20.00 Compound Compound oo oo (a) (a) 10 10 4.6. 4.6. Figure Figure oo oo 'o 'o I I "'+-----'--,-----,------r-----r----~----r---~---~---~.-----..-----..------, "'+-----'--,-----,------r-----r----~----r---~---~---~.-----..-----..------, 0 0 ru ru C' C' C' C' C' C' C' C' C' C' 0 0 0 0 C' C' ru ru '<1 '<1 0 0 C' C' C' C' lD lD - 0 0 C' C' 0 0 C' C' C' C' C' C' CD CD 0 0 0 0 CJ) CJ) (f) (f) _y_ _y_ I I w w "'-..__ "'-..__ ....___., ....___., ---- ,,,..------._ ,,,..------._ ,,,..---... ,,,..---... - 0 0 0 (\J 0 0 0 0 0 0 0 0 0 0 OJ 0 \ 0 0 '\ CD \ 0 \. 0 0 'i r--- \ '\ \\ 8b----.. o.---< \ co*' 0 \( 0 E \ 0 ¾ 0 J ~ tJ, __y '-....,_/ 0 ~ 0 >. 0.,,. en \ L N Q) Cl I C ea. 0 w c:i \ 0 ,,__ 0 (Y) C "Cl C ;:) 0 §: 0. 0 -E :z:: 0 0 ~ 0 u (\J e 0 " 0 0 ..-----~----~----~----~---~----~-----+o oo oi oo s oo g oo r 00·2 oo o oo 2- oo r- ( l_S/(3)>1 ) DL60j 120.00 120.00 (E) (E) (E) (E) ko ko kH kH 110.00 110.00 00 00 100 100 QO QO 90 90 ()0 ()0 ------ 80 80 00 00 70 70 ) ) 00 00 1 1 60 60 ~-:1a1 ~-:1a1 mol- QO QO 50 50 kJ kJ 40.QO 40.QO 2 2 Energy/( Energy/( QO QO y;y-D 30 30 (4): (4): ()0 ()0 20 20 Compound Compound (c) (c) 00 00 4.6. 4.6. 10 10 Figure Figure QO QO 'o 'o I I V V 0 0 0 0 0 0 0 0 V V (\J (\J - 0 0 0 0 0 0 0 0 0 0 CD CD 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (\J (\J - 0 0 0 0 0, 0, (f) (f) I I _y _y w w ,,------..._ ,,------..._ --...___,,o --...___,,o '-...../ '-...../ ,,----... ,,----... "'--0 "'--0 - 00 00 120 120 (E) (E) ko ko 110.00 110.00 00 00 100. 100. 90.00 90.00 00 00 BO BO (E) (E) 00 00 0 0 k 70 70 ) ) 1 1 60.00 60.00 *lCT *lCT n1ol- 00 00 50 50 kJ kJ 00 00 40 40 2 2 Energy/( Energy/( o,&-D 00 00 : : 30 30 (5) (5) (E) (E) 0 k 00 00 20 20 Compound Compound (d) (d) 4.6. 4.6. 10.00 10.00 Figure Figure oo oo 'o 'o I I V V 0 0 0 0 ru ru C C 0 0 0 0 0 0 C C 0 0 V V C\J C\J 0 0 C C 0 0 0 0 C C 0 0 u) u) 0 0 - ll) ll) C C 0 0 - 0 0 CJ) CJ) 0 0 (f) (f) _:::{_ _:::{_ I I w w ,,.---...__ ,,.---...__ ~o ~o '--.../ '--.../ '--.../ '--.../ ,,----... ,,----... - 120.00 120.00 00 00 (E) (E) (E) (E) 0 0 110 110 k ktt ktt 00 00 100 100 00 00 90 90 00 00 80 80 00 00 70 70 ------ ) ) 00 00 1 1 60 60 *lc1 *lc1 mol- 00 00 50 50 kJ kJ 00 00 40 40 3 3 Energy/( Energy/( 00 00 E,E,e-D 30 30 : : (6) (6) 00 00 20 20 Compcrund Compcrund (e) (e) 00 00 4.6. 4.6. 10 10 Figure Figure oo oo 'o 'o 'Sl" 'Sl" 0 0 0 0 r;-i r;-i 0 0 0 0 0 0 0 0 rJ rJ 'Sl" 'Sl" 0 0 0 0 0 0 0 0 0 0 0 0 cD cD 0 0 0 0 CD CD 0 0 0 0 0 0 - 0 0 0 0 01 01 lfl lfl ~ ~ I I w w ~ ~ ---._,,o ---._,,o ----- ,,---.._ ,,---.._ '-...._./ '-...._./ - Table 4.6. Variation of the isotope effect on relative rates, kH(t) / k0 (t), as both time and Emax are varied. (a) Compound (2) : a,a,e,e,e-D5 TIME / s Eaax I (kJ mol-1) 200 400 600 800 1000 1200 1.00E-14 1.1851 1.1055 1.0761 1.0607 1.0512 1.0454 1.00E-13 1.1851 1.1055 1.0761 1.0607 1.0513 1.0454 1.00E-12 1.1851 1.1056 1.0761 1.0609 1.0517 1.0463 1.00E-11 1.1851 1.1056 1.0766 1.0628 1.0570 1.0557 1.00E-10 1.1851 1.1060 1.0816 1.0787 1.0787 1.0787 1 .OOE-09 1.1852 1.1104 1.1053 1.1053 1.1053 1.1053 1.00E-08 1.1853 1.1348 1. 1348 1.1348 1.1348 1.1348 1 .OOE-07 1.1868 1.1665 1.1665 1.1665 1.1665 1.1665 1.00E-06 1.2009 1.2006 1.2006 1.2006 1.2006 1.2006 1 .OOE-05 1.2379 1.2379 1.2379 1.2379 1.2379 1.2379 1.00E-04 1.2796 1.2796 1.2796 1.2796 1 .2796 1.2796 1.00E-03 1.3286 1.3286 1.3286 1.3286 1.3286 1.3286 (b) Compound (3): P,P-D2 TIME / s Emax I (kJ mol-1) 200 400 600 800 1000 1200 1.00E-14 1.4325 1.2045 1. 1335 1.0985 1.0778 1.0647 1 .OOE-13 1.4325 1.2045 1. 1335 1 .0986 1.0779 1 .0649 1.00E-12 1.4325 1.2045 1.1336 1.0990 1.0789 1.0669 1.00E-11 1.4325 1.2046 1. 1347 1.1035 1.0909 1.0884 1.00E-10 1.4325 1.2057 1.1468 1.1407 1.1407 1.1407 1 .OOE-09 1.4325 1.2170 1.2052 1.2052 1.2052 1.2052 1.00E-08 1.4329 1.2816 1.2816 1.2816 1.2816 1.2816 1.00E-07 1.4372 1.3712 1.3712 1 .3712 1.3712 1.3712 1.00E-06 1. 4794 1.4772 1.4772 1.4772 1.4772 1.4772 1 .OOE-05 1.6060 1.6060 1.6060 1 .6060 1.6060 1.6060 1 .OOE-04 1.7694 1. 7694 1.7694 1. 7694 1. 7694 1. 7694 1.00E-03 1.9895 1.9895 1.9895 1.9895 1.9895 1.9895 (c) Compound (4): y,y-D2 TIHE / s Emax I (kJ mol-1) 200 400 600 800 1000 1200 1.00E-14 1.1122 1.0447 1.0184 1.0044 0.9957 0.9903 1.00E-13 1.1122 1.0447 1.0184 1.0044 0.9958 0.9904 1.00E-12 1.1122 1 .0447 1.0184 1.0046 0.9963 0.9913 1.00E-11 1.1122 1.0448 1.0189 1 .0067 1.0017 1.0008 1 .OOE-10 1.1122 1.0452 1.0242 1.0222 1.0222 1.0222 1.00E-09 1. 1122 1.0501 1.0467 1.0467 1.0467 1.0467 1 .OOE-08 1.1124 1.0730 1.0730 1.0730 1.0730 1.0730 1.00E-07 1. 1141 1.1006 1. 1006 1.1006 1.1006 1.1006 1.00E-06 1. 1288 1.1290 1. 1290 1. 1290 1.1290 1.1290 1.00E-05 1.1585 1.1585 1.1585 1.1585 1.1585 1.1585 1.00E-04 1.1893 1.1893 1. 1893 1. 1893 1. 1893 1.1893 1.00E-03 1.2223 1.2223 1.2223 1.2223 1.2223 1.2223 (d) Compound (5) : o,&-D2 TIHE / s Emax I (kJ mol-1) 200 400 600 800 1000 1200 1.00E-14 1.0229 1.0066 o. 9997 0.9958 o. 9933 0.9919 1.00E-13 1.0229 1.0066 0.9997 0.9958 o. 9933 0.9919 1.00E-12 1.0229 1.0066 0.9997 0.9959 0.9935 0.9922 1.00E-11 1.0229 1.0066 0.9999 0.9966 0.9954 0.9951 1.00E-10 1.0229 1.0068 1.0019 1.0015 1.0015 1.0015 1.00E-09 1.0229 1.0088 1.0085 1.0085 1.0085 1.0085 1.00E-08 1.0230 1.0159 1.0159 1.0159 1.0159 1.0159 1 .OOE-07 1.0241 1.0236 1.0236 1.0236 1.0236 1.0236 1.00E-06 1.0314 1.0319 1.0319 1.0319 1.0319 1.0319 1.00E-05 1.0410 1.0410 1.0410 1.0410 1.0410 1.0410 1.00E-04 1.0511 1.0511 1.0511 1.0511 t.051 t 1.0511 1 .OOE-03 1.0628 1.0628 1.0628 1.0628 1.0628 1.0628 (e) Compound (6): £,£,£-D3 TIHE / s E11a:it I (kJ mol-1) 200 400 600 800 1000 1200 1.00E-14 1.0685 1.0299 1.0154 1.0077 1.0028 0.9999 1 .OOE-13 1.0685 1.0299 1.0154 1.0077 1.0029 1.0000 1.00E-12 1.0685 1.0299 1.0155 1.0078 1.0032 1.0005 1 .OOE-11 1.0685 1.0299 1.0158 1.0090 1.0063 1.0058 1.00E-10 1.0685 1.0302 1.0189 1.0178 1.0178 1.0178 1 .OOE-09 1.0685 1.0332 1 .0315 1.0315 1.0315 1.0315 1.00E-08 1.0686 1.0466 1.0466 1.0466 1.0466 1.0466 1 .OOE-07 1.0699 1.0630 1.0630 1.0630 1.0630 1.0630 1.00E-06 1.0806 1.0810 1.0810 1.0810 1.0810 1.0810 1 .OOE-05 1.1013 1.1013 1.1013 1.1013 1.1013 1.1013 1.00E-04 1.1250 1.1250 1.1250 1. 1250 1.1250 1. 1250 1.00E-OJ 1. 1543 1.1543 1.1543 1. 1543 1. 1543 1. 1543 52 number-of-states to the vibrational frequencies and the grain-size for the calculations. As the vibrational frequencies are rounded to multiples of the grain-size, and the critical energy difference (~ZPE) is calculated from these frequencies, the rounding of the frequencies affects ~ZPE. When ~ZPE is increased, the number-of-states effect becomes less significant. Consider, for example, compound (5). Decreasing the grain-size to 0.07 kJ moi-1 for the calculations causes ~ZPE to be set to 0.24 kJ moi-1, as in the calculations using a grain-size of 0.20 kJ moi-1, but the rounding of the frequencies to the smaller grain-size increases G*H(E) more than G*0 (E), resulting in k8 (t) being greater than k0 (t) at all times. Increasing the grain size to 0.13 kJ moi- 1, reduces ~ZPE to 0.12 kJ moi- 1 and causes G*0 (E) to be more competitive with G* 8 (E) at large energies resulting in an inverse isotope effect at short times. When the grain-size for the calculations is such that ~ZPE is large, only normal isotope effects are predicted, but when the grain-size is such that ~ZPE is small, inverse isotope effects are predicted at short times. The implications of the dependence of the isotope effect kH(t) / k0 (t) on the grain-size and Emax highlights two facets of the calculations. First, the accuracy of the FIK results is of paramount importance. If the results are accurate and an inverse isotope effect is present, the model for the transition state must reproduce this effect to be valid. That the FIK results show a large spread is not reassuring for the interpretation of the calculations. Second, the maximum energy considered, Emax• is shown to exert an influ,ence on the calculations at short times. Without knowing the experimental Emax the limit for the rate coefficients k(E) at large energies E, which determine the isotope effect at short times, cannot be assigned. The calculation of reliable ratios, kH(t) / k0 (t), for the FIK time-frame becomes difficult, making the interpretation of the results more complex. 53 4.5. Discussion of the transition state model. In the transition state predicted by the calculations, the C-H frequencies of the pentyl group which undergoes a-cleavage are lower than the corresponding frequencies of the non-reacting pentyl group. The major lowering is at the P-carbon which corresponds to the site of the incipient radical. Considering the vibrations as harmonic oscillators, the frequency, u, is related to the force constant, k, as follows [54]: I 1 k 12 u = 2n c(µ) µ is the reduced mass of the vibration and c is the speed of light. If the reduced mass of the vibration in the transition state is taken to be the same as that in the molecular ion, then as u and k are the only variables. The change in vibrational frequency, .!\u, can be related to the change in the force constant, .!\k, as: 1,2 .!\ u = ( .!\k ) The ~-carbon frequency lowering of 10.32 % corresponds to lowering the force constant by approximately 20%. Herzberg has reported force constants for C-H stretching where the carbon is saturated and where the carbon is a radical [100]. The force constants reported are 4.79x105 and 4.09x105 dynes cm-1 respectively. The force constant for the C-H stretching in the radical is approximately 15% lower than where the carbon is saturated. This lowering of the force constant is similar to the lowering predicted by the calculations for the P carbon in the transition state. On the other hand, Pacansky and co-workers have measured the infrared spectra of alkyl radicals in argon matrix [101,102], and have concluded that C-H stretching and bending frequencies are larger in radicals than in the corresponding alkanes [72). If the 54 reduced masses of the vibrations are considered the same for both radical carbon and saturated carbon, the force constants have also increased. Taking the experiment to be reliable would indicate that the transition state predicted by the calculations does not resemble the reaction products, or at least not the product radical. Recent work has suggested that the a-cleavage reaction of aliphatic amines may not be as simple as previously believed [ 103]. A reverse critical energy for the a-cleavage is proposed [103] which would place the reaction products lower in energy than the transition state. A reverse critical energy would be consistent with the large activation energy for the addition of n-alkyl radicals to alkenes [104]. The presence of a reverse critical energy would not undermine the QET calculations as the calculations depend only upon the forward critical energies. The presence of significant reverse critical energies would, however, lessen any requirements for the transition state to resemble the reaction products. The agreement between the calculated force constant alterations for the P-carbon and the value given by Herzberg would then appear fortuitous, especially as Herzbergs' values for the force constants are in apparent disagreement with the recent experiments. 55 Chapter 5. Investigations of the McLafferty rearrangement. 5.1 Introduction. The McLafferty rearrangement holds a unique place in mass spectrometry. It is probably the most studied reaction in mass spectrometry, but more significantly it is by convention the only "named reaction" in mass spectrometry. The McLafferty rearrangement has been comprehensively reviewed [105,106], and, although these reviews are now dated, they still cover much of the work performed in an effort to elucidate as far as possible the mechanistic details of the reaction. The McLafferty rearrangement is described as y-hydrogen transfer through a six-centred transition state to an unsaturated site, followed by, or concurrent with, 13-bond cleavage [107,108]. The rearrangement has been proposed to occur in a wide range of radical cations including carbonyl systems, their nitrogen and sulfur analogs, long chain olefins and aromatic compounds [105]. Many compounds of biological interest contain these functional groups, and it is therefore not surprising that the McLafferty rearrangement has been proposed to occur in a number of these compounds [109-112]. The rearrangement has recently been proposed to occur in high molecular weight polystyrene radical cations [113]. In principle, the McLafferty rearrangement can proceed in either a concerted or stepwise fashion as depicted in Scheme 5.1 for ionised 2-pentanone. Both mechanisms have received support [i 14,115], but current opinion is that the reaction is best described as occurring in a stepwise fashion. Scheme 5.1. The McLafferty rearrangement in ionised 2-pentanone. (a) concerted model +. +. H OH R_vJ + ~ -~ (b) stepwise model +. H Jj 56 5.2 Evidence for the stepwise nature of the McLafferty rearrangement. Strong evidence supporting the stepwise nature of the McLafferty rearrangement has been obtained for carbonyl containing systems, most notably the molecular ions of butanoic acid, 2-pentanone and n-hexanal. The McLafferty rearrangement in the butanoic acid molecular ion has been proposed to occur in a stepwise fashion primarily because of the extensive hydrogen exchange which has been observed between the ~ and y- hydrogens of the aliphatic chain and the hydroxylic hydrogen. In general, the occurrence of hydrogen exchange is taken as one of the key proofs of the stepwise nature of a reaction. A mechanism for the hydrogen exchange in butanoic acid has been suggested by Weber et al. [116] based upon their own measurements and those of McAdoo et al. [ 117]. The mechanism suggested is reproduced in Scheme 5.2. That the pathways leading to the elimination of some of the C2H4 species are so extensive suggests that species formed by these pathways would not be observed at short reaction times, but would be significant at longer times. FIK experiments [116] show that at times less than 10-10 s only y- hydrogen transfer precedes P- cleavage. At times greater than 10-10 s hydrogen exchange occurs prior to P-cleavage. In all cases, elimination of C2H4 proceeds only as a consequence of hydrogen transfer from the y position. Transfer of hydrogen from the P- position results in hydrogen exchange, but does not lead directly to product formation. The time-dependence of hydrogen exchange observed in the butanoic acid radical cation has also been observed in FIK studies of octan-2-one [ 118] and pheny lhexanone [ 119]. In both of these systems, no hydrogen exchange is observed in product ions formed 0 0 •oH ~OH' ~OH' ·CzH/H" ·CzH/H" 11 11 OH 1r 1r ' ' ~HII ~HII +• +• ow ow Hz"" Hz"" J-OH• J-OH• H"H• H"H• y y J J hydrogen hydrogen rotation rotation ~ ~ -CzHPH -CzHPH 0 0 H H various various OH" OH" •• •• H...._• H...._• Hy~OHY Hy~OHY 11 11 H2Y, H2Y, showing showing H5HYH• H5HYH• H'Hy~ H'Hy~ OHY OHY OHY OHY )-oH acid acid )-oHY )-oHY rotation rotation ~- H H y y • 2 2 • • H P P • • butanoic butanoic [116]) [116]) Ci/i Ci/i 0 0 •· •· II II p p ·C2H2'HYH• ·C2H2'HYH• OH"' OH"' II II y y OH OH OH OH ~OH ~OH • • ionised ionised 0 ~OH ~OH ·~:1l ·~:1l I I ~ ~ HzYW HzYW y y P'Hyl P'Hyl 'o• 'o• H,;.,, HYH~ HYH~ ·•Hyl ·•Hyl H H Hy Hy H H . . from from • • ~ ~ /' /' (Reference (Reference H H 'U 'U Hy Hy H H H y y Hy Hy rototlon rototlon ~ ~ ---"" ---"" r r ethylene ethylene __,_ __,_ ...... - .!..!i.:. .!..!i.:. --- of of OHO OHO 0 0 mechanisms. mechanisms. •· •· II II 0 0 OH OH ~Hy ~Hy 1r~ 1r~ AOH· AOH· ~OHY ~OHY I I UlOHY UlOHY UO UO YH• YH• • • KzYHo KzYHo Hz Hz exchange exchange Elimination Elimination rcrt.atiOQ. rcrt.atiOQ. W- ·C2H2"H2J ·C2H2"H2J rotation rotation ~ ~ 5.2. 5.2. -:0.HH -:0.HH y y 0 0 • • OHO OHO Hz" Hz" • • OHY OHY Hz" Hz" Scheme Scheme 1L. 1L. H!_ H!_ H\r H\r H H r-toH LJ LJ j j ~H• ~H• U 2 HY HY Hz'~~ Hz'~~ ~ ~ . . ~6 ~6 ~ ~ ~'Y, ~'Y, ~ ~ •· •· OHY OHY AOHo AOHo ~OH• ~OH• (~OY,?, (~OY,?, 57 within ?xlo-10 s of ionisation. Product ions formed after 10-6 s have undergone hydrogen exchange to the extent of approximately 70%. Further evidence for the stepwise nature of the McLafferty rearrangement in the butanoic acid radical cation comes from the observation of skeletal rearrangement [120]. Direct elimination of C2H4 results in the loss of (3- and y- carbons. 13c labelling has shown that at times less than 10-9 s after ionisation (3- and y- carbons are lost exclusively, while at times greater than 10-7 s the a.- carbon is lost with the (3- and with the y- carbons. Scheme 5.3 rationalises the loss of C2H4 containing the a.- and (3- carbons through a cyclobutanol intermediate. Loss of C2H4 containing a.- and y carbons may occur through the formation of a cyclopropyl intermediate [120]. The 2-pentanone radical cation eliminates C2H4 in a similar fashion to the butanoic acid radical cation. 2H and 13c labelling by McAdoo et al. [121] has shown that at times greater than 10-6 s extensive hydrogen exchange and skeletal rearrangement have occurred. Reaction mechanisms similar to those suggested for butanoic acid radical cation have been proposed [121]. Perhaps the most compelling evidence for the stepwise nature of the McLafferty rearrangement comes from FIK studies of n-hexanal by Morgan et al. [122] and Derrick et al. [ 123]. McLafferty rearrangement of the n-hexanal radical cation produces one of two Possible product ions ([C2H4O]+. and [C4H8]+·), with their complementary neutral species (C4H8 and C2H40 respectively). The two reactions are depicted in Scheme 5.4(a). At times less than 6x10- 11 s the formation of [C2H4O]+. is favoured, while at longer times the formation of [C4H8]+. is favoured. These observations have been explained on the basis of a common first step in which an intermediate ion is formed (Scheme 5.4(b)). The intermediate ion can react as shown in Scheme 5.4(c) to ...... I I ... u .... .-.u.... I I NU -,,u + + I I +O 0 I I \/ +O 0 ~/ u-...... £ u I UN I/ .... MU I.... I I .. u I nU . M . I l ~u l l I I .;_ I I 0 0 01'. 0 +O I + ~ / ~/ ~ 0 u.... -u.... 'u/ .,,uI I I ...... NU.... '"'U•..... I I :r: .-.u ... u NU ... M M I I I NU .. u -.u 1 l 1 I I +01'. 0 ·oI oI ~/ ·,1 £ u U-UN I I I - ..,u:r: £ -f: / '\...... u-u.., M I ... i5 ""Y Scheme 5.3. Skeletal rearrangement of ionised butanoic acid leading to the elimination of ethylene. (Reference [ 120]) Scheme 5.4. McLafferty rearrangement in ionised n-hexanal showing (a) the two possible sets of reactions, (b) formation of the common intermediate ion and (c) the two reactions of the intermediate. (a) +• a ► C2H4~ +• + C 4H8 0 H ► C 2H40 - + C4H~ +· (b) ► H H (c) +• OH (i) + H ·HA +· (ii) OH 17 • I + H H~ 58 produce either [CH2CHOH]+- and CH 2 CHCH 2 CH 36:°eaction (i)), or CH2CHOH and [ cH 3 CHCHCH 3 +j. (reaction (ii)). Reaction (i) is suggested to have a higher critical energy than reaction (ii), but to be more direct and hence more competitive at short times. At longer times reaction (ii) is suggested to be more competitive due to the lower critical energy [123]. An alternative explanation would be that elimination of C4H8 to form [C2H4O]+- is a concerted reaction, and hence favoured at short times, while the elimination of C2H 4O to produce [C4H 8]+- is a stepwise process, slower than elimination of C4H8, but dominant at longer times. Charge retention by the olefin (c.f. reaction (ii) in Scheme 5.4(c)), has also been observed by Weber et al. [116] in the case of the n-hexanoic acid radical cation. At times less than 4x 10-11 s, the McLafferty rearrangement results in the elimination of neutral I-butene to form [CH2C(OH)2]+-. At longer times, CH2C(OH)2 is eliminated to form [CH3CHCHCH3]+-. These observations have been explained on the basis of the thermochemistry. ~Hr( CH2C(OH)2 ) is taken to be -382 kJ moi- 1, which combined with ~He( [CH3CHCHCH3]+-) of 869 kJ moi-1 predicts that the products in which the charge is retained in the olefin (487 kJ moi-1) will be 15 kJ moi-1 more stable than the products in which the neutral alkene is eliminated. ( Literature values for ~Hr([CH2C(OH)2]+- of 502 kJ moi-1 and for ~Hf(CH3CHCHCH3) of O kJ moi-1 [20,18] give ~r(products) as 502 kJ rnoi-1 .) 5.3. ·3-Ethyl-2-pentanone. In a recent isotopic labelling study of ethyl butyrophenone [124], D and 13c isotope effects have been found for the elimination of ethylene from the molecular ion. These isotope effects were taken as evidence that loss of ethylene from the molecular ion may best be described as a concerted process (Scheme 5.5), although the possibility of a Scheme 5.5. Elimination of ethylene from ionised ethyl butyrophenone by McLafferty rearrangement +. OH • 59 stepwise elimination was not dismissed [124]. In view of this apparent discrepancy with other studies of similar systems (c.f. 2-pentanone), a study of 3-ethyl-2-pentanone extensively labelled with D and 13c was initiated. 3-Ethyl-2-pentanone (7; Figure 5.1) allows both hydrogen transfer and C-C cleavage to be studied in an intramolecular environment. All 3-ethyl-2-pentanones used in this study (identified in Figure 5.1) were synthesised by Dr. M.B. Stringer in Prof. J.H. Bowie's laboratory at the University of Adelaide and were used without further purification. Ionised 2-pentanone decomposes in the metastable time-frame predominantly by loss of methyl radical and elimination of ethylene [121]. The MIKE spectrum of 2-pentanone, recorded using the instrument described in Chapter 3, is shown in Figure 5.2(a). The ratio [M-CH3]+ / [M-C2H4]+. is 1.5, consistent with previous measurements [121]. The MIKE spectrum of 3-ethyl-2-pentanone (Figure 5.2(b)) shows that methyl loss is much less significant in the metastable time-frame for this compound ([M-CH3]+ / [M-C2H4]+. = 0.02). Loss of C3H6 and C2H5· are also evident in the MIKE spectrum of 3-ethyl-2-pentanone. Partial MIKE spectra of the nine isotopically labelled 3-ethyl-2-pentanones are shown in Figure 5.3. No other fragment ions of significant intensity were observed. All MIKE spectra, except for that of compound (16), were obtained using electron ionisation with an electron energy of 70 eV and an accelerating potential of 8 kV. Compound (16) was run under field ionisation with an emitter potential of 10 kV. The normalised intensities for the loss of ethylene and ethyl radical species from compounds (7) - (16) are given in Table 5.1. Table 5.2 gives the normalised intensities for the loss of propene species from the compounds. It is evident from the spectra of the D labelled compounds ((8) - (11) and (15)) that Figure 5.1. 3-Ethyl-2-pentanone (7) and the 9 isotopomers studied. * indicates the location of 13 C. 0 (7) 0 0 0 (9) (8) (10) 0 0 0 * * (11) (12) (13) 0 * 0 *D3 0 *D3 D2 D2 D3 D (14) (15) (16) * D3 D3 (a) (b) Figure 5.2. MIKE spectra of the molecular ions of (a) 2-pentanone, and (b) 3-ethyl-2-pentanone. (a) Compound (8) J 40 30 20 (b) Compound (9) 40 30 20 (c) Compound (10) 40 30 20 Figure 5.3. Partial MIKE spectra of the 9 labelled 3-ethyl-2-pentanones. The x axis indicates the mass of the eliminated neutral (d) Compound (11) _JJJ I 40 30 20 (e) Compound (12) 40 30 20 (t) Compound (13) 30 20 Figure 5.3. (Cont.) (g) Compound ( 14) 30 20 (h) Compound (15) 30 20 (i) Compound (16) ____ ...... ,.__ _ __....,._ __ _ 35 30 Figure 5.3. (Cont.) Table 5.1. Normalised intensities of the [M- ethylene]+- products from (7) - (16). Compound no. M-28 M-29 M-30 M-31 M-32 M-33 M-34 M-35 7 100 9 8 100 20 22 3 9 29 94 100 4 10 100 41 29 2 4 11 100 22 3 12 14 2 12 100 90 7 13 100 96 7 14 100 8 15 1 27 100 67 2 2 16 100 94 2 2 Table 5.2. Normalised intensities of the [M - propene ]+ · products from (7) - ( 16). Compound no. M-42 M-43 M-44 M-45 M-46 M-47 M-48 M-49 7 100 8 57 100 43 9 10 100 80 20 10 91 64 100 91 11 12 29 32 59 100 12 100 100 13 100 100 14 1 100 1 15 5 42 100 89 33 7 16(a) (afNo [M-prcpene]+- products are observed from (16) under field ionisation. 60 hydrogen rearrangement has occurred. The metastable time-frame (observation window) for compound (9), m/z 118, extends from 1.Sxlo-5 s to 4.lxlo-5 sand is similar for the other compounds. The identification of specific hydrogen transfer / C-C bond cleavage paths in this time range is obscured by hydrogen exchange and the appearance of isobaric fragments. In (9), for example, the peak corresponding to loss of 29 u, assigned to loss of C2H5· and C2H3D, is almost as intense as the peak due to loss of 30 u which arises from the direct elimination of C2H2D2. Loss of C2H5· or C2H3D can only proceed after hydrogen exchange has occurred. This suggests that hydrogen exchange is a quite significant process, occurring through a five-membered transition state [125,126] in a manner analogous to that reported for 2-pentanone [121] and butanoic acid (Scheme 5.2) [116]. Elimination of C2H5· can only occur after a series of hydrogen exchange steps and is thus unlikely to be significant in the loss of 29 u from (9). Skeletal rearrangement is not observed in the elimination of ethylene from any of the 13c labelled compounds (12), (13), (14) or (16), although compound (14) is the only compound in which skeletal rearrangement would be clearly evident, either by the loss of 12C2H4 or by the loss of 13C2H4. Neither of these is observed. (Loss of 13c2H4 would give rise to a peak corresponding to loss of 30 u, which is present, but the intensity of this peak is such that it suggests only the loss of 13c12CH5· contributes.) Given that no skeletal rearrangement is observed, the ratios [M-12C2H 4]+. I [M- 13C12CH4]+. for (12) and (13), and the ratio [M-13C2H4]+. I [M-13C12CD4]+. for ( 16) should provide information on the relative importance of the ~- and y- carbons in the McLafferty rearrangement. After correcting for the contribution to [M-29]+. from [M- 12c2H5]+ in (12) and (13), isotope effects on ion abundances of 1.23 for (12) and 1.12 for ( 13) were calculated. For ( 16), all peaks are resolved, which allows an isotope effect on ion abundances of 1.06 to be measured. As the ~- carbon is directly 61 involved in the C-C bond cleavage, it is not surprising that the isotope effect for (12) is greater than that for (13) or (16). Correcting the isotope effects for fractional transmittances makes only a slight change to the values. For (16), T 0_ 5 for [M- 12C2H 4]+. is 17 meV and T0_5 for [M-13c 12CH4]+. is 18 meV. The ratio of fractional transmittances, calculated as described in Chapter 2, is 1.01 in favour of [M-12c2H4]+·. Correction for this gives the isotope effect in (16) as 1.05. Using the same transmittance ratio for (12) and (13) gives 1.22 and 1.11 respectively as the ion abundance ratios. Both of these ratios are significant. That the ratio for (12) is appreciable is expected as the 13c is directly involved in the C-C cleavage, while the ratio for (13) indicates that the terminal carbon is also influential in the reaction. The ratios for (13) and (16), where substitution of deuterium for hydrogen in all positions, reduces the ion abundance ratio from 1.11 to 1.05, provide evidence that the transfer of hydrogen is distinct from the C-C cleavage in the reaction. If the overall reaction were a concerted process the effect of deuteration would be to reduce the amounts of both C2D4 and 13c 12CD4 elimination by similar amounts, which should have little effect upon the ion abundance ratio. That the ratio is reduced is consistent with a stepwise mechanism for the reaction. Substitution of deuterium for hydrogen decreases the rate of hydrogen transfer, but not the rate of C-C cleavage. The rate of ethylene elimination should become less dependent upon the C-C cleavage and become more dependent upon the hydrogen transfer. The rate of hydrogen transfer becomes more similar for the two ethylene losses and the ratio should decrease. The reduction in the ion abundance ratios for (16) and (13) is consistent with this. Although skeletal rearrangement is not observed in the ethylene eliminations, loss of propene arises from skeletal rearrangement. Loss of propene is proposed to occur through a cyclopropyl intermediate as shown in Scheme 5.6. The experimental ( ( H H + + + + 3-ethyl-2-pentanone. 3-ethyl-2-pentanone. • • of of ion ion 4 4 +· +· OH OH molecular molecular • • the the from from • • propene propene of of 4 4 + + OH OH elimination elimination • • the the + + OH OH for for 4 4 • • mechanism mechanism 4 4 +· +· 0 0 Proposed Proposed 5.6. 5.6. +· +· 0 0 Scheme Scheme 62 observations that (12) and (13) both lose 12c3H6 and 13c12CiH6 in equal numbers fits with this scheme, as does the observation that (14) loses only 13c 12C2H 6. Minor contributions corresponding to [M-12C3HJ+. and [M-13Ci12CH6]+. in (14) are most likely due to non-specific labelling in the precursor compounds. Scheme 5.6 shows that there is a reversibility between the [3-ethyl-2-pentanone]+. and [3-methyl-2-hexanone]+. ion structures which implies that if [3-ethyl-2-pentanone]+. were regenerated after skeletal rearrangement, any 13c would be located at its original site. Loss of 13c integrity would not be evident in the loss of ethylene, even though considerable skeletal rearrangement may have occurred. Skeletal rearrangement indicates that the above isotope effects on ion abundances for (12), (13) and (16), need to be considered with caution. Significant preference for the loss of the lighter isotopic species is seen for the ethylene elimination in all compounds although the accurate determination of the isotope effects is not possible. Without reliable experimental isotope effects, the results of QET calculations and kinetic modelling cannot be evaluated and reporting such theoretical results would be misleading. It can be said, however, that the results obtained are similar to those results obtained for 2-pentanone [121], and are consistent with a stepwise model for the loss of ethylene from the 3-ethyl-2-pentanone molecular ion by the McLafferty rearrangement. 5.4. The McLafferty rearrangement in some substituted benzenes. In the case of alkyl benzenes, the McLafferty rearrangement would result in the formation of a [C7H8]+. ion with the methylene cyclohexa-1,3-diene structure (Figure 5.4(a)). The assig-nment of a structure to this [C7H8]+. species has presented a number of problems in that the rearrangement product of the McLafferty rearrangement should Figure 5.4. [C7H8]+. structures. (a) methylene cyclohexa-1,3-diene radical cation (b) toluene radical cation Figure 5.5. Potential energy profile for the stepwise elimination of propene from ionised n-butylbenzene. 929 kJ mor1 824 kJ mor1 63 be the methylene cyclohexa-1,3-diene radical cation [(17)]+., but the most commonly encountered [C7H8]+. species is the toluene radical cation (Figure. 5.4(b)). In a study of the energetics of formation of [C7H 8]+. from the n-butylbenzene and 2-phenylethanol radical cations by Burgers et al. [127], Af-Ii{[C7H8]+·) was estimated to be 960 kJ moi- 1. This ion was considered to have the methylene cyclohexa-1,3-diene structure. In a separate study of n-butylbenzene and 2-phenylethanol by Kuck and Grutzmacher [128], Af-Ii{[C7H8]+·) was estimated to be 900 kJ moi-1 which is very close to the heat of formation of the toluene radical cation (Affi{[toluene]+·) = 901 kJ moi-1 [18] ). A number of calculations have been performed to obtain ,1Hf([C7H 8]+·) for the methylene cyclohexa-1,3-diene radical cation. Dewar and Landman [129] have calculated ,1Hi{[C7H8]+·) for the methylene cyclohexa-1,3-diene radical cation to be 916 kJ moi-1 using the MINDO/3 method. McLoughlin et al. [130] have obtained a value of 966 kJ mol-1 using an ab-initio method. Using ion cyclotron resonance, Bartmess has estimated the methylene cyclohexa-1,4-diene radical cation to be 75 kJ moi-1 Iess stable than the toluene radical cation [131]. Combining this with Chen et al.'s [132] estimate for the methylene cyclohexa-1,3-diene radical cation to be 58 kJ mo1-l more stable than the 1,4- isomer, places ,1Hi{[C7H 8]+•) for the methylene cyclohexa-1,3-diene radical cation 17 kJ moi-1 above that for the toluene radical cation, i.e. 9i8 kJ moi-1. Ion energetics seem unable to identify unambiguously the [C7H8]+. structures, although, if the structure is not the diene, the reaction is not the McLafferty rearrangement. Identification of the [C7H 8] +. product ions formed from n-butylbenzene as the methylene cyclohexa-1,3-diene radical cation relies mainly on CID spectra. 64 CID spectra of the [C7H8]+- molecular ion formed from toluene and the [C7H8]+ fragment ion formed from n-butylbenzene in the ion source exhibit differences in the m/z 75-78 region [127,133]. Loss of CHf is more pronounced from [C7H8]+. ions of toluene, than from the [~H8]+- ions from the n-butylbenzene molecular ions. Further evidence for the existence of a [C7H8]+- structure other than that of toluene comes from ion cyclotron resonance experiments on the reactivity of [C7H8]+- ions. Abstraction of NO2 from neutral alkylnitrites, established as a characteristic ion-molecule reaction of ionised toluene [134], is not observed with [C7H 8]+- ions generated from 2,4-dimethyl-1-octadecyl benzene. The methylene cyclohexa-1,3-diene structure is assigned to these [C7H8]+. ions. Given that the [C7H 8]+. ion generated from n-butylbenzene has the methylene cyclohexa-1,3-diene structure, the rearrangement should involve specific y- hydrogen transfer from the butyl chain. D labelling of n-alkylbenzenes by Lightner et al. [135] has shown that in n-butylbenzene the transferred hydrogen originates mainly from the y- carbon and to a lesser extent from the 13- and o- carbons as well. In higher homologues, hydrogen transfer has been observed from more remote positions. Hydrogen exchange between the ortho positions of the benzene ring and the alkyl chain has also been observed. Wesdemiotis et al. [136] and Borchers et al. [137] have studied the hydrogen transfer in n-butyl and n-pentylbenzene and concluded that hydrogen exchange between the ortho position of the benzene ring and the alkyl chain occurred through five-, six- and seven- membered transition states involving 13-, y- and 6- hydrogens respectively. y- Hydrogen transfer was identified as the major process involved in the formation of [C7H8]+-. Scheme 5.7 depicts a mechanism which explains the hydrogen exchange observed between the ortho position of the ring and the "(- carbon. Hydrogen exchange between the ortho position and the 13- and o- positions could occur in a similar fashion. l+• l~ • 4 • a ~+Hu__ ·H, l l+• 1.::,. • • a , 2 +~ll Scheme 5.7. Hydrogen exchange between they-carbon and the ortho position of the benzene ring in ionised n-butylbenzene, concurrent with propene elimination. 65 The intermediate ion formed after hydrogen transfer is a distonic radical cation [138] which should be a stable species, lower in energy than either transition state involved in the McLafferty rearrangement in the n-butylbenzene radical cation. Wesdemiotis et al. [139] have estimated Af-1/[intermediate]+·) to be 874 kJ moi-1. Chen et al. [132] have estimated the appearance energy of [C7H8]+• from the n-butylbenzene radical cation to be 105 kJ moi-1, which in combination with Af-1/n-butylbenzene]+·) = 824 kJ moi-1 [140], gives the energy of the highest energy transition state as 929 kJ moi-1. If the reaction products are located at this energy, i.e. there is zero non-fixed energy, ~Hi{[C7H 8]+·) = 908 kJ moi-1 using ~H/propene) = 21 kJ moi-1 [15]. If a reverse critical energy,were present for the C-C cleavage, Af-l/[½H8]+·) would be lower than this estimate. Figure 5.5 shows a potential energy profile for the stepwise elimination of propene from the n-butylbenzene radical cation in which there is no reverse critical energy for the reaction, and the non-fixed energy is zero. 5.5. Benzyl ethyl ether and a,a'-diethoxy-p-xylene. MacLeod and Djerassi observed the formation of [C7H 8]+. from ionised benzyl ethyl ether (18) and they tentatively assigned the methylene cyclohexa-1,3-diene structure to this ion [141,142]. Deuterium isotope effects on ion abundances for the loss of C2H40 and C2H3DO from ionised benzyl 1-D-ethyl ether favoured loss of C2H40 by a factor of 2, while in the case of the losses of C3 H 6 and C3 H 5 D from ionised 3-D-n-butylbenzene loss of C 3H 6 was favoured by a factor of 1.2 [142]. With n-butylbenzene hydrogen transfer was observed to be 95 % specific from they- carbon, while for benzyl ethyl ether hydrogen transfer was observed to be> 98 % specific from they- carbon at an ionising energy of 18 eV. Hydrogen transfer in benzyl ethyl ether appears to be more site-specific and more kinetically significant than in n-butylbenzene. 66 5.5.1. Benzyl ethyl ether. The two benzyl ethyl ethers investigated in this study were benzyl ethyl ether (18) and benzyl 1,1-Dz-ethyl ether (19), identified in Figure 5.6. Samples of both of these were supplied by Prof. J.H. Bowie of the University of Adelaide. Samples of benzyl l,l-D2-ethyl ether were also supplied by Prof. F.W. McLafferty of Cornell University. Both samples of the deuterated ether gave identical results. The MIKE spectrum of the molecular ion of (18), m/z 136, is shown in Figure 5.7(a). At the accelerating potential 8 kV used, the MIKE spectrum represents those decompositions occurring from 1.9x10-5 s to 4.4x10-5 s after ionisation. The dominant process is loss of acetaldehyde giving rise to m/z 92 with minor contributions from loss of ethylene and loss of ethyl radical at m/z 108 and m/z 107 respectively. The slight shoulder on the m/z 92 peak (Figure 5.7(a)) was found to increase in height as the accelerating potential was reduced. Reducing the accelerating potential to 2 kV (Figure 5.7(b)) increased the metastable time-frame sampled for this ion to between 3.7x10-5 s and 8.7x10-5 s. At 2 kV the resolution was less ( note the merging of the m/z 107 and m/z I 08 peaks ), and the shoulder of the m/z 92 peak increased. This shoulder proves to be of considerable significance, as will become clear from later discussion of the mechanism of loss of acetaldehyde. The origin of this shoulder will be discussed later. The'structure of the m/z 92 fragment ion from (18) was studied by CID. The m/z 92 fragment ion produced in the ion source was selected with the magnet, and collided with helium gas in the collision cell located at the intermediate-slit. Helium gas was admitted to the collision cell until the m/z 92 ion current was attenuated by about 50 %. The CID spectrum of the m/z 92 ion is shown in Figure 5.8(a). This spectrum is in good agreement with that reported by Burgers et al. [ 127] for the m/z 92 ion formed Figure 5.6. Benzyl ethyl ether (18) and benzyl 1,1-D2-ethyl ether (19). 0 l (18) (19) (a) (b) I I I I I I I I I 0.2 0.3 . 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 5.7. MIKE spectra of the molecular ion of (18) a: (a) 8 kV and (b) :! kV accelerating potentials. (a) 0.1 0.2 0.3 0.1+ 0.5 0.6 0.7 0.8 0.9 1.0 (b) r I I I I I I 0.1 0.2 0.3 ds 0.6 0.7 0.8 0.9 '1.0 Figure 5.8. CID spectra of the m/z 92 ions formed from (a) benzyl ethyl ether and (b) toluene. 67 from n-butylbenzene. For comparison, the CID spectrum of the m/z 92 ion formed by electron ionisation of toluene under identical conditions is shown in Figure 5.8(b). The spectra are very similar, however the more pronounced m/z 77 peak in the toluene spectrum (Figure 5.8(b)) has been considered to be significant by other workers [133]. On this basis, the m/z 92 ion formed from benzyl ethyl ether is assigned the methylene cyclohexa-1,3-diene structure [(17)]+ .. The MIKE spectrum of the molecular ion, rn/z 138, of benzyl 1,1-Drethyl ether (19) is shown in Figure 5.9(a). At the accelerating potential 6 kV used, the metastable time-frame extends from 2.2x10-5 s to 5.lxl0-5 s. The fragment ions m/z 108 and rn/z 107 due to loss of ethylene and ethyl radical respectively contain both D atoms. The rn/z 93 ion due to loss of acetaldehyde contains only one D atom. There is a shoulder on the rn/z 93 peak which increased upon lowering the accelerating potential to 2 kV (Figure 5.9(b)). At 2 kV the metastable time-frame for the rn/z 138 ion extends from 3.7xl0-5 s to 8.8xl0-5 s. Wesderniotis et al. [ 139] have reported the loss of C2H4O from the molecular ion of benzyl l,l-D2-ethyl ether (19). Loss of C2H4O implies hydrogen exchange between the ethyl chain and the benzene ring. The reported ratio [M - C2H4O]+. / [M - C2H3DO]+. increased from 0.02 at 2.5xl0-5 s observation time to 0.10 at 6.0xl0-5 s. The shoulder on the rn/z 93 peak in Figure 5.9(b) is ~ 6 % of the height of the m/z 93 peak. Figure 5.10 shows the expanded m/z 90 - 98 regions of the MIKE spectra of benzyl ethyl ether (18) and benzyl 1,1-Di-ethyl ether (19) recorded at 2 kV accelerating potential. A broad shoulder is evident on the main peak in both spectra. That the same sort of shoulder is observed with both ( 18) and ( 19) indicates that the shoulder on ibe · (a) (b) I I I I I I I I I I I 0.0 0.1 b.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 5.9. MIKE spectra of the molecular ion of (19) at (a) 6 kV and (b) 2 kV accelerating potentials. potential. potential. ion ion accelerating accelerating 95 95 I I kV kV 2 2 fragment fragment at at of of (19) (19) mass mass (b) (b) and and 90 90 18) 18) I I ( ( (a) (a) of of spectra spectra 95 95 I I MIKE MIKE ion ion the the of of fragment fragment of of regions regions 95 95 - mass mass 90 90 I I 90 90 m/z m/z Expanded Expanded 5.10. 5.10. Figure Figure 68 m/z 93 peak from the labelled molecule (19) cannot be due simply to hydrogen exchange with the ring. If there were hydrogen exchange with the ring, this would be strong evidence that the elimination of acetaldehyde occurs in a stepwise fashion as suggested by Wesdemiotis [139]. The experimental results provide no evidence of hydrogen exchange. 5.5.2. a.,a.'-Diethoxy p-xylene. a,a'-Diethoxy p-xylene (20) and four of its isotopomers ( a.-ethoxy a.'-1, 1-D2-ethoxy p-xylene (21 ), a-ethoxy a '-1, 1,2,2,2-D5-ethoxy p-xylene (22), a.-ethoxy a•- 18O-ethoxy p-xylene (23), a-1,1,2,2,2-D5-ethoxy a.'-1,1,2,2,2-D5- 18o-ethoxy p-xylene (24) ) were supplied by Prof. J.H. Bowie of the University of Adelaide. These compounds are identified in Figure 5 .11. The MIKE spectra of the molecular ions of these five compounds are shown in Figure 5.12. The MIKE spectrum of the molecular ion of (20) reveals three major decomposition channels: loss of C2H4O giving rise to an ion at m/z 150; loss of C2H6O giving rise to an ion at m/z 148; and loss of C3H7O· giving rise to m/z 135. Other fragment ions are present at m/z 166, m/z 165, m/z 104 and m/z 103, arising from losses of CiH4, C2H5·, C4H 10O2 and 91u respectively. It is concluded from the MIKE spectra of the labelled compounds, that the loss of 91u from [(20)]+. must represent both the loss of C4H 11O2 and C7Hi in order to rationalise the observed peaks. Loss of acetaldehyde would presumably result in the formation of the 4-ethoxymethyl methylene-cyclohexa-1,3-diene radical cation [(25)]+·. Loss of C2H6O, presumably ethanol, is a process not observed in benzyl ethyl ether. The MIKE spectra (Figure 5.12) show that the five ethoxy hydrogens and a hydrogen either from the ring or from one of the benzyl carbons are eliminated as C2H6O. The peaks due to the loss of Figure 5.11. Cl,Cl-, Die thoxy-p-xylene (20) and th e 5 isotopmers, (21) - (24). (21) (23) (24) E/Eo 1.0 0.9 0.8 ). (24 - 0.7 (20) 0.6 compounds 5 the 0.5 of ions 0.4 molecular 0.3 the of (20) 0.2 spectra MIKE Compound 0.1 (a) .12. 5 0.0 Figure E/Eo E/Eo 1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 (21) (21) 0.2 0.2 (Cont.) (Cont.) Compound Compound 0.1 0.1 (b) (b) 5.12. 5.12. 0.0 0.0 Figure Figure E/Eo 1.0 _ 0.9 ____ , 0.8 ~ 0.7 '-----J 0.6 0.5 0.4 0.3 (22) 0.2 Compound (Cont.) (c) 5.12. 0.1 Figure 0.0 Figure 5.12. (Cont.) (d) Compound (23) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 E/Eo E/Eo E/Eo 1.0 1.0 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 \.. \.. (24) (24) 0.5 0.5 Compound Compound (Cont.) (Cont.) (e) (e) 5.12. 5.12. 0.4 0.4 Figure Figure 69 ethanol interfere with the analyses of the MIKE spectra with respect to the acetaldehyde losses. Expanded regions of the MIKE spectra of the molecular ions of (21 ), (22), (23) and (24) are shown in Figure 5.13. The MIKE spectrum of the molecular ion of (21) (Figure 5.13(a)) shows a broad composite peak arising from the losses of C2H40, C2H3DO, CiH60 and C2H4D20. Peak subtraction enables the individual peaks to be extracted reasonably reliably. The MIKE spectrum of the molecular ion of (22) (Figure 5.13(b)) contains four almost completely resolved peaks. The MIKE spectrum of (23) (Figure 5.13(c)) has three peaks which represent the four losses: C2H40, C2H4180, C2H60 and C2H6180. The peaks due to losses of C2H4180 and C2H 60 are superimposed. To estimate the abundances of these peaks, two assumptions were made. First it was assumed that the ratio [M - C2H40]+. I [M - C2H60 ]+. observed for (23) is the same as the ratio observed for (20), i.e. 5.35. Second it was assumed that the peaks due to losses of C2H60 and C2H618o had the same shape. Making these two assumptions allowed the individual peaks to be constructed. The MIKE spectrum of the molecular ion of (24) (Figure 5.13(d)) shows peaks due to the losses of C2H4 and C2HD/8o which are resolved, and two peaks arising from the losses of C2D418o and C2HD50 which are not resolved. Peak subtraction revealed the two components in the latter case. The MIKE spectra of the molecular ions of (21) - (24) were recorded using both analog and digital recording systems (Chapter 3). Spectra were recorded a number of times, and, when sample supplies allowed, on different days. Peak subtraction was performed for all spectra, with both peak heights and peak widths being measured, enabling ion abundances to be estimated in two ways. First, the ion abundance was considered to be represented by the peak height only, and second, the peak area was considered to represent the ion abundance. The ion abundance ratio, i.e. [M - light figure 5.13. Expanded regions of the MIKE spectra of the molecular ions of (21) - (24) showing the peaks corrsponding to elimination of acetaldehyde and ethanol. (a) Compound (21) (b) Compound (22) ; I I I I I I I ; Figure 5.13. (Cont.) (c) Compound (23) (d) Compound (24) [M - C2D4 18O]+ I 70 acetaldehyde]+- / [M - heavy acetaldehyde]+- , was calculated both ways. The ion abundance ratios were tested for significance using Q-tests, and for confidence limits using t-tests [144,145]. Q-tests indicated that all ion abundance ratios were statistically significant. The mean and standard deviations for each compound are shown in Table 5.3 along with the confidence limits. A confidence limit of 99.5 % indicates that there is a 99.5 % probability that the true average lies within the range given by the mean and standard deviation. The collection efficiency was calculated from the fractional transmittances, calculated using the method described in Chapter 2. The mean energy release, T0_5, was estimated from the peaks in the MIKE spectra. The collection efficiency was obtained as the ratio of the fractional transmittances for the two competing processes. The measured ion abundance ratio, mean energy release T0_5, collection efficiency and corrected ion abundance ratios for (21) - (24) are given in Table 5.4. The ion abundance ratios for (21) and (23) are 3.28 ± 0.19 and 1.09 ± 0.07 respectively. These ratios have been previously reported as 3.2 and 1.15 ± 0.04 [143]. The earlier figures had not been corrected for the fractional transmittances, and were based on fewer measurements. The ion abundance ratios for (22) and (24) are 3.41 ± 0.17 and 1.06 ± 0.05 respectively. The 'magnitude of the ratios for (21) and (22) indicate that hydrogen transfer is a kinetically significant process. The ratios for (23) and (24), although not as large, indicate that the C-O bond cleavage also is kinetically significant, although it must be remembered that the ratio for (23) is less secure, being obtained from peak subtraction. The significance of both hydrogen transfer and C-O cleavage processes suggests two possible kinetic schemes to describe the elimination of acetaldehyde. The first is a Table 5.3. Ion abundance ratios for [M-light acetaldehyde]+- / [M-heavy acetaldehyde]+ for (21) - (24) with confidence limits. Compound Ion abundance ratio Confidence limit % 21 3.33 ± 0.19 99.5 22 3.61 ± 0.07 99.9 23 1.12±0.07 99.9 24 1.09 ± 0.05 99.9 Table 5.4. Ion abundance ratios corrected for the collection efficiency G. Compound Ion abundance ratio To.s G Ion abundance ratio (measured) meV kJmoi-1 (corrected) 21 3.33 ± 0.19 116 11.2 1.015 3.28 ±0.19 22 3.61 ± 0.07 118 11.4 1.058 3.41 ±0.07 23 1.12 ± 0.07 127 12.3 1.029 1.09 ±0.07 24 1.09 ±0.05 126 12.2 1.028 1.06 ±0.05 71 concerted mechanism. The second is a stepwise process with hydrogen transfer and C-O cleavage occurring consecutively. 5.6. Energetics. Energetics for the loss of acetaldehyde from both benzyl ethyl ether (18) and a,a'-diethoxy p-xylene (20) were obtained by combining ionisation and appearance energies, measured using photoionisation mass spectrometry, and heats of formation obtained either from literature, where available, or by estimation using additivity rules. An alternative method of obtaining energies of reactant and product ions, i.e. performing a molecular orbital calculation, was also employed. Photoionisation measurements on benzyl ethyl ether (18) and a,a'-diethoxy p-xylene (20) were made by Dr. J.C. Traeger of LaTrobe University [97]. Photoionisation efficiency curves for [M]+. and [M - C2H4O]+. for both compounds are shown in Figure 5.14. The adiabatic ionisation energies for the two compounds were placed at 8. 74 ± 0.03 e V and 8.42 ± 0.02 e V respectively. The appearance energies of the [M - C2H4O]+. species are 8.91 ± 0.03 eV and 9.16 ± 0.10 eV respectively. Critical energies for the loss of acetaldehyde from the two molecular ions were estimated as 0.17 ± 0.06 eV (16.0 ± 6.0 kJ moi- 1) for benzyl ethyl ether and 0.74 ± 0.12 eV (71.2 ± 12.0 kJ moi- 1) for a,a'-diethoxy p-xylene. The additivity rules of Benson [15] were used to calculate ~Hf298 for benzyl ethyl ether, a,a'-diethoxy p-xylene and acetaldehyde (Table 5.5). Details of the calculations are given in Appendix 4. The calculated value of -165 kJ moi- 1 for acetaldehyde agrees well with the literature value of -166 kJ mol-1 (~Ht298(acetaldehyde: gas phase)[14]). Combining these mr's with the ionisation energies gives mf for the molecular ions ai a, .,; ,.._ .,; "'a, "'.,; > "'.,; 01 > '-.. .., QI ;,- .,; ' on l 'J r a.i rr: l~ w 0:: z .,;~· LJJ Id z. LJJ z z. N 0 \0 .,;o"' .._.. ~- ., 0 - t- 00 :r: 0 ai (L °' :r: . .,; Q_ C . 0 :, :, .,; ' X Q ' 0.' :,' :, X ' 0 IT) X "' .c 0 ai ~ ., .C co ~ D "' ., • U1 ., -.-· (D - ~ m ID L' "' ' u "I L' ai o "'n: " Cl u N (C ''- 0 F. '"'' . '- ~ e u)' ,.._ ai AJtEl I J IJ.::l::J I-JO I U:lZ fNO !Ol □ f-ld Lll'JJ I JI .::l.::lJ NO I ltlZ I NO IOlOl-ld .,;"' ~ N 0 "' "' > > OJ .,; QJ "' '-.. '-.. '" r r I_.:) L'J n::: 0:: L,J ::;: oW. z. ID I.LI "'W a, z z C) 0 r- r- .._.. C) - 0 r--- ::c °'00 I Q_ "' CL 00 ai w ,_ . L .c .c. ~. •· .. a, . CD ,D :, >.c .c ~ ,D ai ... • l"I ~ 0 N s, c,, ID - "'I I :, L :,, I L . 0 N 0 ,.._ "C H J: C .. ,: .... '- I ' OCl. E r-- CD a, ' "' n 1{3 lJ l.:l .:G NO I UJZ I l~O I OiOHd J.JNJ I JI .:LI] NO I 11::JZ IMO I OlOHd Figure 5.14. Photionisation efficiency curves for [M]+- and [M - C2H4O]+- for (18) and (20). Table 5.5. Heats of fonnation, in kJ moi-1, calculated from Bensons rules and photoionisation measurements. Compound Aflr(neutral) Aflr(ion) 18 -113 731 20 -307 506 acetaldehyde -165 Table 5.6. Comparison of ionic heats of fonnation, in kJ mol-1, from different methods. Compound ~Hr(ion)(a) Difference 18 731 718 -13 17 911 901 -10 20 506 502 -4 25 741 687 -54 (a) Using Bensons rules and photoionisation. (b) Using MOPAC package. 72 (18) and (20) as 731 ± 3 kJ moi- 1 and 506 ± 2 kJ moi-1 respectively. If the reverse critical energies for the acetaldehyde eliminations are assumed to be negligible, the measured appearance energies determine the sum of the ~Hr's of the products. For benzyl ethyl ether, this gives mr(products) as 747 kJ moi- 1. Using this value and ~Hr(acetaldehyde) as -165 kJ moi-1 allows ~Hr([(l7)]+·), (methylene cyclohexa-1,3-diene radical cation), to be calculated as 911 ± 9 kJ moi-1. Literature values for ~Hr([(l 7)]+·) range from 900 to 960 kJ mol- 1 (vide supra ). For a.,a.'-diethoxy p-xylene, the sum of the ~Hr's of the products is 577 kJ moi- 1, which gives mr([(25)]+·) as 741 ± 14 kJ moi-1. Heats of formation for the ionic species were also calculated using the MNDO method [146] as implemented in the MOPAC package [147]. Optimised geometries which correspond to stable species, as indicated by the absence of negative eigenvalues in the force constant matrix, were obtained. The geometries obtained for acetaldehyde and the radical cations of (18), (17), (20) and (25) are given in Appendix 5. The calculated heats of formation are compared with the values obtained from the appearance energy measurements in Table 5.6. The agreement between the two methods is good for all species except [(25)]+-, for which the MOPAC value is considerably less than that estimated from the appearance energy. 5.7. Concerted elimination of acetaldehyde from a.,a.'-diethoxy p-xylene. For the labelled a.,a.'-diethoxy p-xylenes, two isotopomeric acetaldehydes can be eliminated from each molecular ion. Assuming that there are no competing reaction channels and that there is no subsequent decomposition of the products, abundance ratios for the product ions can be calculated using the program PEAKS described in 73 Chapter 2. In the calculations of k(E), the maximum internal energy Emax• was in all cases set at 240 kJ moi- 1 on the grounds that k(E) would always be greater than 106 s- 1 once E had reached 240 kJ moi-1. Rate coefficients k(E) > 106 s- 1 do not contribute significantly to decomposition in the metastable time-frame. P(E) for the reactant ion was set to a constant value of 1.0. Various forms of P(E) were tested, but the calculated ion abundance ratios were not affected significantly. The grain-size for the calculations was 0.14 kJ moi-1 (12.0 cm-1) for all reactant ions and transition states. For each of the molecular ions of (21) - (24) the symmetry numbers, er, for competing acetaldehyde eliminations are equal. Consequently er for all reactions was set to 1. The critical energy E0 for the loss of C2H40 from the molecular ion of (20) was estimated as 71.2 ± 12.0 kJ moi-1. The E0's for loss of the isotopomeric acetaldehydes from the molecular ions of (21) - (24) are affected by the zero-point energies (ZPE's) of the various reactant ions and transition states. The vibrational frequencies of the molecular ions were obtained using the MOPAC program and the geometry obtained for the molecular ion of (20) (Appendix 5). The magnitudes of the vibrational frequencies obtained using this method are typically overestimated by about 10 %, except for those of the low frequencies which are underestimated [74]. The vibrational frequencies were corrected for these effects by lowering all frequencies by 10 % and setting the lowest vibrational frequency in each molecular ion to 36 cm-1. The adjusted vibrational frequencies for the molecular ions of (20) - (24) are given in Table 5.7. 5.7.1. Modelling of the transition state. As a starting point, the vibrational frequencies of the transition state were considered to (20) (21) (22) (23) 12q) 36 36 36 36 36 36 36 J6 36 36 36 36 36 36 36 !6 36 J6 36 36 ~•1 q2 q1 qq 36 /0 ·10 68 70 66 // .,, 7'1 75 71 "/8 ·,a 77 ·15 /'I 111? 1110 110 1q I 108 1q6 1'15 100 1"5 111 1•9 11113 1q·1 1119 IH 212 211 206 211 199 251 2q5 2qa 250 243 253 252 252 252 2q9 285 285 285 285 285 123 320 3 I 1 320 309 352 352 J'19 352 3'17 q19 11111 395 418 389 q20 q 19 1119 '"9 397 q75 1176 1.1711 •in 1168 516 536 532 5J5 52 / 5'10 'jllU sn ~110 538 639 638 585 6 39 58'1 719 659 63, "/19 585 791 I 18 717 791 632 '/92 791 ·122 rn 716 '/93 793 791 793 719 806 806 793 806 723 BJ'• OJJ 805 BJ•• 791 881 8% 818 881 8oq 087 BIO 835 887 818 921 881 8116 917 818 921 88/ 867 921 835 928 921 881 92/ 8116 9311 926 0B7 93q B'16 9•19 9JU 921 9•19 865 970 9311 925 969 867 1050 9•17 9311 1050 881 1051 969 911 J 1051 887 1076 978 9118 1076 923 1077 1007 955 10n 93q 1088 1051 968 1088 941 1115 1076 970 1114 91lll 1116 1088 1013 1116 9118 1152 1116 1051 1152 955 11511 1123 1076 1153 955 1157 1139 1088 1155 96'1 1157 1153 1116 1157 968 1202 1155 112 3 1185 970 1205 11s·, 1152 1203 1012 1223 1203 11511 122 J 1013 1251 1222 l 156 1256 1088 12/2 1221 1157 1265 1121 12/9 1258 1203 12"/6 1123 1288 1275 1221 1281 1145 1293 1287 12211 1293 1152 12911 12911 1258 12911 115q 1294 12911 1275 12911 1156 1294 12911 1287 12911 1211 I J0 11 1295 129q 1303 1221 1309 1301 129•1 1306 12211 1310 1 JOB 1 JO I 13 10 1257 131q 13 12 1 JUB 1313 1285 1JJ5 1 323 1 J 12 1335 1296 133/ 1336 1)22 1337 I J03 I JJB 1 JJ7 1 336 1337 1 J 10 1355 1338 1 J JI 1)116 1 321 1360 1357 1 351 1)55 13)7 1 367 I 366 I 365 1 J60 13116 1 J7 J l J72 1 J/2 1370 1367 1)78 1 317 1377 1 l71 1 H6 1'1/9 1'179 11q9 )1179 1'H9 2858 2112 2111 2858 2111 2858 2115 21111 2858 2111 2867 2858 2160 2867 2113 2867 2858 2177 2867 211 11 2916 2867 21·19 2916 2159 291"/ 2917 2858 2917 2160 2920 2918 2858 2920 2176 2920 2920 2867 2920 2177 2948 29118 2911 29110 2178 2949 29q9 2918 2949 2180 2950 2950 2920 2950 2858 295) 2953 29 119 2953 2858 JO 1q JO 14 2952 301'1 2917 3016 JO 16 3015 )016 2918 30J8 3038 3038 3038 JOJ8 3039 3039 3039 3039 3039 3048 3oq9 3048 10,9 3oq9 3052 3052 3052 3052 3052 Table 5.7. Vibrational frequencies for the molecular ions of (20) - (24). (Frequencies are in cm- 1.) 74 be the same as those of the reactant ion, i.e. the vibrational frequencies of the reactant ion were used for the transition state except that one C-0 stretching frequency was removed as the reaction coordinate. Three groups of vibrational frequencies considered to be sensitive in the transition state were identified. One group consisted of six frequencies ( 2 C-H stretches and 4 C-H bends) associated with C(l) of the ethyl group ( group 1 ). A second group consisted of three frequencies ( 1 C-C bend and 2 C-0 stretches, one of which was the reaction coordinate) representing a COC linkage ( group 2 ). A third group contained seven frequencies ( 2 C-H stretches, 4 C-H bends and 1 C-C stretch ) representing a benzyl group ( group 3). The frequencies in each group for each of the acetaldehyde eliminations are given in Table 5.8. The transition state was modified so as to fit the ion abundance ratios for (21) and (23) only. The ion abundance ratios for (22) and (24) were used as a test of the transition state model. 5.7.2. Results. The calculations described here were performed using energy units of cm-I and have been converted to kJ moi- I_ The resultant values thus have more significant figures and may appear to disagree slightly with the values given earlier. In the first series of calculations, vibrational frequencies of the transition state were adjusted so as to reproduce the measured ion abundance ratios in the cases of (21) and (23). E0's for the loss of C2H40 from the molecular ions of (21) and (23) were set to 7 f.17 kJ mo1-I. Considering (21 ), the two transition states differed only as regards the reaction coordinate ( 1202 cm-I for the loss of C2H40 and 1155 cm-I for the loss of C2H3DO ). The zero-point energy (ZPE) difference between the two transition states is Table 5.8. Three groups of vibrational frequencies, in cm-1, for the labelled reactant (molecular) ions considered sensitive in the transition state for the concerted model. Compound 21 22 23 24 2916 2114 2916 2113 2916 2916 2113 2113 2867 2112 2867 2111 2867 2867 2111 2111 Group 1 1274 978 1274 970 1274 1274 968 968 1116 930 1116 846 1116 1116 846 846 1076 870 1076 835 1076 1076 819 819 792 659 792 585 792 792 585 585 1355 1323 1355 1322 1360 1346 1365 1346 Group 2 1366 1366 1366 1366 1370 1355 1323 1310 1202* 1155* 1202* 1153* 1202* 1185* 1155* 1145* 2918 2918 2920 2918 2858 2858 2858 2858 1308 1308 1306 1303 Group 3t 1287 1287 1287 1285 1258 1258 1255 1257 1157 1157 1157 1123 1050 1050 1050 1013 * This frequency is the reaction coordinate for the elimination. t Group_ 3 frequencies are the same for both acetaldehyde eliminations from the same reactant ion. 75 0.28 kJ moi-1, which puts E0 for the loss of C2H3DO at 71.45 kJ moi-1 . Similarly the reaction coordinates for the loss of C2H40 and CiH/80 from (23) were 1202 cm-1 and 1185 cm-1, leading to a value of71.27 kJ moi-1 forE0 for the loss ofC2H/8o. Using these transition states in which only the reaction coordinate was removed, the ion abundance ratio for (21) was calculated to be 1.12, c.f. the measure value of 3.28. The vibrational frequencies of the ethyl C(l) group ( group 1 ) were lowered by 5 %, giving a ZPE difference of 0. 99 kJ mol-1 and putting E0 for the loss of C2H3DO at 72.16 kJ moi- 1. The calculated ion abundance ratio was then 1.49. Further 5 % lowerings of the group 1 frequencies to a maximum value of 25 % were made. For a 25 % lowering, E0 for C2H3DO loss was 75.01 kJ moi-1 and the calculated ion abundance ratio was 4.51. ZPE differences, E0's for loss of C2H3DO and the calculated ion abundance ratios are given in Table 5.9. Best agreement between the calculated and measured ion abundance ratios occurs when the group 1 frequencies are lowered by between 15 and 20 %. For (23), the transition states from which only the reaction coordinates had been removed gave an ion abundance ratio of 1.04 ( c.f. 1.09 measured ). The vibrational frequencies associated with the COC linkage (group 2) were lowered by 20 %, giving E0 for the loss of C2H/8o as 71.30 kJ moi-1 and an ion abundance ratio of 1.06. Further 20 % lowerings of the group 2 frequencies were made to a maximum of 60 %, at which point E0 for the loss of C2H418o was 71.37 kJ moi-1 and the calculated ion abundance ratio was 1.08. Lowering of the group 2 frequencies by more than 60 % did not greatly increase the ion abundance ratio. The ZPE differences, E0's for C2H4180 loss and the calculated ion abundance ratios are given in Table 5.10. Best agreement occurs when the group 2 frequencies are lowered by between 40 and 60 %. The lowering of group 1 and group 2 frequencies were then combined to calculate fresh Table 5.9. Lowerings of the group 1 frequencies and the ion abundance ratios calculated for (21). Energies are in kJ moi-1. % lowering L\ZPE Calculated ion abundance ratio * 0 0.29 71.46 1.12 5 0.99 72.16 1.49 10 1.70 72.87 1.97 15 2.42 73.59 2.63 20 3.12 74.29 3.41 25 3.84 75.01 4.51 t E0 for loss of CzH3DO. E0 for loss of CzH4O constant at 71.17 kJ moi-1. * Measured ion abundance ratio: 3.28 ± 0.19. Table 5.10. Lowerings of the group 2 frequencies and the ion abundance ratios calculated for (23). Energies are in kJ mol. % lowering ~PE Calculated ion abundance ratio * 0 0.10 71.27 1.04 20 0.13 71.30 1.06 40 0.17 71.34 1.08 60 0.20 71.37 1.08 t E0 for loss of C2H418O. E0 for loss of C2H4O constant at 71.17 kJ mol-1. * Measured ion abundance ratio : 1.09 ± 0.07. 76 ion abundance ratios for both (21) and (23). The group 1 frequencies were lowered from 15 % to 20 % in 1 % steps, while the group 2 frequencies were lowered from 40 % to 60 % in 5 % steps. The ZPE difference for (21) varied from 2.49 kJ moi-1 to 3.24 kJ moi- 1, and that for (23) varied from 0.18 kJ moi-1 to 0.22 kJ moi-1. The ion abundance ratios are given in Table 5.ll(a) and 5.ll(b). Agreement between the calculated and measured ion abundance ratios is best when the group 1 frequencies are lowered by 18 (± 1) % and the group 2 frequencies are lowered by 55 (±5) %. For these lowerings, the ZPE difference for (21) was 2.94 kJ moi-1 and that for (23) was 0.20 kJ moi- 1. The influence of the benzyl associated frequencies (group 3) was tested next by lowering them in steps of 20 % to a maximum of 60 %, while keeping the group 1 frequencies at the 18 % lowering and the group 2 frequencies at the 55 % lowering. The calculated ion abundance ratios for (21) and (23) are given in Table 5.12. Lowering the group 3 frequencies did not affect the ZPE differences of the transition states, hence the ZPE differences for the lowerings of group 3 frequencies for (21) and (23) are 2.94 kJ moi- 1 and 0.20 kJ moi- 1. The group 3 frequencies exert an effect on the calculated ion abundance ratios, but no experimental observations are available to determine the significance of the effect. In view of this, a lowering of 10 (±10) % was chosen for the group 3 frequencies. For lowerings of 18 (±1) %, 55 (±5) % and 10 (±10) % for the frequencies in groups 1, 2 and 3 respectively, the calculated ion abun.dance ratios for (21) and (23) were 3.29 and 1.08. To test this model of the transition state, ion abundance ratios for (22) losing CiH40 and C2D40 and for (24) losing C2D40 and C2D/8o were calculated. For (22) the ZPE difference was 3.17 kJ moi-1 which, with E0 for loss of C2H40 as 71.17 kJ moi-1 gave E0 for the loss of C2D40 as 74.34 kJ moi- 1. The ion abundance ratio was Table 5 .11. Calculated ion abundance ratios for lowerings of both group 1 and group 2 frequencies for (a) (21) and (b) (23). (a) % lowering (group 1) 15 16 17 18 19 20 % lowering group 2 40 2.69 2.85 3.01 3.19 3.38 3.57 45 2.71 2.86 3.02 3.21 3.40 3.59 50 2.75 2.88 3.04 3.26 3.41 3.62 55 2.77 2.93 3.10 3.28 3.48 3.69 60 2.79 2.95 3.12 3.31 3.51 3.72 Measured ion abundance ratio : 3.28 ± 0.19. (b) % lowering (group 1) 15 16 17 18 19 20 % lowering group 2 40 1.08 1.08 1.08 1.08 1.08 1.08 45 1.08 1.08 1.08 1.08 1.08 1.08 50 1.08 1.08 1.08 1.08 1.08 1.08 55 1.08 1.08 1.08 1.08 1.10 1.10 60 · 1.10 1.10 1.10 1.10 1.10 1.10 Measured ion abundance ratio : 1.09 ± 0.07. Table 5.12. Effect of group 3 frequencies upon the ion abundance ratios for (21) and (23) with a group I lowering of 18 % and a group 2 lowering of 55 %. Frequency lowering Ion abundance ratios % (21) (23) 0 3.28 1.08 20 3.31 1.09 40 3.39 1.09 60 3.61 1.09 Table 5.13. Measured and calculated ion abundance ratios. Compound Measured ion abundance ratio Calculated ion abundance ratio (21) 3.28 ± 0.19 3.29 (22) 3.41 ± 0.07 3.46 (23) 1.09 ± 0.07 1.08 (24) 1.06 ± 0.05 1.07 77 calculated to be 3.46. For (24) the ZPE difference was 0.17 kJ moi-1, however E0 for the loss of C2D40 was 74.34 kJ moi-1 (as for (22)), which gave E0 for loss of c2o/8o as 74.51 kJ moi-1. The ion abundance ratio was calculated to be 1.07. The calculated and observed ion abundance ratios for (21), (22), (23) and (24) are given in Table 5.13. In all cases agreement is considered to be satisfactory. The abundances of the molecular ions and the product ions for the elimination of acetaldehydes from (21), (22), (23) and (24) are shown in Figure 5.15 for the optimised transition state lowerings of 18 %, 55 % and 10 %. Consideration was given to the uncertainty (± 12.0 kJ moi-1 ) in the critical energy for the loss of C2H40 from the molecular ion of (20). For the lower limit of 59.21 kJ moi-1, the ion abundance ratios were recalculated as 4.67, 5.00, 1.11 and 1.09 for (21), (22), (23) and (24) respectively. For the upper limit of 83.13 kJ moi-1 the ion abundance ratios were recalculated to be 2.63, 2.73, 1.07 and 1.06 for (21), (22), (23) and (24) respectively. Changing the critical energy clearly has little effect upon the calculated ratios in the case of (23) and (24), while those for (21) and (22) are altered by up to 30 %. In view of the ease with which the calculated ion abundance ratios for (21) and (22) can be altered by changing the group 1 frequencies (see Table 5.9(a)), these transition state frequencies would need to be adjusted by a modest amount to reattain reasonable agreement 5.8. Hydrogen exchange in the benzyl 1, 1-Di-ethyl ether radical cation. One of the more convincing pieces of experimental evidence for a stepwise reaction mechanism for the loss of acetaldehyde would be the observation of hydrogen exchange in the reaction products. For benzyl 1,1-D2-ethyl ether (19), hydrogen exchange would be manifested as the appearance of a fragment ion at m/z 94 arising from the loss 100.00 oo ao /~ ;t the oo .Q so II,) N Q, - -- ...... ._,, f -- \l abundance from oo ~o oo relative 20 /\i \'\ acetaldehydes oo ,o .. 0 0 0 ,, ,, " 0 0 "' = ~~ 0 0 3 0 0 g &, 0 i,, "' 0 '" s 0 g of i5 (1) 3 VJ - ~ '° '-- '-- 'J? 101'., elimination ;t ~ 'Jlj i-1'.i the for 1)1) :i: n 0 + ~ N ,. bi) C Q ::, Q. N er ._,, 'N ~ ._,, ions abundance l)IJ states. 40 product l):J relative 20 two transition ')0 the _0 ~ ~ ,, _, 0 -· -· .;., = " ,, ~ "" 0 ,, c- 0 ,> ,·, ~ _, 3 and .., :l ::7. (1) (/) - ~ '-- -_,, ':S optimised ion IJ': 1 the ;t ')C ~ a-:i using molecular 1)0 the f:j(/ 3 ::, C Q Q. (.,J ~ 'N ._,, '8 for abundance obtained 'JO 4C (24) + curves - !\I ')0 re1at,ve z,;,,. (21) time of ')C vs. r .. :, _, 0 0 " 0 :: " '-' j. 0 " o 3 0 0 A 0 "' "" " 0 ~ .;, 3 ions - C :::; (1) l' 0 -.:i '-- ";~ I':: abundance molecular Ion 0~ ~ + cl': 5.15. ')~ Q. ::, C i ._,, ~ '8 e bC' aDundance 0 ~ + ,.o .P 'i': Figure 4:; ;; 0 + 'j0 relat,ve 20 110 s :0 g ;::; g g g 0 ;. g 8 ~i' (1) VJ ,.. '- '--' 78 of C2H 4O. The partial MIKE spectrum of [(19)]+. (Figure 5.lO(b)) shows an asymmetric peak with a maximum at rn/z 93 and which tails to rn/z 98. The partial MIKE spectrum of [(18)]+. (Figure 5.lO(a)) shows a similar peak with a maximum at m/z 92 and which tails to rn/z 96. It is apparent that the same processes are operative for both ions, giving rise to the peak tail. Hydrogen exchange in the elimination of acetaldehyde cannot be the sole cause of the tailing as only rn/z 92 can be formed from [(18)]+. by acetaldehyde loss. The tail must be either an instrumental artefact, or due to a process or processes which compete with the acetaldehyde elimination. While it is generally accepted that decomposition inside the electric sector of a reverse-geometry double-focusing mass spectrometer gives rise to a tail on the high mass side of a peak in the MIKE spectrum, the extent of the tailing on the peaks in Figures 5. lO(a) and (b) is greater than that usually observed and suggests that the tail may be chemical in nature. As the tail extends over several mass units, no definite mass assignment is possible, but two processes which could contribute are loss of C2H2O and loss of C2H3O·. From [(18)]+·, loss of C2H2O would give rise to a product ion at rn/z 94, and loss of C2H3O· would give rise to an ion at rn/z 93. From [(19)]+. loss of C2H2O would give rise to m/z 96, loss of C2HDO would give rise to rn/z 95, and loss of C2D2O would give rise to m/z 94. Loss of C2H 3O· would give rise to an ion at m/z 95, loss of C2H2DO· would give rise to an ion at m/z 94, and loss of C2HD2O· would give rise to an ion at m/z 93. Any of these processes could be operative which would result in the presence of m/z 94, due to loss of C2H4O from [(19)]+·, being impossible to detect. Calculations were, therefore, performed to establish whether the absence of hydrogen exchange would exclude the possibility of a stepwise mechanism. 79 5.8.1. The kinetic model for hydrogen exchange. A reaction scheme describing hydrogen exchange and the stepwise elimination of acetaldehyde from benzyl 1, 1-Di-ethyl ether (19) following ionisation is shown in Scheme 5 .8. Transfer of D from the reactant ion results in the formation of the dis tonic ion intermediate [(26)]+·. This intermediate may then undergo C-0 cleavage to eliminate C2H3DO giving rise to the ion [(27)]+•, D transfer to regenerate [(19)]+·, or H transfer to form [(28)]+·. This new reactant ion [(28)]+. can then undergo similar hydrogen transfer to [(19)]+., except that two distinct hydrogen isotopes are available for transfer. Transfer of H results in the reformation of [ (26)]+. which may eliminate C2H3DO. D transfer results in the formation of [(29)]+., which can undergo either reverse hydrogen transfer giving [(28)]+. or C-0 cleavage to eliminate C2H40 resulting in the ion [(30)]+.. This reaction scheme may be represented kinetically as shown in Scheme 5.9. In order to solve the kinetic scheme, the method of Laplace transforms as described in Appendix 6 was used. For the situation in which there are no zero roots (see Appendix 6), the abundance of each species at time t, [ lt, is given as: [MIJ - [exp (-R(n)t) [ ~-l)m R(n)m a(3-m) ]] dE 1 1-;~) t 1, 0 II (R(l) - R(n)) 1~1 ,l.,t"' E""'°ll [M2Ji = !P(E) kl k4 !,[exp(-R(n)t) [ 4-j3 - R(n) ] IdE II (R(l) - R(n)) ]=\ e-. J.:i!n [Il)i = J P(E) kl ~ [exp(-R(n)t)[I: \l)m R(n)m a(2-m) dE 0 1 ]I II (R(l) - R(n)) 1,;1 1"'" 0 0 0 0 D~ D~ H~ H~ + + + + 7+• 7+• 2 2 . . D D ~ ~ [(27)]+ [(27)]+ [(30)]+. [(30)]+. (19). (19). 0 of of ion ion .. .. molecular molecular the the from from 0 0 D~ D~ [(26)]+• [(26)]+• H H acetaldehyde acetaldehyde D D of of [(29)]+. [(29)]+. elimination elimination Stepwise Stepwise 5.8. 5.8. Scheme Scheme _[(19)]+• _[(19)]+• Scheme 5.9. Kinetic scheme representing the reactions shown in Scheme 5.8. Ml k3(E) 11 ------~Pt TS4 M2 kg(E) 12 ------~P2 TS5 80 c_.,. [l2Jt = f P(E) kl k4 k6 :Xp(-R(n)t) ] dE 0 t. [ IT (R(l) - R(n)) 1.~1 l.1'T'I [P!Ji = [:~) kl k3 [ :(2) _~ [exp(-R(n)t) [ i<-l~m R(n)m a(2- ml dE TI R(n) L R(n) IT (R(l) - R(n)) "•r '"~ , 'l,: I 1#n E..,..,. [P2lt = f P(E) kl k4 k6 k8 _4-_1_ exp(-R(n)t) dE 0 I 4- [ TIR(n) - R(n) ~n(R(l) - R(n)) II "~ I ~ P(E) is the energy deposition function, which was kept constant in the calculations, Emax is the maximum internal energy and ki represent ki{E), the rate coefficients in Scheme 5.9. The terms R(n), a(n) andj3 are defined in Appendix 6. A computer program, CALKT, was written incorporating this solution of the kinetic scheme. The program calculates the abundance of each ionic species over the time range 10- 14 s to 10-3 s. These abundances are plotted against log10 (time / s). The extent of hydrogen exchange is evaluated by calculating the change in abundance of [(30)]+·, with respect to the change in the sum of the abundances of both products, [(27)]+. and [(30)]+·. The extent of decomposition in the time-frame sampled by the MIKE experiment is given by the changes in abundances of the products from the entrance-time, t1, and the exit time, t2, for the second field-free region of the mass spectrometer. For the benzyl l,l-D2-ethyl ether radical cation, m/z 138, at an accelerating potential of 8 kV, t1 and 12 are 1.9x10-5 s and 4.4xio-5 s. Program CALKT is listed in Appendix 7. Input to program CALKT consists of the rate coefficients, ki(E), for each of the eight 81 reactions, the maximum internal energy of the reactant ion and the symmetry numbers, CJ, for each reaction. For the reactions shown in Scheme 5.9 the symmetry numbers are 2, 1, 1, 1, 1, 1, 2 and 1 for reactions 1 through 8 respectively. In steps 1 and 7, two D atoms are available for transfer giving CJ= 2. For all other steps only one pathway for reaction is available which gives CJ= 1 for all other steps. 5.8.2. Ion structures and energetics. The heat of formation of the molecular ion of benzyl ethyl ether has been estimated as 731 kJ mol- 1 (Section 5.6) and the combined heats of formation of [C7H8]+- and acetaldehyde have been placed at 747 kJ moi- 1 from appearance energy measurements which allowed the energy barrier to reaction to be placed at 16.0 kJ moi- 1. Using a thermochemical calculation the heat of formation of the intermediate distonic ion [(31)]+- was calculated to be 774 kJ mol-1 (see Appendix 8). As the energy barrier to reaction has been measured to be 16.0 kJ moi- 1, the estimate of the heat of formation of the intermediate ion rules out the possibility of the elimination being stepwise with this intermediate. Wesdemiotis et al. [139] using a different thermochemical calculation have estimated Afft<[(18)]+·) as 765 kJ moi-1, Afft<[31)]+·) as 761 kJ moi- 1 and the barrier to reaction as 26 kJ mo1- 1. Given that the barrier to reaction has been measured to be 16.0 kJ mo1; 1, the heats of formation of the products [139] are too high in energy by 10 kJ moi- 1. If ~Hf for the products is reduced to 781 kJ moi- 1, these estimates for the energetics become consistent with a stepwise mechanism which occurs via [(31)]+-. As an alternative; heats of formation for all species have been calculated using the MOPAC program. The heat of formation of [(18)]+- was calculated as 718 kJ moi- 1, 82 that for [C7H8]+. as 901 kJ moi-1, acetaldehyde as -176 kJ moi-1 and that for [(31)]+. as 704 kJ moi-1. Combined with the energy barrier to reaction, the energetics calculated are consistent with a stepwise mechanism. The geometries used to calculate heats of formation for these species are given in Appendix 5. The two approaches to estimating the energetics appropriate to the stepwise mechanism are similar when the heats of formation and the energy barrier observed from the appearance energy are combined to place constraints on the critical energies, E0's, for each reaction. The thermochemical estimations based on those of Wesdemiotis et al. [ 139] give the following limits for the critical energies: E0 ( hydrogen transfer ) between O and 16 kJ moi-1 E0 ( reverse hydrogen transfer ) : between 4 and 16 kJ moi- 1 E0 ( C-O cleavage ) less than 20 k J moi- 1. For the MOP AC estimates, the critical energies are within the following limits: E0 ( hydrogen transfer) between O and 16 kJ moi-1 E0 ( reverse hydrogen transfer ) : between 14 and 30 kJ moi-1 E0 ( C-O cleavage) between 20 and 30 kJ moi-1. The two sets of energetics differ only in that the value of Aflr ([intermediate]+·) from the MOPAC calculation is lower than that obtained from the thermochemical estimate. The MOPAC calculation also places a lower limit on the critical energy for the C-O cleavage. Eight different E0's are involved in the kinetic scheme shown in Scheme 5.9. These E0's have been obtained from those given above for the unlabelled ions through 83 consideration of the effects of isotopic substitution upon the zero-point energies of the various species. 5.8.3. Vibrational frequencies and critical energies. Four stable ions, [(19)]+., [(28)]+., [(26)]+. and [(29)]+. (Ml, M2, 11 and 12 in Scheme 5.9) are involved in the kinetic model (Scheme 5.8). The MOPAC calculations on [(19)]+. and [(26)]+. were extended to obtain vibrational frequencies for the four labelled ions. These frequencies are given in Table 5.14. Five transition states are involved in the kinetic model, three represent hydrogen transfer (TSl, TS2 and TS3) and two represent C-O cleavage (TS4 and TS5). The hydrogen transfer transition states were constructed by averaging the vibrational frequencies of the two stable ions connected by the transition state. One C-H stretch was chosen to represent the reaction coordinate. The C-O cleavage transition states were constructed from the vibrational frequencies of the appropriately labelled intermediate by removing one C-O stretch as the reaction coordinate. The vibrational frequencies for the five transition states are given in Table 5.15. Table 5.16 gives the zero-point energies (ZPE's) for the species involved in the reaction mechanism shown in Scheme 5.9, and also for the unlabelled reactant ion, [(18)]+., and intermediate ion [(31)]+·. The critical energies for the eight reactions can be obtained from the ZPE's and the critical energies for the corresponding steps in the unlabelled reaction. Let E0 for the hydrogen transfer from [(18)]+. be X, that for reverse hydrogen transfer be Y and that for C-O cleavage be Z. The critical energies can be related as follows: Ml I 1 H2 12 36 36 36 36 37 42 39 42 72 57 72 57 84 65 84 69 138 11 1 135 108 143 201 144 197 265 217 265 219 285 275 274 276 307 325 308 322 336 359 333 357 435 405 427 400 468 468 466 469 519 518 1198 514 557 555 547 551 658 666 100 653 725 698 701 723 761 730 752 739 795 71~ 3 760 780 845 799 816 801 860 814 840 806 869 833 873 817 878 869 878 833 893 870 891 867 928 894 910 880 930 910 929 905 937 949 938 913 973 977 969 975 978 1001 986 1021 984 1025 996 1035 1006 1036 1073 1038 1054 1087 1088 1069 11 14 1112 111 4 1076 1139 1121 1139 11 14 1140 1142 1177 1120 1154 1147 1194 1158 1192 1176 1204 1202 1226 1216 1218 1241 1238 1241 1265 1254 1270 1259 1273 1259 1292 1274 1291 1277 1294 1287 1294 1289 1295 1297 1295 1295 1315 1301 1318 1301 1336 1308 1337 1308 1347 1322 1346 1318 1361 1402 1352 1402 1387 1409 1385 1409 1481 1668 1478 1679 2112 2124 2113 2120 2115 2175 2228 2125 2836 2890 2836 2890 2901 2905 2893 2921 2948 2921 2901 2932 2950 2932 2949 2938 3015 2939 2951 2956 3032 2997 3015 2997 3039 3049 3039 3049 3044 3063 3044 3063 3048 3069 3048 3069 3055 3074 3054 3074 Table 5.14. Vibrational frequencies for the 4 reactant ions (Ml, 11, M2 and 12) of Scheme 5.9. (Frequencies are in cm-1.) TS1 TS2 TS3 TS4 TS5 36 36 36 36 36 110 41 41 42 42 65 65 65 57 57 75 75 77 65 69 1211 123 122 111 108 172 173 171 201 19'7 241 241 .242 217 219 281 275 275 275 276 316 317 315 325 322 3117 3117 3116 359 357 420 416 413 405 400 1168 467 1.168 468 469 519 509 506 518 514 556 552 5119 555 551 662 683 677 666 653 712 700 712 698 723 745 741 745 730 739 770 752 770 743 780 823 808 809 799 801 837 827 823 814 806 851 853 8115 833 817 873 814 856 869 833 882 881 879 870 867 911 902 896 894 880 920 920 917 910 905 943 943 925 949 913 975 973 972 977 975 990 99 11 1004 1001 1021 1004 1011 1016 1025 1035 1022 1055 1056 1087 1069 1071 1088 1079 1112 1076 1113 1113 1095 1121 1114 1130 1130 1127 1142 1120 114 1 1160 1148 1147 1158 1150 1171 1176 1176 1202 11811 1191 1203 1216 1241 1221 1217 1229 1241 1254 1239 1254 1260 1259 1259 1265 1266 1266 12711 1277 1283 1283 1284 1287 1289 1291 1291 1292 1297 1295 1296 1296 1295 1301 1301 1308 1310 1310 1308 1308 1322 1322 1322 1322 1318 1335 1335 1332 1402 1402 1382 137'7 13Tr 1409 1409 1399 1398 1397 1668 1679 1574 1573 1579 2124 2120 2146 2202 2177 2175 2125 2863 2119 2863 2890 2890 2903 2899 2907 2905 2921 2935 2911 2917 2921 2932 2941 29 111 29114 2932 2938 2977 2946 2954 2939 2956 3015 3006 3006 2997 2997 30115 3045 3045 3049 3049 3054 30511 30511 3063 3063 3059 3059 3059 3069 3069 3065 3065 3065 3074 3074 Table 5.15. Vibrational frequencies for the 5 transition states (TSl, TS2, TS3, TS4 and TS5) of Scheme 5.9. (Frequencies are in cm- 1.) Table 5.16. Zero-point energies (ZPE's) for the stable ions and transition states of scheme 5.9. Also given are the ZPE's of the corresponding unlabelled species. Ion ZPE / kJ mol-1 (i) [C6H5CH20CH2CH3]+- : [(18)]+- 461.66 (ii) [C6H6CH20CHCH3]+. : [(31)]+- 460.57 (iii) TS ( (i) H (ii) ) 443.60 (iv) TS ((ii) ➔ products) 454.37 (v) [(C6H5CH20CD2CH3)]+. : [(19)]+- 445.15 (vi) [C6H5DCH2OCDCH3]+- : [(26)]+- 444.69 (vii) [(C6H4DCH2OCHDCH3]+- : [(28)]+- 445.43 (viii) [C6H4D2OCHCH3]+- : [(29)]+- 444.41 (ix) TS ( (v) H (vi) ) 432.32 (x) TS ( (vi) H (vii) ) 428.07 (xi) TS ( (vii) H (viii) ) 432.34 (xii) TS ( (vi) ➔ products) 438.49 (xiii) TS ( (viii) ➔ products) 438.20 84 E0(1) = X + ZPE(i) - ZPE(v) - ZPE(iii) + ZPE(ix) Eo(2) = Y + ZPE(ii) - ZPE(vi) - ZPE(iii) + ZPE(ix) E0(3) = Z + ZPE(ii) - ZPE(vi) - ZPE(iv) + ZPE(xii) E0(4) = Y + ZPE(ii) - ZPE(vi) - ZPE(iii) + ZPE(x) E0(5) = X + ZPE(i) - ZPE(vii) - ZPE(iii) + ZPE(x) E0(6) = X + ZPE(i) - ZPE(vii) - ZPE(iii) + ZPE(xi) E0(7) = Y + ZPE(ii) - ZPE(viii) - ZPE(iii) + ZPE(xi) E0(8) = Z + ZPE(ii) - ZPE(viii) - ZPE(iv) + ZPE(xiii) ZPE(i) is the zero-point energy of species (i) in Table 5.16, etc ... (As the ZPE's are calculated in energy units of cm-1, and the calculations were actually performed using units of cm- 1, the following energies have more significant figures than, and may disagree slightly with, the values given above.) Taking the values for ZPE's in Table 5.16, the critical energies, in kJ moi- 1, are: E0(1) = X + 16.51 - 11.28 = X + 5.23 E0(2) = Y + 15.88 - 11.28 =Y + 4.60 E0(3) = Z + 15.88 - 15.88 = z E0(4) = Y + 15.88 - 15.53 =Y + 0.35 E0(5) = X + 16.22 - 15.53 = X + 0.69 E0(6) = X + 16.22 - 11.26 = X + 4.96 Eo(7) = Y + 16.16 - 11.26 =Y + 4.90 E0(8) = Z + 16.16 - 16.17 =Z - 0.01 . E0's are greater for reactions involving D transfer (steps 1, 2, 6 and 7) as compared with their corresponding reactions which involve H transfer. Similarly, C-0 cleavage which eliminates C2H3DO (step 3) proceeds with a larger critical energy than cleavage 85 to eliminate <;H4O (step 8). From the range in E0's given in Section 5.6 and the corrections calculated above, the following ranges for E0, in kJ moi-1, are obtained for the thermochemically estimated energetics: E0(1) ranges between 5.23 and 21.22 E0(2) ranges between 18.59 and 24.54 E0(3) less than 20.00 E0( 4) ranges between 4.29 and 20.29 E0(5) ranges between 0.69 and 16.69 E0(6) ranges between 4.96 and 20.96 E0(7) ranges between 8.85 and 24.60 E0(8) less than 19.99 . Using the MOPAC energetics the critical energies, in kJ moi-1, are: E0(1) ranges between 5.23 and 21.22 E0(2) ranges between 18.65 and 34.59 E0(3) ranges between 20.00 and 29.99 E0(4) ranges between 14.34 and 30.33 E0(5) ranges between 0.69 and 16.69 E0(6) ranges between 4.96 and 20.96 E0(7) ranges between 21.05 and 34.89 E0(8) ranges between 19.99 and 29.98 86 5.8.4. Details of the calculations. Further requirements for the calculation other than the symmetry numbers, vibrational frequencies and critical energies described above, are the grain-size for the k(E) calculation and the maximum energy Emax· Typical values for the grain-sizes were 0.14 kJ moi-1 (12.0 cm-1) in the case of a stable ion and 0.16 kJ moi-1 (13.3 cm-1) in the case of a transition state. Emax was initially set at 240 kJ mol-1, as the values of k(E) at energies above 240 kJ moi-1 were greater than 1010 s- 1, i.e. much greater than the maximum k(E) which contributes to the ion decomposition in the metastable time-frame. Reducing Emax to 60 kJ moi-1 did not affect the values of k(E) at low energy and did not alter the calculated ion abundances for the MIKE experiment, although the total product ion formation was reduced. The values ofk(E) at 60 kJ moi-1 were greater than 106 s- 1 for all calculations. Emax was set to 60 kJ moi-1. The effect of lowering the transition state vibrational frequencies was tested as follows. For the hydrogen transfer transition states (TSl, TS2 and TS3), five vibrational frequencies (one C-H stretch and four C-H bends) were lowered by 0, 5, 10, 15 and 20 %. These vibrational frequencies are given in Table 5.17. For the C-O cleavage transition states (TS4 and TS5), seven vibrational frequencies (two benzyl C-H stretches, four benzyl C-H bends and one C6Hs-C-O bend) were lowered by 0, 5, 10, 15 and 20 %. These vibrational frequencies are given in Table 5.18. Lowering of these vibrational frequencies affects the ZPE of the transition state and thus affects the critical energy for the reaction. The critical energies for each of the eight reactions for the five adjusted transition states are given in Table 5.19. Table 5.17. Altered transition state frequencies, in cm-1, for the hydrogen transfer transition states TS 1, TS2 and TS3. % lowering 0 5 10 15 20 2146 2039 1931 1824 1717 975 926 878 829 780 TSl 920 874 828 782 736 873 829 786 742 698 662 629 596 563 530 2119 2013 1907 1801 1695 973 924 876 827 778 TS2 920 874 828 782 736 874 830 787 743 699 683 649 615 581 546 2863 2720 2577 2434 2290 972 923 875 826 778 TS3 925 879 833 786 740 879 835 791 747 703 677 643 609 575 542 Table 5.18. Altered transition state frequencies, in cm-1, for the C-0 cleavage transition states TS4 andTS5. % lowering 0 5 10 15 20 2932 2785 2639 2492 2346 2890 2746 2601 2457 2312 1308 1243 1177 1112 1046 TS4 1287 1223 1158 1094 1030 1176 1117 1058 1000 941 1112 1056 1001 945 890 217 206 195 184 174 2938 2791 2644 2350 2350 2890 2746 2601 2457 2312 1308 1243 1177 1112 1046 TS5 1289 1225 1160 1096 1031 1202 1142 1082 1022 962 1114 1058 1003 947 891 219 208 198 186 175 Table 5.19. Critical energies, in kJ moI-1, for the adjusted transition states. X,Y and Z are described in Section 5.8.3. Reaction no. Transition state vibrational frequency lowering 0 5 10 15 20 1 X + 5.23 X+ 5.71 X + 6.18 X + 6.66 X + 7.14 2 Y +4.60 Y + 5.08 Y + 5.56 Y + 6.03 Y + 6.51 3 z Z + 0.01 Z + 0.03 Z +0.04 Z + 0.06 4 Y + 0.11 Y + 0.83 Y + 1.31 Y + 1.79 Y + 2.28 5 X + 0.69 X + 1.18 X + 1.66 X + 2.14 X+ 2.63 6 X + 4.96 X+ 5.00 X+5.04 X + 5.07 X + 5.11 7 Y +4.90 Y + 4.93 Y +4.97 Y + 5.01 Y + 5.04 8 Z - 0.01 Z -0.02 Z - 0.03 Z -0.04 Z -0.04 87 5.8.5. Results obtained using the thermochemical energetics. The thermochemical energetics are highly restrictive, in that Af--4(products) is placed at the same energy as the thermochemical limit for reaction derived from experiment. E0 for C-0 cleavage is constrained to be less than 20.00 kJ moi-1. Only E0 for hydrogen transfer and the transition state vibrational frequencies are variable. E0(1) for D transfer from [(19)]+. was varied from the minimum value of 5.23 kJ moi-1 to the maximum value of 21.22 kJ moi-1, with E0(3) for the C-0 cleavage from [(26)]+. set at 20.00 kJ moi-1. E0 for the other reactions were fixed as shown in Table 5 .19. Initially the transition states with only the reaction coordinate removed were used with E0(1) set at 5.23 kJ moi-1. The calculated ion abundances are shown in Figure 5.16(a). 90 % of the reactant ions have decomposed within 10-6 s (ion source residence time). 34 % of the product ions formed within 10-6 s are due to the loss of C2H40. No decomposition is predicted for the metastable time-frame sampled by the MIKE experiment. The extent of hydrogen exchange is depicted in Figure 5.16(b), which shows as a functrion of time the proportion of the total rate of formation of product ions which is due to elimination of C2H40. Elimination of C2H40 rises quickly to 30 % of the total product ion formation, then increases further until C2H40 elimination accounts for 100 % of the product ions formed at 10-5 s. No product ions are formed at times greater than 10-5 s. Increasing E0( 1) to 17 .19 kJ moi-1 increases decomposition within 1o- 6 s to 100 %, with 20 % of the product ions being due to the elimination of C2H40 (Figure 5.17). On increasing E0(1) to 21.22 kJ moi-1, decomposition within 10-6 s remains at 100 %, but the amount of C2H40 product formed drops to 7 % (Figure 5.18). With E0(1) set at its maximum value of 21.22 kJ moi-1, lowering of E0 for C-0 cleavage does not alter (b) 8 g (a) e I ~ ' g I 8 £ 'i g 8 g ~ 3 8 ~ ~ vi...., P1 u 8 ~ 8 .g ~ ~ 0 L. 0. 0 ' '----' ~ ~ g C: ii -0 " '-- -0" § 8 ±. : ..c i o• " .... P2 ~ ~ ~ 8 y ~ ::: ~ ., I :,; u I .. 8 ~~ ,: ~ I I g I' s 2 :J\◄.00f,IJO ·14! ')0 -'t1.'lO ·10 'JO .ij 'JO .a')0 -fn -i 'JO :§:.,o .too .~ 00 ~-[,o- Figure 5.16. (a) Ion abundance curves and (b) extent of C2H4O elimination from [(19)]+. using the thermochemical energetics. 1 E0 (D-transfer) = 5.23 kJ rnoi- 1 E0 (C-O cleavage)= 20.00 kJ rnoi- (b) (a) g i §,--_ ~, ~ ~ ~ • " ' I p 1 8 ~ i ~ g g e ~ 'E <.) ;: 8 ;:l ~ 'j "O ~ 0 '- ... c.. 0 V i: '-' ~ <.) "O i ~ ' '- -g" I ;: ;:l ~ + 11 ' ' ~. .&J I I I ~ " :r: <' V UN ,. ~ ' > I ~ .., ~ I ~ :::s V P2 .. g ~ ;: 'ii II I I g 8 2 2 I I I I g ~ ., 00 o_•,...o, -10. ')0 oo ·7.00 -II •• ➔ .oo .. ~.00 0 . _')') - ~, ·b ':lO -12. ':l'J .'lt.lJO -a ·• ,o -14 ')') " . ' ·4.00 log,. ( time / s ) log 10 ( time I s ) Figure 5.17. (a) Ion abundance curves and (b) extent of C2H40 elimination from [(19)]+. using the thermochemical energetics. E0 CD-transfer) = 17.19 kJ mol· 1 E0 (C-0 cleavage)= 20.00 kJ moI·1 -~.O? - -:..4.~0 - -~.O') 00 -Ii the .$.,o using ?0 .,1i / s ) 1 ?O -5 [(19))+. time mo1· ( (b) '.lO 10 ·10 kJ from log ':? .'11 20.00 ·12.'J? elimination ')0 0 4 ··13 H cleavage) 2 ?O ◄ C 1 1 ~ 0 ~ :;, 'ii ~ 0 i g ~ ~ 1: " g SJ ~- : :, : " of I Vl u ::, 0. 0 ... (C-0 + "O ..., '- "O u 0 -c, :i ,-, ._, 2, ~. 0 E extent 1 (b) moi- :i(l and -~ kJ 1 p P2 ')O _. _ curves 21.22 energetics. ·" -,5';C' abundance Ion CD-transfer)= 0 thermochemical (a) E / s ) 5.18. time (a) ( log., Figure '}0 ·13 ?O ."14 8 T--~, 8 ~ 8 ll ~ 8 ~ e II ~ g ~ ~ 2 g 0 ~ ~ g 8 'i u " C: "' lJ ,:: C: .. ::, ~ '- .0 "O - .~ .; g ~ (a) . (b) ~ I I p 1 I 8 \1 ~ !j I I I I g ~j I I I \ I I • I I I g ~ ~j \ I ,....., ..,"' (,) I ,: ~ I I ;l ~ :. I j I "Cl 0 I '- I I a.'" ~ I ._.. ~ u ~ (,) I "Cl ;g ::: ~ I :: I '- -g I g ;l • I + i .Q i I Cl I s I ::i;, g u 8 I I > I I u S?i _, ~ I ,: I :::, I v "Cl g ... 8 ,: \l I I I 8 I ~ ~ ~ P2 g ~ 0 -10 ')') -8.l)0 -,i ')') 00 ?0 -~.oo -4.00 -:1 'JO ')'J •.,,')() -·1◄.00 ·13 1)0 ·I~. ?O 0 -14.:'. •t:! ';0 ·12. ?I} -!I JO -10.n - ·9 'l? ·II 'l'J " ·o ·◄')') ·- ':') log10 ( time / s ) log 10 ( time I s ) Figure 5.19. (a) Ion abundance curves and (b) extent of C2H4O elimination from [(19)]+. using the thermochemical energetics. 1 E0 (D-transfer) = 21.22 kJ moi- 1 E0 (C-O cleavage)= 15.21 kJ moi- Figure 5.20. Potential energy profiles for the extreme cases (a) E0(H-transfer) = 5.23 kJ mo1- 1 ; E0(C-O cleavage)= 20.00 kJ moi-1, and 1 1 (b) E0(H-transfer) = 21.22 kJ mol- ; E0 (C-O cleavage)= 15.21 kJ mor . (a) 20.00 (b) 21.22 15.21 4.00 88 the extent of decomposition occurring within 10-6 s but does decrease the amount of C2H4O eliminated. With E0(3) reduced to 15.21 kJ moi-1, and E0(1) set to 21.22 kJ moi-1, elimination of C2H4O accounts for only 2 % of the total decomposition (Figure 5.19). The potential energy profiles which describe the extreme situations, i.e. the abundances shown in Figures 5.16 and 5.19, are shown in Figure 5.20. Varying the vibrational frequencies of the transition states for hydrogen transfer and C-O cleavage has negligible effect on the calculated ion abundances. With both hydrogen transfer and C-O cleavage transition state vibrational frequencies lowered by 20 %, E0(1) increased to 7.14 kJ moi-1, due to the ZPE changes, and E0(3) increased to 20.06 kJ moi-1. Decomposition within 10-6 s was 94 %, with 34 % of the product ions being due to the elimination of C2H4O. No decomposition is predicted for times greater than 10-5 s. Increasing E0(1) to 23.13 kJ moi-1, decomposition within 10-6 s increased to 100 %, with 5 % of the product ions being due to the elimination of C2H4O. These results are almost identical to those obtained without adjustment of the vibrational frequencies of the transition states. 5.8.6. Results obtained using the MOPAC energetics. The MOPAC energetics place only moderate constraints on the critical energies for hydrogen transfer and C-O cleavage. To investigate the effect of the energetic data on the ion abundances, the transition states from which only the reaction coordinate was removed were chosen initially. E0(3) (C-O cleavage) was set to the maximum value of 29.99 kJ moi- 1 and E0(1) (D transfer) was set to the minimum value of 5.23 kJ moi-1. The other E0 were set as described by the relationships given in Table 5.19. The calculated ion abundances are shown in Figure 5.21. Within 10-6 s, 87 % of the reactant ions have decomposed to products, with 34 % of these product ions being due ,. ·) .?O ◄ · ... -~ _.lo the -ho using ) s -~,,- I 1 ?o -h [(19)]+. time (b) moi- ( 'JO ·10 kJ from log10 ·11.':l'J 29.99 12.')'J 1 - elimination 0 -IJ'J'J 4 H 2 cleavage)= C g ~, s -- • . g ~ g s i " :;; g ~ g I: g ~ g o},,._n of "' () 0 ;::, Cl. l- I .... -c:, + -c:, ,-, '- 3 r.J '-' ~ ~ (C-0 Ea extent 1 (b) mo1· :JO and .~ kJ P1 ?2 -4.'JO 5.23 curves I I I I I I I I I I I I I I I I I I I I '° = -5 I I I I energetics. -li" abundance -ho Ion (D-transfer) ) (a) MOPAC Ea s -El?'J ':'J I ➔ 5.21. time/ ( (a) 0 -1:J'JC log1 Figure 1 I ·ll.1'J \ -12.?i:I M2 -=0~ ,~ ·1:! \ M1 \ i ~1 ;; :i 6 ~ ~ ~ ~ 1 'I i l' g ~ 8 'il l' ~ l'r o.U.:J"; () ., ~ " C: ,:: ::, ., ::, ~ .. '- -c:, .0 .... .:: "ii -~.'lo -~.'lo -~.'JO -~.'JO I I :too :too 'lo 'lo ' ' -i -i the the -S.,o -S.,o using using 'llJ 'llJ -9 -9 / / s ) 1 1 ,:,o ,:,o -9 -9 [(19)]+. [(19)]+. time time (b) (b) moi- ( ( 10 10 10 10 -10 -10 kJ kJ from from log -'11.,:,0 -'11.,:,0 29.99 29.99 ·12.')0 ·12.')0 =< =< elimination elimination rio rio 0 0 -1::i -1::i 4 H cleavage)= cleavage)= 2 C -u.oo -u.oo ~ ~ g g p p S! S! 0 0 0 0 s s ~ ~ : : g g g g ,; ,; :;, :;, g g 0 0 g g 'il 'il 0 gl gl of of I I <.) <.) Ill Ill 0 0 .. .. ::, ::, + + _, _, (C-0 (C-0 -0 -0 -0 -0 u u ' "Q "Q :!: :!: ,...... ,...... ..::; ..::; s~ s~ ~ ~ Ea Ea extent extent 1 1 (b) (b) moi- 00 00 and and -J -J kJ kJ l l P2 P2 p p --4.0'J --4.0'J 7.62 7.62 curves curves ,. ,. co co energetics. energetics. ...... abundance abundance AC AC -1,')Q -1,')Q CD-transfer)= CD-transfer)= Ion Ion Ea Ea MOP MOP (a) (a) -11')0 -11')0 / / s ) ')' ')' 'I. 'I. -9 -9 5.22. 5.22. time time ( ( (a) (a) ".)Q ".)Q \ \ 10 10 -10 -10 log Figure Figure •11.')0 •11.')0 I I I I ·l2.'>Q ·l2.'>Q \ \ I I 'J? 'J? ·13 ·13 111 111 I I 8 8 21 21 ~ ~ s s § § g g C: C: .. .. "' "' <.) <.) " " " " C C ; ; :, :, ~ ~ "ii "ii ~ ~ .r-. .r-. -0 -0 '- (a) g (b) !!~8 § M1 g 'I 8 I Ii ' " ' Pl g & I ~ i I' I g 8 I '< e 'I ,- ;/l .... (.) g ;:l 8 ' j; ~ ' "O i I I 0 I .. '- 0.. 0 ' ...... ,~ u 8 I (.) -0 :;, i::: :;, "' '- -g g & + ' ~ ~ I 0 "' " ~ g u !! I u I ~ :: t! I ;; 6 'ii P2 "O g .. 8 lo I I 'l I g I I I I ~j ~ /_ - I ~ ~ I I I I I I I g .,:: =;:::::::::::: :--- •10 ')0 -i:HlO ·• .00 - .oo -4. 'J? -l oo ~~ .• ~'JO 0-1◄.':').. I:) 'JO · 12. ')') ·ll .'JO "I0 _")':) ~o .:t'JO -5.-;o -◄,':') . ... -u.,o •1J n ·12 -ll '' -10 -5 " =r:=-I.co " " log., ( lime s ) log 10 ( lime I s ) I Figure 5.23. (a) Ion abundance curves and (b) extent of C2H40 elimination from [( 19)]+. using the MOPAC energetics. 1 E0 CD-transfer)= 17.19 kJ moi- 1 E0 (C-0 cleavage) = 29. 99 kJ moJ· :1.'lO :1.'lO .. .. ,00 ,00 "◄ I I ' ' I I I I I I I I ' ' I I I .?0 .?0 " " the the using using / / s ) ------ 1 1 10 10 [(19)]+. [(19)]+. lime lime (b) (b) ( ( mo1· 10 10 ·IO'JO ·IO'JO kJ kJ from from log ·IS.?O ·IS.?O 29.99 29.99 'J'J 'J'J I~ I~ . . elimination elimination O O •1:}')i;) •1:}')i;) 4 H cleavage)= cleavage)= 2 .')'J .')'J C ~ ~ g g g g 0 0 ~ ~ § § Ii' Ii' ,: ,: 8 8 g g ~ ~ IZ IZ j, j, 0 0 ii! ii! ~ ~ - g g 0~·1 ◄ g g g g :a: :a: 'ii 'ii of of u u 0.. 0.. ...... ;::l ;::l V) V) 0 0 (C-O (C-O _, _, '-. '-. " " .,, .,, q_ q_ ~ ~ y y :::;i :::;i ,....., ,....., ~ ~ '-' '-' ,±.. ,±.. 0 0 E extent extent 1 1 (b) (b) moi- ')j ')j and and -~ -~ kJ kJ 1 1 p p P2 P2 ·• curves curves ' ' I I I I 21.22 21.22 I I ' ' I I ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ...... , ...... , " " ~o ~o energetics. energetics. .;; .;; abundance abundance Ion Ion CD-transfer)= CD-transfer)= 0 0 (a) (a) E MOPAC MOPAC / / s ) 5.24. 5.24. time time ( ( (a) (a) 10 10 log Figure Figure ·\300 ·\300 ~'J ~'J o.-,. o.-,. g g ~ ~ g g ~ ~ i i !! !! s s ~ ~ g g " " i i 8 8 '3 '3 g g 8 8 8 8 g g Ii' Ii' ~ ~ ii' ii' " " -~Ml -~Ml " " u u " " ...... " " ~ ~ u u u u ~ ~ ;:, ;:, ~ ~ '"ii '"ii .t::; .t::; .,, .,, Is! Is! '- (b) 8 (a) 8 � § �, p 1 8 s fi! :. g g ll :;; 8 :; I I e " 8 8 ;:l � i � '- " 0.. u l! g C jl ':o' " ,: ....._,_ 8 g ..0 i + � IJ 8 g I u � I I I � I I I I 8 I I g I I ,;: I � I I I I I 8 I I :;; I I I I I I � I I � I I I I 8 I I P2 g I 0 : o.,, �f;l -I:!':') -ti')'3 .,, -•.IJO -J')') -1,.0') -131)1) -12')0 ·U 'JO •10 ,o -9 'JO ,. - .00 00 .00 .,..00 . 00 log 10 ( time / s ) log 1 0 ( lime I s ) + Figure 5.25. (a) Ion abundance curvesand (b) extent of C2H4O elimination from [(19)] . using the MOPAC energetics. 1 1 E0 CD-transfer)= 21.22 kJ moi- E0 (C-O cleavage)= 20.00 kJ moi- 89 to the elimination of C2H4O. In the metastable time-frame sampled by the MIKE experiment, 81 % of the decomposing ions eliminate C2H4O. The extent of product ion formation is represented in Figure 5.21(b). At 104 s, all product ions being formed are due to the elimination of <;H4O. Increasing E0(1) to 7.62 kJ moi-1 (and other hydrogen transfer E0's accordingly) did not greatly affect the results of the calculation (Figure 5.22). Little effect on the results was observed until E0(1) was increased to 17.19 kJ moi- 1 (Figure 5.23). Decomposition after 10-6 s increased to 97 %, with 20 % of the product ions being due to the elimination of C2H4O. Within the metastable time-frame, product ions formed by the elimination of C2H4O accounted for 27 % of all product ions formed. Further increasing E0(1) resulted in decomposition within 10-6 s increasing to 100 %. For E0(1) set at 21.22 kJ moi-1, i.e hydrogen transfer and C-O cleavage transition states equal in energy, decomposition within 10-6 s was 100 % with 7% of the product ions being due to the elimination of C2H4O (Figure 5.24). With E0(1) set to the maximum value of 21.22 kJ moi-1, E0(3) (and E0(8)) were reduced. Decomposition within 10-6 s remained at 100 %. Reducing E0(3) to 20.00 kJ moi-1 did, however, decrease the amount of product ion formed by the elimination of C2H4O within 10-6 s to 0.4 % (Figure 5.25). Next the effect of lowering the transition state vibrational frequencies was tested. Lowering both hydrogen transfer and C-O cleavage transition state frequencies by 20 % produced negligible change in the ion abundances. ForE0(1) set at 7.14 kJ moi-1 and E0(3) set at 30.05 kJ moi-1, decomposition within 10-6 s was 91 % with 34 % of the product ions formed being due to the elimination of C2H4O. For E0( 1) set at 23.13 kJ mol-1 and E0(3) set at 30.05 kJ moi- 1, decomposition within 10-6 s was 100 %, with 90 8% of the product ions being due to the elimination of C2H4O. Reducing E0(3) to 19.94 kJ moi- 1, decomposition within 10-6 s remained at 100 % with the amount of product being formed due to the elimination of CiH4O decreasing to 0.5 %. 5.8.7. Discussion. The results of the calculation of the ion abundances need to be considered in the light of the experimental results. The MIKE spectra of [(18)]+. and [(19)]+. establish that the loss of acetaldehyde does occur in the time range 1.9x10-5 s to 4.4x10-5 s. Using the thermochemical energetics, the calculations predict that neither C2H4O nor C2H3DO will be eliminated at times greater than 1o- 5 s. On the basis of the thermochemical energetics, therefore, the stepwise model fails to account for the basic fact that acetaldehyde is lost from the benzyl ethyl ether radical cation at times greater than 10-5 s. Using the MOPAC energetics, the calculations indicate loss of C2H3DO and C2H4O from [(19)]+. when E0(1). for D transfer, is less than or equal to 12 kJ mol- 1, and E0(3), for C-O cleavage, is 30 kJ moi- 1. When E0(1) is greater than 12 kJ moi- 1 (Figure 5.26(a)) no ion decomposition is predicted to occur at times greater than 10-5 s. For E0(1) at 12 kJ moi-1 (Figure 5.26(b)) 30 % of the product ions formed in the time range l.9x10-5 s to 4.4x10-5 s are due to the elimination of C2H4O. When E0(1) is less than '12 kJ moi-1 (Figure 5.26(c)) 80 % of the product ions formed in the time range above are due to the elimination ofC2H4O. On the basis of the MOPAC energetics, the stepwise model predicts that elimination of C2H4O from [(19)]+. is a significant process. Based on the observation that there is very little, if any, hydrogen exchange and the prediction that hydrogen exchange should be a significant process, it is concluded that there is no available evidence which favours a stepwise mechanism, over Figure 5 .26. Potential energy profiles for the stepwise elimination of C 2H 3DO from [(19)] +. using the MOPAC energetics, for the situations: (a) E0 (D-transfer) > 12 kJ mo1-1 ; E0 (C-O cleavage)= 30 kJ mor1, (b) E0 (D-transfer) = 12 kJ mor1 ; E0 (C-O cleavage) = 30 kJ mor1, (c) E0 (D-transfer) < 12 kJ mo1-1 ; E0 (C-O cleavage)= 30 kJ mor1. 30 kJ mo1- 1 14 kJ mo1- 1 (b) 12 kJ mol-1 30 kJ mol-1 14 kJ moi-1 (c) 12 kJ mo1- 1 30 kJ mo1- 1 14 kJ moi-1 91 a concerted mechanism, for the elimination of acetaldehyde from the benzyl ethyl ether radical cation. 5.9. Stepwise elimination of acetaldehyde from a.,a.'-diethoxy p-xylene. A stepwise model for the elimination of acetaldehyde from the molecular ion of a.,a.'-diethoxy p-xylene (20) should be similar to that for the elimination of acetaldehyde from the benzyl ethyl ether molecular ion. The question of hydrogen exchange was addressed above, with the conclusion being reached that no justification for the hydrogen exchange process was available. Accordingly, the model for the stepwise elimination of acetaldehyde from a.,a.'-diethoxy p-xylene does not include hydrogen exchange. 5.9.1. The kinetic model. Transfer of hydrogen from the ethyl group to the ring results in the formation of the dis tonic ion [32]+-. As hydrogen exchange has been ruled out, the hydrogen transferred to the ring is considered to be distinguishable from the hydrogen initially located at the ortho position. Only the transferred hydrogen may undergo reverse transfer to reform the molecular ion. For the labelled compounds (21) - (24), one of two isotopomeric acetaldehydes can be eliminated from the molecular ion. Scheme 5.10 depicts the loss of C2H4O and C2H3DO from the molecular ion of (21 ), and the kinetic scheme which describes these reactions is shown in Scheme 5.11. M represents the molecular ion, PI is the product formed by loss of the lighter acetaldehyde, P2 is the product ion f?rmed by loss of the heavier acetaldehyde, and 11 and 12 are the distonic intermediate ions formed in the eliminations to produce Pl and P2 respectively. ki(E) are the rate coefficients describing each step. Scheme 5.10. Stepwise elimination of C2H40 and CiH300 from the molecular ion of (21 ). ad 0 + . + I • . + 0--4, 0 . + o---h o---l 0 0 :rN + Scheme 5.11. Kinetic scheme representing the reactions shown in Scheme 5.10. M II TS4 klE) PI P2 92 This reaction scheme was solved using the method of Laplace transforms as described in Appendix 6. The abundance of each species at time t, [ lt, is given as follows for the case of no zero roots: [Ml1 - t(E) t, fxp(-R(n)t) [ ~ 2(n) - R(n)(K2 + K3) + K2 K3 ll dE rr (R(l) - R(n)) l•• l;ft\ E....,..y [Illt = J P(E) kl [exp(-R(n)t)[ (K3 - R(n)) ]] dE 0 t, 3 IT (R(l) - R(n)) l•I :i.ln E,_,. [l2lt = J P(E) k4 [exp(-R(n)t)[ (K2 - R(n)) ]] dE 0 t ~ IT (R(l) - R(n)) l.:1 1""' klk3[ K3 + f, [ exp(-~(n)t) (R(n) - K3) ]] dE 3 IT R(n) L R(n) rr (R(l) - R(n)) n,1 n:: I 1.~1 "l.1" e,.,,..._. [P2Jt = f P(E) k4 k6[ K2 + [ exp(-~(n)t) (R(n) - K2) ]] dE 0 3 ~ IT R(n) L R(n) IT (R(l) - R(n)) ll• I n::. I 'lei '11' V'\ ki represent ki(E), the rate coefficients in Scheme 5.11, Emax is the maximum internal energy of the reactant ion and P(E) is the energy deposition function. R(i), K2 and K3 are defined in Appendix 6. The rate coefficients ki(E) have been corrected for the symmetry numbers for each reaction. The symmetry numbers are the same for each of the compounds (21) - (24). In the hydrogen transfer steps, 1 and 4, either one of two hydrogens can be transferred to form the intermediate, therefore cr = 2. The intermediate can transfer only one hydrogen back to reform the molecular ion, giving cr = 1 for steps 2 and 5. C-O cleavage occurs in one way from the intermediate ion, 93 giving cr = 1 for steps 3 and 6. A computer program, NKT, was written incorporating this solution of the kinetic scheme. The program, listed in Appendix 7, evaluates the ion abundances over the time-range 10-14 s to 10-3 s. The decomposition in the metastable time-frame sampled by the MIKE experiment is determined by calculating the change in abundance of the product ions over the time-range defined by the entrance and exit times, t1 and ½_, for the molecular ion of interest, for the second field-free region of the mass spectrometer. The ratio of these changes in abundance is equivalent to the ion abundance ratio measured in the MIKE experiment. 5.9.2. Ion structures and energetics. The heat of formation of the distonic ion [(32)]+- was estimated using the thermochemical calculation described in Appendix 8 . .1.IIc([(32)]+·) was calculated to be 565 kJ moi- 1, which is 59 kJ moi- 1 greater than the heat of formation of the molecular ion (506 kJ moi-1) estimated from photoionisation (PI) measurements of the ionisation energy (Section 5.6). The measured appearance energy of the product ion places the barrier to reaction at 71.2 ± 12.0 kJ moi-1, so that the potential energy profile for the stepwise model is as shown in Figure 5.27(a). The distonic ion [(32)]+- is between the reactant and products in energy. Combining these £'.\Hf values with the energy barrier for the reaction (neglecting the uncertainty) gives the following limits upon critical energies of the individual steps: Eo ( hydrogen transfer ) between 59 and 71 kJ moi- 1 Eo ( reverse hydrogen transfer ) between 0 and 12 kJ moi-1 E0 ( C-O cleavage ) between 0 and 12 kJ moi- 1. Figure 5.27. Potential energy profiles for the stepwise elimination of C2H40 from the molecular ion of (20) obtained using (a) the thermochemical energetics, and (b) the MOPAC energetics. (a) - tE 0 between _rndl2kJmor1 E 0 between 59 and 71 kJ mor1 59 kJ mol-1 (b) Eobetween 59 and 71 kJ mol-l Eo between 25 and 86 kJ mol-l 15kJmor,I 94 Aflt{[(32)]+·) was also calculated using the MOPAC package. The optimised geometry obtained for [(32)]+- (Appendix 5) gives Mli{[(32)]+·) as 487 kJ moi- 1. With the MOPAC calculated values for Mli{[(20)]+·) (502 kJ moi-1 ) and Mli{products) (512 kJ moi- 1 ), and the experimental barrier to reaction of 71 kJ moi-1, the potential energy profile shown in Figure 5.27(b) is obtained. The distonic ion is lower in energy and is more stable than the molecular ion. Combining these ~Hr values with the energy barrier to reaction (neglecting the uncertainty) gives the limits: E0 ( hydrogen transfer ) between O and 71 kJ moi- 1 E0 ( reverse hydrogen transfer ) between 15 and 86 kJ moi- 1 E0 ( C-O cleavage ) between 25 and 86 kJ moi- 1. E0's for the reactions involved in the kinetic scheme (Scheme 5.11) have been obtained from the limits given above through consideration of the effects of isotopic substitution upon the zero-point energies of the various species. 5.9.3. Vibrational frequencies and critical energies. For each of (21) - (24), three stable ions, namely the molecular ion, M, and the two distonic intermediate ions, 11 and 12, are involved in the kinetic scheme. The vibrational frequencies of the molecular ions of (21) - (24) have been given previously (Table 5.7). The MOPAC calculation on the distonic ion [(32)]+- was extended to obtain vibrational frequencies. Two distonic ions are formed in the reactions of each of the labelled species, giving a total of nine distonic ions. The vibrational frequencies for the nine distonic i?ns are given in Table 5.20. For each of the compounds (21) - (24), four transition states are required. Two (20) (21) (22) (23 l (2q) 11,12 11 12 I I 12 II 12 11 12 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 3~ 36 36 36 36 36 36 35 36 36 36 36 36 36 36 36 117 47 "6 115 411 47 117 4] 43 50 50 50 50 116 50 50 115 115 68 68 63 68 51 68 68 57 57 98 98 97 96 93 98 97 92 92 IJ? 136 135 10'1 135 138 I 39 104 104 150 111'} 1"6 147 1116 1119 150 138 140 199 198 19'/ 197 19] 198 196 190 188 215 2111 ·212 212 209 213 213 205 206 230 229 230 228 228 229 227 223 221 248 246 2117 237 211 J 2116 247 229 230 292 290 291 285 289 291 291 281 281 329 328 326 327 323 327 327 319 320 365 365 362 365 362 365 365 361 361 399 399 392 397 388 399 399 387 387 444 442 437 432 428 443 1143 420 422 1165 1161 11611 1155 1161 463 464 440 442 533 533 527 532 525 533 533 524 522 561 561 5116 559 536 560 554 533 530 581 579 578 573 5"16 577 580 567 570 707 661 687 587 602 ·101 707 586 587 730 707 693 101 684 730 730 602 59? 771 "/30 719 716 706 771 771 683 683 793 771 769 732 715 792 793 705 705 797 797 793 172 770 797 796 711 711 824 824 797 797 793 824 824 719 721 869 851 812 822 798 869 867 770 770 877 868 832 8211 811 877 877 798 797 8911 ij72 851 8117 850 893 893 811 811 906 877 879 869 853 905 905 822 822 910 896 900 8"/1 863 909 909 8116 8116 931 908 905 877 880 929 931 851 851 936 915 931 896 901 935 936 853 853 1035 932 9111 908 905 1035 1019 863 862 1038 935 951 915 919 1038 1035 869 871 1052 962 976 9311 929 1052 1052 880 880 1061 1007 1035 946 932 1060 1061 901 901 1063 1035 1052 956 9111 1063 1063 915 915 1076 1038 1059 962 969 1076 1076 920 920 1077 1063 1076 1011 1011 1077 1077 92J 929 1115 106'1 1080 1035 10:,2 1113 1113 939 9)9 1116 I0"/7 1114 1038 1059 1116 1115 9115 9116 1145 1100 11311 1063 1076 11115 1145 956 956 1154 1116 II 38 1064 1099 1153 11511 959 963 1159 1145 11'15 1077 11111 1157 1159 969 967 1195 1156 1156 1103 11311 1195 1195 1U09 995 1223 1191 1164 1116 1138 1209 1223 1012 1011 1233 1197 11911 I \115 1148 1231 1233 1061 1061 1241 1226 1218 1157 1156 1241 1241 1099 1094 1253 12311 1228 1195 11611 125 3 1248 1102 1103 1259 1241 1241 1203 11911 1259 1259 1133 1133 1272 1253 1259 1224 1218 1272 1272 1138 1139 1279 1259 1265 1233 1228 1279 1279 1148 1142 1285 1272 1278 1241 !265 1284 1283 1161 1162 129?. 1279 1288 1253 12"/8 1290 1292 1193 1193 1296 1285 1296 1259 1287 1294 1296 1197 1203 1296 1292 1296 1272 1296 1296 1296 1213 1218 1297 1296 1297 1279 1296 1296 1297 1221 1229 1301 1297 1301 1285 1297 1301 1301 1265 1265 1302 1300 1301 1292 1301 1301 1302 1278 1278 I 310 l 30 l 1309 1300 1~09 1310 1310 1288 1288 1326 1310 1326 \JO 1 1326 1315 1326 1300 1300 1337 1337 1332 I 310 1332 1337 1337 1308 1308 I 3111 1339 I 3110 1337 I 3110 1339 I 3111 1331 1331 1361 I 348 I 36 I 13117 1 36 I 1352 1361 1335 1346 1388 1380 1388 I 378 1388 1371 1388 1350 1378 11111 1411 11110 1lt11 11110 11111 14 11 1410 IU 10 \1131 14 3 1 \1131 1113 I 11131 14 3 1 11131 1113 I 103 I 1679 1679 166"/ 1679 16(,5 1679 16113 1665 1628 2850 2097 212U 2096 2120 2850 2850 2096 2096 2670 2108 2175 2108 21 n 2669 2870 2108 2100 28Vi 2870 2650 2159 2155 2875 2875 2120 2120 2890 2875 2870 2174 2 171 2890 2890 21 39 2139 2909 269U 2890 2178 2182 2909 2909 2155 2155 2920 2920 29011 2870 2850 2920 2920 2159 2159 2926 2926 2909 2875 2870 2926 2926 2171 2171 2933 2933 2920 2690 2890 293) 2933 2174 2174 2933 2933 2926 2920 29011 293) 2933 21H 2176 2938 2938 2933 2926 2909 29)8 2938 2182 2181 29116 2945 2940 2933 2926 29 116 29•6 2670 2870 295U 2950 29"5 2933 2933 2950 2950 2890 2890 2957 295"/ 2950 29 38 2946 2957 2957 2900 2904 2997 2997 2997 295"/ 2950 2997 2997 2926 2926 301 11 301'1 30111 299/ 301'1 301'1 30\11 2933 2933 30 117 30~7 30 117 30117 3011"/ 30117 3047 3047 30~7 3062 3062 3062 3062 JU&2 3062 3062 306? 3062 3065 3065 3065 3065 3065 3065 3065 3065 3065 Table 5.20. Vibrational frequencies for the 9 distonic ions involved in the stepwise elimination of acetaldehyde from the molecular ions of (20) - (24). (Frequencies are in cm- 1.) 95 represent hydrogen transfer (TSl and TS2; Scheme 5.11) and two represent C-0 cleavage (TS3 and TS4; Scheme 5.11). Each hydrogen transfer transition state was constructed by averaging the vibrational frequencies of the two stable ions connected by that transition state. A C-H stretch was chosen to be the reaction coordinate. Each C-0 cleavage transition state was constructed from the vibrational frequencies of the appropriate distonic ion by removing one C-0 stretch as the reaction coordinate. The vibrational frequencies of the 18 transition states are given in Table 5.21. The zero-point energies (ZPE's) of the species involved are given in Table 5.22. The critical energies for the six reactions were obtained from these ZPE's and the critical energies for the corresponding steps in the unlabelled compound (20). Let the critical energy E0 for hydrogen transfer from the molecular ion of (20) be X and E0 for the reverse hydrogen transfer be Y (Y is related to X by the difference between the Llllf's of the molecular ion of (20) and the distonic ion [(32)]+·). Let E0 for the C-0 cleavage be Z. The various critical energies can be related as follows: E0(1) = X + ZPE(M(20)) - ZPE(i) - ZPE(TS1(20)) - ZPE(iv) E0(2) = Y + ZPE(l1(20)) - ZPE(ii) - ZPE(TS1(20)) - ZPE(iv) E0(3) = Z + ZPE(l1(20)) - ZPE(ii) - ZPE(TS3(20)) - ZPE(vi) Eo(4) = X + ZPE(M(20)) - ZPE(i) - ZPE(TS2(20)) - ZPE(v) E0(5) = Y + ZPE(l2(20)) - ZPE(iii)- ZPE(TS2(20)) - ZPE(v) E0(6) = Z + ZPE(l2(20)) - ZPE(iii)- ZPE(TS4(20)) - ZPE(vii) Zf>E(i) is the zero-point energy of species (i) in Table 5.22 for (21) - (24), and so on. ZPE(M(20)) is th~ zero-point energy of the molecular ion of (20), ZPE(l1(20))) is the zero-point energy of the intermediate species formed by hydrogen transfer from [(20)]+., and so on. (As the ZPE's are calculated in energy units of cm-1, and the (20) (21 l (22) (23) (2q l TS1 TS3 TS1 TS2 TS3 rs11 TS1 TS2 TS3 TS4 TS1 TS2 TS] rsq TSl TS2 TS] rs, 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 36 116 q7 46 q3 117 116 45 111 115 lfJI 116 116 47 47 40 QO 43 q3 60 50 60 60 50 50 59 58 50 116 60 60 50 50 56 56 q5 q5 73 66 73 69 68 63 71 65 66 57 72 71 66 68 611 M 57 57 86 98 86 68 98 97 86 86 96 93 66 66 96 97 83 83 92 92 1qo 139 138 139 136 135 106 123 10•1 I 35 1qo 140 138 139 105 106 1oq 1oq 1'19 150 1q7 1'16 1q9 1116 lll't IQ] 147 1116 1Q7 1115 1q9 150 12Q 125 136 1Q0 1·1q 199 174 172 198 197 173 168 197 193 17q 17 3 198 196 164 163 190 186 213 215 212 212 21'1 212 210 208 212 209 212 212 213 213 203 203 205 206 2Q 1 230 239 239 229 230 238 238 228 228 239 239 229 227 2)3 232 223 221 251 2Q8 2q9 2119 2q6 2q7 21111 2Q8 237 2113 2Q9 2119 2% 2Q7 239 239 229 230 289 292 288 286 290 n, 265 287 265 289 266 266 291 291 28Q 28Q 281 281 326 329 324 323 328 326 322 320 327 323 3211 32•1 327 32'/ J 1Q )IQ 319 320 359 365 358 357 365 362 357 356 365 362 358 358 3~5 365 35Q 35Q 361 361 1109 399 •107 Q03 399 392 39'1 392 3,7 388 1109 1106 399 399 388 388 387 387 QJ2 llllll 1130 1,20 11112 1137 Q25 112ll •132 1120 QJ1 •132 QQ3 flQ3 •109 Q10 q20 Q22 Q72 1165 469 1171 1161 116'1 11&11 1167 Q55 1161 Q70 '171 1163 116'1 •155 Q55 4QO qq2 535 53] 5311 531 53] 527 532 528 532 525 53•1 53,1 53] 533 526 525 52Q 522 551 561 551 51111 561 5'16 5'19 537 559 536 550 5117 560 55•1 536 535 533 530 610 581 608 608 579 578 579 581 57 3 516 608 60~ 577 580 575 517 567 570 7 lb ·101 660 673 661 687 611 619 587 602 7111 71'1 707 707 586 5% 586 587 761 730 ., 13 706 707 693 713 701 707 68 11 761 761 730 730 617 616 602 599 782 771 761 755 730 719 718 71'1 716 706 781 782 771 771 699 699 683 683 793 793 782 780 771 769 761 753 732 715 793 793 792 793 712 712 705 705 802 797 801 799 797 793 783 781 112 770 801 801 797 796 717 717 711 711 830 a2q 829 815 824 797 801 799 797 793 830 830 824 824 755 756 719 721 875 869 9119 829 851 812 820 808 822 798 875 874 869 867 788 788 770 770 882 877 869 851 B6B 832 830 0211 82Q 811 882 882 an 877 808 808 798 797 907 8911 6n 866 872 851 8117 8Q8 a•11 850 905 905 893 893 B15 815 811 811 91Q 906 0B2 88Q 877 B79 868 B60 869 B53 914 91 q 905 905 829 829 822 822 919 910 908 911 896 900 611 B72 871 863 918 918 909 909 846 8Q6 8Q6 846 932 931 917 915 908 905 882 8811 877 880 932 932 929 931 849 8Q9 851 851 942 936 923 931 915 931 908 911 896 901 942 9112 935 936 860 860 853 853 1003 1035 933 938 932 9111 917 915 908 905 1003 995 1035 1035 865 865 863 862 10•10 1052 911 I 950 935 951 925 927 915 919 10 114 10113 1052 1052 875 877 869 871 1052 1061 966 973 962 916 939 936 93'1 929 1052 1052 1060 1061 88• 880 880 880 1068 1063 993 1007 1007 1052 947 9•11 946 932 106/ 1068 1063 1063 913 913 901 901 1070 1076 1021 1030 1035 1059 956 9•19 956 9•11 1070 1070 1076 1076 925 925 915 915 1063 1077 1oq5 1055 1063 1076 965 968 962 969 1082 1082 1077 1077 931 931 920 920 1096 1115 1070 1076 10611 1086 991 991 1011 1052 1096 1096 1113 1113 937 937 929 929 1116 1116 1076 1088 1077 1114 10211 1033 1035 1059 111 q 1115 1116 1115 9113 9113 939 939 1134 11'15 1096 1115 1100 113•1 1045 1055 1063 1076 11311 1134 11115 1145 950 950 9q5 946 11'19 115•1 1112 1129 1116 1138 1070 1076 106 11 1099 I 1'19 1109 1153 1154 956 956 956 956 1156 1159 1128 1139 1105 11'15 1076 10911 1077 11111 1154 1155 1157 1159 962 9611 959 963 1158 1195 11119 1146 1156 1156 1096 1115 1103 1130 1157 1158 1195 1195 968 968 969 967 1199 1223 1155 1156 1191 11611 1113 1129 1116 1138 1191 1191 1209 1223 9?0 983 1009 1011 1214 1233 1175 1161 1197 1194 11311 11•15 11115 11118 1206 1213 1231 1233 1012 1012 1061 1061 1229 12b I 1201 1199 1226 1218 1149 1151 1151 115~ 1228 1229 1201 1241 1038 1038 1099 109q 12Q9 1253 1224 1220 1234 1228 1157 1156 1195 116•1 12 119 1208 1253 1206 1094 1091 1102 1103 1263 1259 1230 1220 1201 12Q 1 1176 1161 1203 119q 1259 1257 1259 1259 1112 1112 1133 1133 1269 1272 1250 1250 1253 1259 1203 1199 122•1 1218 1268 1268 1272 1272 1128 1129 1138 1139 1280 1279 1265 1267 1259 1265 1223 1220 1233 1228 1280 1280 1279 1279 11 Q1 I 1 Q2 1148 11Q2 1286 1285 12711 1276 1272 1278 1229 1226 12111 1265 1286 1286 1284 1263 1150 1107 1161 1162 1290 1292 1263 1286 1279 1288 1250 1262 1253 1279 1290 1289 1290 1292 1157 1158 1193 1193 1293 1296 126/ 1292 1285 1296 1265 1276 1259 1287 1292 1293 12911 1296 1175 1175 1197 1203 1295 1296 1290 1295 1292 1296 1274 1287 1272 1296 12911 1295 1296 1296 12011 1208 1213 1216 I 301 1297 1294 1296 1296 1297 1263 1295 1279 1216 1300 1300 1296 1297 1218 1220 1221 1229 I 303 1301 1299 1300 1297 1301 1267 1295 1285 1297 1301 1301 1301 1301 1223 1227 1265 1265 1306 1302 1302 13011 1300 1301 1293 1300 1292 1301 1305 1306 1301 1302 1262 1262 12·15 1278 1309 1310 1306 1307 1301 1309 1301 1305 1300 1309 1308 1308 1310 1310 1282 1282 1266 1286 1322 1326 1312 1316 1310 1326 1306 1310 1301 I 326 1322 1322 1315 1326 1292 1292 1300 1300 1331 1H7 1323 1330 1337 1332 1311 132Q 1310 1332 1326 1331 1337 1337 1301 1301 1306 1306 1338 1341 1337 1335 1339 13 110 1323 13311 1 337 1300 1337 1337 1339 1341 1309 1309 1331 1331 1306 1361 1339 1339 13110 1361 1337 1339 13117 1361 13 112 1J4Q 1352 1361 1327 1327 1335 1346 1361 1368 1353 1359 1380 1388 1353 1359 1 376 1388 1354 1356 1371 1368 1336 1JQ2 1350 1378 I 37B 1411 1373 lff/ 1q 11 10 10 1372 1377 1411 11110 1 365 1370 1411 1411 13'16 1363 1q 10 1410 1 392 I o 31 1 391 1391 11131 1'131 1391 1391 11131 1'13 I 1391 1391 1431 1431 1369 1369 14 31 1431 1•105 1679 1•1011 111011 1679 1667 1404 111011 1679 1665 11101' 1•011 1679 160 3 1•0• 1'104 1665 1626 1580 2850 1500 15'7] 2097 2120 1560 1572 2096 2120 1580 1561 2850 2650 1572 1553 2096 2096 265• 2870 2105 2145 2106 2175 2100 2127 2106 2139 265• 285Q 2869 2670 2104 2104 2106 2106 2664 28'75 2111 2854 2870 2850 2111 2157 2159 2155 28611 2864 2875 2875 2110 2110 2120 2120 2879 2890 28611 28611 2875 2870 2160 217 11 217Q 2171 2871 2879 2890 2890 2117 2117 2139 2139 2912 2909 2667 2679 2890 2890 2175 2180 2176 2182 2912 2912 2909 2909 2127 2127 2155 2155 2919 2920 2919 2911 2920 29011 2179 26511 2870 2850 2919 2919 2320 2920 2157 2160 2159 2159 2923 2926 2921 2913 2926 2909 2864 266• 2875 2870 2923 2923 2926 2926 2174 2174 2171 2171 2927 2933 292'/ 2920 2933 2920 2867 2879 2890 2090 2927 2921 2933 2933 2175 2175 217q 21H 2941 2933 2941 2937 2933 2926 2919 2911 2920 29011 2941 2941 2933 2933 2178 2178 2176 2176 29411 2938 2943 29111 2938 2933 2921 2913 2926 2909 2944 29114 2938 2936 2161 2181 2182 2161 29b8 2946 29118 2945 29•5 29QO 2927 2923 2933 2926 29118 2946 2946 2946 2864 286• 2670 2870 2952 2950 2951 2949 2950 29116 29•1 2941 2933 2933 2952 2952 2950 2950 287Q 2874 2690 2890 2985 · 2957 2965 2983 2957 2950 29114 29119 2938 29"6 2965 2965 2957 2957 2911 2911 29oq 2904 3007 2997 3007 3007 2997 2997 2985 2964 2957 2950 3007 3007 2997 2997 2922 2922 2926 2926 )027 30 I q 3027 3027 30 lQ 3010 3016 3027 2997 301'1 3021 3027 3014 301Q 2966 2966 2933 2933 3003 30117 3003 3043 30117 3007 3003 3043 30 117 30'17 30•13 3oq3 30117 3047 3043 JOQ3 30117 30117 3056 3062 3056 3056 3062 3062 3056 3056 3062 3062 3056 3056 3062 3062 3056 3056 3062 3062 3059 3065 3059 3059 3065 3065 3059 3059 3065 3065 3059 3059 3065 3065 3059 3059 3065 3065 Table 5.21. Vibrational frequencies for the 18 transition states involved in the stepwise elimination of acetaldehyde from the molecular ions of (20) - (24). (Frequencies are in cm-1.) Table 5.22. Zero-point energies (ZPE's), in kJ mol-1, for the ions involved in the stepwise model for the elimination of acetaldehyde from a,a'-diethoxy p-xylene. Ion 20 21 22 23 24 (i) [M]+. 684.59 668.14 644.37 684.07 603.57 (ii) [It]+. 683.49 667.02 643.27 682.92 603.63 (iii) [12]+- 683.49 667.63 644.43 682.93 603.65 (iv) TSl 666.99 650.46 626.70 666.38 590.82 (v) TS2 666.99 655.31 631.83 666.44 590.88 (vi) TS3 677.28 660.81 637.06 676.71 597.58 (vii) TS4 677.28 661.44 638.39 676.83 597.70 96 calculations were actually performed using these enrgy units, the following energies have more significant figures than, and may slightly disagree with, the values given above.) Using the values of ZPE for (21) in Table 5.22, the critical energies, in kJ moi- 1, are obtained as: E0(1) = X + 16.45 - 16.53 = X - 0.08 E0(2) = Y + 16.47 - 16.53 =Y - 0.06 E0(3) = Z + 16.47 - 16.47 = Z E0(4) = X + 16.45 - 11.69 = X + 4.76 E0(5) = Y + 15.86 - 11.69 = Y + 4.17 E0(6) = Z + 15.86 - 15.84 = Z + 0.02 From the ranges given for the E0's and the corrections for the ZPE's, the ranges for E0's, in kJ moi-1, for the reactions of [(21)]+. using the thermochemical energetics are: E0(1) between 58.92 and 70.92 E0(2) between -0.06 and 11.94 E0(3) between 0.00 and 12.00 E0(4) between 63.76 and 75.76 E0(5) between 4.17 and 16.17 E0(6) between 0.02 and 12.02 Using the MOPAC energetics, the E0's, in kJ moi-1, are obtained as: E0(1) between -0.08 and 70.92 Eo(2) between 14.94 and 85.94 97 E0(3) between 25.00 and 86.00 E0(4) between 4.76 and 75.76 E0(5) between 19.17 and 90.17 E0(6) between 25.02 and 86.02 Similar analyses give the range of E0's for the other isotopomers (22), (23) and (24). A negative value for an E0 implies an unstable ion; all zero and negative E0's have been treated as being slightly greater than zero (+0.6 kJ moi-1). 5.9.4. Details of the calculation. The grain-size used in these calculations was 0.14 kJ moi-1 (12.0 cm-1) for a reactant ion (whether a molecular ion or intermediate ion) and 0.16 kJ moi-1 (13.3 cm-1) for a transition state. Emax was set to 240 kJ moi- 1 as in the calculations using the concerted model. The overall energy barrier to elimination of acetaldehyde from the molecular ion of (20) was fixed at 71.2 kJ moi-1 as in the calculations using the concerted model. The consequences of the error limits (± 12.0 kJ moi- 1 ) on this barrier will be discussed later. E0 for the C-0 cleavage was first set to the maximum value (12.0 kJ moi-1 for the series of caJculations based on the thermochemical energetics and 85.8 kJ mol-1 for the MOPAC series). In both series of calculations, E0 for hydrogen transfer was initially set to 59.2 kJ moi- 1~ and gradually increased to 71.2 kJ moi-1. Then, with E0 for hyq.rogen transfer at 71.2 kJ moi-1, E0 for the C-0 cleavage was gradually reduced to 0.0 kJ moi-1 (actually 0.6 kJ moi-1) in the case of the series of calculations based on the thermochemical energetics and 73.8 kJ moi-1 in the case of the MOPAC series. For the calculations involving isotopically labelled compounds (21) - (24), critical energies 98 were determined from the value of Eo for the corresponding step in the elimination of acetaldehyde from the unlabelled ion [(20)]+- described above, and the ZPE adjustments given in Section 5.9.3. In modelling the hydrogen transfer transition state, five frequencies( 4 C-H bends and 1 C-H stretch ) were considered sensitive to the progress of the reaction. For the C-O cleavage transition state, seven frequencies ( 4 C-H bends, 2 C-H stretches and 1 C 6H 5-C stretch ) were considered sensitive. The sensitive frequencies in both transition states were each lowered by amounts of 5, 10, 15 and 20 %. These frequencies are given in Table 5.23. Lowering the sensitive frequencies associated ' with the hydrogen transfer transition state results in the critical energy for the D transfer steps to increase with respect to the critical energy for the comparable H transfer steps. Lowering the frequencies associated with a transition state in a labelled species reduces the ZPE of that transition state, but as the frequencies associated with the comparable transition state in the unlabelled species are also lowered, it is necessary to increase all H and D transfer critical energies by the ZPE lowering of the unlabelled species in order to maintain the reference critical energy. As the sensitive frequencies associated with the D transfer transition state are smaller than those associated with the H transfer transition state, they are reduced by a lesser amount than the H transfer transition state and correction for the ZPE lowering of the reference species effectively increases the critieal energy for D transfer with respect to that for H transfer which remains unaffected. Lowering the hydrogen transfer transition state frequencies by 5 % caused E0 for D transfer to increase by 0.5 kJ moi-1 with respect to that for H transfer. Subsequent 5 % lowerings further increased E0 for D transfer by 0.5 kJ moi-1, i.e. for a 20 % lowering of the sensitive frequencies, E0 for D transfer was increased by 2.0 kJ mol- 1 with respect to the critical energy for the H transfer which remained constant. Table 5.23. Vibrational frequencies, in cm-1, considered sensitive in the transition states for hydrogen transfer and C-0 cleavage. Compound 20 21 22 23 24 793 801 673 801 619 793 617 616 616 1068 1076 866 1076 860 1067 1068 860 860 H-transfer 1116 1112 931 1113 927 1114 1115 931 931 1280 1274 973 1274 968 1280 1280 968 968 2941 2867 2145 2867 2127 2941 2941 2175 2174 248 246 247 237 243 246 247 229 230 1116 1116 1114 1116 1114 1116 1115 1102 1103 1195 1197 1194 1195 1194 1195 1195 1197 1193 C-0 cleavage 1296 1292 1288 1292 1287 1294 1292 1288 1288 1310 1310 1309 1310 1309 1310 1310 1308 1308 2890 2890 2890 2890 2890 2890 2890 2890 2890 2933 2933 2933 2933 2933 2933 2933 2933 2933 99 For the C-0 cleavage transition state, a similar effect is observed, but, as the effects of 180 substitution are less than those of D substitution, the effect is less pronounced. The magnitude of the effect is of the order of 0.01 kJ moi-1, effectively insignificant. In the following discussion of the results obtained for the two series of calculations based on the different energetic schemes, the values of the critical energies mentioned are those for the elimination of acetaldehyde from the molecular ion of the unlabelled compound (20). It is emphasised that the necessary corrections for isotopic substitution were made in the calculations involving isotopically substituted species. 5.9.5. Results of the calculations based on the thermochemical energetics. Ion abundances were calculated for the loss of C2H40 and C2H3DO from the molecular ion of (21) using transition states in which only the reaction coordinate had been removed, i.e. vibrational frequencies were as given in Table 5.21. E0 for hydrogen transfer was set to 59.2 kJ moi-1 and E0 for C-0 cleavage was set to 12.0 kJ moi-1 (E0 for reverse hydrogen transfer was 0.6 kJ moi-1 ). The ion abundances as a function of time are shown for (21) in Figure 5.28(a). No intermediate ions are present at significant abundance at any time. The relative rates of formation of the products are show.n in Figure 5.28(b). Loss of C2H40 is faster than loss of C2H3DO. For the metastable time-frame sampled by the MIKE experiment (2.24x10-5 s to 5.27x10-5 s for m/z 196 at an accelerating potential of 8 kV), the ion abundance ratio was 1.14. On increasing E0 (or hydrogen transfer, while keeping E0 for C-0 cleavage fixed, the ion abundance ratio for loss of C2H 40 relative to loss of C2H 3DO increased. Increasing E0 for hydrogen transfer to 71.2 kJ moi-1, the ion abundance ratio increased (b) (a) g C> 2 M I I ;j I Ii' \ ....,V) (J :, ,, ~ :1 \ I 0 .... I c. le! ;j \ 0 0 ::: ,: .S...., s ~ 0 iJ s" I 1-, I 'Ill ,: ..s ~ <.) C: :;, I P1 .,," I 0 C: ,: ....,Ill .. ;:, " IIJ Ill g > ,;; .~ :;:; ~ .... ~ d Cl 4) 'ii 0:: .. g 'il 'il 0 g Sl Sl 0 g ~ _,,-,,;') -~, 'J') -2 ::o 0 ·U.'JO -1J ?O -12. ')') :-'10 ')IJ -ho -aoo -bo -~... -4.')0 -l.oo Q,t4,";l') -13 :::i -IZ. '!') ·U ':? •10 ')'j " -- 'l'J - '')'J -i;l'JC - .?? ◄" log10 ( time / s ) log10 ( time/ s ) Figure 5.28. (a) Ion abundance curves and (b) relative rate of formation of products for (21) using the thermochemical energetics. E0 CH-transfer)= 59.2 kJ mo1·1 E0 (C-O cleavage)= 12.6 kJ mo1· 1 (a) (b) g g ~ -1 P1 I'\ I \ ~ ~ ~ ~ ..,Ill u ::, -0 ~ 0 :i \ ... c.. >i! ;j \ 0 •' C I P1 .s I !< .., '.i! I I ~ 0 I i I s"' \! ... ('\ P2 I 0 ,. ' !< I " I I g ;, I I - 0 "C I I I I ...,V § !< i ., 0 .c i ... V V" ~ :;;> ~ ..,~ 0 " d ,., v 0:: .. 8 ~ ll 0 :;: 2 2 0 I I 1/ I I ~1 ~I o.!4.0':> 'JO ·JO 'lO -b'lO ·d ?:> .\ ,,0 :L, .§ 00 --~ ')0 ')!) ., ':)') -h ')') -,.-::o ·J ')') -L1'lO ·12. •U.'JOL ·•.oo g-1, ·I:! 'JC ·12 'JIJ ·II. ':'J ·IO'l~O .\." -a" log10 ( time / s ) \oglO ( time / s ) Figure 5.29. (a) Ion abundance curves and (b) relative rate of formation of products for (21) using the thermochemical energetics. 1 E0 (I-I-transfer) = 71.2 kJ mo1·1 E0 (C-0 cleavage)= 12.6 kJ moi- (b) (a) ~..------3 g M Pl i • ~ ..,"' v ;:J • ~ "O i 0 ... 0. ~ ~ Pl -0 0 ~ I I c:: I 0 ~ ~ $ ~ 'ii d o c:: .: '., s .E ~ '.:: ~ ;; -0 -c, ., 0 c:: ~ .., . P2 d o ] ¥ ... d ., ~ 8 .?: ~ .., 0 ~ ~ d d v ~ g p,:: lo ~ ~ ~ ~ :a: I I «::.: . ~1 ------::::::;: ' 0 -1-4.'JO ~13 ')0 -\?..')':I -11.~'l ·l'J ':? -5'lO -5'>0 •l.$ 'J') 4j ,. I 0 -14-:, -12')0 -12.,? -11n -'io\:.~'j') -d'.O .. ?.n -6~'> -k-~o ·4.'J'J -:~n l log 10 ( time / s ) log 10 ( time / s ) Figure 5.30. (a) Ion abundance curves and (b) relative rate of formation of products for (21) using the thermochemical energetics. 1 E0 CH-transfer)= 71.2 kJ moi- 1 E0 (C-0 cleavage)= 0.6 kJ moi- 100 to 4.16. It can be seen from the ion abundance vs. time and relative rate curves shown in Figure 5.29 that the rate of elimination of C2H4O has increased considerably. The ion abundance ratio increased further as E0 for C-O cleavage was reduced, with E0 for hydrogen transfer held at 71.2 kJ moi-1. With E0 for C-O cleavage at 0.6 kJ moi-1 the ion abundance ratio was 7.55. Loss of C2H4O is much faster than loss of C2H3DO (Figure 5.30). The transition state frequencies were lowered, and the ion abundance ratios for loss of C2H4O and C2H3DO recalculated. The ion abundance ratio increased as the vibrational frequencies for both hydrogen transfer and C-O cleavage transition states were lowered. The ion abundance vs. time, and relative rate vs. time, curves were essentially similar to those obtained above and are not shown. The ion abundance ratios calculated for the various combinations of transition state frequencies and critical energies are given in Table 5.24. The measured ion abundance ratio of 3.28 ± 0.19 is easily reproduced by a number of transition state and critical energy combinations. Ion abundances were calculated for the loss of C2H4O and C2H4 18O from the molecular ion of (23) in parallel fashion. With E0 for hydrogen transfer at 59.2 kJ moi- 1 and E0 for C-O cleavage at 12.0 kJ moi- 1, the ion abundance ratio calculated using the transition states with only the reaction coordinates removed was 1.08. The ion abundance vs. time and the relative rate curves for the losses of C2H4O and C2H418O are shown in Figure 5.31. Loss of C2H4O is slightly faster than loss of C2H418o. On increasing E0 for hydrogen transfer to 71.2 kJ moi-1, the ion abundance ratio decreased to 1.03. The ion abundance vs. time and the relative rate curves (Figure 5.32) show that loss of C2H4O and loss of CiH418O are comparable. Lowering E0 for C-O cleavage decreases the ion abundance ratio calculated for (23) further. With E0 for C-O cleavage at 0.0 kJ moi- 1 the ion abundance ratio was calculated to be 0.98. The Table 5.24. Ion abundance ratios calculated for (21) using the thermochemical energetics. kJ mol·1 E0 for H-tramrcr 59.2 61.6 64.0 66. 4 68.8 71.2 71.2 71.2 71.2 71.2 71.2 Eo for C-0 cleavage 12.6 12.6 12.6 12.6 12.6 12.6 10.2 7.8 S.Q 3.0 0.6 H-transfer CO cleavage lowering lowering I l 00 1. 14 1.22 1. 40 1. 87 2.84 4.16 5.69 6.69 7.21 7.4Q 7.55 OS 1. 18 1.25 1.45 1.94 2.94 4.27 5.76 6.74 7.23 7.46 7.55 00 10 1. 18 1.26 1.46 1.97 2.99 4,33 5.81 6.77 7.25 7 .46 7.56 15 1. 19 1 .27 1.47 1.99 3.04 4.40 5.86 6.80 7.26 7.47 7.56 20 1.16 1.24 1.46 2.00 3.08 4.46 5.91 6.83 7.28 7.48 7.56 00 1. 14 I. 23 1.46 2.05 3.24 4.86 6.74 8.03 8.69 9.00 9. 13 05 1. 17 1 .26 1.51 2. 12 3.36 4.99 6.85 8.10 8.73 9.01 9.14 05 10 1. 18 1.27 1.53 2.15 3.43 5.07 6.92 8. 13 8.75 9.02 9.lli 15 1. 18 1.28 1.54 2. 19 3.49 5.15 6.99 8. 17 8.77 9.03 9.15 20 1. 16 1. 26 1. 53 2.20 3.5li 5.23 7.06 8.21 8. 79 9.04 9.15 00 1. 17 1.28 1 .58 2.31 3. 81 5.80 8. 17 9.80 10.64 11.02 11. 19 05 1. 20 1, 32 1.63 2.40 3,95 5.97 8.31 9.88 10.68 11.04 11.20 10 10 1. 21 1.33 1.65 2.44 4.03 6.07 8. 39 9.93 10. 70 11.05 11.21 15 1.22 1. 34 1.49 2. 49 4. 11 6. 17 8.48 9.98 10. 73 11.07 11.21 20 1. 19 1.32 1.66 2.51 4. 19 6.28 8.57 10.03 10.76 11.08 11.22 00 1. 17 1.30 2.31 2.55 4.36 6. 77 9,70 11.73 12.79 13.28 13,50 05 1,20 1.34 2.38 2.66 4.53 6,91 9.87 11.84 12.85 13.31 13.51 15 10 1.21 1. 35 2.41 2.71 4.63 7.09 9.97 11.90 12.88 13.32 13,52 15 1.21 1. 36 2.44 2.76 4.73 7.23 10.09 11.96 12.91 13.34 13,52 20 1. 19 1.34 2. 42 2.80 4. 84 7.36 10.20 12.03 12,94 13. 35 13.53 00 1.20 1.36 2.46 2.89 5. 10 8.05 11.74 14,33 15,70 16,33 16. 61 05 1. 23 1. 40 2.54 3.01 5.30 8.30 11.96 14.47 15. 77 16. 37 16,63 20 10 1.24 1.42 2.58 3.07 5. 39 8.45 12.08 14.54 15. 81 16.38 16.64 15 1. 25 1.li3 2.61 3. 14 5.55 8.62 12.23 14.63 15.85 16. 40 16.65 20 1.22 1. 41 2.59 3.20 5.69 8.79 12.38 14. 72 15.90 16.43 16.66 Table 5.25. Ion abundance ratios calculated for (23) using the thermochemical energetics. kJ mo1· 1 E0 for H-transfer 59.2 61.6 64. 0 66.4 68.8 71.2 71.2 71.2 71.2 71.2 71.2 E0 for C-0 cleavage 12.6 12,6 12.6 12.6 12.6 12.6 10.2 7.8 5.4 3.0 0.6 H-transfer CO cleavage lowering lowering l l 00 1. 08 1.08 1. 07 1. 07 1.05 1. 03 1.00 0.99 0.98 0.98 0.98 05 1.11 1.11 1.10 1. 09 1. 07 1.04 1.01 0,99 0.99 0.98 0.98 00 10 1. 11 1.11 1. 10 1. 09 1.07 1. 04 1.01 0.99 0.99 0.98 0.98 15 1.11 1. 11 1. 11 1. 10 1.07 1.04 1.01 0.99 0.99 0.98 0,98 20 1. 08 1.08 1.07 1.07 1.05 1.02 1.00 0.99 0,98 0.98 0.98 00 1.08 1.08 1,07 1. 07 1.05 1.02 1.00 0,99 0.98 0.98 0.98 05 1. 11 1.10 1. 10 1. 09 1.07 1. 04 1.01 0.99 0,98 0,98 0.98 05 10 1. 11 1. 11 1. 10 1.09 1.07 1. 04 1.01 0.99 0.98 0,98 0,98 15 1.11 1. 11 1. 11 1. 09 1,07 1. 04 1,01 0.99 0.98 0,98 0.98 20 1. 08 1.08 1.07 1. 06 1,05 1.02 1 .oo 0.99 0.98 0,98 0.98 00 1. 10 1.09 1.09 1.08 1.06 1.03 1.01 0,99 0.98 0.98 0.98 05 1. 12 1. 12 1. 12 1. 11 1.08 1.05 1.01 0.99 0.98 0.98 0,98 10 10 1. 13 1. 13 1.08 1.11 1 .08 1 .05 1.01 0.99 0.98 0.98 0.98 15 1. 13 1, 13 1. 12 1.11 1.08 1.05 1.01 0.99 0.98 0.98 0.98 20 1. 10 1. 10 1 .09 1.08 1.06 1.03 1.00 0,99 0.98 0.98 0.98 00 1.08 1 .08 1.07 1.07 1.05 1.02 1.00 0.99 0.98 0.98 0.98 05 1. 11 1.10 1. 10 1 .09 1 .07 1 .OQ 1.01 0.99 0.98 0.98 0.98 15 10 1.11 1.11 1.10 1.09 1.07 1,04 1.01 0.99 0.98 0,98 0.98 15 1.11 1.11 1.11 1.09 1.07 1.0Q 1,01 0.99 0.98 0.98 0.98 20 1.08 1.08 1.07 1.06 1.05 1.02 1.00 0.99 0.98 0.98 0.98 00 1.08 1 .08 1.07 1.07 1.05 1.03 1.00 0.99 0.98 0.98 0.98 05 1.11 1.10 1. 10 1.09 1.07 1.04 1.01 0.99 0,98 0.98 0.98 20 10 1. 11 1.11 1. 10 1 .09 1.07 1.04 1.01 0.99 0.98 0.98 0.98 15 1.11 1.11 1. 11 1.09 1.07 1.04 1.01 0.99 0.98 0.98 0.98 20 1.08 1 .08 1.07 1.07 1.05 1.02 1.00 0.99 0.98 0,98 0.98 (b) § (a) ~ g : I M ~ ~ ~ ...,"' u ::, ~ ~ i "O ..0 0.. ,~ ~ 0 Q .§ ~ g ..., :ii i d Q ..6 ' 0 ~ ~ .... ~ - ~ .,," p 1 0 Q,J § ~ .., ~ .:i i d Q '" " Q,J ~ g > ~ :.::: ~ :;:; Q d "ii" v i:t! .. 8 ,;i 'I ~ ,: 0 ~ ~I - , ' 0 1 ·14 •.:, •l;! -:i; •t2.':C ·ll ·;-:: ·\';. ~J -:i ~') .ij ')'J .s_';'J -~ 'JO -fi.')O ·4 ':J ~·)') ~-1◄.0Q-b '),J ·l?..'l'l ~')') -'t'J ')0 -h 'lO ., 'JO .s_')'J ..t'l? -t,o .~_O'J .~ ')') log 10 ( time / s ) log 10 ( time / s ) Figure 5 .31. (a) Ion abundance curves and (b) relative rate of formation of products for (23) using the thermochemical energetics. E0 CH-transfer);;:;;59.2 kJ moi- 1 E0 (C-O cleavage);;:;; 12.6 kJ moi- 1 ·l.oo ·l.oo P2 P2 ?O ?O -4 -4 I I I I I I I I I -!J.oo -!J.oo I I I I I I ◄" ◄" .).?O .).?O using using ) ) s s (23) (23) -8" -8" for for 1 1 time/ time/ I I ( ( 10 10 (b) (b) mo1· .fo-'lo---.a·?O .fo-'lo---.a·?O products products log kJ kJ of of ·lt.'lO ·lt.'lO 12.6 12.6 ')0 ')0 -,~. -,~. 00 00 formation formation ~12 ~12 of of cleavage)= cleavage)= ~t4.'l0 ~t4.'l0 0 'ii 'ii l, l, ~ ~ el el 0 0 0 0 ~ ~ C C ,i ,i 0 0 ~ ~ 0 0 ~ ~ " " 8 8 • • rate rate d d d d Q.) Q.) d d Q.) Q.) ,. ,. ...... ...... Vl Vl ...... i:: i:: > > C: C: ::, ::, u u c.. c.. 0 0 ~ ~ ..., ..., 0 0 0 0 .E .E :;:; :;:; ..., ..., ..., ..., v v .s .s "O "O (C-0 (C-0 0 0 E relative relative 1 1 (b) (b) 00 00 moi- and and P1 P1 .. .. kJ kJ 00 00 energetics. energetics. ~ curves curves I I I I I I I I I I I I ' ' I I 71.2 71.2 I I I I ' ' I I I I I I I I I I I I I I ' ' I I I I I I I I I ·" ·" " " I I I I I I I I I I I I I I I I I I ' ' abundance abundance ,f)O ,f)O • • thermochemical thermochemical (I-I-transfer)= (I-I-transfer)= Ion Ion 0 0 ) ) the the E (a) (a) ?O ?O s s / / !-I !-I " " (a) (a) 5.32. 5.32. time time ( ( ')0 ')0 10 10 ·10 ·10 log Figure Figure '!') '!') ·11 ·11 -\2.?':. -\2.?':. ')0 ')0 ·tJ ·tJ ,o ,o ·14 ·14 0 8 8 e e ~ ~ ~ ~ g g 'l 'l ll ll ~ ~ j j :;: :;: g g ll ll ~ ~ i1 i1 g g i1 i1 ~ ~ :1 :1 ;j ;j ,:: ,:: Q.) Q.) .. .. ., ., " " u u " " i:: i:: ::, ::, i:: i:: .::: .::: ; ; ..:, ..:, - ~ ~ "O "O '- 00 -~ ,'JO ◄ • .§,oo 'JI) , _, .$.'JO using 'j'J (23) ., / s ) for ')Q 1 g .• time ( 'JO moi- 10 -~o products kJ log 'lO (b) of ·ll 0.6 =;:::::::: ·12,0:,') 'JO formation ·l::? of cleavage)= I rate 0 0 ~ j 0 :;: q . 2 • 0 ~ 0 'l ~ Q·14.0";l ,- V: u u d ii) > t. 0 " ::, f: .. .. " ...., 0 ...., 2 .9 0 -0 ...., z A:: v (C-0 0 E relative 1 (b) ')0 moi- and P2 ·•.1 kJ ,o energetics. -4. curves I ' 71.2 " . " abundance thermochemical Ion -1,Zl CH-transfer)= 0 the (a) E ·l!'lO / s ) " (a) M 5.33. time ( ')~ 10 -10 log Figure •II.'~: -·12'J'J 'lO -·r2 ':') ~ § 8 ¥ ~ 8 w 8 ~ s j! II :;: g ,; :;; lil 8 2 8 8 o,i4 u u C " ;:, " g .. ~ ci .:: .:, ,:, u '-- 101 ion abundance vs. time and relative rate curves (Figure 5.33) show that loss of C2H/8o becomes the faster process. Lowering the transition state frequencies for hydrogen transfer and C-O cleavage gave the ion abundance ratios in Table 5.25. The measured ion abundance ratio of 1.09 ± 0.07 is reproduced for all the transition state combinations when E0 for C-O cleavage is 12. 0 kJ moi-1 and E0 for hydrogen transfer is less than, or equal to, 71.2 kJ moi-1. Considering (21) and (23) together, best agreement between the calculated and measured ratios occurred when both hydrogen transfer and C-O cleavage transition state frequencies were lowered by 10 %, and when E0 for hydrogen transfer was slightly less than the maximum value, i.e. 67.9 kJ moi-1, and E0 for C-O cleavage was at the maximum of 12.0 kJ moi-1. The calculated ion abundance ratios were 3.33 and 1.08 for (21) and (23) respectively. To test this model, ion abundance ratios for (22) and (24) were calculated. The ratio obtained for the losses of CiH4O and CiD40 from the molecular ion of (22) was 3.07, and the ratio for the losses of C2D4O and c2o/8o from the molecular ion of (24) was 1.03. These ratios do not compare well with the experimental values of 3.41 ± 0.07 for (22) and 1.06 ± 0.05 for (24). 5.9.6. Results of the calculations based on the MOPAC energetics. Using the MOPAC energetics, E0 for the C-O cleavage and E0 for the reverse hydrogen transfer are each 73.2 kJ mo1· 1 greater than the corresponding value based on the thermochemical energetics. These increases in E0's result in the rate coefficients for these steps decreasing, which leads to an increase in the concentration of the intermediate species. 102 Considering (21), the transition states with only the reaction coordinate removed were selected and the critical energies were set to 85.8 kJ moi-1 and 59.2 kJ moi-1 for C-O cleavage and hydrogen transfer steps respectively (E0 for the reverse hydrogen transfer was 73.8 kJ moi- 1). The calculated ion abundances as a function of time are shown in Figure 5.34(a). The relative rates of formation of products (Figure 5.34(b)) are similar, however, to those obtained for the comparable calculation using the thermochemical estimates. The ion abundance ratio was calculated to be 1.03. Increasing E0 for hydrogen transfer had the effect of increasing the ion abundance ratio for loss of C2H4O to loss of C2H3DO. With E0 for hydrogen transfer at 71.2 kJ moi- 1, the ion abundance and relative rate curves shown in Figure 5.35 were obtained. The abundances of the intermediate ions decreased and the relative rate of C2H 40 loss increased, giving an ion abundance ratio of 2.57. Reducing E0 for C-O cleavage, with E0 for hydrogen transfer at 71.2 kJ moi-1, increased the calculated ion abundance ratio. With E 0 for C-O cleavage at 73.8 kJ moi-1 (Figure 5.36), the ion abundance ratio was 6.46. Lowering the transition state vibrational frequencies caused the calculated ion abundance ratios for loss of C2H4O and C2H3DO to increase as shown in Table 5.26. The ion abundance ratios calculated for (21) using the MOPAC energetics are consistently lower than the comparable ratios calculated using the thermochemical energetics (compare Tables 5.24 and 5.26), even though the relative rates are similar. The ion abundance ratios calculated for (23) are also consistently lower on the basis of the MOP AC energetics than those calculated on the basis of the thermochemical energetics, as shown in Table 5.27. The ion abundance and relative rate curves for the situations: (i) E0(1) = 59.2 kJ moi- 1 and Eo(3) = 85.8 kJ moi- 1; (ii) E0 1) = 71.2 kJ moi- 1 and E0(3) = 85.8 kJ moi- 1; and (iii) E0(1) = 71.2 kJ moi- 1 and E0(3) = 73.8 kJ ?o ·l -4.IJ~ n .§ 'llJ , -l ,o -$ 1 p using 'l'l -b (21) / s ) for time 1 ~ ~ ( ,o (b) -10 moi- log" products ')? kJ -11 of 'l'J 85.8 -1c o? -1~ formation of o? -1• cleavage)== ~1 ~ s 0 • ~ 0 j 0 $ 0 ;, 'l 'l 0 2 s1 0 rate "' ::, 0 V c.. .. 0 c " 0 s .. QJ I., QJ " :! _, " -c ~ - ..s 0 _, _, ci p;: (C-O 0 E relative 1 (b) ')') 1 p -J moi- and kJ ·"·';':I I I ' I curves I I 59.2 ..\o, energetics. 'JO I ' ' I -Ii, abundance .\." MOPAC Ion CH-transfer)== ) 0 -:a s (a) the E .a / \ ')Q (a) •9 time \ 5.34. M ( ')') 10 ·I~ log ~ ':0 Figure ·U ':.J ·I~. ·,-: -1~ ')'; g & :1 i g ~ • ~ g $ jl g 8 $ g ~ g 2 'l ~ g o_lu QJ () C: " QJ C: " :, > " .. ~ -c ' .!l :.::: u -~.?O -~.?O ·4.00 ·4.00 I I _§_')Q _§_')Q 'lO 'lO ' ' ~, ~, .$_,:,Q .$_,:,Q using using 'JO 'JO (21) (21) -~ -~ / / s ) for for ~ ~ ~~ ~~ 1 1 time time ( ( ')!) ')!) moi- 10 10 (b) (b) ·10 ·10 products products log kJ kJ ')0 ')0 of of .'11 .'11 85.8 85.8 •l;?.'JI) •l;?.'JI) 00 00 formation formation ·13 ·13 of of cleavage)= cleavage)= -U.?O -U.?O 0 ;; ;; 0 0 ~1 ~1 2 2 ~ ~ $ $ 0 0 I! I! - rate rate ~ ~ 0 0 0 0 i i g g 0 0 ~ ~ " " ...... "' "' V V 0 0 ., ., u u 0 0 .. .. C C "- E E > > ::i ::i .. .. ., ., " " ..., ..., 0 0 0: 0: 0 0 "O "O ~ ~ :.;:; :.;:; ...., ...., ...., ...., v v (C-0 (C-0 0 0 E relative relative 1 1 (b) (b) moi- and and 1 1 p p kJ kJ --~ --~ / / curves curves I I I I 71.2 71.2 - l, l, energetics. energetics. l. l. abundance abundance -- ( ( MOPAC MOPAC CH-transfer)= CH-transfer)= Ion Ion 0 0 E the the (a) (a) ~ ~ / / s ) M M (a) (a) 5.35. 5.35. time time ( ( ,A~-- 10 10 log Figure Figure )~ )~ ·11 ·11 ')I) ')I) ·le! ·le! ';C ';C ·1~ ·1~ 2 2 ~ ~ ~ ~ ;_L-:, ;_L-:, ~ ~ 'l 'l i i g g ~ ~ ~ ~ ~ ~ g g ll ll ~ ~ ~ ~ 8 8 ~ ~ 8 8 g g § § " " ~ ~ '-' '-' ,: ,: .. .. u u = = " " ; ; ~ ~ ~ ~ -,:: -,:: .c .c v v ' ' . (a) (b) g g ~ - 11 ~ Pl i ~j ...."' u ;:, ~ s; "C i ..0 a. ~ p 1 ,: 0 -" ::: £ ~ .... :il ~ s ~ " ..::: 'C) ,, u .2 ii' C ~ :: 0 "C I I P2 C ;:, ~ ...."' ~ .c i ..d 0 " ~ >"' 'I .~"' :;:; 'I 0 ] ~ .."' g P2 p::"' ,. 'il !' 2 2 gl ---~, log10 ( time / s ) logio ( time / s ) Figure 5.36. (a) Ion abundance curves and (b) relative rate of formation of products for (21) using the MOPAC energetics. E0 CH-transfer)= 71.2 kJ mo1· 1 E0 (C-0 cleavage)= 73.8 kJ moi- 1 Table 5.26. Ion abundance ratios calculated for (21) using the M0PAC energetics. kJmo1·1 66.4 71.2 71.2 71.2 71.2 71.2 71.2 E0 for H-transfer 59.2 61.6 M.O 68.8 Ea for C-0 cleavage 85.8 85.8 85.8 85.8 85.8 85.8 83.4 81.0 78.6 76.2 73.8 H-transrer CO cleavage lowering lowering s s 00 1.03 1. 11 1.27 1. 56 2.00 2.57 3.36 4.21 5.05 5.80 6.46 05 1.07 1. 15 1. 32 1.62 2.08 2.65 3. 43 4.27 5.09 5.82 6.47 00 10 1. 07 1. 16 1.34 1.65 2. 12 2.71 3.49 4.33 5.13 5.86 6.49 15 1 .00 1.17 1 .35 1 .68 2. 17 2.11 3.56 4.39 5.18 5.90 6.53 20 1.05 1.15 1. 35 1. 70 1.90 2.84 3.65 4.49 5.29 6.00 6.61 00 1 .03 1. 12 1.31 1. 64 2.16 2.83 3. 78 4.84 5.89 6.86 1.12 05 1 .07 1. 16 1.36 1. 71 2,24 2.93 3.87 4.91 5.94 6.89 1.13 05 10 1.08 1. 18 1.38 1. 74 2.29 2.99 3.94 4.98 6.00 6.94 7.76 15 1.08 1. 19 1. 40 1. 78 2.35 3.06 4.02 5.05 6.06 7.00 7.81 20 1 .06 1.17 1. 40 1. 81 2.41 3. 15 4. 13 5.18 6.19 1.12 7.92 00 1 .05 1. 16 1.38 1. 76 2.36 3. 16 4.31 5.62 6.95 8.21 9.33 05 1.09 1 .20 1. 43 1. 84 2.46 3.27 4.42 5.71 7.02 8.25 9.35 10 10 1 .10 1 .22 1 .46 1.88 2.52 3.35 4.50 5.79 7.09 8. 31 9. 39 15 1. 11 1 .23 1 .48 1.93 2.59 3.44 4.60 5.89 7. 17 8.38 9.45 20 1 .08 1.22 1.49 1.96 2.67 3.55 4.74 6.05 7.34 8.55 9.60 00 1 .06 1. 18 1.42 1. 86 2.54 3.45 4.81 6,41 8.06 9.66 11.10 05 1.09 1.22 1 .48 1. 94 2.65 3.58 4.95 6.52 8. 15 9,71 11. 13 15 10 1. 10 1. 24 1 .51 1.99 2.72 3.67 5.05 6.63 8.24 9.79 11. 19 15 1. 11 1 .25 1.54 2.04 2.80 3.78 5. 17 6.74 8.35 9.89 11. 27 20 1 .09 1 .24 1. 55 2.08 2.89 3.91 5.34 6.94 8.56 10.09 11.45 00 1 .08 1.22 1.49 1 .98 2.76 3.82 5.45 7.41 9.51 11.57 13.46 05 1. 12 1 .26 1 .56 2.07 2.89 3.97 5.61 7.56 9.62 11. 64 13.49 20 10 1. 13 1. 28 1 .59 2. 13 2.97 4.08 5.74 7.69 9.73 11. 74 13.58 15 1. 14 1. 30 1 .62 2. 19 3.06 4.20 5.89 7.84 9.87 11. 87 13.68 20 1. 11 1.29 1. 63 2.24 3. 17 4.36 6. 10 8.09 10. 14 12. 13 13.92 Table 5.27. Ion abundance ratios calculated for (23) using the MOPAC energetics. kJmoI·1 E0 for I I-transfer 59.2 61. 6 64.0 66.4 68.8 71.2 71.2 71.2 71.2 71.2 71.2 Ea for C-0 cleavage 85.8 85.8 85.8 85.8 85.8 85.8 83.4 81.0 78.6 76.2 73.8 H-transfer CO cleavage lowering lowering 1 s 00 1 .06 1. 05 1.05 1. 04 1.03 1.01 1.00 0.98 0.98 0.97 0.97 05 1 .09 1 .09 1.08 1.07 1 .05 1.03 1.00 0.99 0.98 0.97 0.97 00 10 1. 09 1 .09 1 .08 1. 07 1.05 1 .03 1.oo 0.99 0.97 0,97 0.97 15 1. 10 1 .09 1. 08 1. 07 1.05 1 .02 1 .oo 0.98 0.97 0.97 0.97 20 1 .06 1 .05 1. 05 1.04 1 .02 1.01 0.99 0.98 0.98 0.97 0.97 00 1 .06 1 .05 1.05 1.04 1.03 1.01 1 .oo 0.98 0.98 0.97 0.97 05 1 .09 1 .09 1.08 1, 07 1 .05 1 .03 1 .oo 0.98 0.97 0.97 0.97 05 10 1.09 1 .09 1. 08 1.07 1.05 1.03 1 .oo 0.98 0.97 0.97 0.97 15 1. 10 1.09 1.08 1.07 1.05 1.02 1,00 0.98 0.97 0.97 0.97 20 1.06 1.05 1 .05 1 .04 1.02 1.01 0.99 0.98 0.97 0.97 0.97 00 1.07 1 .06 1 .06 1. 05 1.04 1 .02 1.oo 0.99 0.98 0.97 0.97 05 1.10 1. 10 1.09 1 .08 1.06 1.03 1.01 0.99 0.97 0,97 0.97 10 10 1. 10 1.10 1.09 1.08 1.06 1.03 1.01 0.99 0.97 0.97 0.97 15 1. 11 1. 10 1 .09 1.08 1.06 1.03 1.00 0.99 0.97 0.97 0.97 20 1.07 1.06 1.06 1.05 1. 03 1 .01 1.00 0.98 0.98 0.97 0.97 00 1 .06 1.05 1 .05 1.04 1.03 1.01 1.00 0.98 0.98 0.97 0.97 05 1.09 1.09 1 .08 1.07 1.05 1. 03 1.00 0.99 0.97 0.97 0.97 15 10 1 .09 1 .09 1 .00 1 .07 1.05 1 .03 1.00 0.99 0.97 0.97 0.97 15 1.10 1 .09 1.09 1. 07 1.05 1.03 1.00 0.98 0.97 0.97 0.97 20 1 .06 1.05 1 .05 1.04 1.03 1.01 0.99 0.98 0.97 0.97 0.97 00 1.06 1. 05 1.05 1 .04 1.03 1.02 1.00 0.99 0.98 0.97 0.97 05 1.09 1.09 1 .00 1 .07 1.05 1 .03 1.01 0.99 0.97 0.97 0.97 20 10 1. 09 1.09 1. 08 1. 07 1.05 1. 03 1. 01 0.99 0.97 0.97 0.97 15 1. 10 1. 09 1.09 1, 07 1 .05 1 .03 1.00 0,99 0.97 0.97 o. 97 20 1.06 1 .06 1 .05 1.04 1.03 1.01 0.99 0,98 0,98 0.97 0.97 .. .. -\ -\ - •4,')0 •4,')0 -5.00 -5.00 00 00 -i -i -S.,o -S.,o using using ) ) P1 P1 " " s s -I -I (23) (23) I I for for -boo -boo 1 1 time time ( ( ')0 ')0 10 10 •10 •10 mol- (b) (b) log products products kJ kJ ')I) ')I) of of -'it -'it 85.8 85.8 ·12';') ·12';') formation formation •IJ'l'J •IJ'l'J of of cleavage)= cleavage)= ~ ~ o_ll-4.~".) o_ll-4.~".) 2 2 0 0 0 0 i i ~ ~ 0 0 ~ ~ • • '! '! ~ ~ ~ ~ ~j ~j " " :;: :;: rate rate d d c:: c:: V V 0 0 ...... V V 0.. 0.. a a " " 0 0 ...... 0 0 ;/) ;/) > > u u ...... ::, ::, 0 0 .., .., .., .., .2 .2 p:: p:: :;:; :;:; :;:; :;:; v v 0 0 -u -u (C-0 (C-0 Ea Ea relative relative 1 1 (b) (b) 'JO 'JO and and moi- P1 P1 kJ kJ --1.n --1.n ------ ' ' I I I I ' ' curves curves I I 59.2 59.2 I I I I I I I I I I I I I I I I I I oa oa "'--1... "'--1... ,; ,; energetics. energetics. ,o ,o ' ' I I ' ' I I I I I I I I AC AC -d -d abundance abundance / / .s.n .s.n MOP MOP Ion Ion CH-transfer):;:; CH-transfer):;:; ) ) (a) (a) the the Ea Ea s s < < / / (a) (a) •9')0 •9')0 5.37. 5.37. \ \ time time / / ' ' \ \ ( ( ,a ,a 10 10 -1; -1; log Figure Figure n n ;=:::::::::::: ;=:::::::::::: -11 -11 -12':j -12':j ·12':'; ·12':'; 'JC 'JC g g ~j ~j j j ~ ~ :i :i ~ ~ $ $ 8 8 I; I; !' !' 8 8 8 8 '! '! ;1 ;1 ;I\• ;I\• ~1 ~1 o_U o_U u u c:: c:: V V " " c:: c:: " " ::, ::, V V > > ., ., ...... ~ ~ ~ ~ :;:; :;:; .....__ .....__ .0 .0 v v ~~_.,o .,.ao .§.'JO ?O , ~ using -5.'Jo 'JG (23) -~ / s ) for 1 zf;o time ( rio moi- -10 products (b) log" kJ n of -ll 85.8 'JO .'12 10 formation -1~ of cleavage)= I -1.t.O'J rate ~ ~ ~ ~ 0 g ~ 0 :;: ~ ~ 0 ~ ~ l! l' u Ill d ::., 0 I., a. I., C: :, I., ~ E :l ..., 0 "O _, 0 _, .9 .E 0:: :;:; v (C-O 0 E relative 1 (b) ~') moi- and 1 .~ p kJ •4,'J'; curves :~P2 71.2 r r r r r r ' r , ';'; -~ energetics. ')') ·i:i abundance -::o MOPAC Ion -S CH-transfer)= 0 (a) the E I)';: ·:! =::;:::::::::: / s ) (a) ')') M .5 5.38. time '\. ( ------ 0 ·I?,::, ==---== log, Figure '.:'J ·II •I.'.!')') ·I.'.!~ ~'J ··14 -I ~ • s ,; g i ~ ~ , ~ g ~ 'l ~ ~ ! 0 g i s1 8 ' :: u.. ::., V ... :, :, _, ii Ix .,:, 'ii .~ '- ,o ·l .,o ◄ • -~.00 ')') ➔ using -)')') ) ,~ (23) s -:i I for '°JI) 1 -~ time ( ~o moi- -"10 products log" kJ of (b) -'11.1,; 73.8 .'1 formation -'t:!'l? of :;'J cleavage)= rate :;: i s; Si! 0 0 .. ~ ~ 0 ~ 0 ~ 0 0 °" )0 U.'14 (J "' 0 ::, Q. :) L, ::: rj ... L, 0 a V V :) ..., -0 0 ..., .s .... 0 ..., ..., .~ .; 0:: (C-0 0 E relative 1 (b) and moi- P1 P2 -!','j kJ -4:,', curves ' ' ' ' ' . 71.2 'Zl ~ ., energetics. ~'j -,;: abundance :, MOPAC Ion CH-transfer)= 0 ) ~ (a) the E s \ I \ (a) \ M 5.39. \ time ( -,~ \oglO Figure ·II::, :o -1~ ·.-:: -1:• ~:; j ~ 8 2 jl ~ ~1 ~ ~ ~j ~ • ~ ~ ., ~ g .:>_14 1' ~ 8 ~ :) V :) V ;:, > L, :-.: .I:> -g .., ' .; 103 moi-1, are given in Figures 5.37, 5.38 and 5.39 respectively. The rate of loss of c;tt4O gradually decreases until loss of C2H418O becomes faster than loss of C2H4O, as was seen with the calculations using the thermochemical energetics. Considering the ion abundances calculated for (21) and (23), best agreement with the measured ratios occurred when the transition state frequencies were lowered by 10 % for hydrogen transfer and 5 % for C-O cleavage, and when the E0's for hydrogen transfer and C-O cleavage were 71.2 kJ moi-1 and 85.8 kJ moi-1 respectively. The calculated ion abundance ratios were 3.27 and 1.03 respectively. The agreement for (21) is much better than for (23). To test this model, ion abundance ratios for the loss of C2H4O and C2D4O from the molecular ion of (22), and the loss of C2D4O and c;o418o from the molecular ion of (24) were calculated. The calculated values of 2.89 and 1.01 for (22) and (24) respectively do not agree well with the measured values of 3.41 ± 0.07 and 1.06 ± 0.05. 5.9.7. Discussion of the stepwise models. Error limits on the energy barrier were neglected above, as the consequences of varying the overall energy barrier are predictable. Reducing the energy barrier to that given by the lower limit to reaction, corresponds to lowering E0 for all reactions, which results in an increase for all k(E)'s for the reactions. An increase in the k(E)'s will result in increases in the ion abundance ratio. Increasing the energy barrier to that given by the upper limit, increases all E0's for the reactions which results in reduced values for k(E)'s for all reactions. Reducing the k(E)'s results in a decrease of the ion abundance ratios. Consider, for example, the calculations using the transition states from in which only the reaction coordinates are removed. For the thermochemical energetics with E0 for hydrogen transfer at 71.2 kJ moi-1 and Eo for C-O cleavage at 12.6 kJ moi-1, the 104 calculated ion abundance ratios for (21), (22), (23) and (24) are 4.16, 3.47, 1.03 and 1.01 respectively. Reducing the values of E0 to 59.2 kJ moi-1 and 0.6 kJ moi- 1 increases the calculated ion abundance ratios to 6.24, 5.04, 1.04 and 1.02 .. Increasing the values for E0 to 83.2 kJ moi- 1 and 24.6 kJ moi- 1 decreases the calculated ion abundance ratios to 3.02, 2.59, 1.02 and 1.01 respectively. The magnitudes of the calculated ion abundance ratios vary, but the relative magnitudes of the ratios remain in the order (21) > (22) > (23) > (24), c.f. the measured ion abundance ratios (22) > (21) > (23) > (24). For the MOP AC energetics, similar behaviour is expected and observed. With E0 for hydrogen transfer and C-O cleavage at 71.2 kJ moi-1 and 86.2 kJ mol- 1 respectively and using the transition states with only the reaction coordinates removed, the calculated ion abundance ratios for (21), (22), (23) and (24) are 2.57, 2.34, 1.01 and 1.01 respectively. Decreasing the values of E0 to 59.2 kJ moi- 1 and 74.2 kJ moi- 1 respectively, increases the calculated ion abundance ratios to 3.33, 2.99, 1.02 and 1.01 respectively. Increasing the values of E0 to 83.2 kJ moi- 1 and 98.2 kJ moi-1 decreases the calculated ion abundance ratios to 2.17, 1.98, 1.01 and 1.00 respectively. The relative magnitudes of the ion abundance ratios remain in the same order as predicted from the thermochemical energetics, i.e. (21) > (22) > (23) > (24). 5.10. Conclusion. As said earlier, reaction mechanisms can be disproved but not proved. The concerted ml?chanism for the McLafferty rearrangement in the cx.,cx.'-diethoxy p-xylene molecular ion explains the isotope effects upon ion abundances in the absence of hydrogen scrambling. The stepwise mechanism involving a distonic ion intermediate fails in two respects. 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Traeger, Org. Mass Spectrom., .U, 483 (1978). 131. J.E. Bartmess, J. Amer. Chem. Soc., .l!M, 335 (1982). 132. J.H. Chen, J.D. Hays and R.C. Dunbar, J. Phys. Chem., ER, 4759 (1984). 133. F.W. McLafferty, R. Kornfeld, W.F. Haddon, K. Levsen, I. Sakai, P.F. Bento III, S.C. Tsai and H.D.R. Schuddemage, J. Amer. Chem. Soc., .25., 3886 (1973). 134. M.M. Bursey, M.K. Hoffman and S.A. Beneza, J. Chem. Soc., Chem. Commun., 1417 (1971). 135. D.A. Lightner, G.B. Quistall and E. Irwin, Appl. Spectrosc., 25., 253 (1971). 136. C. Wesdemiotis, H. Schwarz, F. Borchers, H. Heinbach and K. Levsen, Z. N aturforsch., fil, 1150 ( 1978). 137. F. Borchers, K. Levsen, C. Wesdemiotis and H. Schwarz, Int. J. Mass Spectrom. Ion Phys., 18., 203 (1978). 138. B.F. Yates, W.J. Bouma and L. Radom, J. Amer. Chem. Soc., 106, 5805 (1984). 139. C. Wesdemiotis, R. Feng and F.W. McLafferty, J. Amer. Chem. Soc., 107, 715 (1985). 140. R.G. McLoughlin, J.D. Morrison and J.C. Traeger, Org. Mass Spectrom., 14, 104 (1979). 124. D.J. Underwood, Ph.D. Thesis, University of Adelaide (1981). 125. D.J. McAdoo, D.N. Witiak and F.W. McLafferty, J. Amer. Chem. Soc., 99, 7265 (1977). 126. F.P. Boer, T.W. Shannon and F.W. McLafferty, J. Amer. Chem. Soc., .2Q, 7239 (1968). 141. J.K. MacLeod and C. Djerassi, Tet. Lett., 12., 2183 (1966). 142. J.K. MacLeod and C. Djerassi, J. Amer. Chem. Soc., B..2, 5182 (1967). 143. D.J.M. Stone, J.H. Bowie, D.J. Underwood, K.F. Donchi, C.E. Allison and P.J. Derrick, J. Amer. Chem. Soc., W, 1688 (1983). 144. D.P. Shoemaker, C.W. Garland and J.I. Steinfield, "Experiments in Physical Chemistry", 3rd Edition, McGraw-Hill, New York (1974). 145. R.E. Walpole and R.H. Myers, "Probability and Statistics for Engineers and Scientists", 2nd Edition, Macmillan, New York (1972). 146. M.J.S. Dewar and W. Thiel, J. Amer. Chem. Soc., .2.2, 4899 (1977). 113 147. J.J.P. Stewart, QCPE Program no. 455, Chemistry Department, Indianna University. Al. I Appendix 1. Computer program QET. N N > > - I, I, 1 1 CM-1 ' ' F, F, ,FI) ,FI) IE IE THE THE =', =', =' =' ') ') ERROR ERROR GIVING GIVING REACTION, REACTION, INCREMENT INCREMENT IS',I) IS',I) IS',I) IS',I) 'I, 'I, THE THE •••• •••• PATHWAYS PATHWAYS OF OF '//) '//) ETC ETC FREQUENCY FREQUENCY CH-1 CH-1 ENERGY ENERGY NUTOT NUTOT HENRG HENRG 1 1 /' /' 1 FACTOR FACTOR =',F, =',F, MAXIMUM MAXIMUM CH-1 SYHFAC:2 SYHFAC:2 SYMFAC:4 SYMFAC:4 EXCEEDED. EXCEEDED. EXCEEDED. EXCEEDED. FREQUENCIES FREQUENCIES FRAGMENTATION FRAGMENTATION ,F,/' ,F,/' ENERGY ENERGY 1 => => SYMMETRY SYMMETRY THEN THEN ARTS(60000) ARTS(60000) =',F,' =',F,' ARH0(60000) ARH0(60000) : 0.00/ 0.00/ 0.00/ 0.00/ TWO TWO =',I6) =',I6) BOUND BOUND BOUND BOUND 1 1 DELTAE=',FB.1,'CH-1'/) DELTAE=',FB.1,'CH-1'/) =',16//) =',16//) F,G,H F,G,H THE THE ) ) ) ) ARE ARE TITLE(60) TITLE(60) REACTANT REACTANT 1 (FREQ(J),J:1,N) (FREQ(J),J:1,N) (FRTS(J),J:1,NTS) (FRTS(J),J:1,NTS) MAXIMUM MAXIMUM N N ( 1 1 FOR FOR ENERGY ENERGY FACTOR FACTOR ARRAY ARRAY MENRG MENRG ARRAY ARRAY 1 1 ACTEN+ZPE ACTEN+ZPE ENHAX,RINT,ACTEN,ZPE,ERROR,SYMFAC: ENHAX,RINT,ACTEN,ZPE,ERROR,SYMFAC: NUTOT NUTOT ,ENHAX,RINT,ACTEN,ZPE,ERROR,SYMFAC ,ENHAX,RINT,ACTEN,ZPE,ERROR,SYMFAC PATHWAYS PATHWAYS PRODUCTS, PRODUCTS, NU(150),NUTS(150) NU(150),NUTS(150) 1)(TITLE(J),J:1,60) 1)(TITLE(J),J:1,60) 1031)(TITLE(J),J:1,60),DELTAE 1031)(TITLE(J),J:1,60),DELTAE I I ) 60A1 60A1 PRECISION PRECISION PRECISION PRECISION 1 : : (' (' (' (' THERE THERE 1000 1000 (2,2) (2,2) (2,•) (2,•) (2,•) (2,•) (2, (2, ARTS/60000 ARH0/60000 FREQ(150),FRTS(150) FREQ(150),FRTS(150) INIT(N,FREQ,NU,EINT,ERROR) INIT(N,FREQ,NU,EINT,ERROR) 1:2,N 1:2,N ALLOWS ALLOWS IF IF FOUR FOUR SAHE SAHE 4 4 CRITICAL CRITICAL SYMMETRY SYMMETRY INTEGER INTEGER REAL REAL DOUBLE DOUBLE DOUBLE DOUBLE DATA DATA DATA DATA FORMAT(//' FORMAT(//' CHARACTER DELTAE DELTAE ACCEPT ACCEPT FORMAT(' FORMAT(' CHARACTER READ READ READ READ OPEN(UNIT:2,FILE:F,STATUS:'OLD') OPEN(UNIT:2,FILE:F,STATUS:'OLD') READ READ NTS=N-1 NTS=N-1 READ READ TYPE TYPE WRITE(1,1008)ENMAX,DELTAE,RINT,SYMFAC,ERROR WRITE(1,1008)ENMAX,DELTAE,RINT,SYMFAC,ERROR OPEN(UNIT=7,FILE='WORK.F1',STATUS='NEW') OPEN(UNIT=7,FILE='WORK.F1',STATUS='NEW') OPEN(UNIT:1,FILE:G,STATUS:'NEW') OPEN(UNIT:1,FILE:G,STATUS:'NEW') FORMAT( FORMAT( FORMAT(//,' FORMAT(//,' FORHAT(//60A1' FORHAT(//60A1' IF(MENRG.GT.59999)WRITE(6,601)MENRG IF(MENRG.GT.59999)WRITE(6,601)MENRG FORMAT( FORMAT( NUTOT:NU(l) NUTOT:NU(l) FORMAT(/' FORMAT(/' NUTOT:NUTOT+NU(I) NUTOT:NUTOT+NU(I) WRITE(1,5601)MENRG WRITE(1,5601)MENRG MENRG:ENMAX/EINT MENRG:ENMAX/EINT DO DO CALL CALL FORMAT FORMAT WRITE(l,1002) WRITE(l,1002) WRITE(l, WRITE(l, FORMAT FORMAT IF(NUTOT.GT.59999)WRITE(6,610)NUTOT IF(NUTOT.GT.59999)WRITE(6,610)NUTOT FORMAT(' FORMAT(' WRITE(1,5610)NUTOT WRITE(1,5610)NUTOT 1' 1' 2' 2' SYMFAC: SYMFAC: 1008 1008 C C 1000 1000 C C C C C C 1 1 C C C C 1031 1031 C C 1002 1002 2 2 601 601 5601 5601 610 610 4 4 5610 5610 OF OF VALUE. VALUE. E E OF OF EINTS. EINTS. AS AS TRANSTION TRANSTION MULTIPLES MULTIPLES OF OF OF OF CURRENT CURRENT INIT. INIT. REACTANTS REACTANTS FORM FORM INTERVALS. INTERVALS. ROUNDING ROUNDING INIT. INIT. REQUIRED. REQUIRED. BY BY UNITS UNITS EINT. EINT. THE THE INTEGER INTEGER CM-1 CM-1 IS IS NUMBER NUMBER IN IN TO TO OF OF HEADING. HEADING. AS AS FUNCTION. FUNCTION. DIFFERENT DIFFERENT RINT RINT ROUTINE ROUTINE ALLOWABLE ALLOWABLE ,$) ,$) INTEGERISED INTEGERISED KA(E) KA(E) ION ION UNITS UNITS OPTIMISED OPTIMISED IN IN IN IN ENERGY ENERGY FINALLY FINALLY =' =' IN IN OUTPUT OUTPUT FROM FROM IN IN STATES STATES AND AND WHICH WHICH FREQUENCES FREQUENCES MAXIMUM MAXIMUM FREQUENCIES. FREQUENCIES. K(E) K(E) CORRESPONDING CORRESPONDING REACTION. REACTION. CHOSEN CHOSEN REACTANT REACTANT TO TO INTERVAL INTERVAL ENMAX ENMAX K(E)'S K(E)'S INTERNAL INTERNAL THE THE OF OF MOLECULE MOLECULE ONLY ONLY INCREMENTS INCREMENTS CALCULATION. CALCULATION. STATE STATE REACTANT REACTANT THE THE STATE STATE =',$) =',$) DATA DATA ARRAYS: ARRAYS: THE THE K(E) K(E) CONVENIENT CONVENIENT DENSITY DENSITY FREQUENCIES FREQUENCIES OF OF FREQUENCIES FREQUENCIES FREQUENCIES. FREQUENCIES. ENERGY ENERGY A A OF OF OF OF OF OF THE THE ENERGY ENERGY FOR FOR ENERGY ENERGY STATE STATE ANO ANO DOUBLE DOUBLE INTERVAL INTERVAL FILE FILE THE THE FILE=',$) FILE=',$) WHICH WHICH =',$) =',$) OUTPUT OUTPUT FOR FOR CORRELATE CORRELATE OUTPUT OUTPUT FOR FOR ENERGY ENERGY VALUE VALUE STATE STATE ENERGY ENERGY DATA DATA TO TO RATE RATE CALCULATION CALCULATION REACTANT REACTANT INITIALLY INITIALLY DENSITY DENSITY ENERGY ENERGY REACTANT REACTANT TRANSITION TRANSITION ALL ALL TRANSITION TRANSITION K(E) K(E) FILE FILE INTERNAL INTERNAL ENERGY ENERGY FREQUENCIES. FREQUENCIES. INTERNAL INTERNAL F,G,H F,G,H STRING STRING VARIABLES VARIABLES VARIABLE VARIABLE TRANSITION TRANSITION USED USED OF OF QET(F,G,H,RELEN) QET(F,G,H,RELEN) HOLD HOLD HOLD HOLD HOLD HOLD HOLD HOLD HOLD HOLD HOLD HOLD 12 12 CONTAINING CONTAINING 1 INPUT INPUT RELATIVE RELATIVE OF OF OUTPUT OUTPUT THEORY THEORY OUTPUT OUTPUT (NUTS(!)) (NUTS(!)) (NU(I)) (NU(I)) INCREMENT INCREMENT )RELEN )RELEN IS IS 1010) 1010) 1030) 1030) 1000) 1000) 1020) 1020) FUNCTION. FUNCTION. QET QET RELATIVE RELATIVE TO TO TO TO 1 TO TO CONTAIN CONTAIN TO TO CONTAIN CONTAIN TO TO TO TO MAX. MAX. CRITICAL CRITICAL MAXIMUM MAXIMUM CALCULATED CALCULATED MAXIMUM MAXIMUM (6, (6, (6, (6, TRkNSITION TRkNSITION FILE FILE QET(F,G,H,RELEN) QET(F,G,H,RELEN) PARAMETER-APPROX PARAMETER-APPROX ENERGY ENERGY RATE RATE OPTIMISED OPTIMISED TO TO TO TO THE THE THIS THIS ALOG10(RATE) ALOG10(RATE) SIGMA SIGMA VIBRATIONAL VIBRATIONAL A A THE THE THE THE HOLLERITH HOLLERITH THE THE ENERGY ENERGY AN AN ARRAY ARRAY ARRAY ARRAY ARRAY ARRAY ARRAY ARRAY AN AN ARRAY ARRAY ARRAY ARRAY MULTIPLES MULTIPLES STATES STATES MEANING MEANING END END REA0(5, READ(5,2000)H READ(5,2000)H FORMAT(' FORMAT(' SUBROUTINE SUBROUTINE READ(5,2000)G READ(5,2000)G PROGRAM PROGRAM FORMAT(' FORMAT(' STOP STOP FORMAT(' FORMAT(' FORHAT(A12) FORHAT(A12) REA0(5,2000)F REA0(5,2000)F FORMAT(' FORMAT(' CALL CALL WRitE WRitE CHARACTER WRITE(6, WRITE(6, WRITE WRITE WRITE(6, WRITE(6, INPUT INPUT FILE FILE FILE FILE RELEN: RELEN: H: H: F: F: G: G: RRATE: RRATE: RINT: RINT: NUTOT: NUTOT: E: E: OELTAE:THE OELTAE:THE NUTOTS:SIGMA NUTOTS:SIGMA ERROR: ERROR: HENTS: HENTS: NU: NU: ENHAX: ENHAX: NUTS: NUTS: EINTS:THE EINTS:THE HENRG:THE HENRG:THE EINT: EINT: FRTS: FRTS: FREQ: FREQ: ARHO: ARHO: TITLE: TITLE: QET/RRKH QET/RRKH C C CRATE: CRATE: C C C C C C C C C C C C C C C C C C C C C C 1020 1020 C C C C C C C C 1010 1010 C C 1030 1030 C C 1000 1000 C C C C CARTS: CARTS: C C C C C C C C C C 2000 2000 C C C C C C C C CALL DENS(N,EINT,NU,NUTOT,MENRG,ARHO) CALL SETUP(ACTEN,RINT,H,RELEN) WRITE( 1, 1004) RETURN 1004 FORMAT(//' TRANSITIONSTATE FREQUENCIES'//) END CALL INIT(NTS,FRTS,NUTS,EINTS,ERROR) C SUBROUTINEINIT MENTS:(ENMAX-DELTAE)/EINTS C OPTIMISES THE GRAINSIZE (EINT OR EINTS) WRITE(1,5611)MENTS C ANDROUND THE VIBRATIONALFREQUENCIES 5611 FORMAT(/' MENTS=',I6) SUBROUTINEINIT(N,FREQ,NU,EINT,ERR) IF(MENTS.GT.59999)WRITE(6,611)MENTS DIMENSIONFREQ(N),NU(N} 611 FORMAT(' ARRAYBOUND EXCEEDED. MENTS IS',I) DO 1 I:2,N NUTOTS:NUTS( 1) L:I DO 602 I:2, NTS 2 M:L-1 602 NUTOTS:NUTOTS+NUTS(I) IF (FREQ(M).LE.FREQ(L)) GO TO 1 WRITE(1,5612)NUTOTS T:FREQ(M) 5612 FORMAT(' NUTOTS=',I6//) FREQ(M):rREQ(L) IF(NUTOTS.GT.59999)WRITE(6,612)NUTOTS FREQ(L}:T 612 FORMAT(' ARRAYBOUND EXCEEDED. NUTOTS IS',I) IF (M,EQ,1) GO TO CALL DENS(NTS,EINTS,NUTS,NUTOTS,MENTS,ARTS) L:M GO TO 2 DO 603 I:2,MENTS 603 ARTS(I):ARTS(I)+ARTS(I-1) CONTINUE K:DELTAE/EINT E2:ERR+ERR RH:1./(ARHO(K}•1.D35) REML::1. RMIN:RM•2.997E10•SYMFAC DO 3 I:1,N TRHIN:ALOG10(RMIN) F=rREQ(I) WRITE(1,13073)RMIN,TRMIN,RM NMAX:F/E2+ 1 13073 FORMAT(' MINIMUMRATE =',E14.5,'S(-1)',2E14.5/) NMIN:(NMAX-1)/2 WRITE(1,1006) J:NMAX 1006 FORMAT(//' THE DENSITYOF STATES ANDNUMBER OF STATES ', DO 4 K:NMIN,NHAX 1'FUNCTIONSNEED TO BE MULTIPLIED'/' BY 1.0035 ', TRY:F/FLOAT(J) 2'TO BE RETURNEDTO THE ACTUALVALUES'/, REM::O. 3' THE DENSITYOF STATES, NUMBEROF STATES ANDTHE ACTUALRATE', DO 5 L:1,N 4' HAVEALL BEEN AVERAGED'/) ITRY:FREQ(L)/TRY + 0,5 WRITE(l,5613) REM:REM+ ABS(1,-ITRY*TRY/FREQ(L)) . , IF (REM.GE.REHL)GO TO 40 5613 FORMAT(1 ENERGY RATE LOG(RATE) .. 1' DENSOF STTS: NO OF STTS ') 5 CONTINUE WRITE(1,•)DELTAE,RMIN,TRMIN,ARHO(K) REML:REM WRITE(7,•)0ELTAE,RMIN,TRMIN,ARHO(K) EINT:TRY DO 604 E:DELTAE,ENMAX,RINT 40 J:J-1 K::E/EitlT 4 CONTINUE KTS:(E-DELTAE)/EINTS 3 CONTINUE IF(KTS,LE.2)GOTO 604 WRITE (1,8) EINT ARBAR:ARHO(K)+ARHO(K+1)+ARHO(K+2)+ARHO(K+3)+ARHO(K-1)+ARHO(K-2) 8 FORMAT(' OUTPUTOF INIT',20X,' OPTIMUMENERGY UNIT: ',F// ARTBAR:ARTS(KTS)+ARTS(KTS+1}+ARTS(KTS+2)+ARTS(KTS+3)+ARTS(KTS-1) ' UNITS FREQUENCYROUNDED TO') l+ARTS(KTS-2) DO 6 I:1,N RATE:(ARTBAR/ARBAR•2.997E101 EINT)*SYMFAC NU(I}:FREQ(I)/EINT + 0,5 RRATE: ALOG10( RATE ) RrRQ:KU(I) *EINT > WRITE(1, 1 )E,RATE,RRATE,ARBAR,ARTBAR WRITE (1,7) NU(I),FREQ(I),RFRQ WRITE(7,•)E,RATE,RRATE,ARBAR,ARTBAR 7 FORMAT(1X,I4,2F11,2) w- 604 CONTINUE 6 CONTINUE CLOSE(UNIT:1) RETURN CLOSE(UNIT:7) END > +:>, - **RATE 999 7 IF READ(11,*,END=999)X,Y,Z WRITE(3,*)X,Y,Z READ(11,*,END:999)X1,Y1,Z1 READ(11,*,END:999)X2,Y2,Z2 END EN:X1+GAP RATE:Z1+GAP*(Z2-Z1)/RINT GOTO EN:ANINT(EN+RINT) GAP:RINT+GAP TRATE:10. 21:22 READ(11,*,END:999)X2,Y2,Z2 WRITE(3,*)EN,TRATE,RATE RETURN CLOSE(UNIT:11) CLOSE(UNIT=3) GOTO END 200 7 999 ) ACTEN - NEW') ALGORITHM ,STATUS:'OLD') 1 1 LEVELS B.S. .5)/10. WORK.F2',STATUS:'NEW') BADENS THE ARHO(MENRG) 1 ROUNDED + ( TO H ' TITLE (N) DENS(N,EINT,NU,NUTOT,MENRG,ARHO) SETUP(ACTEN,RINT,H,RELEN) 1000 OENS NU BADENS 1000 SETUP = PRECISION PRECISION OUTPUT I=l, STATES I=l, 1=1,MENRG I=l,N J:JJJ,MENRG (UNIT:7) 1 2 2 10 200 DOUBLE DOUBLE SUBROUTINE DIMENSION DO BADENS:1.D-35/EINT DO ARHO(I):O.DO JJ:NU(I) JJJ:JJ+l DO RETURN ARHO(J):ARHO(J)+ARHO(J-JJ) ARHO(JJJ):ARHO(JJJ)+BADENS ARHO(l) END SUBROUTINE CHARACTER*80 DO CHARACTER*(*) READ(7,*,END:2)EN,RATE,RRATE EN:EN+RELEN OPEN(UNIT:8,FILE: ACTEN:ACTEN+RELEN OPEN(UNIT=7,FILE='WORK.F1 CLOSE WRITE(8,*)EN,RATE,RRATE CLOSE(UNIT:8) IF(GAP,EQ,0,)THEN GAP:(INT(ACTEN/RINT))*RINT GAP=INT(GAP*10. DO OPEN(UNIT:11,FILE='WORK.F2',STATUS:'OLD') OPEN(UNIT=3,FILE:H,STATUS: COUNTS SUBROUTINE SUBROUTINE ADJUSTS C C C 2 10 C C 2 A2.1 Appendix 2. Calculation of metastable ion abundances. The method used for the integration of k(t) (defined in Section 2.6.3) relies upon k(t) being an exponential function oft. A plot of ln(k(t)) vs. ln(t) yields a straight line. It then follows that ln(k(t)) = m ln(t) + c and k(t) = exp(m ln(t) + c). m and c are constants which can be calculated for given values oft and k(t). The ion abundance is given as the integral of k(t), with respect to time, over the observation window. t2 I= J k(t).dt t 1 To solve this integral, let t = exp(x) which gives dt = exp(x).dx and ln(t) = x. Substitution gives exp(x 2 ) . I= J exp(mx + c) exp(x).dx exp(x 1) A2.2 exp(x 2 ) = J exp(c) exp((m + l)x).dx exp(x 1) exp(x 2 ) = exp(c) J exp(x(m + 1)).dx exp(x 1) exp(x 2 ) exp(x(m + 1)) = exp(c) m+ 1 Back substitution for x gives t2 (m + 1) t I= exp(c) m+l Ion abundances can be calculated and compared for competing reactions, which, in the case of isotopically distinguished reactions, gives the isotope effect. Program PEAKS which uses this method to calculate the ion abundances is listed below. Also listed is program TIME, which calculates k(t) as described in Section 2.6.3., but calculates k(t) over the time range 10-14 s to 10-3 s for both reactions and the ratio of the two k(t)'s over this time range. FLAG:2 PROGRAMPEAKS GOTO15 CHARACTER*80TITLE(2) 14 RK2:0. CHARACTER*14F(2),G FLAG:1 INTEGERFLAG 15 IF(RK1.LT.O. )RK1:10.HRK1 WRITE(6, 100) IF(RK2.LT.O.)RK2:10.••RK2 1 1 100 FORMAT( OUTPUTFILE : ,$) SRK=RK1+RK2 READ(5,200)G ESRK:SRK •TI 200 FORMAT(A14) IF(ESRK.GT.87.5)GOTO 40 OPEN(UNIT:13,FILE=G,STATUS:'NEW') ESRK:EXP(-ESRK) , WRITE(6, 120) SUM1:SUM1+RK1•ESRK WRITE( 13,120) SUM2:SUM2+RK2*ESRK 120 FORMAT(' PARENTMASS,ACCELERATING VOLTAGE ',$) IF(FLAG.EQ. 1)GOTO 17 READ(5,*)MASSP,VLTAGE IF(FLAG.EQ.2)GOTO 18 WRITE(13,105)MASSP,VLTAGE GOTO20 105 FORMAT(4X,I4,8X,F8.1) READ(11,•,END:40)EN1,RK1 VEL1:SQRT(1.929734E8*VLTAGE/MASSP) 17 GOTO19 T1:1.89931/VEL1 + 1.E-6 18 READ(12,•,END:40)EN2,RK2 T2=4.59199/VEL1 + 1.E-6 19 FLAG:O T1LOG:ALOG10(T1) GOTO12 T2LOG:ALOG10 (T2) 20 CONTINUE DELT:T2-T1 40 WRITE(13,•)TI,SUM1,SUM2 FT1:T1*1.E6 CLOSE(UNIT:11) FT2:T2*1.E6 CLOSE(UNIT:12) FDT:FT2-FT1 IF(TI.EQ.T1)T1S1:SUM1 WRITE(13,170)FT1,T1LOG,FT2,T2LOG,FDT IF(TI.EQ.T1)T1S2:SUM2 FFR ENTRANCE MICROSECONDS ALOG10= 170 FORMAT(//,' TIME: ',F8.3,' ( IF(TI.EQ.T2)T2S1:SUM1 1F, 1 ) 1 • FFR EXIT TIME : 1 ,FS. 3, 1 MICROSECONDS( ALOG10= ,/, IF(TI.EQ.T2)T2S2:SUM2 2F, 1 ) 1 1 /, 1 FFR RESIDENCETIME= ',F8.3,' MICROSECONDS') 30 CONTINUE DO 10 !=1,2 WRITE(13,140)(TITLE(1)) WRITE(6,130)I 140 FORMAT(//, 1 REACTION1 : 1 ,ABO ) 130 FORMAT(1 FILE ( 1 ,12, 1 1 ,$) ): WRITE(13,150)(TITLE(2)) READ(5,200)F(I) 150 FORMAT(' REACTION2: ',ABO ) 10 CONTINUE C CALCULATETHE ION ABUNDANCERATIO ( AREAUNDER K(T) CURVES) WRITE( 13, 11 0 ) AS1 :ALOG (T2S 1 ) 110 FORMAT(II,' TIME K(T) FOR RXN1 K(T) FOR RXN2 1 ) AS2:ALOG(T1S1) DO 30 I:1,2 AS3:ALOG(T2) IF (I. EQ. 1 )TI :T 1 AS4:ALOG(T1) IF(I.EQ.2)TI=T2 EM1:(AS1-AS2)/(AS3-AS4) OPEN(UNIT:11,FILE:F(1),STATUS: 10LD') C1 :AS2-EM 1 *AS4 OPEN(UNIT:12,FILE:F(2),STATUS:'0LD') AREA1:(T2**(EM1+1.)-T1**(EM1+1.))*EXP(C1)/(EM1+1.) READ(11,210)(TITLE(1)) AS5:ALOG(T2S2) READ(12,210)(TITLE(2)) AS6:ALOG(T1S2) 210 FORMAT(A80) EM2:(AS5-AS6)/(AS3-AS4) SUM1:0. C2:AS6-EM2•AS4 SUM2=0. AREA2:(T2•*(EM2+1. )-T1*•(EM2+1. ))*EXP(C2)/(EM2+1.) FLAG:O RATIO:AREA1/AREA2 DO 20 J:1, 1000 WRITE(13,160)AREA1,AREA2,RATIO READ(11,*,END:40)EN1,RK1 160 FORMAT( AREAUNDER RXN1 CURVE= 1F READ(12,*,END:40)EN2,RK2 II,' ', ,/, !'-)> 1 AREAUNDER RXN2 CURVE = 1 , 1F ,I, C THIS SECTION ALLOWSFOR THE INPUT DATAFILES TO COMMENCE \.,J 2 1 RATIO OF AREAS= 1F ) C AT DIFFERENTENERGIES. THE ENERGIESMUST HAVE A COMMON ', WRITE(6,•)RATIO C INCREMENT. STOP 12 IF(EN1.EQ.EN2)GOTO 15 END IF(EN2.GT.EN1)GOTO 14 RK1:0. > ~ !'-l ' • I . RXN2 FOR K(T) 1 RXN ',$) FOR : :',$) ',$) : (T) ; K 20 "*RK1 50 FILENAME FILENAME FILENAME KH KD F,G,H TITLE(2) TIME )RKl:10, l 2000 0, INPUT INPUT OUTPUT (K1/K2)') , 1000) TIME 1010) 1 1 1 ABO = NE, HTL ( ( J (6, 1. (UNIT:12) 30 RATIO 20 PROGRAM CHARACTER•12 READ(5,2000)F FORMAT(' CHARACTER•SO WRITE(6, FORMAT(A12) READ(5,2000)G FORMAT(' FORMAT(' WRITE READ(5,2000)H WRITE(6,1020) IF(TL.LE.-14.)WRITE(13,210)TITLE(1) IF(TL.LE.-14.)WRITE(13,210)TITLE(2) OPEN(UNIT:13,FILE:F,STATUS:'NEW') TL:-14. OPEN(UNIT:12,FILE:H,STATUS:'OLD') TI:10. READ(11,210)TITLE(1) OPEN(UNIT:11,FILE:G,STATUS:'OLD') READ(12,210)TITLE(2) FORMAT IF(TL.LE.-14)WRITE(13,110) FORMAT READ(11,•,END:40)EN1,RK1 SUMl:O. READ(12,•,END:40)EN2,RK2 DO IFCRK SUM2:0. IF(RK2,NE.O,)RK2:10.••RK2 IF(ESRK.GT.87,5)GOTO ESRK:EXP(-ESRK) SRK:RK1+RK2 ESRK:SRK•TI RATEO:SUM1/SUM2 SUMl:SUMl+RKl•ESRK FORMAT(X,E10.3,E14.5,X,E15.5,4X,E10.3) SUM2:SUM2+RK2•ESRK CONTINUE IF(TL.GE.-2.9)GOTO WRITE(13,*)TI,SUM1,SUM2,RATEO CLOSE(UNIT:11) CLOSE TL:TL+O. GOTO END STOP 1 1 1000 1010 1020 2000 30 110 210 120 40 20 50 A3. 1 Appendix 3. Adjustment of the vibrational frequencies for the N-methyl-dipentylamine molecular ion. The vibrational frequencies chosen to represent the molecular ion of N-methyl dipentylamine (1) can be grouped as follows. Frequencies associated with the N-CH3 group: 2985 2962 2820 1485 1474 1430 1195 1130 1000 1000 1000 500 400 350 120. Frequencies associated with the carbon skeleton of eae:h pentyl side-chains: 1026 999 926 881 842 762 725 500 491 355 286 188 100 95 82. Frequencies associated with individual carbons in each of the pentyl side-chains: a 2928 2855 1475 1465 1446 1107 13 2933 2853 1462 1317 1274 1209 y 2928 2855 1446 1416 1305 1224 8 2933 2858 1453 1364 1122 1074 E 2962 2961 2883 1467 1460 1053. The frequency alterations for isotopic substitution are constrained by the Teller-Redlich product rule as: 3N-6 IT k=l Primed quantites refer to the isotopically substituted molecule, mi are the masses of the individual atoms, Mis the molecular mass, Ialblc are the moments of inertia around the principal axes, N is the number of atoms in the molecule which has 3N-6 vibrational degrees of freedom if the molecule is non-linear (3N-5 if the molecule is linear). A3. 2 Assuming that l 3lblc = l3'Ib'Ic' and that only K vibrational frequencies are affected by isotopic substitution of I atoms gives: K I u:: ~ u[::r [:·r For compound (2), there are 5 isotopically substituted atoms (I), and the molecular masses are 171 (M) and 176 (M'). Assuming 6 sensitive frequencies at each substituted carbon gives K=12 which gives: 12 w' II k = IT [-+]3/2 [~13/2 k=l rok i=l 171 J = 0.0057 If isotopic substitution affects all frequencies equally, then: = ( 0.0057 )11 12 = 0.6501 For the 6 frequencies associated with the a carbon become: 1904 1856 959 953 940 720 , and the 6 frequencies associated with the E carbon become: 1928 1927 1876 955 950 685 . For compounds (3), (4) and (5), K=6, 1=2, M' =173, therefore: 6 w' 3/2 II k = [~l k=l rok 171 = 0.1272 ⇒ = 0.7092 A3. 3 The frequencies obtained are: (3 2080 2023 1037 934 904 857 y 2077 2025 1026 1004 926 868 6 208020271030967796762. For compound (6), K=6, 1=3, M"=l 74; 6 2 ~l 3/2 g :: - u[+]3'2 [171 = 0.1272 ⇒ = 0.7092 The 6 frequencies associated with the E carbon for (6) are: 1769 1768 1722 876 872 629 . The Teller-Redlich product rule also holds for transition states. As the transition state frequencies used in this study were obtained by lowering the molecular ion frequencies by certain proportions, and these proportional lowerings were applied to both the labelled and unlabelled species, the transition state frequencies were automatically constrained within the Teller-Redlich product rule. A4.l Appendix 4. Estimation of heats of formation for neutral species using Benson's rules. Benson's rules [ 1] can be used to estimate L\H O r298 for a molecule from the contributions of groups. Groups are defined as a polyvalent atom, with more than 2 ligands, together with all of its ligands. The nomenclature for groups is to identify first the polyvalent atom and then its ligands, e.g. C - (C)(H)3 represents a primary methyl group and C - (C8 )(O)(H)2 represents a methylene group between a benzene ring and an oxygen atom. Estimates of L\H 0 f298 were made for 3 compounds; acetaldehyde, benzyl ethyl ether and a,a'-diethoxy p-xylene. Acetaldhyde: Group Contribution / kJ moi- 1 C - (C0 )(H)3 -43 CO- (C)(H) -122 L\H0 r29s -165 A4.2 Benzyl ethyl ether: Group Contribution / kJ moi-I 5 x CB - (H) 69 C8 -C 23 C - (C8 )(O)(H)2 -34 0- (C)2 -97 C - (C)(O)(H)2 -34 C - (C)(H)3 -43 1 gauche interaction 3 ~H0 f298 -113 a,a'-Diethoxy p-xylene Group Contribution / kJ moi- 1 4 x CB - (H) 55 2 x CB - C 46 2 x C - (CB)(O)(H)2 -68 2 x 0- (C)2 -194 2 x C - (C)(O)(H)2 -68 2 x C - (C)(H)3 -85 2 gauche interactions 7 ~Ho f298 -307 REFERENCE 1 S.W. Benson, "Thermochemical Kinetics", 2nd Ed., Wiley-Interscience, New York(1976) A5.l Appendix 5. Optimised ion geometries. The optimised geometries obtained from the calculations performed with the MOPAC package are given below for acetaldehyde, the benzyl ethyl ether radical cation [(18)]+ and its distonic isomer [(31)]+-, the a,a'-diethoxy p-xylene radical cation [(20)]+. and its distonic isomer [(32)]+., the methylene cyclohexa-1,3-diene radical cation [(17)]+. and the 4-ethoxymethyl methylene cyclohexa-1,3-diene radical cation [(25)]+·. The geometries are defined in terms of internal coordinates [1]. For any atom i this consists of an interatomic distance (r) in angstroms from an already defined atom j, an interatomic angle (0) in degrees between atoms i and j and an already defined atom k (k and j must be different atoms) and a torsional angle () in degrees between i, j, k and an already defined atom 1 (I cannot be the same as j or k). The first 3 atoms are defined by 2 interatomic distances and 1 interatomic angle only. For the acetaldehyde molecule; Atom r e J k 1 1 0 2 C 1.221386 3 C 1.516742 124.965851 4 H 1.107351 113.140905 0.155344 3 2 1 5 H 1.109024 109.578703 120.880785 3 2 1 6 H 1.109045 109.567753 -120.566539 3 2 1 7 H 1.111595 114.016444 -179.855551 2 3 4 REFERENCE 1. J.J.P. Stewart, MOPAC manual, QCPE Program no. 455, Indiana University. The benzyl ethyl ether radical cation. The benzyl ethyl ether distonic radical cation. Atom r 6 $ j k l i Atom r 6 $ j k l 1 C 1 C 2 C 1.426761 2 C 1.422360 3 C 1.394680 121. 132224 3 C 1.424783 120.438779 4 C 1.455459 120.360100 -0.676452 3 2 1 4 C 1.393222 120.362979 0,403877 3 2 1 5 C 1.407544 119.517684 -0.318127 4 3 2 5 C 1,501433 122.536063 0.584463 4 3 2 6 C 1.402591 120.471574 o. 100210 5 4 3 6 C 1.404293 121.895231 -0.006419 1 2 3 7 C 1.527668 120.299646 -179.832328 1 2 3 7 C 1,507283 121.054976 -176.793497 6 1 2 8 0 1.392459 109.208204 144.830036 7 1 2 8 0 1,462443 111.353130 78.086708 7 6 5 9 C 1.413362 119.893502 -178.632637 8 7 1 9 C 1.275653 129.202984 71.494306 8 7 6 10 C 1.539979 109.273987 179.160304 9 8 7 10 C 1.508265 119.868432 179.856219 9 8 7 11 H 1.093591 118.937069 179. 984240 2 3 4 11 H 1.092701 117.582006 -179.288898 1 2 3 12 H 1.092930 118.765890 179. 860307 3 4 5 12 H 1.089475 120.016783 -179.298196 2 3 4 13 H 1,092271 121. 143764 179. 890719 4 5 6 13 H 1.090766 119.243185 -179.755995 3 2 1 14 H 1.093185 119.752170 -179.224415 5 6 1 14 H 1.089379 120.983515 179.923176 4 3 2 15 H 1.093546 119.814196 177. 618978 6 1 2 15 H 1. 117334 108.404211 120.362138 5 4 3 16 H 1.126424 112.562517 63. 178424 7 8 9 16 H 1. 116929 108.527523 -124.619858 5 4 3 17 H 1. 122073 111.868552 -56.919957 7 8 9 17 H 1. 119048 112.516154 -37.320160 7 6 5 18 H 1. 121246 110.593437 58.220382 9 8 7 18 H 1. 118245 113.257884 -160.082530 7 6 5 19 H 1. 121399 110.470965 -59.897314 9 8 7 19 H 1. 106971 121.725193 0,134907 9 8 7 20 H 1. 108060 111.865346 60. 937004 10 9 8 20 H 1. 107905 114.003794 -179.786375 10 9 19 21 H 1. 108777 108.752253 -179.901157 10 9 8 21 H 1. 112051 108.610483 58.872629 10 9 19 22 H 1. 107986 111.876235 -60.695812 10 9 8 22 H 1. 112088 108.502010 -58.572123 10 9 19 ~ N The a,a'-diethoxy p-xylene radical cation. The a,a'-diethoxy p-xylene distonic radical cation. i Atom r e q, j k l i Atom r e q, j k l 1 C 1 C 2 C 1.419320 2 C 1.421889 3 C 1.399850 120.900958 3 C 1.434392 120.699104 4 C 1.465450 120.480927 0.055361 3 2 1 4 C 1.400110 119.376688 o. 682734 3 2 1 5 C 1.417337 118.933603 -0.034954 4 3 2 5 C 1.501650 122.897970 0.367473 4 3 2 6 C 1.398767 120.315097 0.248577 5 4 3 6 C 1.402610 122.312017 0.210179 1 2 3 7 C 1.533628 121.547343 -179.635191 1 2 3 7 C 1.507134 121.209614 -177.444092 6 1 2 8 0 1.394663 106.621479 104.390177 7 1 2 8 0 1.462437 111.694490 78. 114287 7 6 5 9 C 1.412449 119.815214 -179.570403 8 7 1 9 C 1.275850 129.214029 67. 162859 8 7 6 10 C 1. 540211 1[}9. 234860 179.770663 9 8 7 10 C 1.508355 119,849829 179. 810628 9 8 7 11 C 1.534854 121.015143 -180,538189 4 5 6 11 C 1.532990 119.497385 179. 136936 3 2 1 12 0 1.394661 106.700805 -107.271399 11 4 5 12 0 1.399027 112.065288 66.601125 11 3 2 13 C 1.412598 119.810131 -179.032649 12 11 4 13 C 1.404315 120.729133 95.242894 12 11 3 14 C 1.540566 109.270522 179.853030 13 12 11 14 C 1.540159 109.515300 177.713309 13 12 11 15 H 1.093467 118.979191 -179.721025 2 3 4 15 H 1,093134 117.329211 -179.290159 1 2 3 16 H 1.092960 119.904930 -179.932132 3 4 5 16 H 1.090170 120,707633 -179. 118925 2 3 4 17 H 1.093000 119. 166514 179.475210 5 6 1 17 H 1,089775 121.778268 179.723656 4 3 2 18 H 1.093205 119, 138767 -178.884193 6 1 2 18 H 1. 117829 108,268515 120.284185 5 4 3 19 H 1. 122942 112.784632 61.168663 7 8 9 19 H 1. 117326 108,321472 -124.983329 5 4 3 20 H 1 .120565 112.474904 -58.892395 1 8 9 20 H 1.119143 112,458898 -37.520504 1 6 5 21 H 1, 121449 110,635698 58.817481 9 8 7 21 H 1, 118051 113, 197203 -160.143137 1 6 5 22 H 1.121194 110,574437 -59.318499 9 8 7 22 H 1.106904 121.775319 -0.090499 9 8 7 23 H 1,108042 111.872601 60,806661 10 9 8 23 H 1, 111598 108.703991 -181,317580 10 9 19 24 H 1, 108800 108.749440 179.980977 10 9 8 24 H 1.112521 108.425227 61.316956 10 9 19 25 H 1. 108472 111,872487 -60,812373 10 9 8 25 H 1. 107915 114,006846 -59.861430 10 9 19 26 H 1, 120727 112,280167 60, 184434 11 12 13 26 H 1.119993 112.907868 -29.783265 11 12 13 27 H 1,122821 112,883697 -59.922377 11 12 13 27 H 1, 122487 106.871023 -146.052604 11 12 13 28 H 1, 121426 110,576524 58.880110 13 12 11 28 H 1, 122125 111.270161 56.653364 13 12 11 29 H 1, 121006 110,622646 -59,245773 13 12 11 29 H 1.123645 111.033766 -61.470218 13 12 11 30 H 1,107570 111.868532 60. 746394 14 13 12 30 H 1, 108327 111,855201 61,096577 14 13 12 31 H 1, 108860 108.769563 179.962952 14 13 12 31 H 1, 109199 108.971774 180.200547 14 13 12 32 H 1. 108049 111.868980 -60.878803 14 13 12 32 H 1. 107818 111,914550 -60.468899 14 13 12 ~ uJ The methylene cyclohexa-1,3-diene radical cation. The 4-ethoxymethyl methylene cyclohexa-1,3-diene radical cation. k l Atom r 0 4> j k l i Atom r 0 4> j l 1 C 1 C 2 C 1.415629 2 C 1.356807 3 C 1.424997 122.364571 3 C 1.460746 120.966318 4 C 1.404427 117.966124 0.771264 1 4 C 1. 352431 120.504823 0.251626 3 2 1 3 2 C 1.499203 123.718596 0.511306 4 5 C 1.505263 123.129143 -0.083712 4 3 2 5 3 2 C 6 C 1.475286 122.670347 -0.087467 1 2 3 6 1.428744 122.207284 -0.990340 1 2 3 C 1 7 C 1.352345 121.670916 179.765476 6 1 2 7 1.397114 121.692613 179.988349 6 2 8 H 1.092385 120.111177 179.942055 1 2 3 8 C 1.534905 120.566372 -178.628844 3 2 1 0 9 H 1.090749 117.614141 -179.785154 2 3 4 9 1.395446 107.057572 75.890341 8 3 2 10 H 1.090687 117.700723 -179.757830 3 2 1 10 C 1.410755 119.982885 162.403287 9 8 3 11 11 H 1.091482 121.406294 179.950937 4 3 2 C 1.539962 109.309946 178.547113 10 9 8 12 H 1. 116916 108.242237 122.400035 5 4 3 12 H 1.094541 118.021077 178.876980 1 2 3 13 H 1. 116800 108.249832 -122.861603 5 4 3 13 H 1.095640 119.700743 -179.965385 2 3 4 14 H 1.088958 123.550026 0.037441 7 6 1 14 H 1.093504 120.968596 -179.384127 4 3 2 15 H 1.089124 123.465238 -0.007303 1 6 5 15 H 1. 118536 107.918743 121. 185226 5 4 3 16 H 1. 118387 107.858408 -124.209629 5 4 3 17 H 1.088989 122.699056 0.048602 7 6 1 18 H 1.088903 122.619176 0.055932 1 6 5 19 H 1.120600 113.124956 41.201278 8 9 10 20 H 1. 122386 111.637221 -78.379213 8 9 10 21 H 1. 121437 110. 744736 57.530748 10 9 8 22 H 1. 121904 110.588905 -60.581889 10 9 8 23 H 1. 108078 111.863789 60.838474 11 10 9 24 H 1. 108852 108.821195 180.017988 11 10 9 25 H 1. 108028 111.862089 -60.746067 11 10 9 ~ :i:,. A6.l Appendix 6. Solution of kinetic schemes using Laplace transformation. Laplace transformation is an operational method for solving linear differential equations with constant coefficients. Both sides of the differential equation are transformed by means of certain functions and operator pairs, resulting in an algebraic equation which may be manipulated by standard algebraic methods. The transform solution must then be inverted ( or inversely tranformed) to obtain the desired solution. To apply the method of Laplace transformation to the solution of a differential equation (or series of differential equations), four distinct steps are required [1]. 1. Express the differential equation in terms of its Laplace transform. 2. Insert the initial conditions. 3. Rearrange the equation to give the transform of the solution. 4. Determine the inverse transform to obtain the solution. Application of this method to problems in kinetics is described in detail elsewhere [2,3]. For a function F(t) and its derivative F'(t), the Laplace transform L{F'(t)} is defined as ~ L{F'(t)} = Jexp(-st) F'(t) dt , 0 which gives L{F'(t)} = -F(0) + s f(s) where F(0) is the value of the function F(t) at t = 0, and f(s) = L{F(t)}, the Laplace transform of F(t). The notation to be used in the solution described here, is obtained as follows [ 1]. Let x = F(t), and at t = 0, x = x0, i.e. F(0) = x0 dx/dt = x1, i.e. F'(0) = x 1. A6.2 Also denote the Laplace transform of x by x, i.e. x = L{x} = L{F(t)} = f(s). Using "dot" notation, i.e. dx/dt = x, L{x} =x L{x} = s:x: - Xo. If F(O) = 0, then L{x} = S:X. Consider the kinetic scheme given in Section 5.8.1 for the stepwise elimination of acetaldehyde from the benzyl ethyl ether radical cation, Scheme 5.9, reproduced below. Ml 1 ) k3(E) ~11 PI TS4 k5(E) M2 ~ ~) ks(E) ~ 12 TS5 P2 Let ki represent the rate coefficients ki(E), then the rate of change in the abundance of each species at time t is given by • Ml= -kl Ml+ k211 . 11. = kl Ml + k5 M2 - ( k2 + k3 + k4) 11 Pl. = k311 M2. = k4 11 + k7 12 - ( k5 + k6 ) M2 12 = k6 M2 - ( k7 + k8 ) 12 . P2=k812 . A6.3 To find the abundance of each species at time t, i.e. Ml, M2, 11, 12, Pl and P2, these differential equations need to be solved. Setting j 1 = k2 .+ k3 + k4, j2 = k5 + k6 and j3 = k7 + k8, and rearranging gives Ml + kl Ml - k2 I1 =0 • 11 + jl 11 - kl Ml - k5 M2 =0 • Pl - k3 11 =0 • M2. + j2 M2 - k4 I1 - k7 12 =0 12. + j3 12 - k6 M2 =0 P2 - k8 12 =0 Transforming these equations and inserting the initial conditions t = 0, Ml(0) = M1 0, 11(0) = 12(0) = M2(0) = Pl(0) = P2(0) = 0, i.e. only Ml is formed initially, and Mio molecular ions are formed per unit time, (S.Ml-M10 )+k1Ml-k2Il =0 (S.11) + jl 11 - kl Ml -k5 M2 = 0 (S.Pl) - k3 I1 = 0 (S.M2) + j2 M2 - k4 I1 - k7 12 = 0 (S.12) + j3 12 - k6 M2 = 0 (S.P2) - k8 12 = 0. Collecting terms; (S + k 1) M 1 - k2 I1 =M10 (1) (S + j 1) I1 - k 1 --M 1 - k5 M2 =0 (2) S.Pl - k3 I1 =0 (3) (S + j2) M2 - k4 I1 - k7 12 =0 (4) (S + j3) 12 - k6 M2 =0 (5) S.P2 - k8 12 =0 (6) A6.4 Equation (5) gives (S + j3) 12 = k6M2 12 = k6M2 (7) (S + j3) Substituting (7) into (4) gives (S + j2) - k6k7 M2 = k4 TI [ l S + j3 (S + j2)(S + j3) - k6k7] M2 = k4 I1 [ (S + j3) -[M2= k4(S+j3) i -I1 (8) (S + j2)(S + j3) - k6 k7 Substituting (8) into (2) gives (S + jl) 11 -kl Ml -r k4 k5 (S + j3) ] (S + j2)(S + j3) - k6 k7 (S + jl)[(S + j2)(S + j3) - k6 k7] - k4 k5 (S + j3)] 11 = kl Ml [ (S + j2)(S + j3) - k6 k7 II - kl [ [(S + j2)(S + j3) - k6 k7] ] Ml (9) (S + jl)[(S + j2)(S + j3) - k6 k7] - k4 k5 (S + j3) Substituting (9) into (1) gives A6.5 (S + kl) Ml -[ kl k2 [(S + j2)(S + j3) - k6 k7 ] Ml = Ml0 (S + jl)[(S + j2)(S + j3) - k6 k7] - k4 k5 (S + j3) ⇒ Ml = M1 0[(S + jl) [(S + j2)(S + j3) - k6 k7] - k4 k5 (S + j3) (S + kl)[(S + jl)[(S + j2)(S + j3) - k6 k7] - k4 k5 (S + j3)] - kl k2[(S + j2)(S + j3) - k6 k7] Let the denominator of this expression be D, which can be expressed as a quadratic equation with negative roots R(l), R(2), R(3) and R(4), i.e. D = (S + R(l))(S + R(2))(S + R(3))(S + R(4)). If there are no zero roots, i.e. R(l), R(2), R(3) and R(4) are non-zero, Ml= [(S + jl)[(S + j2)(S + j3) - k6 k7] - k4 k5 (S + j3)] Ml0 D (10) Substituting (10) into (9) gives 11 = kl [(S + j2)(S + j3) - k6 k7] M1 0 (11) D Substituting (11) into (8) gives M2 = kl k4 M1 0 (S + j3) (12) D Substituting (12) into (7) gives A6.6 12 = kl k4 k6 M1 0 1 (13) D Equation (3) gives Pl = kl k3 M1 0 [(S + j2)(S + j3) - k6 k7] (14) S.D and equation (6) gives P2 = kl k4 k6 k8 M1 0 (15) S.D Six expressions, (10) - (15), for the Laplace transforms have been obtained, and can be expressed in form similar to Ml = M1 0fa(O)S 3 + a(l)S2 + a(2)S + a(3) l [cs + R(l))(S + R(2))(S + R(3))(S + R(4)) where a(O) = 1, a(l) = (il + j2 + j3), a(2) = (jl(j2 + j3) + j2 j3 - k6 k7 - k4 k5) and a(3) = jl(j2 j3 - k6 k7) - j3 k4 k5. Using inverse the Laplace transforms given at the end of this appendix, which were derived from those given in published tables [4,5], !, [exp (-R(n)t) [ ! 0 (-l)m R(n)m a(3-m)] l TI (R(l) - R(n)) hi 1..t-n Similarly, M2 = kl k4 M1 0 l,(exp(-R(n)t) [ j3 - R(n) ]] TI (R(l) - R(n)) ,1: I 1Jln A6.7 The coefficients a(i) now change to a(O) = 1, a(l) = j2 + j3 and a(2) = j2 j3 - k6 k7 11 = kl M1 0 ~ [exp(-R(n)t)fl: (~l)m R(n)m a(2-m) ]] II (R(l) - R(n)) 4- l•I 31 12 = kl k4 k6 Mlo ~rexp(-R(n)t) Itt- 1 ]] II (R(l) - R(n)) ~-,l:~n For P 1 and P2, the denominator changes to S.D, which gives the solutions a different form. Pl = kl k3 M1 0 [ :(2) _ f [exp(-R(n)t) [ ],(-l)m,,_R(n)m a(2 - m) ]] II R(n) L R(n) II (R(l) - R(n)) "': I ~=I hi l~Y\ P2 = kl k4 k6 k8 M1 0 [-~-1__ ~ exp(-R(n)t) ]] II R(n) [R(n) (R(l) - R(n)) n~\ -r & The above solutions to the kinetic scheme hold for the case in which there are no zero roots of the polynomial D. If D has one zero root, then the solutions are based on, Ml = a(O)S3 + a(l)S2 + a(2)S + a(3) S(S + R(l))(S + R(2))(S + R(3)) and become; Ml = Mlo [ a(3) _ f [exp(-R(n)t) [ 1/-l~mR(n)ma(3-m) ]]] I1R(n) L R(n) II (R(l) - R(n)) r-:1 n -1 ~, - Un A6.8 M2 = kl k4 Mlo [-j;- + f [exp(-R(n)t) [ (R(n): j3) ]]] TI R(n) R(n) TI (R(l) - R(n)) rF\ Ln•1 ~, :i..a,V\ 11 = kl Mlo [ a(2) _ f [exp(-R(n)t) [ JJ-l)mR(n)ma(2-m) ]]] TIR(n) L R(n) fI (R(l) - R(n)) I'\= I l.:1 1#"' 12 = kl k4 k6 M1 0 [-1__ exp(-R(n)t) 3 [ ]] 11 R(n) R(n) t(R(l) - R(n)) l'l; I ~ ~ 3 Pl = kl k3 Mlo[ a~2)t + a(l) a(2) L TI R(l) --:it-~-P\:.I 'l•' TIR(n) TIR(n)2 V\:\ "''' + ~ [exp(-R(n)t) [ .t(-l)mR(n)ma(2-m) ]]J L R(n)2 TI (R(l) - R(n)) :l_cl n .. 1 ~n exp(-R(n)t) P2 = kl k4 k6 k8 M1 0 [-~- 3 [ ]] TI R(n) n::\ [. R(n) ~?(1)-R(n)) In practice, the case where D has one zero root arises only if (k3 = 0 OR k5 = 0) AND (k6 = 0 OR k8 = 0). This corresponds to the situation where there are either no products formed, or there is no reaction of M2, i.e. no hydrogen exchange. The computer program written which incororates this solution, program CALKT (listed in Appendix 7), considers the case of zero roots and treats this situation accordingly. In the above solutions there is no allowance for the energy deposition function P(E). Allowance for the energy deposition function is made by defining the energy deposition function such that A6.9 Emax M1 0 = f P(E) dE. 0 The expressions obtained when this is done are given in Section 5.8.1 for the case in which D has no zero roots. The second scheme to be solved is the elimination of acetaldehyde from a.,a.'-diethoxy p-xylene radical cation as described by the scheme below ( Scheme 5.11 ). M 12 TS4 k Pl P2 The solution is worked as above and provides the following results for the case in which the characteristic equation, D, has no zero roots. M = M 0 ! [exp(-R(n)t) [ R2;n) - R(n)(K2 + K3) + K2 K3 ] ] TI (R(l) - R(n)) }:l ~n . 11 = kl M 0 t. [exp(-R(n)t)[ ~3 - R(n)) ]] TI (R(l) - R(n)) 1:1u-n A6.10 12 = k4 M0 !, txp(-R(n)t) [ ~2 - R(n)) ] ] Il (R(l) - R(n)) ];I );#Y' Pl = kl k3 M0 [ ~3 + ~ [ exp(-R(n)t) (R(n) - K3) ]] IlR(n) L R(n) fi (R(l) - R(n)) l'\:I l•I n:1 ~n P2 = k4 k6 M0 [ ~2 + ~ [ exp(-R(n)t) (R(n) - K2) ]] IlR(n) L R(n) n(R(l) - R(n)) n:I l:l n .. , l.fn Where R(l), R(2) and R(3) are the negative roots of D, given as D = S3 + S2(Kl + K2 + K3) + S(Kl(K2 + K3) + K2 K3 - kl k2 - k4 k5 + (Kl K2 K3 - kl k2 k3 - k4 k5 K2 ) and Kl= kl+ k4, K2 = k2 + k3 and K3 = k5 + k6. For the case in which there is one zero root, i.e. only two non-zero roots, in the characteristic equation D, 'l. M=Mo[K2K3 + \ [ (R(n)2 - (K2 + K~ R(n) + K2 K3) exp(-R(n)t) ]] R(l) R(2) L R(n) n (R(l) - R(n)) n:.1 l.:cl 1:tn 11 = kl M0[ _K_3__ (K3 - R(n)) exp(-R(n)t) '1. R(l) R(2) R(n) n (R(l) - R(n)) ]] +t [ )rl "lYn 12 = k4 Mo [-K_2__ (K2 - R(n)) exp(-R(n)t) +[[ -i. R(l) R(2) R(n) Il (R(l) - R(n)) ]] l~• "" I l:#n A6.11 Pl = kl k3 Mo [ (1 + K3 t) K3 (R(l) + R(2)) R(l) R(2) (R(l) R(2))2 + ~ [ (K3- R(~) exp(-R(n)t) ]] L R(n)2 II (R(l) - R(n)) n:::.I };l l~n P2 = k4 k6 M0 [ (1 + K2 t ) K2 (R(l) + R(2)) R(l) R(2) (R(l) R(2))2 + ~ [ (K2- R(:)) exp(-R(n)t) ]] L R(n)2 II (R(l) - R(n)) n~, 1=1 lin The occurrence of zero roots in this solution indicates that no product ions are being formed. The computer program which incorporates this solution, program NKT (listed in Appendix 7), detects the occurrence of zero roots and proceeds accordingly. Allowance for an energy depostion function P(E) is as described above. The solution obtained for the case of no zero roots when the P(E) function is included is given in Section 5.9.1. The transform/original pairs used in this solution were derived from those available from published tables [4,5]. The transform/original pairs used are: TRANSFORM ORIGINAL N M I: b(n) S0 [ I: (-1 )0 b(n) a(i)0 ] exp(-a(i)t) l'\:O ⇒ n:0 M .II (S + a(i)) II (a(l) - a(i)) ,:.\- la: I t[ '1.tn l A6.12 for M ~ N + 1 and a(l):;ta(2):;ta(3) ... :;ta(M), a(i):;tQ 't/ 1, and M N l: b(n) sn .... o :(0) _\ [ c!. (-~ )°;(n) a(i)n ] exp(-a(i)t) ] "" .II a(i) a(i) II (a(l) - a(i)) S ll. (S + a(i)) ~.,., L. l::• L&I 1,~i.. for M ~ N and a(i):;ta(2):;ta(3) ... :;ta(m), a(i):;tQ 't/ 1. REFERENCES 1. K.A. Stroud, "Laplace Transforms", Halsted Press, New York (1973). 2. C. Capellos and B.H. J. Bielski, "Kinetic Systems", Wiley-Interscience, New York (1972). 3. N.M. Rodiguin and E. N. Rodiguina, "Consecutive Chemical Reactions", D. Van Nostrand, Princeton (1963). 4. P.A. McCollum and B.F. Brown, "Laplace Transform Tables and Theorems", Holt, Rinehart and Winston, New York (1965). 5. E.D. Rainville, "The Laplace Transform", Macmillan, New York (1963). A7.1 Appendix 7. Computer programs CALKT and NKT. C PROGRAMCALKT CALLCALCON(TIME) CHARACTER*12NIH 981 CONTINUE DOUBLEPRECISION TIME,TIN1,TIN2,TIN3 CLOSE(UNIT:10) DOUBLEPRECISION EN(1000),RKE(8, 1000),SYM(8) TYPE *,' AFNP RUNNING' COMMON/RATES/ ILIM,EN,RKE CALL AFNP TYPE 6002 CLOSE(UNIT :20) 6002 FORMAT(' MAXENERGY = ',$) 500 TYPE •,• PLOTTING ' ACCEPT*,ENMAX CALL PLOTIT ILIM:-1 END DO 10 I:1, 1000 SUBROUTINECALCON(TIME) ILIM : ILIM + 1 DOUBLEPRECISION RKE(B, 1000),EN(1000),CM1,CM2,CA1,CA2,CB1,CB2, READ(11,*,END:999)EN(l),(RKE(1,I)) 1R(4, 1000),TIME,CM1P(4, 1000),CM2P(4, 1000),CA1P(4, 1000), IF(EN(I),GT.ENMAX)GOTO999 2CA2P(4, 1000),CB1P(5,1000),CB2P(5, 1000),RD(4,1000),EXT 10 CONTINUE COMMON/RATES/ ILIM,EN,RKE 999 CLOSE(UNIT:11) COMMON/CPAR/ CM1P,CM2P,CA1P,CA2P,CB1P,CB2P,R,RD IF(ILIM.LE.O) STOP 'NO DATA ' DO 1 IEN:1, !LIM DO 20 I:2,8 CM1:0. J:!+10 CM2:0. JHI = 0 CA1:0. DO 12 IK:1, !LIM CA2:0. JHI = JHI + 1 CB1:0. READ(J,*,END:20)REN,RAKE CB2=0. 14 IF ( REN.LT.EN(JHI) ) THEN NP = 4 READ(J,*,END:20) REN,RAKE IF(R(4,IEN).EQ.O,) NP = 3 GOTO14 IF(TIME.EQ.O.) THEN ELSE IF ( REN.GT.EN(JHI).AND.JHI.LE.ILIM ) THEN CM1 :1. RKE(I,JHI) = 0.0 GOTO900 JHI: JHI + 1 END IF GOTO14 DO 100 I:1, NP ELSE EXT:DEXP(-R(I,IEN)*TIME) RKE(I,JHI):RAKE CM1:CM1+CM1P(I,IEN)*EXT END IF CM2:CM2+CM2P(I,IEN)*EXT 12 CONTINUE CA1:CA1+CA1P(I,IEN)*EXT 20 CLOSE( UNIT= J ) CA2:CA2+CA2P(I,IEN)*EXT TYPE 8000 CB1:CB1+CB1P(I,IEN)*EXT 8000 FORMAT('SYMMETRY NUMBERS (1-8): ',$) 100 CB2=CB2+CB2P(I,IEN)*EXT ACCEPT*,(SYH(I),I:1,8) IF(NP.EQ.3) THEN DO 777 I:1,8 CM1:CM1P(4,IEN)-CM1 DO 777 J:1,ILIH CM2:CM2P(4,IEN)-CM2 777 RKE(I,J):RKE(I,J)*SYH(I) CA1:CA1P(4,IEN)-CA1 TYPE •,• SETH RUNNING' CA2=CA2P(4,IEN)-CA2 CALL SETH CB1:CB1P(5,IEN)*TIME + CB1P(4,IEN) + CB1 TYPE *,' CALCONRUNNING ' CB2:CB2P(5,IEN)*TIME - CB2P(4,IEN) + CB2 DO 981 TL:-14,-3,0.2 ELSE TIHE:10.**TL CB1=CB1P(5,IEN)-CB1 > :--J N > w :-,J A(1,3-I) A(2,2-I) 1 1 I) I) 1.D-8,1,D-8) 3 11 11 (R(J,IEN)-R(I,IEN)) 1 K5 1 )NP K4 1 30 (R(J,IEN) 1 I)*(R(J,IEN) I) 1) 4 R(I,IEN) 11 11 1 1, I:0,3 J:1,NP I:0,2 J:1,NP J:1,NP I:1,NP I=1,NP I: I:1,4 1./D( 1, 4 HULLER(0,4,G,10000, SORT(G,4) 0):1, ): : 130 10 110 140 120 150 160 30 20 1 1, IF(I.EQ,J)GOTO DO DO D(ID):D(ID)/D(l) R(I,IEN):-G(I) A(1,2):Q2+P2*J1-K4*K5 A(1,3)=Jl*Q2-J3 IF(R(4,IEN),EQ,O, A(2,0):1, D030J:1,NP A(2,1):P2 A(2,2):Q2 NP RD(I,IEN):RD(I,IEN) D( A( DO DO A(1,1}:P2+J1 CONTINUE DO X:0, DO CH2P(J,IEN):X/RD(J,IEN) X:X+((-1) DO CONTINUE X:X+((-1) XPRO:XPR0 CALL DO CH1P(J,IEN):X/RD(J,IEN) X:-R(J,IEN)+J3 CONTINUE X=O. CB2P(J,IEN):CA2P(J,IEN)/R(J,IEN) XPRO: G(I):0.0 DO CALL X2:CB2P(S,IEN) CA2P(J,IEN):1./RD(J,IEN) CA1P(J,IEN):X*CA2P(J,IEN) CB1P(J,IEN):CA1P(J,IEN)/R(J,IEN) CB2P(5,IEN):1,/XPRO 110 130 10 140 120 150 160 707 20 30 1000),EN(lOOO), K4*K5 K5 1 1 K1,K2,K3,K4,K5,K6,K7,K8,RKE(8, ILIM,EN,RKE D CH1P,CH2P,CA1P,CA2P,CB1P,CB2P,R,RD Q1-P3 1 CB2:CB2P(5,IEN)-CB2 SETH 1066)EN(IEN),TIME,CH1,CM2,CA1,CA2,CB1,CB2 IEN) Q2+P2 PRECISION /RATES/ /PARS3/ 3, /CPAR/ 3 1 ID:2,5 ( J3 RD/4000*1,/ IEN:1,ILIM 1 If 1 707 CB1:RKE(1,IEN)*RKE(3,IEN)*CB1 END RETURN CH2:RKE(1,IEN)*RKE(4,IEN)*CH2 CA1=RKE(1,IEN)*CA1 CA2:RKE(1,IEN)*RKE(4,IEN)*RKE(6,IEN)*CA2 WRITE(10, CB2:RKE(1,IEN)*RKE(4,IEN)*RKE(6,IEN)*RKE(8,IEN)*CB2 FORHAT(8(1X,D15,7)) CONTINUE DATA DOUBLE END COMMON DO SUBROUTINE K2:RKE(2,IEN) K3:RKE COMMON COMMON P2:J2+J K7:RKE(7,IEN) P3:K1+J3 K8:RKE(8,IEN) Kl:RKE(l,IEN) J2:K5+K6 K4:RKE(4,IEN) K5:RKE(5,IEN) P1:K1+J1 K6:RKE(6,IEN) D(5):Ql•Q2-Q3*K4*K5 J1:K2+K3+K4 J3:K7+K8 D(2):P2+P1 D(3):Q2+P1*P2+Q1-K4 D(4):P1 D(1):D(2)+D(3)+D(4)+D(5) Q1:Kl*(J1-K2) Q2=J2*J3-K6•K7 00 Q3:K1 1J1,J2,J3,M10,P1,P2,P3,Q1,Q2,Q3,D(5},A(2,3),G(4),RD(4,1000), 3CB1P(5,1000),CB2P(5,1000),X,X2,X3,XPRO 2R(4,1000),CH1P(4,1000),CH2P(4,1000),CA1P(4,1000),CA2P(4,1000), 1 1066 900 x3:x2·•x2 EPS1 DHAX1(EP1, 1, D-12) CB1P(5,IEN):A(2,2)•X2 EPS2 DHAX1(EP2,1,D-20) IF(NP.NE.3)GOTO 1 !BEG = KN + 1 CONLY THREENON-ZERO ROOTS IEND KN + N DO 1000 I:1,NP C CM1P(I,IEN):CH1P(I,IEN)•X2 DO 100 I = IBEG,IEND CM2P(I,IEN):CM2P(I,IEN)•X2 KOUNT= 0 CA1P(I,IEN):CA1P(I,IEN)•X2 COMPUTEFIRST THREEESTIMATES FOR ROOTAS 1000 CA2P(I,IEN):CA2P(I,IEN)•X2 C RTS(I)+,5, RTS(I)-,5, RTS(I). CH1P(4,IEN):CH1P(4,IEN)•X2 H = • 5 CM2P(4,IEN):CH2P(4,IEN)•X2 RT = RTS(I) + H CA1P(4,IEN):CA1P(4,IEN)•X2 ASSIGN 10 TONN CA2P(4,IEN)=CA2P(4,IEN)•X2 GOTO70 CB2P(4,IEN):(R(1,IEN)•(R(2,IEN)+R(3,IEN))+R(2,IEN)•R(3,IEN)) 10 DELFPR= FRTDEF CB1P(4,IEN):A(2, 1)•XPRO - A(2,2)•CB2P(4,IEN) RT : RTS(I) - H CB1P(4,IEN):CB1P(4,IEN)•X3 ASSIGN 20 TONN CB2P(4,IEN):CB2P(4,IEN)•X3 GOTO70 CONTINUE 20 FRTPRV= FRTDEF RETURN DELFPR: FRTPRV- DELFPR END RT = RTS(I) C THIS SUBROUTINEFINDS THE ROOTSOF A POLYNOMIAL. ASSIGN 30 TONN C THE SUBROUTINEUSES HULLER'S METHODAND IS BASEDON AN GOTO70 C ALGORITHMFROM "ELEMENTARY NUMERICAL ANALYSIS" SECOND EDITION 30 ASSIGN 80 TONN C BY S, D. CONTEAND CARL DE BOOR, MCGRAW-HILLBOOK COMPANY. LAMBDA=-.5 C PAGE78 COMPUTENEXT ESTIMATE FOR ROOT C 40 DELF: FRTDEF- FRTPRV C KN NUMBEROF ROOTPREVIOUSLY COMPUTED, NORMALLY ZERO, DFPRLM: DELFPR•LAMBDA C N NUMBEROF ROOTSDESIRED, NUM: -FRTDEF*(1. + LAMBDA)•2 C RTS = AN ARRAYOF INITIAL ESTIMATESOF ALL DESIREDROOTS; GG : (1. + LAMBDA*2,)1 DELF - LAMBDA•DFPRLH C SET TO ZERO IF NO BETTERESTIMATES ARE AVAILABLE. SQR : GG*GG+ 2.*NUM*LAMBDA•(DELF- DFPRLM) C HAXIT : MAXIMUMNUMBER OF ITERATIONSPER ROOTALLOWED, IF (SQR,LT.O,) SQR : 0. C EP1 = RELATIVEERROR TOLERANCE ON X(I), SQR : DSQRT(SQR) C EP2 = ERRORTOLERANCE ON F(X(I)). DEN = GG + SQR C FUNPOL: FUNPOL(Z,FZ) IS A SUBROUTINEWHICH, FOR GIVEN Z, IF (SQR.LT.O.) DEN = GG - SQR C RETURNSFZ = F(Z). IF (DABS(DEN).EQ,O.) DEN : 1. C LAMBDA= NUH/DEN C FRTPRV= FRTDEF C DELFPR = DELF SUBROUTINEHULLER(KN,N,RTS,HAXIT,EP1,EP2) H = H*LAMBDA DOUBLEPRECISION RTS(N) RT : RT + H DOUBLEPRECISION RT,H,DELFPR,FRTDEF,LAHBDA,DELF,DFPRLM,NUH, IF (KOUNT.GT,HAXIT)THEN 1DEN,GG,SQR,FRT,FRTPRV,EP1,EP2 TYPE 11074,KOUNT C 11074 FORMAT(' NOTENOUGH ITERATIONS: ',I,' USED') C GOTO100 C INITIALISATION END IF > :...J ~ C SUMI2=0.0 70 KOUNT: KOUNT+ 1 SUMPl:O.O CALL FUNPOL(RT,FRT) SUMP2:0.0 FRTDEF : FRT IF(LOCK.EQ.O)GOTO 4 IF (I.LT .2) GOTO75 2 READ(10,1066,END:999)X,TIME,M1,M2,l1,I2,P1,P2 DO 71 J = 2,I 1066 FORMAT(8(1X,D15.7)) DEN : RT - RTS(J-1) 4 LOCK:-1 IF (DABS(DEN).LT.EPS2) GOTO79 IF(TIME.GT.OTIME)GOTO 3 71 FRTDEF : FRTDEF/DEN SUMM1:SUMM1+M1 75 GOTONN SUMM2:SUMM2+M2 79 RTS(I) : RT + .001 SUMI1:SUMI1+I1 GOTO SUMI2:SUMI2+I2 CHECKFOR CONVERGENCE SUMP1:SUMP1+P1 GOTO 80 IF (DABS(H).LT.EPS1•DABS(RT)) 100 SUMP2:SUMP2+P2 IF (DMAX1(DABS(FRT),DABS(FRTDEF)).LT.EPS2) GOTO100 GOTO2 CHECKFOR DIVERGENCE 3 SUMX:100./(SUMM1+SUMM2+SUMI1+SUMI2+SUMP1+SUMP2) IF (DABS(FRTDEF).LT. 10.•DABS(FRTPRV)) GOTO40 SUMM1:SUMX•SUMM1 H = H/2. SUMM2:SUMX•SUMM2 LAMBDA: LAMBDA/2. SUMI1:SUMX•SUMI1 RT = RT - H SUMI2:SUMX•SUMI2 GOTO70 SUMP1:SUMX•SUMP1 100 RTS(I) = RT SUMP2:SUMX•SUMP2 RETURN LOCK:O END WRITE(20,•)0TIME,SUMM1,SUMM2,SUMI1,SUMI2,SUMP1,SUMP2 SUBROUTINEFUNPOL(Z,FZ) IF(LOK.GT.O)GOTO 998 DOUBLEPRECISION Z,FZ,D(5) GOTO 1 COMMON/PARS3/ D 999 LOK:999 FZ:D ( 1 ) GOTO 3 DO 10 J:2,5 998 RETURN FZ=FZ•z + D(J) END 10 CONTINUE SUBROUTINESORT(A,N) RETURN DOUBLEPRECISION A(N),T END DO 1 I=2,N SUBROUTINEAFNP L=I DOUBLEPRECISION OTIME,M1,M2,I1,I2,P1,P2,X,TIME,SUMM1,SUMM2, 2 M=L-1 1SUMX,SUMI1,SUMI2,SUMP1,SUMP2 IF(A(M).LE,A(L)) GOTO 1 LOK:-1 T=A(M) LOCK:-1 A(M):A(L) READ(lO, 1067)X,TIME A (L ):T 1067 FORMAT(2(1X,D15.7)) IF(M.EQ. 1) GOTO 1 REWIND(UNIT:10) L:M OTIME:TIME GOTO2 SUMM1:0.0 CONTINUE SUMM2:0.0 RETURN > SUMil:O.O END :....i VI > :...i O'I +2)) (ITTY Z ll"J',90.,18) +1), S',0.,9) ,Z(ITTY / ') 1) 90, 1.0) 1) O., '#SCRAMBLING/ 1 1, ',-1,11.,0,,TIHE(ITTY+1),TIME(ITTY+2)) ', 10.,ITTY, 1 1.0) 10.,ITTY, 13,0, 13.0, 13,0, 0,,' 0 ):-14,000 10, (0., ):0, )= +2 AXIS(0,,0., LINE(TIME,Z,ITTY,1,0,' NGRAPH(1, AXIS SCALEC(Z, CLOSE ORIGNI(0.5,0.2) SYMBEL(4.,0.,0,2,'8TIME SYMBEL(0,,3,,0,2, SCALEC(TIME, ORIGNI(1.0, (ITTY RETURN Z(ITTY+1 Z CALL END TIME(ITTY+1 CALL CALL CALL CONTINUE CALL CALL CALL CALL CALL CALL CALL TIME(ITTY+2):1,000 WRITE(2,•)TIME(I),Z1,Z2,Z(I) 8 ),X6(l) ) +2) ),X5(I ITTY THEN ),X4(I )) !U"J',90.0,18) + 1 ) , X 1 ( 1 X , ) 1 + ),X3(l 8 ITTY ):0, S',D.,10) ) (I / 1 ') ') ') ') ') , X 1 ( 1 X , ITTY:ITTY-1 Z GOTO ),X2(I ),EQ,XS(I-1 (I 90, , O,' 1.0) 1) 1, 1,0,' 1,0,' ),Xl '&#ABUNDANCE/ 10, 1, ITTY, ',-1,11,,0,,-14,0,1.0) ),OR,X5(I ITTY, ITTY, 10,,ITTY,1) 1.0) , ' ' , 10,, X4, 13,0, 13.0, 13,0, 0. , PLOTIT X1(200),X2(200),X3(200),X4(200),X5(200),X6(200) TIME(200),Z(200) 0. ( (TIME, 1,200 l:1,ITTY NGRAPH(1, I: AXIS(0.,0.,' SYMBEL(4.,0.,0.2,'&0TIME AXIS SCALEC(TIME, LINE(TIME,Xl,ITTY,1,0,' I:ITTY,2,-1 ORIGNI(0,5,0,2) ORIGNI(1,0, LINE(TIHE,X2,ITTY,1,0,' LINE(TIHE,X3,ITTY, SYMBEL(0,,3,,0,2, LINE LINE(TIHE,X5,ITTY,1,0,' LINE(TIME,X6, SCALEC(Xl, :ITTY-1 IF 1 334 8 lTTY=I ITTY DIMENSION DIMENSION DO READ(20,•,END:2)TIME(I CALL DO SUBROUTINE X2(ITTY+1):X1(ITTY+1) X3(ITTY+2):X1(ITTY+2) X4(ITTY+2):Xl(ITTY+2) X5(ITTY+1):Xl(ITTY+1) CALL X5(ITTY+2):X1(ITTY+2) CALL X6(ITTY+1):Xl(ITTY+1) X6(ITTY+2):X1(ITTY+2) CALL Z(I):100.•Z2/(Z1+Z2) CALL CALL TIME(ITTY+1):-14,000 CALL CALL X2(ITTY+2):Xl(ITTY+2) DO CALL X3(ITTY+1):Xl(ITTY+1) CALL X4(ITTY+1):Xl(ITTY+1) IF(X6(I).EQ.X6(I-1 TIME(I):ALOGlO(TIME(I)) END Z1:(X5(I)-X5(I-1)) Z2:(X6(I)-X6(I-1)) TIHE(ITTY+2):1,000 CALL CALL CALL CALL CALL 1 334 2 > -.J :--J 125) THEN ) ,5.E3) ,2.) ,o. RAKE ,O. ,O. ',$) 10, ,90, 1),EQ.0,0 ,JHI) ,-6, (1-6): (I ,4,8, 6 'P(E)',-4,8,,90,,0.,0. RKE(K,J) l,ILIM 1, 300 RKE 1 : : : EN(J) ILIH-1 'K(E)' 'ENERGY' : I K 125) = NUMBERS 22 I+1 1,, 1) 1) 1) 1,, P(E) 24 24 1,)GOTO = RUNNING J ) ILIM DO 8KE(K,I) J DO GOTO EN(I) AND 5E3, ILIM 6 6 FUNCTION (SYM(I},I:1,6) 1, 1, SYHHETRY SETH 1 •, I: I= J:1,ILIM J:1,ILIM J:1, UNIT: 1 PLOT(O,,O,, NGRAPH(l,12,,10,,1.) PLOT(XEN,YRK,IPEN) PLOT(O.,O., PLOT(O,,O,, AXIS(2.5E3, AXIS(5.25E4,0,0625, AXIS(2. P(E) K(E)'S 8000 SETPE(ILIM,PEN) ORIGNI(0.5,0,5) SCALE(5.E3,2,) SCALE(5.E3,0, RKE(5,1),EQ,O,O.AND,RKE(6, IF IF 320 300 777 300 777 IPEN:1 ACCEPT DO IF(RKE(I,J),LT, YRK:1,+DLOG10(RKE(I,J)) RKE(I,J}:RKE(I,J)*SYM(I) XEN:2,5E3+EN(J) DO CONTINUE DO FORMAT(' PLOT IPEN:O DO DO IF( CALL CALL CALL CALL CALL CALL CALL !PEN: CALL CONTINUE CALL END CALL CALL CALL TYPE•, TYPE CONTINUE CLOSE( END EVALUATE 12 777 300 C 8000 24 20 22 C 'FOR014.DAT', 1000) THEN 1000) ) REN,RAKE 0,0 1000),CP(2,4, = + 1 1000),RM 1 JHI 14 14 : DATA READ(J,*,END:20) RKE(I,JHI) JHI GOTO GOTO 'FOR012.DAT', 'FOR013.DAT', 1000),CI(2,3, 999 THEN NO ',$) ) I PE(1000) EN(1000),RKE(6, CM(3, TIME,TIN1,TIN2,TIN3,RG(3, IEN,EN,RKE RG,RK2,RK3 PE CM,CI,CP STOP NOM(6) 'FOR016.DAT'/ + 1 1000 REN,GT.EN(JHI).AND.JHI.LE,ILIM + 1 6 1 PEFUN,EMAX: NKT PEN ( PRECISION PRECISION PRECISION PRECISION /RATES/ /PARS2/ •,PEN,EHAX /PARS1/ ILIM /PEFUN/ E-6 I:1, I:2 IK:1,ILIH JHI 0 1. SYH(6) NOM/'FOR011.DAT', 6000 REN.LT.EN(JHI) IF : = 10 ( 12 20 PROGRAH REAL DOUBLE INTEGER DOUBLE DOUBLE DOUBLE COMMON CHARACTER•12 COMMON DATA ILIM: COMMON IF(EN(I),GT,EMAX)GOTO COHMON ILIM:-1 FORMAT(' ACCEPT READ(11,•,END:999)EN(I),(RKE(1,I)) TYPE DO J:!+10 TIN1:0.0 JHI DO TIN2: TIN3:1.E-8 JHI READ(J,•,END:20)REN,RAKE OPEN(UNIT:11,FILE:NOM(1),STATUS:'OLD') CLOSE(UNIT:11) IF(ILIH,LE,0) DO IF CONTINUE OPEN(UNIT:J,FILE:NOH(I),STATUS:'OLD') ELSE ELSE -'FOR015.DAT', C C 10 6000 14 999 XX:2.5E3+EN(J) REAC:0,0 YY=0.0625+PE(J) INT1:0.0 CALL PLOT(XX,YY,IPEN) INT2:0.0 IPEN:O PRD1:0.0 320 CONTINUE PRD2:0.0 DO 30 IK= 1, ILIM DO 100 l:1,3 WRITE(9,1000)EN(IK),(RKE(J,IK),J:1,6) REAC:REAC+CM(l,IEN)*EXPG(l) 1000 FORMAT(7(2X,E16.9)) INT1:INT1+Cl(1,I,IEN)*EXPG(l) ,CALL SETM(IK,RM) INT2:INT2+Cl(2,I,IEN) 1 EXPG(l) 30 CONTINUE 100 CONTINUE TYPE •,• CALCONRUNNING 1 INT1:RKE(1,IEN)*INT1 1 PE(IEN) DO 90 RETIME:-14.,-3.,0. 1 INT2:RKE(4,IEN) 1 INT2*PE(IEN) DO 90 IEN:1,ILIM IF(RG(3,IEN).NE.O.O)THEN TIME:10.**RETIME DO 200 l:1,3 90 CALLCALCON(TIME) PRD1:PRD1+CP(l,I,IEN) 1 EXPG(I) CLOSE(UNIT:10) 200 PRD2:PRD2+CP(2,I,IEN)*EXPG(I) TYPE•, 1 AFNP RUNNING PRDl:RKE(l,IEN)*RKE(3,IEN) 1 (PRDl+CP(1,4,IEN)) CALL AFNP PRD2:RKE(4,IEN)*RKE(6,IEN) 1 (PRD2+CP(2,4,IEN)) CLOSE(UNIT:20) GOTO900 TYPE •, 1 PLOTIT RUNNING1 E~E CALL PLOTIT PRD1:(l.+RK3(IENJ*TIME) 1 CP(1,l,IEN) + CP(1,2,IEN) STOP I RUNCOMPLETED 1 PRD2:(1,+RK2(IEN) 1 TIME)1 CP(2,1,IEN) + CP(2,2,IEN) END DO 300 1:1,2 SUBROUTINECALCON(TIME) J:I+2 DOUBLEPRECISION TIME,REAC,INT1,INT2,PRD1,PRD2,SUM,EXPG(3) PRDl:PRDl+CP(l,J,IEN)*EXPG(l) DOUBLEPRECISION EN(1000),RKE(6,1000) 300 PRD2:PRD2+CP(2,J,IEN) 1 EXPG(l) DOUBLEPRECISION CM(3, 1000),Cl(2,3, 1000),CP(2,4,1000) PRD1:RKE(l,IEN) 1 RKE(3,IEN) 1 PRD1*PE(IEN) DOUBLEPRECISION RG(3, 1000),RK2(1000),RK3(1000) PRD2:RKE(4,IEN) 1 RKE(6,IEN) 1 PRD2*PE(IEN) DOUBLEPRECISION PE(1000) GOTO900 COMMON/RATES/ IEN,EN,RKE END IF COMMON/PARS1/ CM,Cl,CP 900 WRITE(lO, 1066)EN(IEN),TIME,REAC,INT1,INT2,PRD1,PRD2 COMMON/PARS2/ RG,RK2,RK3 1066 FORMAT(7(2X,Dl7,10)) COMMON/PEFUN/ PE RETURN IF(TIME.EQ.O.O)THEN END REAC:1.0DO SUBROUTINESETM(IK,RM) INT1:0.0 DOUBLEPRECISION RK1,RK12,RK23,RK45,XO,X1,X2,X3,X4,X5 INT2:0.0 DOUBLEPRECISION RK2G1,RK2G2,RK2G3,RK3G1,RK3G2,RK3G3 PRD1:0.0 DOUBLEPRECISION RG1K2,RG1K3,RG2K2,RG2K3,G(3),A(4) PRD2:0.0 DOUBLEPRECISION EN(1000),RKE(6, 1000),RM,AVA,AZY GOTO900 DOUBLEPRECISION CM(3, 1000),Cl(2,3, 1000),CP(2,4,1000) END IF DOUBLEPRECISION RK2(1000),RK3(1000),RG(3, 1000) DO 20 l:1,3 COMMON/RATES/ IEN,EN,RKE EXPG(l):RG(l,IEN)*TIME COMMON/PARS1/ CM,CI,CP IF(EXPG(l).GT.-709)EXPG(l)=DEXP(-EXPG(l)) COMMON/PARS2/ RG,RK2,RK3 20 CONTINUE COMMON > /PARS3/ A ;---1 00 > ;-...J "° PROBLEM' X4) X4) 1 X1) 1 X1) X5) 1 1 1 'ROOTS 100 RG(1,IK):RG(3,IK) RG(3,IK):0,0 RG(3,IK):0,0 RG(2,IK):RG(3,IK) STOP GOTO RG2K3/X4 RG1K3/X5 1 RK2(IK)/(X1 1 1 X2 RK3(1K) 1 RG(2,IK) 1 X2 1 1 lf(RG(2,IK).EQ,0,0)THEN lf(RG(l,IK).EQ,O.O)THEN 1,IK):-CI(2,1,IK)/RG(1,IK) 900 IF RETURN lf(RG(3,IK).EQ,0,0)THEN RG2K2:RG(2,IK)-RK2(IK) RG2K3:RG(2,IK)-RK3(1K) Xl:RG(1,IK) RK23:RK2(IK) RG1K2:RG(1,IK)-RK2(1K) RG1K3:RG(1,IK)-RK3(IK) ELSE ELSE END X2:RG(1,IK)-RG(2,IK) X3:RG(1,IK)+RG(2,IK) X4:RG(2,IK) X5:-RG(1,IK) ELSE CM(1,IK):-RG1K2 CM(2,IK):-RG2K2 CP(2, Cl(1,1,IK):RG1K3/X5 Cl(1,2,IK):RG2K3/X4 Cl(1,3,IK)=RK3(IK)/X1 CP(2,3,IK):-CI(2,3,IK)/RG(3,IK) CP(2,4,IK):RK2(IK)/XO Cl(2,1,IK):RG1K2/X5 CI(2,2,IK):RG2K2/X4 Cl(2,3,IK):RK2(IK)/X1 CP(1,1,IK):1,/X1 CP(1,2,IK):-X3*RK3(IK)/(X1 CP(1,3,IK):RG1K3/(RG(1,IK) CP(1,4,IK):-RG2K3/(RG(2,IK) CP(2,1,IK):1,/X1 CP(2,2,IK)=-X3 CP(2,3,IK):RG1K2/(RG(1,IK)*X5) END GOTO CM(3,IK):RK23/X1 CP(2,4,IK):-RG2K2/(RG(2,IK) CP(2,2,IK):-CI(2,2,IK)/RG(2,IK) 100 900 999 D-10) 1. K) I D-10, 0.25 1, 1 ( 1. (RG(3,IK)-RG(2,IK)) (RG(3,IK)-RG(1,IK)) (RG(2,IK)-RG(3,IK)) 1 1 1 /RG 1,IK) IK) 50000, 999 1, G, 1, Cl(1,2,IK) CI(1,3,IK) Cl(1, 3, ( 1 1 1 0, ( ):-Cl )/AlY IK l:1,3 IK):-G(l) IK):-G(3) 1,IK):RK2G1/X1 MULLER ):1,0 ):A(l 10 1, 1, 1, 1 ( RK1:RKE(1,IK)+RKE(4,IK) RK2(1K):RKE(2,IK)+RKE(3,IK) RK3(IK):RKE(5,IK)+RKE(6,IK) RK12:RKE(1,IK)*RKE(2,IK) RK45:RKE(4,IK)*RKE(5,IK) A( A(2):RK1+RK2(IK)+RK3(IK) RG(1, A(3):RK1*(RK2(1K)+RK3(IK))+(RK2(IK)*RK3(IK))-RK12-RK45 A(4):RK1*RK2(1K)*RK3(1K)-RK12*RK3(IK)-RK45*RK2(IK) AZY:(A(l)+A(2)+A(3)+A(4)) A(l IF(XO,EQ,0,0)GOTO A(2):A(2)/AlY A(3):A(3)/AZY A(4):A(4)/AlY DO RG(2,IK):-G(2) RG(3, RK2G1:RK2(IK)-RG(1,IK) RK2G2:RK2(IK)-RG(2,IK) CONTINUE X1:(RG(2,IK)-RG(1,IK)) X2:(RG(1,IK)-RG(2,IK)) X3:(RG(1,IK)-RG(3,IK)) RK2G3:RK2(IK)-RG(3,IK) RK3G1:RK3(IK)-RG(1,IK) RK3G2:RK3(1K)-RG(2,IK) RK3G3:RK3(IK)-RG(3,IK) CALL XO:RG(1,IK)*RG(2,IK)*RG(3,IK) G(I):0,0 Cl(1,1,IK):RK3G1/X1 CI(1,2,IK):RK3G2/X2 CI(1,3,IK)=RK3G3/X3 CI(2, CI(2,2,IK):RK2G2/X2 CI(2,3,IK):RK2G3/X3 CM(1,IK):RK2G1 CM(2,IK):RK2G2 CM(3,IK):RK2G3 CP CP(1,2,IK):-CI(1,2,IK)/RG(2,IK) CP(1,3,IK):-Cl(1,3,IK)/RG(3,IK) CP(1,4,IK):RK3(IK)/XO 10 20 > 0 - :-J 100 70 100 40 NN 100 75 79 GOTO GOTO GOTO GOTO GOTO GOTO GOTO GOTO GOTO RETURN USED') 1 DFPRLM) ',I, - LAMBDA*DFPRLM 1, - SQR = - ITERATIONS DEN O. GG 10.•DABS(FRTPRV)) THEN LAMBDA)*2 = ROOT + FRTPRV DEN= SQR FOR ,001 + 1 ENOUGH 2,*NUM*LAMBDA*(DELF - 11074,KOUNT + RTS(J-1) SQR LAMBDA*2.)*DELF H 2,I NOT TONN RT LAMBDA/2. FRTDEF/DEN - FRT + FRTDEF DELF NUM/DEN + DELFPR*LAMBDA -.5 (I.LT.2) - = KOUNT + H = = =RT+ = = TYPE : : = = FRTDEF 80 RT = J DSQRT(SQR) -FRTDEF*(l. GG GG*GG ESTIMATE RT IF FUNPOL(RT,FRT) RT (1. = = H/2, ff = = H*LAMBDA = (DMAX1(DABS(FRT),DABS(FRTDEF)).LT.EPS2) (DABS(FRTDEF).LT. (DABS(H).LT.EPS1*DABS(RT)) = (DABS(DEN).LT.EPS2) DIVERGENCE (KOUNT.GT,MAXIT) CONVERGENCE 71 (SQR.LT.O.) (DABS(DEN).EQ,0,) = (SQR.LT.O.) = = : FORMAT(' NEXT RT RTS(I) IF LAMBDA IF IF RTS(I) H DO DEN IF FRTDEF IF END KOUNT FRTDEF CALL RT IF FRTPRV DELFPR H IF DEN IF LAMBDA= SQR NUM: DELF DFPRLM SQR ASSIGN GG LAMBDA FOR FOR 100 80 71 79 CHECK CHECK 75 11074 70 40 30 COMPUTE EDITION Z, AN COMPANY. ZERO. ROOTS; ON 70 70 70 GIVEN SECOND BOOK FOR GOTO GOTO GOTO BASED AVAILABLE. ALLOWED. DESIRED IS NORMALLY. ARE POLYNOMIAL. ANALYSIS" ROOT ALL WHICH, AND A OF PER OF MCGRAW-HILL X(Il, AS COMPUTED, METHOD ESTIMATES ON NUMERICAL ROOT ROOTS SUBROUTINE BOOR. l. ESTIMATES A F(X(I)). FOR DE TijE ITERATIONS BETTER RTS(I). IS F(Z DESIRED. HULLER'S ON PREVIOUSLY NO = OF TOLERANCE 1.D-20) DELFPR CARL RTS(N} RT,H,DELFPR,FRTDEF,LAMBDA,DELF,DFPRLM,NUM, FINDS INITIAL IF FZ - "ELEMENTARY USES ROOT ROOTS AND (EP1,1.D-12) (EP2, ERROR OF ESTIMATES· NUMBER N + H - H OF OF ZERO MULLER(KN,N,RTS,MAXIT,EP1,EP2) IBEG,IEND RTS(I)-.5, FROM ) TOLERANCE + 1 TONN TONN TONN = FRTDEF FRTDEF FRTPRV RETURNS (I TO FUNPOL(Z,FZ) 0 ARRAY PRECISION PRECISION 10 = DMAX1 DMAX1 = KN+ = I KN THREE 30 20 = SUBROUTINE 78 = : RTS RTS(I) RTS(I) NUMBER NUMBER SUBROUTINE RELATIVE AN MAXIMUM ERROR SET .5 S.D.CONTE 100 : = : = FIRST ALGORITHM PAGE THIS BY THE DOUBLE DOUBLE SUBROUTINE !BEG !END: RTS(I)+.5, DELFPR EPS1 EPS2: DO KOUNT RT FRTPRV RT ASSIGN ASSIGN RT= DELFPR ASSIGN KN FUNPOL: RTS: 1DEN,GG,SQR,FRT,FRTPRV,EP1,EP2 EP2: EP1 MAXIT INITIALISATION C C C C C C C C N C C C 1 H C C C C C C 10 C C C COMPUTE 20 C GOTO2 END SUBROUTINEFUNPOL(Z,FZ) 999 LOCK:999 GOTO3 DOUBLEPRECISION Z,FZ,A(4) 998 RETURN COMMON/PARS3/ A FZ:A( 1) END SUBROUTINEPLOTIT DO 10 J:2,4 DIMENSIONTIME(300),Z(300),Z1(300),Z2(300) 10 FZ=FZ*Z + A(J) DIMENSION RETURN X1(300),X2(300),X3(300),X4(300),X5(300) CALL NGRAPH(l,13.0, 13.0, 1.0) . END DO 1 1=1,300 SUBROUTINEAFNP DOUBLEPRECISION OTIME,M,I1,I2,P1,P2,X,TIME,SUHH,SUHX,SUHI1, ITTY:I 1 1SUMI2,SUMP1,SUMP2 REA0(20,*,END:2)TIME(I),Xl(I),X2(I),X3(I),X4(I),X5(I) LOCK:-1 2 ITTY:ITTY-1 READ(10,1067)X,TIHE DO 334 I:1,ITTY 1067 FORMAT(2(2X,017.10)) 334 TIME(l):ALOG10(TIME(I)) REWIND(UNIT:10) CALLORIGNI(0.5,0.2) OTIME:TIME CALL SYMBEL(4.,0.,0.2, '&ULOG\1\0 (TIME/ S )',0.,22) SUHM:0.0 CALL SYMBEL(0.,3.,0.2, 1 &URELATIVEABUNDANCE/ !IAJ',90.0,27) SUMI1:0.0 CALL ORIGNI(l.O, 1.0) SUMI2=0.0 CALL SCALEC(TIME,10.,ITTY,1) SUMP1:0.0 TIME(ITTY+l):-14.000 SUMP2:0.0 TIHE(ITTY+2):1.000 2 REA0(10,1066,END=999)X,TIHE,M,11,I2,P1,P2 X1(ITTY+1):0. 1066 FORMAT(7(2X,017.10)) CALL SCALEC(X1,10.,ITTY+1, 1) IF(TIME.GT.OTIME)GOTO3 X1(ITTY+1):X1(ITTY+2) SUMH:SUHM+H Xl(ITTY+2):X1(ITTY+3) SUMI1:SUMI1+I1 CALL AXIS(O.,O., 1 ',-1,11.,0.,-14.0,1.0) SUMI2:SUMI2+12 CALL AXIS(O.,O.,' ',1,10.,90.,X1(ITTY+1),X1(ITTY+2)) SUHP1:SUMP1+P1 CALL LINE(TIME,X1,ITTY, 1,0, 1 ') SUMP2:SUMP2+P2 X2(ITTY+1):Xl(ITTY+1) GOTO2 X2(ITTY+2):X1(ITTY+2) 3 SUMX:100./(SUHM+SUHI1+SUHI2+SUHP1+SUHP2) X3(ITTY+1):X1(ITTY+1) SUMH:SUMH*SUMX X3(ITTY+2):X1(ITTY+2) SUMI1:SUMI1*SUMX X4(ITTY+1):X1(1TTY+1) SUMI2:SUMI2*SUMX X4(ITTY+2):X1(ITTY+2) SUMP1:SUMP1*SUMX X5(ITTY+1):X1(ITTY+1) SUMP2:SUMP2*SUMX X5(ITTY+2)=X1(ITTY+2) WRITE(20,*)0TIME,SUHM,SUMI1,SUMI2,SUMP1,SUMP2 CALL LINE(TIME,X2,ITTY, 1,0,' 1 ) IF(LOCK.GT.O)GOTO998 CALL LINE(TIME,X3,ITTY, 1,0,' ') SUMM:M CALL LINE(TIME,X4,ITTY,1,0,' ') SUMil:11 CALL LINE(TIME,X5,ITTY, 1,0,' 1 ) SUMI2=12 RLM:(100-Xl(ITTY+1))/X1(ITTY+2) SUHP1:P1 CALL DASHLN(B.0,0.,8.0,RLM,2.5) SUMP2:P2 CALL DASHLN(9.34,0.,9.34,RLM,2.5) CALL DASHLN(9.71,0.,9.71,RLM,2.5) > OTIME:TIME -:-.J DO 7 I= 1,300 Z2(ITTY+1 )=0. 7 Z (I ):0. 21 (ITTY+2}:0. 1 DO 8 I =1 , ITTY, 1 22 ( ITTY+2) :0. 1 IF(X5(I).NE.O.) Z(I):X4(I)/X5(I) CALL AXIS(O.,O.,' ',-1, 11.,0.,TIME(ITTY+1),TIME(ITTY+2)) 8 CONTINUE CALL AXIS(O.,O.,' ',1,10.,90.,0.,0.1) 10 CALL NGRAPH(l,13.0, 13.0, 1.0) CALL LINE(TIME,21,ITTY,1,0,' ') CALL ORIGNI(0.5,0.2) CALL LINE(TIME,22,ITTY,1,0,' ') CALL SYMBEL(4.,0.,0.2, '&#LOG\1\0 (TIME/ S )',0.,22) CALL DASHLN(8.0,0.,8.0,RLM,2.5) , CALL SYMBEL(0.,3.,0.2, '&@I#ON ABUNDANCERATI0',90.,22) CALL DASHLN(9.25,0.,9.25,RLM,2.5) CALL ORIGNI(1.0, 1.0) CALL DASHLN(9.64,0.,9.64,RLM,2.5) CALL SCALEC(TIME,10.,ITTY, 1) CALL CLOSE TIME(ITTY+1):-14.000 RETURN TIME(ITTY+2)=1.000 END CALL SCALEC(Z,10.,ITTY, 1) SUBROUTINESETPE(ILIM,PEN) Z(ITTY+l ):0. DOUBLEPRECISION PE(1000),XX Z(ITTY+2 )= 10. 0 INTEGER PEN CALL AXIS(O., o.,' ',-1, 11., o., TIME(ITTY+1), TIME(ITTY+2)) COMMON/PEFUN/ PE CALL AXIS(O.,O.,' ', 1, 10.,90.,Z(ITTY+1),Z(ITTY+2)) XX:-2.5DO/ILIM CALL LINE(TIME,Z,ITTY,1,0,' ') GOT0(10,20,30)PEN CALL DASHLN(8.0,0.,8.0,RLM,2.5) STOP ' NO P(E) ' CALL DASHLN(9.34,0.,9.34,RLM,2.5) C PIE) = 1.0 CALL DASHLN(9.71,0.,9.71,RLM,2.5) 10 DO 15 J:1, !LIM DO 18 1:2, ITTY 15 PE(J) =1. 0 Z1(I}:X4(I)-X4(I-1) GOTO50 22(I):X5(I)-X5(I-1) C P(E):EXP(-ENERGY) IF(21(I).GT.2M)2M:21(I) 20 DO 25 J:1,ILIM IF(22(I).GT.2M)2M=Z2(I) 25 PE(J):DEXP(XX 1 J) 18 CONTINUE GOTO40 2M: 1. /2M C P(E):EXP(MAXENERGY-ENERGY) ITTYN:ITTY 30 DO 35 J:1,ILIM DO 19 I:2, ITTYN JJ:ILIM+1-J 21 (I}:21 (I)•ZM 35 PE(J):DEXP(XX•JJ) 22(l):22(I)•ZM 40 XX:1.IDEXP (XX) IF(21(I).EQ.O •• AND.Z2(I).EQ.O.)ITTY:ITTY-1 DO 45 J:1,ILIM 19 WRITE(2,•)TIME(I),21(I),22(I) 45 PE(J):PE(J) 1 XX CALL NGRAPH(l,13.0,13.0,1.0) 50 RETURN CALL ORIGNI(0.5,0.2) END CALL SYMBEL(4.,0.,0.2, '&HLOG\1\0 (TIME/ S )',0.,22) CALL SYMBEL(0.,3.,0.2, l'&@RfELATIVE RATE OF FORMATIONOF PRODUCTS',90.,41) CALL ORIGNI(1.0, 1.0) CALL SCALEC(TIME,10.,ITTY, 1) TIME(ITTY+l):-14.000 TIME(ITTY+2):1.000 21 (ITTY+l ):0. ;..J> -N A8.l Appendix 8. Estimation of heats of formation for dis tonic ions. The heat of formation for the distonic ion [(31)]+. can be estimated from the following scheme and relationship [1] : C('l H Affr(distonic ion)= Affr(neutral) - PA(neutral) + ~Hr(H+) - ~Hr(H·) + D(R-H). Affr(neutral) is the heat of formation of the neutral, PA(neutral) is the proton affinity of the neutral, ~Hr(H+) is the heat of formation of the proton, ~Hr(H·) is the heat of formation of the hydrogen atom and D(R-H) is the bond dissociation energy of the R-H bond. For benzyl ethyl ether the following values are obtained [2-6] ; Affr(neutral) -113 kJ moi- 1 PA(neutral) 809 kJ moi- 1 ~Hr(H+) 1537 kJ moi- 1 [4] ~Hr(H·) 218 kJ moi- 1 [5] D(R-H) 377 kJ moi- 1 This gives the heat of formation of the distonic form of the benzyl ethyl ether radical cation [(31))+. as 774kJ moi- 1. A8.2 The value of PA(neutral) was estimated to be the same as that for n-butyl benzene, obtained from [2] as ; PA(n-butylbenzene) = PA(toluene) + 9 kJ moi-1 = 809 kJ moi-1 . The value of PA(toluene) was taken as 800 kJ moi-1 [3]. D(R-H) was estimated to be the same as that for diethyl ether, i.e. D(CH3CH20CH(CH3)-H) [6]. For a,a'-diethoxy p-xylene the same procedure was used with the value of PA(neutral) being estimated as being equal to PA(p-di-n-butylbenzene), estimated as; PA(p-di-n-butylbenzene) = PA(p-xylene) + 2 x 9 kJ moi-1 = PA(toluene) + 6 + 18 kJ moi-1 = 824 kJ moi-1 . For a,a'-diethoxy p-xylene the following values were obtained ; Lllir( neutral) -307 kJ moi- 1 PA(neutral) 824 kJ mol-1 Lllir(H+) 1537 kJ moi- 1 [4] ~Hc(H·) 218 kJ moi-1 [5] D(R-H) 377 kJ moi-1 The value obtained for ~Hr([(32)]+·) using these values was 565 kJ moi- 1. REFERENCES 1. · C. Wesdemiotis, R. Feng and F.W. McLafferty, J. Am. Chem. Soc., l.Ql, 715 (1985). A8.3 2. W.J. Hehre, R.T. Mciver, J.A. Pople and P.v.R. Schleyer, J. Am. Chem. Soc., .2,Q, 7162 (1974 ). 3. A.G. Harrison, "Chemical Ionization Mass Spectrometry", C.R.C. Press, Florida (1983). 4. Natl. Bur. Stand. (U.S.), Tech. Note, No. 270-3 (1968). 5. H.M. Rosenstock, K. Draxl, B.W. Steiner and J.T. Herron, J. Phys. Chem. Ref. Data, Q, Suppl. 1 (1977). 6. R.C. Weast (Ed.), "C.R.C. Handbook of Chemistry and Physics", 58th Ed., C.R.C. Press, Florida (1977).