JANUAR11981
ON THE DERIVATION OF A CREEP LAW FROM ISOTHERMAL BORE HOLE CONVERGENCE
BY
J. PRIJ J.H.J. MENGELERS ECN does not assume any liability with respect to the use of, or for damages resulting from the use of any information, apparatus, method or process disclosed in this document.
Netherlands Energy Research Foundation ECN
P.O. Box 1
1755 ZG Petten INH) The Netherlands Telephone (0)2246 - 6262 Telex 57211 ECN-87
SEPTEMBER 1980
DISPLACEMENT AND EXCHANGE REACTIONS
IN RADIOANALYSIS
BY
P.C.A. OOMS
Proefschrift Vrije Universiteit Amsterdam, 30 september I980
- J
CONTENTS
Page
CHAPTER 1. INTRODUCTION 9
1.1. Classification of separation methods 10 1.7.1. Reaction with a free reagent 10 1.1.2. Displacement 11 1.1.3. Isotopic exchange 12
1.2. Analytical criteria 12 1.2.1. Activation analysis with a spe cific reagent 13 1.2.2. Isotope dilution analysis with a specific reagent 14 1.2.3. Activation analysis with a non specific reagent 14 1.2.4. Isotope dilution with a non-speci fic reagent 14 1.2.5. Isotopic exchange 14
1.3. Selection of procedures 15 1.3.1. Liquid-liquid extraction using a non-specific chelating agent 16 1.3.2. Amalgams 16 1.3.3. Loaded active carton 18 1.3.4. Coulometry 19
1.4. List of symbols 21
1.5. Summary of contents 23
1.6. References 25
CHAPTER 2. APPLICATION OF DISPLACMENT - AND EXCHANGE REACTIONS IN NEUTRON ACTIVATION AND ISOTOPE DILUTION ANALYSIS 27
2.1. Introduction 28
2.2. Calculations of equilibrium concentrations 28 2.2.1. Definitions 28 2.2.2. Theoretical considerations 29 - 6 -
Page
2.2.3. Numerical calculations 31 2.2.3.!. Programmes 31 2.2.3.2. Precision of the calculated q-values 33 2.2.4. Application 34 2.2.4.1. Activation analysis 34 2.2.4.2. Isotope dilution analysis 34
2.3. Experimental 35 2.3.1. Chemicals and apparatus 35 2.3.2. Procedure 35
2.4. Conclusions 36
2.5. References 37
CHAPTER 3. RADIOMETRIC DETERMINATION OF THE EXTRACTION CONSTANTS OF SOME METAL DIETHYLDITHIOCARBAMATES IN THE SYSTEM CHLOROFORM/WATER 38
3.1. Introduction 39
3.2. Theory 40
3.3. Experimental 41 3.3.1. Chemicals and equipment 41
3.3.2. Procedure 42
3.4. Results 42
3.5. Discussion 45
3.6. References 47
CHAPTER 4 CALCULATIONS ON PARTITION EQUILIBRIA IN LT/^JID- LIQUID EXTRACTION SYSTEMS, APPLIED TO THE SEPARATION OF METAL IONS FROM AQUEOUS SOLUTIONS 48
4.1. Introduction 49
4.2. Theory 51 4.2.1. Definitions 51 4.2.2. Theoretical considerations 51 - 7 -
Page
4.3. Calculations 54 4.3.1. Computer programmes 54 4.3.2. Applications in activation analysis 54 4.3.3. Application in isotope dilution analysis 55
4.4. References 56
CHAPTER 5. MULTIELEMENT ISOTOPE DILUTION ANALYSIS BY
MEANS OF RADIOMETRIC TITRATION 71
5.1. Introduction 72
5.2. Principle 73 5.2.1. Definitions 73 5.2.2. Theoretical considerations 73 5.2.3. Optimization of the Kl -values 75 5.3. Experimental 76 5.3.1. Chemicals and equipment 76 5.3.2. The titration vessel 76 5.3.3. Procedure 78
5.4. Results 79 5.4.1. Experiments to check the apparatus 79 5.4.2. Radiometric titrations of some solutions of metal ions 81 5.4.2.1. Titration of a 3.2xl0~4 M Cu(II) solution with a
2 l.Oxlo" M Zn(DDC)2 solution in chloroform 81 5.4.2.2. Titration of a mixed solution of 3.15xl0~6 M Cu(II) and l.OxlO-1 M Pb(II) with
4 2.5xlO~ M Zn(DDC)2 dissolved in chloroform 81
5.5. References 84 CHAPTER 6. ISOTOPE DILUTION ELECTROANALYSIS 85
6.1. Introduction 86
6.2. Principle 86 6.2.1. Procedure I: two electrolytic cells connected in series 86 6.2.2. Procedure II: electroysis in one single cell 87
6.3. Experimental 88 6.3.1. Apparatus 88 6.3.2. Reagents 91 6.3.3. Procedures 91
6.4. Results 92 6.4.1. Results from Procedure I 92 6.4.2. Results from Procedure II 92
6.5. Discussion 93
6.6. References 93
APPENDIX I DESCRIPTION OF THE COMPUTER PROGRAMMES QFORW
AND QPLOT 95
APPENDIX II TEXT OF THE COMPUTER PROGRAMMES QFORW AND QPLOT 111
APPENDIX III DESCRIPTION OF THE COMPUTER PROGRAMME QBACK 139
APPENDIX IV TEXT OF THE COMPUTER PROGRAMMA QBACK 147
APPENDIX V DESCRIPTION OF THE APPARATUS FOR SEMIAUTOMATIC LIQUID-LIQUID EXTRACTION 159
SAMENVATTING 163
CURRICULUM VITAE 1 - 9 -
CHAPTER 1. INTRODUCTION
SUMMARY
The application of displacement and exchange reactions in radioanalysis is discussed. A few cases are selected for further investigation. The contents of the other chapters are suimarized. - 10 -
1.1. Classification of separation methods
Many analytical procedures involve an isolation of the compound of inte rest effected by its transfer to another phase. Usually this implies dissolution of the sample. One might then summarize these determinations by the following scheme: sampling —• dissolution —> separation —•• measurement
The separation procedures, based on reaction between the compound to be determined and a reagent, have been distinghuished into three types accor ding to the "reagents" which are used: a) reaction with a free reagent (section 1.1.1.) b) displacement reac tions (section 1.1.2.) c) isotopic exchange reactions (section 1.1.3.). From the point of view of radioanalysis one should make a second dis tinction by the amount of reagent which may be present in excess or in a substoichiometric quantity. Both situations are met in activation ana lysis. Isotope dilution methods are based on the addition of a sub stoichiometric amount of reagent. In this thesis attention will be restricted to one step separation pro cedures.
2i]>iJK_Reaction_with_a_free_reagent
The reagent R may react with the compound C according to
aR + C t R C (1) where a represents the stoichiometric ratio. An equilibrium constant is defined by equation (2) [RC] a = K (2) [Rf[C] eq
Separations based on adsorption belong to this class, in that sense that a is no longer the stoichiometric ratio, but the (variable) ratio of the amount of adsorbent and the component to be adsorbed. Usually the product K [R] is very large and thus the recovery of C vir tually complete. Examples of this situation are found in the determina tion of mercury in air by adsorption to active carbon |l|, or Mn02 |2J, and that of trace elements in dry biological material by dissolution, - 11 -
dilution and adsorption to active carbon |3|. Adsorption has not been applied in isotope dilution analysis since the reproducibility in taking substoichiometric amounts of the adsorbent in case of very low concentrations of the compound to be adsorbed is very poor. Electrodeposition belongs also to this type of reaction. It has been applied in activation analysis for the determination of |4|. The use of coulometry has been reported for the isotope dilution analysis of Cd |5|. lili^ J)is£lacement
The reagent R is added as R A which is present in a separate phase. A h^s to be displaced by the analyte B; that is:
|RQA + B i RgB + §A (3)
Again one can define an equilibrium constant.
[RHB][A]I 3 = K (4) I eq [R A]°[B] a
There are many applications of displacement reactions in chemical sepa ration. A further classification may be brought about by defining the reagent phase. The most common case is addition of the reagent dissolved in an organic liquid, immiscible with the aqueous phase containing the element(s) to be determined. A survey of the procedures is given in ref. |6|. In radioanalysis this approach is used in activation analysis as well as in isotope dilution methods. In the latter case the reagent should be specific or the interferences should be removed first |7|. In activation analysis specificity is not imperative but it may be so if it is necessary to obtain a radiochemically pure counting aliquot or at least a convenient group separation. Another group of displacement reactions is based on ion-exchange. Again the applications in activation analysis are numerous. A survey may be found in refs. |8| and |9|. The use in isotope dilution has been restricted to the determination of a single species |10|. - 12 -
A third class of displacement reactions is presented by the isomorphous exchange between an aqueous solution and a insoluble salt. The experi ments in this domain belong to the oldest part of radiochemistry |il, 12f. Applications in activation analysis have been reported for the lanthanides |13|. The possibility of separating manganese has been discussed [I4|. This principle cannot be used in isotope dilution.
1.1.3. Isotopic exchange
The interest of this group of separations is restricted to analytical methods with the aid of which isotopes of a nuclide can be disting uished, i.e. radioanalysis and mass spectrometry. It is characterized by the reaction
C + C* t C* + C (5) where the bar denotes the second phase and the asterisk the species with a particular isotopic composition. The equilibrium constant is about equal to unity. Small but distinct differences, known as the isotopic effect, may exist for light elements |l5|. The application of isotopic exchange in radioanalysis is mainly found in activation analysis. The isolation of iodine from irradiated rainwa ter may be cited |l6|. The exchange between an aqueous solution of a lanthanide and one of its insoluble salts, mentioned above, is another example. The interaction between an aqueous solution of a (heavy) metal ion and an amalgam presents a third type of exchange reactions |17|.
1.2. Analytical criteria
The result of a separation can be expressed in terms of the chemical yield q, defined as the proportion isolated from the total amount of the substance to be determined. This quantity is a function of the relative amount of reagent added and thus of the ratio p given by
s - . amount of reagent /Mol/ /^\ a amount of compound / Mol/
Again a stands for the stoichiometric ratio in the compound CR . From the practical point of view one must, from here, make a distinc- - 13 -
tion between activation analysis and isotope dilution as well as be tween a specific and a non-specific reagent. Specificity/selectivity (i.e. the decontamination factor > 1000) of a reagent depends among ot her things on equilibrium constants and the relative amounts of two or more competitive substances. > In activation analysis q should have a constant and large value. In clas sical isotope dilution q is measured experimentally and p is derived from it using the simple relation q = ap. From p the amount of the compound of interest is calculated easily. In more complicated cases of isotope dilution, q = f(p), where f(p) depends on the kind of separation meth- hod !18].
1.2^1•_Activation_analYsis_with_a_sgecific reagent
For an aliquot of the activated element of interest, separated according to the principle described in section 1.1.2., the number of counts ob tained is given by
A = q.a .V.t (7) sp
The sensitivity is thus directly proportional to q. The relative statis tical error consists of combinations of the natural background, B, and the variation of q. If one writes
B = b.t (8) one arrives at
! 2l (9) sp
Invariably the function q = f(p) is quite simple since the reagent used is specific, i.e. K defined by equation (4) is large. One obtains eq q = ap (10)
Consequently the accuracy and precision depends on the uncertainty in the amount of reagent added. - 14 -
1.2.2. Isotope dilution analysis with_a_sgecific reagent
The value of q is obtained by dividing A by A0, the count-rate of the added spike measured under identical conditions. Hence
From equations (10) and (6) it fellows that this is also the relative error in the final result. More complicated cases (imperfect liquid-li quid extractions, adsorption) have been considered in ref. |l8|.
\.2.3. Activation analysis with a_non-s£ecific_reagent
The function q = f(p) now includes the K -values of the compounds as well as the ratios of their masses. This function becomes awkward and implicit. This makes the calculation of Aq/q complicated. In general Aq/q will decrease when p increases. A quantitative descrip tion implies knowledge of the K -values and a computer programme to obtain q = f(p) numerically. With a non-specific reagent, multi-element analysis becomes possible. Then there will be a compton continuum which depends on the q-values of the other radioactive compounds. Thus the specific count-rate of the background will tend to increase with p and so does the term b/q.a (eq. 9). To apply multi-channel spectrometry, it is however common prac tice to choose a high p-value [19[ as this keeps Aq/q low.
As in the previous case, knowledge of all K valuess as well as a com- puter programme is required in order to evalili ate q - f(p) and to calcu late the variation of cq with the K 's. Th'j procedure has not yet been reported in literature. li^i5i_ïS2£22i£_ë5£!}§iSSf
For isotopic exchange one has
cV (,2) q- cV lev c,v, i- c2v2 - 15 -
In activation analysis c^V is taken so large compared to c.V that q is practically equal to unity. In isotope dilution c. follows from (13) and the relative statistical error in c from (14).
V2 1-q (13) Cl ~ C2*V.
(U) c 0 \ q / 1-
For an analysis based on isotopic exchange only, the reaction between the exchangeable element and the reagent must be specific, since other wise displacement reactions can occur.
1.3. Selection of procedures
Simple separation procedures using non-specific reagents combined with y-ray spectrometry may be of advantage in elemental analysis by neutron activation and isotope dilution. To this end the collection of trace elements on active carbon for activation analysis and the isotopic exchange of iodine between two immiscible phases have been developed into routine procedures in the radiochemical laboratory of E.C.N. |l6(. In this thesis the following separation systems are studied in more detail.
Group of reactions Application
neutron activation isotope dilution reaction with a free ———— coulometry reagent displacement reactions liquid-liquid extraction using non-specific chelating reagents isotopic exchange amalgams active carbon, loaded with a metal chelate
The ideas which led to this selection are summarized briefly now. - 16 -
1.3.1. L^uid-liquid extraction_usi^_a_non-specif icchelating agent
From the points of view of quantitative description and practical applicability the technique of substoichiotnetric liquid-liquid extrac tion with a non-specific reagent is quite attractive. The description involves the introduction of conditional extraction constants and pro ceeds along the lines described in ref. |lo|. In activation analysis the effects of large differences between the extraction constants can be counterbalanced or balanced by the addition of carriers. In isotope dilution analysis it is not possible to calculate the concen trations of the various reacting compounds from the data obtained by extraction after a single addition of a substoichiometric amount of re agent, even if radioactive spikes of all compounds are added. The num ber of extractions, each time with a fresh (substoichiometric) amount of reagent should at least equal the number of compounds. This implies the corstruction of a convenient titration vessel equipped with a by pass for measurement after each addition. Besides enabling the actual analysis, computerization of the calcula tions should open up the possibility of optimizing the procedure for activation analysis as well as to arrive at the exact values of the extraction constants from approximate data. li^i^i.Amalgaros
The application of amalgams for the determination of ions in aqueous solution has been discussed in radio-analytical literature |l7J. The system can be described very easily if no net mass transport takes place. One then has:
m S m + c_V (15) s 0
If m •» c V, then q a* 1 s 0 Two aspects complicate this picture a) Mercury and all eler 2nts which are more noble then mercury are col lected quantitatively. b) The amalgam is not stable; it oxidizes rapidly. Thus the value of q will initially rise and then decrease again. This limits the applica- - 17 -
bility of the procedure to activation analysis where a large excess of the reagent can be used. To a first approximation the variation of q with time can be described by
,, -at. -Bt q " -at, The reaction with the amalgam is rapid; that is (1 - e )= I. The oxidation rate of the amalgam mainly depends on the partial pressure of oxygen and the intensity of stirring. It follows that Aq/q = -6t.((At/t)2 + (AB/B)2]* (17) The main source of error is AB/B. From equation (17) the optimal value of Bt may be calculated. For reasons mentioned above, the amalgam procedure gives a chemical yield which varies considerably with the shaking time. It can be applied in activation analysis only, using a large excess of reagent. This is not attractive. Consequently the procedure will not be discussed in de tail in this thesis. Results of experiments with copper amalgam at pH = i.l * 0.1 and 23 t 1 C using Cu (T^= J2.8h) as the radiotracer can be summarized as follows. -1 Composition of Conditions B (in min ) amalgam 10~3% Cu in Hg without N - 0.09 ± 0.01 of "pro analysi" flushing grade with N - 0.06 ± 0.01 flushing The uncertainty in the contact time t varies from 1 to 5 L The resul ting relative error in q as defined by equation (17) will thus be of the order of 10 % which is too high for quantitative analysis. - 18 - In principle active carbon loaded with a metal chelate can be used for substoichiometric exchange reactions. It could be dosed by simple weigh ing or pipetting from a suspension. This picture implies the irreversi bility of the adsorption of the metal chelate while the metal self should be completely exchangeable. Actually this is unlikely and a completely reversible system is a more probable supposition. If an excess of carbon and complexing agent are present, *_he recovery q will be constant. In this case the proceJ- r*> vil! not D» applicable in isotope dilution ana lysis. If the amount of carbon is substoichiometric but the chelating agent R is present in excess it easily follows that ''«-'-foc^r (,8) where f (RJO describes the adsorption isotherm of the chetate-complex. For a linear isotherm q becomes constant again. If the isotherm obeys the Freundlich equation (19), log y/m = a .log c + a (19) 2 c0 = lO/q " l).ffl/Vj ' + m/V.10 .[(l/q - l).m/v] ' (20) By considering c as a function of o. it can be seen that application of adsorption on active carbon in isotope dilution analysis is of inte rest only if a is substantially smaller than 1. The adsorption of Pb(II) and Hg (II) and that of Pb- and Hg-chelate - analogous to the liquid-liquid extraction DDC is selected as the chela ting agent - on active carbon suspended in water an'' rhlo^oform, res pectively yields reversible systems which obey Freundlich isotherms. The presence of other adsorbable constituents does not change the ad sorption behaviour. For 'Merck 2184' active carbon at pH » 2.2 * 0.2 (in water) the follo wing values of a. apply: - 10 - Pb(II) 0.677 ± 0.005 Hg(II) 0.99 ± 0.08 Pb(DDC)2 0.70 ± 0.05 Hg(DDC), 0.38 ± 0.05 Thesa results point to the fact that adsorption of the metal-DDC com plexes may well lead to convenient isotope dilution procedures. Because of the difficulties inherent in taking reproducible amounts of the ad sorbent in the case of very low metal ion concentrations (see section 1.1.1.) the possible application in isotope dilution analysis has not further been explored. 2^3i4;_Coulometr2 Specific electrodeposition of the element to be determined leads to the isolation of a fixed amount, a, of the element. Thus one has the sim plest case of isotope dilution analysis: The value of q is determined by a radioactive spike. Usually the fraction which remains in the solution is determined. A counting ali quot of v ml is taken out of the total V ml. If the number of counts of the added spike is A- one arrives for Aq at: Aq -^ . -^ . [1 + (l-q).v/V] (22) It is cbvious that 2 2 Ac/c = [(Aq/q)[(Aq/q) +• (Aa/arJ(Aa/a) ]2* (23) The electrodeposition can be calibrated by using two identical vessels in series |3|- If the suffixes s and x refer to the standard and the sample respectively one has c = f.-^ . c (24) x 'q s The factor f stands for the possible systematic difference between the - 20 - two vessels. Its value can be determined by a standard addition experi ment. Equation (24) holds for the case of a single electroactive compound. If more than one electroactive compound (indicated by suffix i) is pre sent and there is a "blank" current, i, , the current balance becomes b k T ^=(Ci)o.V.(.-e- i )+ib (25) The experimentally determinable q.-values are given by q. = Thus for precisely defined conditions the k. values and i may be de termined from radiotracer experitants. The calibrated vessel can then be used for actual determinations. The two-vessel procedure is of no use here. The main practical difficulty is the elimination of oxygen from the vessels which is necessary to prevent spurious currents. - 21 - 1.4. LIST OF SYMBOLS Svmbo) Definition Dimension A number of counts — A number of counts for the added spike substoichiometric amount, taken M by coulometry -1 specific count-rate mm sp B background -1 b specific count-rate of the nun background -1 original concentration in the M.l liquid sample -1 original concentration of compound M.l (ci>o i in the liquid sample -1 c and c concentrations in two liquid phases M.l which are in equilibrium -I c and c concentrations of standard and M.l s x sample solutions in coulometry -1 F Faraday's equivalent CM f factor of difference between the two vessels in coulometry blank current Amp. conditional equilibrium constant various eq . -1 rate constant in coulometry mm m mass of active carbon g m mass of added metal in amalgam M original value of m M S O valency ratio reagent/reactive compound - 22 - Q charge q chemical yield q and q chemical yields of standard and s *» sample solution in coulometry T time of electrolysis t time V volume of sample V and V_ volumes of two liquid phases which are in equilibrium v volume of counting aliquot y -adsorptio stoichiometrin load c ofrati activo e carbon {: rate constant for isotopic exchange between a solution and an amalgam a. and a„ constants in Freundlich-isotherm - stoichiometric ratio f rate constant for dissolution of the added metal from an amalgam - 23 - 1.5. Summary of contents — Ml I - I • I—•— —,— -•••Hi . The use of a non-specific reagent in liquid-liquid extraction proved to be convenient for activation analysis as well as for isotope dilu tion measurements. It has been chosen as the main theme of this thesis. Analogous to the substoichiometric isotope dilution analysis with a non specific reagent in liquid-liquid extraction , possibilities have been investigated to set up a model which could be applied in substoichio metric electroanalysis. For this coulometry was found to be feasible if the vessels are continuously flushed with nitrogen. Chapter 2 discusses the description of liquid-liquid extraction with a non-specific reagent. The pertinent equations are derived and checked by radio-tracer experiments. Chapter 3 describes the radiometric determination of the extraction constants of some metal-diethyldithiocarbamates (M(DDC) ) m in the system chloroform-water. Data are given and compared to literature values. Chapter 4 deals with the calculation of the extraction yields from the known initial concentrations. A computer programme is developed which is described in Appendices_l_and_2. The application to the extraction of M(DDC) complexes in the m system chloroform-water is investigated. Titration curves are given. Chapter 5 discusses the calculation of the initial concentrations form the results of a radiometric titration in a two phase system. A computer programme is developed which is descri bed in Aggendices_3 and 4. A titration device is construc ted which permits continuous measurement of the chemical yield during the titration. Experimental data are presented for the titration of metal ions in an aqueous solution with Pb(EDC). dissolved in chloroform. The possibility of optimizing the values of the conditional extraction constants is considered. Chapter 6 deals with the detei.nination of metal ions by means of cou lometry. The construction of a suitable apparatus and its - 24 - application to the determination of Cd(II) in aqueous solu tions, using Cd (Ti = 53.5 h) as a radiotracer is des cribed. Appendix 5 discusses the feasibility of an apparatus for semi-auto matic liquid-liquid extraction. Its applicability is demon strated in a simple case. - 25 - 1.6. References |l| Moffit, A.E. and Kupel R.E. Amer. Ind. Hyg. Ass. J. (1971), 614 [2f Jansen, J.H. et al. Anal. Chim. Acta j)2, (1977), 71 |3| van der Sloot, H.A. et al. The simultaneous elimination of 24 42„ 82_ . 32,, . . NaM , K, ^r and p in fc the determination of trace elements in biological material by neu tron activation analysis. Anal. Chem. _52 (1980), 112 |4| De Soete, D.et al. Neutron activation analysis, Wiley-Interscience, New York, 1972, p 358 - 362 13 I Landgrebe, A.R. et al. Anal. Chim. Acta 39, (1967), 151 |6| Morrison, G.H. and Freis?r, H. Solvent extraction in analytical chemistry J. Wiley Inc., New York, 1966 |7| Ruzicka, J. and Stary, J. Substoichiometry in radiochemi cal analysis Pergamon Press, Oxford, 1968 42 - 58 |8| Rieman, W. and Walton, H.F. Ion exchange in analytical che mistry Pergamon Press, Oxford, |9| Korkisch, J. Modern methods for the separa tion of rare metal ions Pergamon Press Oxford, 1969 |10| Ruzicka, J. and Stary, J. Substoichiometry in radiochemi cal analysis Pergamon Press, Oxford, 1968 59 - 114 |11| Starik, J.J. Grundlagen der Radiochemie Aka- demie-Verlag, Berlin, 1963 260 - 316 - 26 - |l2[ Haïssinsky, M. Nuclear chemistry and its appli cations Addison-Wesley Publishing Co. Reading, ?4, 597 - 631 |13| Csajka, M. Radioch JI. Radioanal. Lett. J_3/2/, (1973), 151 11A J Basak, B. et al. J. Radioanal. Chem. 42_, (1978), 35 | 13J Ha'issinsky, M. Nuclear chemistry and its appli cations Addison-Wesley Publishing Co. Reading, 1964, 242 - 275 |16| Luten, J.B. et al. J. Radioanal. Chem. 4_3, (1978), 175 |l7| Ruch, R.R. and DeVoe, J.R. Anal. Chem. 39_, (1967), 1333 |18| Das, H.A. et al. J. Radioanal. Chem. 24_, (1975), 383 |19| Bajo, S. Anal. Chim. Acta 105, (1979), 28 - 27 - CHAPTER 2. APPLICATION OF DISPLACEMENT AND EXCHANGE REACTION? IN NEUTRON ACTIVATION AND ISOTOPE DILUTION ANALYSIS*) SUMMARY The efficiency of a separation of metal ions by combining a displacement or exchange reaction and a liquid-liquid extraction procedure is calculated as a function of parameters such as pH, the ratio of the concentrations of reagent and analyte and the ratio of the concentrations of reagent and interfering ions. The resulting equations can be used to optimize separation conditions in activation analysis and to calculate favourable conditions for multi-element isotope dilution analysis, the application of which is dis cussed. *) Published by P.C.A. Ooms, U.A.Th. Brinkman, H.A. Das, J. Radioanal, Chem. 46 (1978) 255-264, - 28 - 2.1. INTRODUCTION Rapid separation of trace amounts of metal ions from aqueous solutions can be achieved by displacement reactions, combined with liquid-liquid extraction. r+ The organic phase contains a complex RX , from which the metal ion R is (partly) displaced by metal ions present in the aqueous solution. Two types of application can be envisaged: - in activation analysis for the separation of single metal ions or a group of metal ions |l-5|. - in isotope dilution analysis to achieve a multi-element determination, due to the poor selectivity of a properly chosen complexant |6|. In both cases, the extraction efficiency of every single metal ion will be determined by the value of its conditional extraction constant, the pH of the aqueous phase and the concentration of the reagent RX as well as by the conditional extraction constants and concentrations of all other metal ions, present in the system under investigation. This text presents the quantitative description of such separation systems and the principles of their applications in activation analysis and isotope dilution analysis. 2.2. CALCULATIONS OF EQUILIBRIUM CONCENTRATIONS 2.2.1. Definitions Y(y) - metal Y with valency y C* ,C * - initial concentrations of the metal ions in the aqueous phase C ,C - initial concentrations of R and X respectively K X in the organic phase V,V - volumes of the aqueous and organic phase res pectively Equilibrium concentrations of species are indicated by the symbols in square brackets. - 29 - 2.2.2. Theoretical considerations The following displacement is supposed to be in equilibrium: rY + y(RX ) ^ r(YX ) + yR (1) r org y org where Y is written for A, B,... , N respectively (Y ^ R). The equilibrium constant K for this reaction can be written in terms of the conditional e extraction constants Kj, Y and Kj, » (K >r K = "*»'k Y — (2) e |YX | .|H|y ' y org ' ' "Èx,^ (3) |Y| . |HX|y ' ' ' 'org |RX | .|H|r r or "Éx.I ' e ' ' (4) |HX|r 1 'org For the sake of simplicity, valency indicatiors are omitted. The extraction efficiency, q, for Y is defined by the ratio of the amount of Y in the organic phase and the total amount of Y present in both phases: |YX I .V m I yiprg org ( . qY C .V v ; y In practice, it is well possible that the metal ion R, which is incorpo rated in the complex reagent RX , is originally present in the sample solution too. In such a case, isotopic exchange will occur next to displace ment and a q -value can be defined: K |RX I .V = L^rs 2Eê (6) CR ' V + CR • Vg In the following derivations of suitable equations for qv and q three i R assumptions have been made: - 30 - - The distribution coefficients of HX and the complexes YX and RX are y r infinitely large. - The distribution coefficients of the ions themselves are zero *) - No other complexmg agents are present. The mass balance equations are: |Y| . v + |YX | .v = cv . v (7) 1 ' ' y org org Y |R| . v + IRX I .v = c* . v + c . v (8) 1 ' ' r'org org R R org iHXl .V + ylYX I .V + rlRX I .V = C .V (9) 1 'org org •" y'org org ' r'org org X org Combining Eqs. (7) and (8) with (3) and (4) respectively, yields . . Cv y'org (10) "org , |H" V ^x.rl^org » V /V CR + CR. or/ (11) 1 Vorg , |H|r r'org " ^x.R'^lorg Substituting these equations into Eq. (9), one obtains V /V r.(C*R + CR. org ) *<*>-»<»> y.Cv R R Y iHXl1 = C„ - " (i: org X V H Y(y)=A(a) !§!& + -J» v r v Y K: D.|Hx| ie V.|HX| Ex,R ' 'org Tx,Y ' 'org *) ... If another complexmg agent, L, is present in the aqueous phase, this will influence the equilibrium concentrations of the respective species |7|. Eq. (3) then becomes I YX I I Hi n ylorg K^XY= {. * 1MB, • ... +|M -Bn> . ' 'y' O') |Y| . I HXI y I I I I org where 6, 6 are the formation constants of the subsequent adducts. 1 n - 3! - Since the extraction efficiencies q can be written as V /V qY=—^ (.3) V /V + |Hl or 8 VK v- |KXluvi| y and ^X'Y or8 v___/v qR - —-2£B — (14) V /V + - IH or r 8 i^' luvlHX ^x.R' org - as is easily shown by substituting Eqs. (10) and (11) into Eqs. (5) and (6) respectively - it is obvious that calculation of all q values n is possible at any pH for given values of C,., CL, C_, V and V In r ' v ° Y R R org. other words, qY values can be calculated for known values of the so- called p-parameter |8|, defined as Cv V X org , . p = (15) Y Y^y- 2.2.3 Numerical calculations 2iJ_.J_-J_ Programmes - If for a givt;n system all metal ion concentrations are known, all p-values are defined as well. In such a case, the equations given in section 2.2 enable one to predict the q-values at equilibrium. These calculations are necessary in order to evaluate the proper conditions for quantitative extraction in activation analysis. - It is not possible to calculate the original p-values from a single set of q-data. If n metal ions are present, n such sets are required to solve the pertinent equations. These sets have to be determined for the same system at n different C values. This procedure caK n be applied in isotope dilution analysis. Computer programmes have been developed for both types of calculation* ) They were used to study the complexation of metal ions with diethyldithio- carbamate. The 1/n log K' values were either taken from literature |l| or determined by radiotracer experiments |°|. They are given in Table 2.1. Programmes are described in Apnendix I. The text: of the programmes is gi ven in Appendix II. Table 2.1. Values of log Kl for complexation of metal ions with DDC in CHC1 Metal (a) (b) i (a) Data taken from A. Wyttenbach | 1 ' (b) P. Ooms et al.J9J Hg (ID - 26.92 + 0.05 Ag (I) 12.6 - Cu (II) 13.2 12.81 + 0.07 i t' Bi (III) - 15.7 + 0.2 I i In (III) - 9.95 • 0.08 ! Pb (II) 6.8 7.0 + 0.1 Cd (II) 5.6 5.50 + 0.05 Zn (II) 2.6 2.3 + 0.2 - 33 - 2^2^J^2_Prec^s2pn_of_the_calculated_q-yalues Every equilibrium value of q is a function of all Kl values of the elements pertaining to the system under consideration. Precision of the results is thus governed by that of the Kl values used. The statistical error in the results, due to the uncertainty in the K' values is given by Eq. (16). Aq = (16) Ex,i In view of the complexity of this expression, the most convenient way to determine the effect of variations in the Kl values on q, is to repeat the calculation with values of Kl + AKl and K' - Alt' . Fig. 2.\ gives the results of such a calculation for the displacement reaction 2+ of Pb with varying amounts of Cd(DDC) , dissolved in CHC1 . FIG 1 INFLUENCE OF THE UNCERTAINTY IN K' .VALUES OF Pb AND Cd ON THE Ex q VS. D CURVE . log K , _, =70;0.1 9K 5C r '° tx.Cd^ -' CURVE 1 log K^_ : log Kp pb Ex.Cfi lo K S5t CURN.E2 logKEx pb =7C 9 rxCcT CURVE 3 iogK; _. =6 9 logf Fx.Cd 1.0 15 meqDDC P"meq Pb - 34 - 2.2.4 Applications 2.2^4^2 Activation analysis Ir. activation analysis use can be made of the technique of substoichio- metric solvent extraction. This involves the use of a chelating agent, dissolved in an organic phase, the quantity of which is chosen so as to extract only part of the metal ion of interest. It is required that the reagent reacts quantitatively with this metal ion (i.e. > 99.9%) |l0[. The separation of a known amount of a metal ion A, without contamination by a metal ion B, by extraction with a RX solution depends on the pH of the aqueous phase and the ratio C /C . Conditions for optimal separation can be calculated by computer. In practice, the ratio C./C will be ad justed by adding a proper quantity of carrier of metal ion A to the aqueous phase. In Fig. 2.2 - which is discussed in more detail below - 2+ 2+ the influence of the presence of varying amounts of Zn and Cd on the 2+ extraction of Pb with a stoichiometric amount of Zn(DDC) is demon- 2+ strated. To obtain a quantitative separation of Pb from the interfering 2+ metal ions, addition of Pb carrier would be necessary. 2. 2i4^_IsotO£e-dilution_analyj3is Using isotope dilution analysis, it is not always possible to find proper conditions which allow a substoichiometric amount of a complexant to react specifically and quantitatively with just one ion involved. However, multi element isotope dilution analysis will be possible then by means of radiometric titration. In such a case, the metal ions which are able to react with the complexant used, must be spiked with proper radio isotopes. Subsequently, the aqueous phase, which contains e.g. n different me*"al ions, is titrated with an organic solution of RX , using n subsequent additions of reagent. After each addition a set of q values is measured. This provides n sets of n data which are fed into the computer for calcu lation of the original concentrations. The titration must be carried out in a pH region, in which the reaction between the complexant and the metal ions is quantitative. If a rough estimate of the concentrations of the ions involved is known, this region can be calculated using the computer programme mentioned in section 2.2.3.1, Fig. 2.3 demonstrates the influence of the pH on the radiometric titration + of Pb with Zn(DDC)2. - 35 - n a er 08S, meg DOC I! i.10'2*Zr.c =101910"' c meqPb 4 2 FIG 3 2) n«101910" M Cd; cZn=lO" INFLUENCE OF THE pH ON THE FIG. 2 EXTRACTION OF Pb WITH ZnlDOOj c ^iDca-j2'0"3*"" Vors/V = C5 CALCULATED CURVES EXPERIMENTAL CLRJES 2.3 EXPERIMENTAL The calculations made by computer can be checked by experiments in which the q values are determined with the aid of radiotracers. Two sets of experiments 2+ were performed to serve as test cases: Pb was extracted with Zn(DDC) , in 2+ 2+ the presence of varying amounts of Zn and Cd As is shown in Fig. 2.2, the correlation between theory and practical work turns out to be excellent. 2.3.1 Chemicals and apparatus 212. All chemicals used were of analytical purity. The Pb spiking solution was carrier free. A "Schuttelmaschine W-3" of Hormuth Vetter (Germany) was used for the experiments.Counting was performed on a 3 x 3" Nal well-type detector connected to a 400-channel analyzer with automatic read-out. The metal diethyldithiocarbamates were prepared according to the procedure described by Wyttenbach et al J2|. Analysis of these products for their metal content was carried out by complexometric titration. 2.3.2 Procedure Extractions were performed by shaking 50 ml of an aqueous solution containing 2+ 2+ 2+ Pb , Zn and Cd for 5 min. with 25 ml of a solution of Zn(DDC) in CHC1 . The concentrations of the respective ions in the aqueous phase were: - 36 - 3 2,2 CDV = 10~ M (spiked with Pb) 'Pb -9 O, - n x 10 ' M (n=i, , 10) en , Ccd = 1.02 x 10~ M pH =3.3 3 212 II. CD1_ = 10~ M (spiked with Pb) Pb _2 CZn = ,0 M . C„, = n x 1.02 x 10 M (n=l, , 10) Ld pH = 3.8 In all experiments the concentration of Zn(I)DC)- in the organic phase was 2 x I0~3 M. All extractions were carried out in 100 ml. polyethene bottles. The extraction efficiency of Pb(qp, ) was determined by pipetpipettint g 5 ml of the aqueous and organic phases and measuring their activities. 2.4 CONCLUSIONS With respect to the analytical applications of the quantitative description of the present extraction system, the following remarks can be made. - The optimal conditions for c antitative displacement extractions of metal ions with respect to pH can be read from graphs like those given in Fig. 2.3. - Calculations by means of the equations given in section 2.2.2 allow the determination of traces of metal ions by isotope dilution analysis. That is non-selectivity of the reagent is no longer a limiting factor for application of this technique tor multi-element analysis. - The uncertainties in the Kj, values introduce errors into the calculations of the optimal pH ranges (activation analysis) and the original metal ion concentrations (multi-element isotope dilution analysis), which can be deduced from graphs such as drawn in Fig. 2.1. - 37 - 2.5 REFERENCES [l[ A. Wyttenbach, S. Bajo, Anal. Chem. 47 (1975), 2. |2| A. Wyttenbach, S. Bajo, Anal. Chem. 47 (1975), 1813. |3| S. Bajo, A. Wyttenbach, Anal. Chem. 48 (1976), 902. |4| S. Bajo, A. Wyttenbach, Anal. Chem. 49 (1977), 158. |5| S. Bajo, A. Wyttenbach, Anal. Chem. 49 (1977), 1771. |6| T. Braun, J. Tolgyessi, Radiometric Titrations, Pergamon, Oxford 1976. |7| J. Stary, K. Kratzer, Anal. Chim. Acta 40 (1968), 93. |8| H.A. Das, H. Kohnemann, W. van der Mark, J. Radioanal. Chem. 24 (1975) 383. J9| P.C.A. Ooms, U.A.Th. Brinkman, H.A. Das, Radiochem. Radioanal. Lett. 31 (1977), 317. f 101 J. Ruzicka, J. Stary, Substoichiometry in Radiochemical Analysis, Pergamon, Oxford 1968. - 38 - CHAPTER 3. RADIOMETRIC DETERMINATION OF THE EXTRACTION CONSTANTS OF SOME METAL DIETHYLDITHIOCARBAMATES IN THE SYSTEM CHLOROFORM/WATER*) SUMMARY The extraction constants of Zn(II), Cd(II), Pb(II), In(III), Cu(II), Bi(III) as well as Hg(II) diethyldithiocarbamate complexes have been determined for the chloroform/water system. Published by P.C.A. Ooms, U.A.Th. Brinkman, H.A. Das, Radiochem. Radioanal. Letters 31 (1977) 317-322. - 39 - 3.1. Introduction The diethyldithiocarbamate anion (C-H^.NCS. (in the following denoted by DDC) is a common group reagent in analytical chemistry for all those elements that from insoluble precipitates in aqueous solutions with sulfide [1]. Generally, the complexes formed are extractable into several organic solvents. In many cases, DDC is preferred to the more selective dithizone, because of its greater stability against oxidation and acid decompcsi- tior as well as the greater solubility of its complexes in organic solvents. Fast decomposition of the DDC-anion can take place upon acidification of an aqueous solution of NaDDC [2]-[6]. + (C2H5)2NCS2 + H ->• (C2H5)2NH + CS2+ (I) Using M(DDC) in chloroform, the reagent will be partially protonated and thus react as an organic solution of HDDC. Its stability against acid decomposition is much greater than that of NaDDC and depends on the reagent metal M used, the concentration of M(DDC) in chloroform and of m the pH of the aqueous phase. Since DDC is a multielement reagent, trace amounts of metal ions n can be separated from an aqueous solution by reaction with a suitable metal-DDC complex (R(DDC) ) and subsequent extraction into e.f,. chloroform. r Mm+ + m (R(DDC) ) t r (M(DDC) ) + m Rr+ (2) r'org v m org v Here the suffix 'org' refers to the organic phase. The equilibrium constant, K , of this displacement reaction can be written in terms of the extraction constants of the two complexes, VM and *E*,r - 40 - f K with [M(DDC)J .[H+]m m ofg (4) KEX'M [M]. [HDDC]m org and [R(DDC)J .[H+]r E-2EÊ (5) ^x.R [R].[HDDC]' org The sepaiation method based on displacement reactions can be applied in isotopic dilution analysis of aqueous solutions of metal ions, if all extraction constants of the elements under investigation are known [7], Therefore, the K values of the DDC-complexes of several metals had to Ex be determined, since information in literature is scanty [8]-[l3], 3.2. Theory The following methods for determination of the extraction constants were used [i4]. The first method consists of determining the distribution of the metal in question between the chloroform and aqueous phase at various pH values and equilibrium concentrations of HDDC in the organic phase. If the metal is originally present as M(DDC) , dissolved in the organic phase, the following reaction will take place. + (M(DDC) ) o + m H t M""* + m (HDDC) (6) m org org In the case when in equilibrium the metal is present in the aqueous phase exclusively as the metal ion (formation of hydroxo or other com plexes can be neglected) the extraction constant IC, „ can be calculated Ex,M from equation (4). - 41 - Other methods for determination of extraction constants are based on displacement reactions between the metal ion in the aqueous phase and another metal R, dissolved as R(DDC) in chloroform, according to equation (2). From equation (3) it can be seen that, measuring the distributions of M and R over both phases, the extraction constant IC „ can be determined if IC „is known and vice versa. Ex,M bX,K However, if r[M(DDC) ]„ .V = m[Rr+].V (7) m org org aq the calculation of 1L can be done after determining the distribution ratio of one of the metals. In the presence of a masking agent vnieh forms non-extractable complexes ML.,ML2,.•••»ML with metal M, che expression for IC becomes more complicated. [M(DDC) ] / rol v - M + a ninï m org / [R] ^x.M " ° + 3ntL] }' [ST * [R(DDC) ] \ r org 3.3. Experimental 3.3.1. Chemicals and equipment A Kick digital pH meter with glass electrode, a "Schuttelmaschine W-3" of Hormuth Vetter and a 3 x 3 Nal well-type detector connected to a 400 channel analyser and read out were used. All chemicals used were of analytical purity. The radioisotopes Zn, 115m„, 203 207„. . 212^, , . . . 116m , Cd, Hgu , Bi and Pb were ofc radiochemica,. l purity. TIn and 64 Cu were obtained by irradiation of small amounts of In (NO-)., and 13 J ±2 -1 Cu-foil respectively at a thermal neutron flux of 5 x 10 cm ,s The crystallized reagents Zn(DDC)„, Pb(DDC)2 and Cu(DDC)2 were prepared according to the procedure described by Wyttenbach et al. [15]. Analysis of these products for their metal content was carried out both by complexometrie titration after their destruction in HNO, and by radio metric titration of their organic solutions. - 42 - 3.3.2. Procedures Procedure I: This procedure was applied only in tne determination of Ex,Zn* Twenty ml of a solution of Zn(DDC)? in chloroform was shaken with 50 ml of an aqueous phase, containing no metals able to react with DDC. In several experiments the pH was adjusted at different values. A situation during 2-5 minutes was ample Lime for equilibrium to be reached. After separation of the phases, the distribution of Zn was determined radiometrically, measuring the activity of Zn in 5 ml aliquots taken from both the aqueous and the organic phase. The determination of [HDDC] was carried out by shaking 10 ml of the organic extract with 25 ml of an aqueous solution of Cu(II). From literature [8] it is known that HDDC as well as Zn(DDC)2 will react quantitatively with a suprastoichiometric amount of Cu(II). So, from the distribution ratio of Cu, determined radio-metrically too, [HDDC] present after the previous reaction with Zn(DDC)_ could be deduced. Then, the extraction constant IL, was calculated according to Ex,Zn equation (4) . Procedure II: An aqueous solution of M(m+) was shaken for at most 30 minutes with 25 ml of a solution of R(DDC) in chloroform. This solution did not contain any free HDDC. In equilibrium state the distribution ratio(s) of one or both metals were determined radio- metrically. Knowing either the value of IC, or K_ M (one of them determined Ex, K Ex , M previously) the unknown extraction constant K_ or IC, _. respectively Ex,M Ex,K was calculated from equation (3). With a masking agent present in the aqueous phase, equation (7) had to be used. 3.4. Results The extraction constant of th Zn-complex with DDC was determined accord ing to Procedure I (Cf. section 3.3.2.). - 43 - The values of K^^ (with R = Zn), K^^ and ^^ (with R = Pb) «Ex.Cu (R = Cu, M = Pb) as well as K^^ and K^. (R = Cu). were determined using the principle of Procedure II (Cf. section 3.3.2.). In the determination of K_ „ , chloride ions were used as complexing fcx,ng agent in order to lower the extremely high value of the distribution coefficient of Hg appreciably. Equation (8) now applies. 2+ [Hg(DDC)2] .[Cu ] ^x He = ° + V[C1 ] >' S ÏT ' «Ex Cu (8) g + EX CU ' [Cu(DDC)2]org.[Hg^ ] ' The value of the conditional complexation constant B, was taken from ref. [16]. In this work, the use of iodide instead of chloride ions, which has been recommended by Stary et al. [14] was rejected because of the inter ferences due to the reaction (9). 2 Cu2+ + 2 i" -*• 2 Cu+ + I (9) The experimental conditions as well as the IC values are recorded in Table 3.1. For Cu, Pb, Cd and Zn, the present results are seen to be in good agreement with the data from literature. The IC, ., values given in Ex,N ref. 8 have been calculated by converting two-phase stability constants 3 into extraction constants using 2.3 x 10 as the partition coefficient -4 of HDDC in chloroform and 4.5 x 10 as the dissociation constant of HDDC [17]. - 44 - TABLE 3.1 Values of conditional extraction constants of metal diethyl- dithiocarbamates, determined for the system CHCl->/H_0. Data about experimental conditions. Metal Composition of l0* «EX.M ion aqueous phase organic phase present literature \rork /ref. V =50 ml V =20 ml Zn(II) org *q 65„ Zn(DDC)=2.5xl0_4M Zn spike 1) pH=2.97 2) pH=4.01 After the first extractions, second 2.3 jf 0.2 2.6 IB/ ones were carried out to determine 2.39+0.02 /13/ HDDC. V =25 ml V =10 ml aq org 64., Cu contains extracts ph=3.76 of the former 2+ -4 Cu =lxJ0 M extractions Cd(II) V =50 ml V =25 ml aq org 115ra„, ., Cd spike Pb(DDC) =2.0x10~5M 5.50_+0.05 4.4 /8/ pH=3.67 5.6 /9/ Cd2+=1.78xl0"5M 5.77+^0.05 /13/ Pb(II) V =50 ml V =25 ml aq org 21 2 5 Zn(DDC)2=1.0xl0~ M 7.0 +_ 0.1 6.8 /8/ Pb spike 7.94+^0.09 /13/ pH=2.97 Pb2+=I.0xl0~5M In(III) V =50 ml V =25 ml aq org ll6m_ In Pb(DDC)2=2.0xl0~ M pH=3.35 1) 9.95+_0.08 In3+=1.49xI0~5M 2) In3+=1.49xl0"5M Pb2+=5.44xl0_5M - 45 - TABLE J.I Continued Cu(II) V =50 ml V =25 ml aq org 64 , pH=2.97 Cu in Cu(DDC). 2+ -2 12.81+0.07 13.2 /8/ 1) Pt =10 M Cu(DDC)2=9.81xl0 M 2) Pb2+=2xl0~2H Bi(III) V =50 ml V =25 ml aq org 4 Bi spike Cu(DDC)2=4.29xlO~ M 15.7+_0.2 — pH=1.28 i3+=4.96xl0~7M Hg(II) V =50 ml V =25 ml aq org 2 °3HU g spik-ue 1) 5 pH=2.76 Cu(DDC)2=4.29xlO~ M Hg2+=4.49xlO~6M 2) 26.92+0.05 2 5 5 Cu =2.78xi0~ M Cu(DDC)2=1.72xlO~ M Cl~=8.97xl0~2M 3.5. Discussion For the determination of the extraction constants, experimental conditions were chosen in such a way that no side-reactions in one of both phases could occur (e.g. formation of hydroxo complexes if the pH is too high). However, if other forms of M and R occur in the aqueous phase, the conditional extraction constants can be defined by [M(bDC) L .[H+]m m orr g 'TXjM (10) [M'l.tHDDcfV org and [R(DDC) ] .[H+]r r org (11) ^x.R = [R'].[HDDC]r org - 46 - [M'] and [R'] are the concentrations of all forms of M and R respective ly in the aqueous phase. Kl and K_. are related by *L " hx,a (,2) The a coefficient [18] is defined by equation (13) a = [M']/[M] (13) - 47 - 3.6. References [ !] Trofinov, A., Zeitschrift fur Anal. Chemie 132 (1951) 282. [ 2] Bode, H., Zeitschrift fur Anal. Chemie 142 (1954) 414. [ 3] Joris, J., Aspila, K.I., Chakrabarti, C.L., Anal. Chem. 41 (1969) 1441. [ 4] Joris, J., Aspila, K.I., Chakrabarti, C.L., J. of Phys. Chem. 74 (1070) 860. [ 5] Martin, A.E., Anal. Chem. 25 (1953) 1260. [ 6] Bhatt, I.M., Soni, K.P., Trivedi, A.M., J. Indian Chem. Soc. 44 (1967) 802. [ 7] Ooms, P.C.A., Brinkman, U.A.Th., Das, H.A., J. Radioanal. Chem. 46 (1978) 255. [ 8] Wyttenbach, \., Bajo, A., Anal. Chem. 47 (1975) 2. [ 9] Bajo, S., Wyttenbach, A., Anal. Chem. 49 (1977) 158. [10] Chermette, H., Colonat, J.F., Tousset, J., Anal. Chim. Acta 80 (1975) 335. [Il] Chermette, H., Colonat, J.F., Tousset, J., Anal. Chim. Acta 88 (1977) 331. [12] Chermette, H., Colonat, J.F., Tousset, J., Anal. Chim. Acta 88 (1977) 339. [13] Bajo, S., Wyttenbach, A., Anal. Chem. 51 (1979) 376. [14j Stary, J., Kratzer, K., Anal. Chim. Acta 40 (1968) 93. [15] Wyttenbach, A., Bajo, S., Amal. Chem. 47 (1975)1813. [16] Marcus, Y., Acta Chem. Scand. 11 (1957) 599. [17] Still, E., Finska Kemisteamfundets Medd. 73 (1964) 90. [18] Ringbom, A., Complexation in analytical chemistry, Interscience Publishers, 1963, New York/London. - 48 - CHAPTER 4. CALCULATIONS ON PARTITION EQUILIBRIA IN LIQUID-LIQUID EXTRACTION SYSTEMS, APPLIED TO THE SEPARATION OF METAL IONS FROM AQUEOUS SOLUTIONS*) SUMMARY A programme is described for the calculation of partition equilibria in the case when metal ions are extracted from an aqueous phase into an immiscible organic solvent containing a chelating reagent. Given the extraction constants and the initial concentrations, titra tion curves are constructed which show the extraction percentage of each ion in the presence of the others. The programme is applied to hypothetical situations both in activation analysis and isotope dilution analysis. Submitted for publication. - 49 - 4.1 INTRODUCTION In chapter 2 [l] theoretical aspects have been discussed concer ning the application of displacement and exchange reactions combined with liquid-liquid extraction in neutron activation analysis as well as in isotope dilution analysis. Separation methods based on that principle can be very convenient to isolate trace amounts of metal icns from aqueous solutions. The extraction percentage of every single metal ion Y depends on: - the conditional extraction constants - the pH - the ratio of the amount of Y to that of the reagent RX - the ratios of the concentration of Y to that of all other metal ions, competing for the chelating agent X. The application of the quantitative description of the extraction equilibria is possible only if a computer programme is available which - for the various elements involved - provides the percentages of extrac tion as a function of the amount of reagent added. Preferably, the results should be presented in tabular as well as in graphical form. The treatment given in ref. [1] assumes constant volumes of both phases. As the reagent is often added stepwise as an organic solution, the re sulting increase in the volume of the organic phase should be taken into consideration. In case of substoichiometric solvent extraction, the calculations may be used to indicate optimal extraction conditions in terms of yield and specificity. This is of interest for applications in activation analysis as well as in isotope dilution analysis. In activation analysis, the addition of carrier after irradiation im proves the selectivity of such a separation. The requirement of quan titative reaction between the reagent and the element of interest (i.e.> 99.9% [2]), implies that there should be a minimum amount of carrier to be added, sufficient to fulfill that requirement. Calculations have been carried out for a hypothetical system chosen in such a way that the effect of an initially large excess in concen trations of elements with low extraction constants bale os the high - 50 - value of the extraction constant for the minor element. The influence of variations in the relative ratios of a set of IC, values has been investigated too. In isotope dilution analysis the present programme facilitates the choice of a convenient chelating agent for the extractions. To demonstrate these applications, the programme has again been applied to hypothetical cases. - 51 - 4.2 THEORY 4.2.1 Definitions Y(y) - metal Y with valency y, where Y = A,B.,....,R X - monovalent chelating agent, used as reagent ï*cR - initial concentrations of the metal ions in the aqueous phase cD,cY - initial concentrations of R (as RX ) and X respectively in the organic solution of reagent to be added. V - volume of the aqueous phase V - initial volume of the organic phase, org AV - volume of the separate additions of the reagent solution n - number of additions a,..y,r - valencies of A,..,Y and R respectively. 4.2.2 Theoretical considerations When an aqueous solution of metal ions is titrated with a chelating agent dissolved in an organic solvent, after each addition the follo wing reaction will reach equilibrium. y+ r+ r Y + y (RX ) J r (YX„) rn + y R (1) J v r'org •*• y org Assuming no other complexing agents being present , the equilibrium constant K for this reaction can be given in terms of the extraction e constants ^^ and K^^. (K )r K = EExx?. YY (2) ' «Ex,*'" where y [YXlnr.tH] c 2L°£S (3) ' CY].[HX]y org - 52 - and [RX ] .[H!r r rg ^x.I ° r (4) org For the sake of simplicity, valency indications are omitted. The ex - traction efficiency, q, for Y is defined by the ratio of the amount of Y in the organic phase to the total amount of Y present in the system. [YX ] .(V° + n.AV ) _ y org org org (5) ^ cy . V In practice it is possible that the metal ion R, which is incorporated in the reagent RX , is originally present in the sample solution too. In such a case, a q value can be defined. K [RXJ .(V° + n.AV ) = rorg ors org_ (6) VV + Vn-AVorg Assuming that the distribution coefficients of HX and the complexes YX and RX are infinitely large, whereas those for the ions themselves are zero, the mass balances are: [Y].V + [YX ] .(V° + n.AV ) = cv.V (7) y org org org Y [R].V + [RX ] .(V° + n.AV ) = c* V + cD.n.AV ,_. r org org org R R org (8) | [HX] + y[YX ] + r[RX ] L(V° + n.AV ) = c .n.AV [ org •"• y org r org J orc~~g or---g' Xv org (9) Combining equations (3) with (7) and (4; with (8) respectively and substituting the result into equation (9), one obtains: - 53 - n.AV r n.AV fcB + c„. ( or6 ) — [HXlorg = CX V + n.AV ' V° + n_ AV org org org org + tHf [Ya] hx,*' l org Y(y) = N(n) y cY I Y(y) = A(a) V° + n.AV [H]y (10) org org t y V K_ v.[HX] Ex,Y org In a similar way it can be shown that the extraction efficiencies q can be written as (V° + n.AV )/V q - -f* 2ES ) /V° + n.AV \ (11 Y ('" °or g " orgX J [H]y y K,Ex,, vY.[HX] org and (V° + n.AV )/V qB = -^B 2» (12) R A/V°° + n.A.AV \\ [H]r (jorg____orIj + K, .[HX]r ix, DR org From this, it is obvious that calculation of all q values is y * o n A possible at any pH for given values of c , cR, c , V, V and ' ^ „• - 54 - i 4.3 CALCULATIONS 4.3.1 Computer programmes Computer programmes have been developed to calculate as well as to plot the q values of the elements involved in the liquid-liquid extraction system as a function of the amount of reagent added. As an additional option, calculated statistical deviations in the q values, due to the uncertainties in the IC, values can be represented graphically (Fig. 4.6b) The principle of the calculations is based on resolving equations (10) - (12). The description of the programmes is given in appendix I, whereas the text of both computer programmes is presented in appendix II. 4.3.2 Applications in activation analysis The aim of calculating partition equilibria in substoichiometric liquid-liquid extraction in activation analysis is a predictive one. More specifically such calculations are of great use to demonstrate effects caused by variations in relative ratios of concentrations in a system with a given set of K^ values. To illustrate this, a hypothetical system was cons 4 consisting of an aqueous solution of three metal ions (A,B and C; in contact with an immiscible organic phase. These ions react with the chelating agent X, added stepwise as a dilute solution of RX in the same organic solvent. Initially, B and C - though having low extraction constants with respect to that of the minor element A - prevent quantitative reaction between A and X since they are present in large excess. In the present example, the concentration of A was varied with respect to that of B and C by adding carrier of A, which improves the selec tivity of the reaction. For concentration ratios A/B and A/C which affect optimal selectivity, the influence of variations in the relative ratios of the set of K_ Ex values was investigated. To this end the 1C values of B, C and R were varied with respect to that of A. The cases tested are summarized in Tables 4.1 en 4.II respectively. The resulting titration curves are plotted in Figures 4.1 - 4.7, - 55 - 4.3.3 Application in isotope dilution analysis Until now, substoichiometric isotope dilution analysis is applicable with severe restrictions only. In case of liquid-liquid extraction it is not always possible to fulfill the requirement of quantitative reaction between the element of interest anda substoichiometric amount of reagent, dissolved in an organic solvent. In most cases the concentrations of the ions to be extracted are low. Consequently, if use is made of the chelating agent X, added as RX - a chelate derived from the weak organic acid HX - interferences will already occur at moderately low pH values [13, depending on the order of magnitude of the respective IC, values. Moreover, since addition of carrier is out of order, selectivity will be seriously affected by the relative ratios of the set of IC, values. As a result, in practice it is rather difficult to select a system in which the various conditions for optimum performance are met. To illustrate this, calculations were again performed for hypothetical cases. Keeping the concentrations of A, B and C constant, the order of magnitude as well as the relative ratios of the K^ values were varied. The results are listed in Table 4.Ill and plotted in Figures 4.8 - 4.13. - 56 - 4.4 REFERENCES [1] Ooms, P.C.A., Brinkman, U.A.Th., Das, H.A., J. Radioanal. Chem., 46_ (1978) 255. [2] Ruzicka, J., Stary, J., Substoichiometric Analysis, Pergamon, Oxford (1968). - 59 - Table I-d (Figure 4) V - 20.0 ml av - 2.0 ml pH » 4 2.5x10*' c„ = I .OxlO-3 K , K_ ,. = I .Oxlll org ' _., C Ex,A Ex,I . J I.Ox 10 K - 3.l6xi0 V = 10.0 ml n = I.. .20 a =• b =• 1.0x10 - c - 2.5x10* Kr „ Ex, Rr org R Ex.B n.iV c.n.&V qA in Z Mol of A Mol of B Mol of C org R org extracted extracted extracted 4 2.0 5.0xlD" 10.00 5.000x10-* 0.01 2.222xl0-8 < O.OI 2.22 xlO"10 < 0.01 3 ,--8 4.0 I.OxlO" 20.00 l.OOOxlO*3 0.02 5.000x10 • 0.01 5.00 xll)"1() < 0.01 3 -3 -8 •10 6.0 I.5xl0~ 30.00 1.500x10 0.04 8.566x10 < 0.01 8.57 xlO < 0.01 3 8.0 2.OxlO* 40.00 2.000x10* 0.07 1.332x10" < 0.01 1.333x10" <• o.oi I 2.5x10 49.99 2.499x10* 0.10 1.998x10" < (..01 1.999x10* < 0.01 10.0 -3 3.0x10 59.99 2.999x10* 0.15 2.994x10" 0.01 2.998x10* ' 0.01 12.0 3.5x10 69.99 3.499x10* 0.23 4.652x10" 0.02 4.662x10* < 0.01 14.0 4.0xl(~ 79.97 3.998x10" 0.40 7.956x10" 0.04 7.984x10* 0.01 I 16.0 -3 4.5x10 89.94 4.497x10 0.89 I.772xlO~ 0.09 1.786x10" 0.03 ! 18.0 -5 j 5.0x10* 99.18 4.959x10" 10.81 2.162x10 1.20 2.394x10* 0.38 -3 | 20.0 5.5x10 99.95 4.997x10 66.14 1.323x10" 16.34 3.268x10* 5.82 j 22.0 6.0x10* 99.98 4.999x10* 83.94 I.b79xl0" 33.59 6.7 i 8x10* 13.79 -3 24.0 6.5x10 99.99 4.999x10" 88.90 1.778x10* 44.48 8.896x10* 70.21 | 26.0 7.0xl0~ 99.99 4.999x10" 91.67 1.833x10" 52.38 .048x10" 25.81 -3 3 ! 28.0 7.5x10 99.99 4.999x10f 93.34 I.867x10" 58.34 . 168x10" 30.69 -3 30.0 8.0xl0~ 99.99 4.999x10 94.45 1 .889x10" 62.99 .260x10* 34.98 32.0 8.5x10 100.00 5.000x10" 95.25 I.905x10" 66.70 .334x10* 38.78 -3 100.00 34.0 9.0x10 I 5.000x10' 95.84 I.917x10" 69.74 .395x10* 42.16 -3 36.0 9.5x10 100.00 5.000x10* 96.31 I .9?'>xlU* 72.28 .446x10* 45.19 -2 3 38.0 1.0x10 100.00 5.000x!0~ 96.68 I. '34x10" 7^ .42 .488x10* 47.92 40.0 Figure 4 15.0 20.0 25.0 10.0 ml of titratif i.qA 2/ia 3.q( - 61 - TABLE II Radiometric titrations of an aqueous solution containing metal ions A, B and ''• with the organic reagent RX Keeping the concentrations constant, the IL values of B, C and R are varied with respect to that of A. Table Il-a (Figure 5) V 20.0 ml iV - 2.0 ml pH - 4 c. » 2.5xlO~' c_ ' l.üxlO-3 K - l.OxlO10 K_ ,. = 3 I6xl07 orj A C Ex,A 1 7 10.0 ml n « 1...20 a-b-c-r-2c- I.Oxl O"" cR = 2.5x10 ^ =• 3.16x10 K. * 1 OxIO org fcx.R n.;v c„.n.:>V qA in Z Mol of A q in 7. Mol of B Mol of C org R org B "c in~ qR in X extracted extracted extracted 4 4 2.0 S.OxlO" 9.98 4.992xl0" 0.35 6.990xl0"7 0.03 7.012xIO"9 0.01 4.0 I.OxlO"3 19.96 9.980xl0"4 0.78 1.565xl0~b 0.08 I.576xl0"8 0.02 6.0 I.5xl0"3 29.93 I.496xl0~3 1.33 2.666xl0-6 0.13 2.698xl0"8 0.04 8.0 2.0xlO-3 39.89 I.994xl0~3 2.05 4.1IOxlO"6 0.21 4.188xl0"8 0.07 1 10.0 2.5xl0-3 49.83 2.49lxl0"3 3.04 6.090xl0"6 0.31 6.262xl0-8 0.10 f 12.0 3.0xl0~3 59.73 2.986xl0~3 4.48 8.960xl0~6 0.47 9.338x10"* 0.15 3 I4.i 3.5xIO* 69.57 3.478xl0~3 6.74 I.348xl0~5 0.72 I.435xl0~7 0.23 1 3 16.0 4.0xl0~ 79.26 3.963xl0-3 10.78 2.156xlO~5 1.19 2.388xl0"7 0.38 -3 18.0 4.5x10 88.52 4.426xIO 19.60 3.920xI0~5 2.38 4.760xl0~7 i'-76 3 20.0 5.0x10 95.95 4.797xl0" 42.81 8.562xlO~5 6.96 1.393xI0~6 2.31 3 22.0 5.5x10 98.78 4.939xl0~ 71.96 1.439x10"4 20.42 4.O84xl0"6 7.5C 24.0 6.0x10 99.40 4.970x10 84.01 I.680xl0~4 34.45 h.890xl0"6 14.25 3 26.0 6.5xl0~ 99.61 4.980xl0~3 89.07 1.781x1o"4 44.91 8.982xl0"6 20.50 ' 28.0 7.0x10 99.72 4.986xl0~3 91.74 I.835xl0-4 52.63 f.053x10"3 26.00 30.0 7.5xl0"3 99.78 4.989xl0-3 93.38 I.868xl0~4 58.50 I.l70xl0-5 30.84 32.0 8.0x10 99.82 4.991xl0"3 94.47 I.889xl0~4 63.10 I.262xIO~5 35.10 3 34.0 8.5xl0" 99.84 4.992x10 95.26 I.905xl0"4 66.79 1.336xl0"5 38.87 36.0 9.0xI0"3 99.86 4.993xl0-3 95.85 I.917xl0"4 69.81 I.396xl0"5 42.24 3 38.0 9.5xl0~ 99.88 4.994x10 96.32 1.926xl0-4 72.33 1.447xl0"5 45.25 2 40.0 l.OxlO" 99.89 4.994xl0~3 96.68 I.934xl0~4 74.46 1.489x10"'' 4/.97 . Figure 5 0.0 5.0 10.0 15.0 20.0 25.0 30.0 ml of tit rant i.qA 2.qB 3.qf - 62 - Table II-b (Figures 6a and 6b) V - 2U.U mi flV 2.Ü ml pH - 4 c. * 2.5xl0*! c„ * I.OxlO"3 * I.OxlO10 K, = I.OxlO8 org A _•> L -I 'S*, A 9 tx,L 7 V° - 10.0 ml n 1...20 R-=b»c«r = 2 c = I.OxlO " c = 2.5x10 - I.OxlO K^ - 3.16x10 org g R Sx.B x R n.aV c_.n.AV q inZ Mol of A Mol of B 7, Mol of C q in 7. R org A «B in Z qc in R org extracted extracted extracted 4 2.0 5.0xl0" 9.92 4.962xl0"4 1.09 2.186xl0"6 0.11 2.208x1 o"8 0.03 3 4 4.0 l.Oxlo" 19.89 9.945xl0~ 2.42 4.844x10-6 0.25 4.952xl0~8 0.J8 3 3 6.0 I.5xl0~ 29.80 1.490x1 O* 4.07 8.142xl0"6 0.42 8.452xlO-8 0.13 3 8.0 2.0xl0~ 39.67 1.983x10-3 6.16 1.234xl0"5 0.65 1.306xl0~7 0.21 3 3 5 10.0 2.5xl0~ 49.48 2.474xIO~ 8.92 1.784xlo" 0.97 I.940xl0"7 0.31 3 12.0 3.0xl0~ 59.21 2.960x10"3 12.68 2.536x1 O*5 1.43 2.862xl0"7 0.46 3 14.0 3.5xlO~ 68.78 3.439xl0~3 18.06 3.6l2xI0"5 2.15 4.312x10"7 0.69 16.0 4.0xl0"3 78.05 3.902x10"3 26.23 5.246xI0_5 3.43 6.866xlü" l.ll 18.0 4.5xl0~3 86.60 4.330x10"3 39.26 7.852xl0"5 6.07 1.2I4xl0"6 2.00 20.0 5.0xl0"3 93.36 4.668x1ü"3 58.43 1.169x1 O"4 12.32 2.464xl0"6 4.25 22.0 5.5xl0~3 96.92 4.846x10~3 75.88 I.518x10-4 23.93 4.786x10 ' 9.04 24.0 6.0xl0~3 98.26 4.913xl0"3 84.97 1.699x1 O"4 36.11 7.222xl0"6 15.16 26.0 6.5x10"3 98.83 4.941xI0"3 89.42 1 .788x1 O"4 45.80 9.160x10~6 21.08 28.0 ;.Od0"3 99.13 4.956xl0"3 91.90 1.838x1 O"4 53.17 1.063xl0"5 26.42 30.0 7.5xl0~3 99.31 4.965xIO-3 93.47 1.869xl0"4 58.86 1.177x1 O"5 31 .15 32.0 8.0xl0"3 99.42 4.97Ixl0~3 94.53 1.89lxl0"4 63.35 1.267x10"3 35.34 34.0 8.5xl0"3 99.;: 4.975xI0-3 95.30 1.906x10"4 o6.97 1 .339x1 O-5 39.07 36.0 9.0x10 99.57 4.M78XI0~3 90.88 1.918x1 o"4 69.95 1.399x10 42.40 -4 38.0 9. 5x10"3 99.62 4.98lxl0"3 96.33 1.927x10 72.44 1.449xlü"5 45.39 40.0 I.OxlO-2 99.66 4.983x10"3 96.70 1.934xl0~4 74.55 1.491xI(T5 48.09 i Figure 6 a 0.8 0.6 - q 3 0.4 •' i ' ' i ' • • 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 ml of titrant. 1.qA 2.qB 3- Figure 6 b 1.0 i 0.8 ml of titrant qA % ^c Figure 6b. The uncertainties in tha respective q's due to variations of 0.15 in the log K^ values of both A, B, C and R are shown by the shaded zones. - 64 - Table II-c (Figure 7) 10 8 20.0 ml 2.0 ml PH 2.5x10 c » 1.0x10 ^x.A 1.0x10 org -I 10.0 ml I...20 r - 2 1.0x10" c„ = 2.5x10 3.I6X1C 0X1 org tx,B Sx.R" '• ° c„.n.AV qA in I Mol of A in Z Mol of B qc inZ Mol of C q in 7. org R org R extracted extracted extracted 6 2.0 5.'3x10 9.85 4.927x10" 3.34 6.682xl0" 0.34 6.890x10 - 5 -7 4.0 l.OxIO 19.66 9.830x10" 7.18 1.437xI0" 0.77 1.536x10 0 . 24 6.0 1.5x10 29.40 I.470xl0~ 11.64 2.328xl0"5 1. 30 2.600x1O" 0 .41 -5 8.0 2.0xl0~ 39.06 1.953x10" 16.85 3.370x10 1.98 3.974x10 0 .64 10.0 2.5x10 48.60 2.430x10" 23.02 4.604x1o" 2.90 5.806x10" 0 .94 12.0 -3 -5 -7 3.0x10 57.95 2.897x10" 30.36 6.072x10 4.17 8.354x10 1 36 14.0 -3 3.5x10 67.02 3.351x10 39.12 7.824xlü" 6.03 1.207x10" 1 99 16.0 75.58 4.0x10" 3.779xl0~ 49.47 9.894x10" 8.91 1.7G3xlO~ 3 00 18.0 -3 83.24 -3 -4 4.5x10 4.162x10 61.10 I.222x10 iI3.58 2.716x10" 4 73 20.0 89.31 3 5.0x10" 4.465x10r 72.54 I .451x10" 20.89 4.178x10" 7 70 22.0 -3 93.26 -3 5.5x10 4.663x10 81.39 1 .628x10" 30.43 6.086x10" 12 15 -3 24.0 6.0x10 95.47 -4 I 4.773x10I 86.95 1.739x10 .0011x10 17 41 26.0 -3 96.70 -3 4 40.00 6.5x10 4.83 xiu 90.27 I .805xH)" 62bxlo" 22 69 28.0 97.44 4 K.8.I3 7.0x10 4.872xIO~ 92.34 1 .847xl0~ 093x1O" 27 60 30.0 -3 97.92 4 JS4.66 7.5x10 4.896x10" 93.72 1 .874xl(l" I97xl0~ 32 06 32.0 98.26 -3 4 ^9.87 8.0x10 4.913x10 94.69 l.894xlo" 281x10" 36 06 34.0 98.50 -3 4 jb4.07 8.5x10" 4.925x10 95.41 I .908xlo" 350x10 39 65 36.0 98.69 4 I&7.5I -5 9.0x10 4.934x10" 95.96 1 .9!9xl()" 407x10 U J_ 89 38.0 98.83 -3 4 70.37 9.5x10' 4.941x10 96.39 I .928xl()" .455x10 45 80 40.0 98.95 I.0x10 4.947x10" 96.74 1.935x10-4 72.77 .496x10-5 48 44 74.82 Figure 7 15.0 20.0 25.0 10.0 nil of titrant 1.'U 2.qB 3.qf - 65 - TABLE III Radiometric titrations of an aqu.'ous solution containing metal-ions A,B and C with the organic reagent RX . Keeping the concentrations constant, both the order of magnitude and the relative ratios of the k values are varied. Table III- a (Figure 8) V =- 20.0 ml = 2.0 ml pH - 2.0 5 6 4 \V cA = 2.0x10" cc = 6.0xl0~ K 0 K _ = 3. 1623xl0 o orS «.A* '-° * c tx,C 5 V - 1...20 a " b » c -• r * 2 org « 2 cB = 4.0xl0~ CR = .0x10 ' K K 3 I623xl0 0.0 ml " x.B* '-7783X ° Ex,R * " n.W org c„.n.5V iaZ Mol of A Mol of B q . in 7. Mol of C q in Z R org \ «B inZ ( R extracted extracted extracted 8 8 -9 9 2.0 4.0xl0~ 4.09 1.636xl0" 0.75 6.022x10 0.13 1 .616xl0~ < 0.01 8 8 8 4.0 8.0xl0~ 9.0! 3.6Q6xlo" 1.73 1.386xl0~ 0.31 3.749x10"° < 0.01 8 8 8 6.0 I.2xl0~ 13.90 5.560x1o" 2.79 2.334xl0" 0.51 b.097xlo"9 < 0.01 8 8 8.0 1.6xlQ~ 18.65 7.460xlo" 3.91 3.l34xl0"8 0.72 8.636xl0"9 < 0.01 _Q 8 8 8 10.0 2.0xl0~ 23.21 9.284x1O" 5.10 4.08lxl0~ 0.95 1.136x10 < 0.01 8 7 8 12.0 2.4xl0" 27.57 I.l03xl0" 6.34 5.072xlO~ 1.18 1.427xl0"8 0.01 8 7 8 14.0 2.8xl0" 31.72 1.269xl0" 7.63 6.l05xlO~ 1.44 1.738xlo"8 0.01 8 7 8 16.0 3.2xl0" 35.66 I.426xl0~ 8.97 7.l77xlO~ 1 .72 2.066x!0~8 0.02 -7 8 18.00 3.6xl0" 39.38 1.575x10 10.36 8.288xl0"8 2.01 2.416x10 0.02 8 20.0 4.0xlC~ 42.90 !.7l6xl0"7 11.79 9.432xl0"8 2.32 2.785xl0~8 0.02 8 22.0 4.4xl0" 46.21 l.848xI0"7 13.25 I.060xl0~7 2.64 3.174xI0~8 0.03 3 24.0 4.8xl0" 49.33 1.9?3xlo"7 14.76 i.iaixio"7 2.98 3.584xlo"8 0.03 26.0 5.2xl0~8 52.25 2.090xlo"7 16.29 1.303xl0-7 3.34 4.0l4xlo"8 0.03 8 28.0 5.6xl0~ 54.99 2. I99xl0"7 17.85 1.428xl0"7 3.72 4.464x10 0.04 8 7 30.0 6.0xl0~ 57.56 2.302X1O"7 19.43 I.554xl0~ 4.1 1 4.936xl0~8 0.04 32.0 6.4xI0~8 59.96 2.398xl0"7 21.03 I.682xl0"7 4.52 5.426xl0~8 0.05 34.0 6.8xl0"8 62.21 2.488xl0"7 22.64 1.81lxl0"7 4.94 5.93Sxlo"8 0.05 8 0.06 36.0 7.2xlO~ 7 7 64.31 2.572xlO" 24.26 I.941xl0" 5.39 6.468xl0"8 7 -7 0.06 38.0 7.6x10- s 66.27 2.651x10 25.89 2.071x10 5.84 7.019x10 — ~. 7 8 0.07 4n n 8.n«m-8 68.1(1 2.724xl0~7 27.52 2.202xlO~ 6.32 7.589xl0" Figure 8 1.0 T 0.B 0.6 1 q 0.» 0.2 2 3 0.0 *- 1 1 1 1 1 1 f 0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0 nil of litrant i.qA 2.qB 3.qc - 67 - Table III- c (Figure 10) 5 5 6 V » 20.0 ml 4V » 2.0 ml pH « 2.0 c. ' 2.0xl0~ c„ • 6.0xl0" K , « I.OxlO K or r EX." ' 1.0 xlO o * A -5 ** Ex, A , V - 10.0 ml n -1...20 a-b»c«r» K 3.1623xl02 org 2 cg - 4.0x10 cR • 2.0x10 IL • 1.0x10* Ex.R * n,AV c -n.AV qA in X Mol of A qBinZ Mol of B ". in J Mol of C org R org qR in X extracted extracted extracted 2.0 8 8 4.0x!0" >.29 2.ll7xl0" 0.06 4.469xl0~'° < 0.01 6.707xlO~" < 0.01 S 8 4.0 8.0xl0" 11.97 4.788xl0" 0.13 !.086xl0~ 0.01 1.W2X10-10 < 0.01 7 8 9 6.0 1.2xl0" 18.83 7.492xlo" 0.23 I.840xl0~ 0.02 2.767xl0~'° < 0.01 9 1 8.0 1.6x10 25.38 1.052x10 0.34 2.7I2xlO~ 0.03 4.081xl0"'° 0.01 7 9 10.0 2.0x10 31.84 1.274x10 0.46 3.7l9xlo" 0.05 5.603x10 '° 0.01 12.0 2.4x10 38.04 1.522x10 ' 0.61 4.882xl0"9 0.06 7.363xl0"'° 0.02 7 14.0 2.8x10 43.94 1.758x10 0.78 6.222xlo"9 0.08 9.399x10 Ü 0.02 7 16.0 3.2x10 49.51 l.»80xl0~ 0.97 7.769xl0"9 0.10 1.I76x)0~9 0.03 7 18.0 3.6x10 54.71 2.188x13~ 1.19 9.544xl0~9 0.12 1.447xl0~9 0.04 7 20.0 4.0xI0~ 59.50 2.380xl0~ 1.44 I.l58xl0"3 0.15 1.760x10 9 0.05 7 22.0 4.4x10 63.88 2.555xl0" 1.73 l.390xlo"8 0.18 2.118xl0~9 0.06 7 7 -9 24.0 4.8xl0" 67.83 2.713xI0" 2.06 I.652xl0~8 0.21 2.525x10 ' 0.07 7 26.0 5.2x10 71.35 2.854xl0" 2.43 1.946xl0"8 (1.25 2.982xlo"9 0.08 7 8 28.0 5.6x10 74.47 2.979xlO~ 2.83 2.268xlo" 0.29 3.49lxl0"9 0.09 30.0 6.0x10 77.21 3.088x10 3.27 2.621xlo"3 0.34 4.05lxl0~9 0.1 1 32.0 6.4xlO~ 79.59 3.l84xl0~ 3.75 3.002xlo"8 0.39 4.661x10 0.12 34.0 6.8x10 81.66 3.266x10" 4.26 3.410x10 0.44 5.320xl0~9 0.14 7 36.0 7.2xIO~ 83.45 3.342xl0" 4.80 3.842xl0~8 0.50 6.023xl0"9 0.16 ! (1.56 -9 0.18 33.0 7.6x10-7 85.01 3.400x10 5.36 4.293xl0" ' 6.767xl0 7 7 8 9 0.20 40.0 B.OxlO" 86.36 3.454x10 4.762X10" 0.63 7.548xl0" Figure 10 l.U U.B ./-''I 0.6 0.4 ,./•''' 0.2 , 2 0.0 3 0.0 5.0 10.0 15.0 20.0 25.0 30.0 :C..O JO.O rni of tit rant i.qA 2.qB 3.qf - 68 - Table III- & (Figure 11) V 5 12 20.0 ml flV - 2.0 ml pH = 2.0 c, = 2.0x10* c = 6.0x!0* K , - 1.0 x 0 K =3. 1623xlOlü org A _c C E x,A ii Ex.B v° - * I... 20 a»b"c-r* 2 cg ' 4.0x10* cR - 2.0XI0*' K^ „ •' 1.7783x 8 10.0 ml n x,B 0 K =3. I623xl0 org Ex.R n..lV c_.n.AV in Z Mol of A in Z Mol of B in Z Mol of C inZ org R org *A "B \ \ extracted extracted extracted 8 8 8 2.0 4.0xlO~ 6.76 2.705x:0* 1.27 I.018xl0* 0.23 2.74ÓX10*9 < 0.01 8 8 3 4.0 8.0xl0~ 13.26 5.304xl(.~ 2.64 2.117xl0* 0.48 5.771xiO~9 < j.01 8 8 6.0 1.2x10* 19.47 7.788xl0" 4.12 3.293xl0* U.75 9.l03xIU~9 < 0.01 7 ! 8.0 1.6x10 25.38 1.0I5xlO~ 5.70 4.563xIO*8 1.06 1.277x10*° 0.01 7 10.0 2.0x10*' 30.99 1.204xlO* 7.39 5.9l5xlO*S i .40 1.680xl0*8 0.01 7 7 12.0 2.4xl0~ 36.28 1.451xl0~ 9.19 7.354xlO*8 1.76 2.122xl0*8 0.02 7 14.0 2.8x10 ' 41.25 1.650xl0* 1 1.10 8.380xl0*8 2.17 2.606xl0~8 0.02 7 7 16.0 3.2xl0" 45.90 I.836xl0* 13.11 l.049xI0*7 2.61 3. Ij6xl0*8 0.03 7 7 18.0 3.6x10* 50.23 2.0I0X10* 15.22 1.218xl0* 3.09 3.7l2xlO*8 0.03 7 7 20.0 4.0-10 54.25 2-I70xl0* 17.42 I.394xl0* 3.61 4.33C.-10*8 0.04 7 22.0 4.4x10 57.98 2.319xl0* 19.70 I.576xl0*7 4.13 5.016x10*° 0.04 7 24.0 4.8x10 61.41 2.456xl0* 22.06 1.765x10* 4.7" 5.749x10** 0.05 7 26.0 5.2x10 64.57 2.383xlO~ 24.48 l.958xl()"7 5.45 6.540xl()"8 0.06 7 7 28.0 5.6xl0" 67.48 2.699xl0~ 26.95 2. I56xl0*7 6.15 7.390xl()"8 0.07 7 30.0 6.0x10 70.15 2.806xlO* 29.47 2.358xlO*7 6.91 8.299xl'J*3 0.07 32.0 6.4x10* 72. 59 2.904x10* 32.02 2.562X10*7 7.72 9.274xl0*a 0.08 7 7 7 34.0 6.8x10 74.33 2.993xl0~ 34.58 2.766xl0~ 8- ;. •> 1 .Ollxlo" 0.09 36.0 7.2x10 76.38 3.075xlO*7 37.16 2.973xl0*7 9.51 I.142x10* 0.10 38.0 7.6x10* 78.76 3. ISOxlO*7 39.73 3.l78xl0"7 10.49 1 .239xlO~7 0.12 7 7 7 40.0 8-OxlO* 80.48 3.2l9xlO*7 4.2.11) 3.384xl0* 11.53 1-J84.xi;j" 0.13 Figure 11 i.o -| 1 0.8 1 0.6 q 0.4 2 . 0.2 - 3 0.0 i " " ' | - •—— 1 : , : . 1 , 0.0 5.0 10.0 15.0 20.0 25.0 30.0 .)5.0 40.0 ml of til rant i.qA 2.qB 3.qc - 69 - Table 111-•e ( Fisure 12 1 V 20.0 ml iV = 2.0 ml pH = 2.3 c, = 2.0x10 c. = 5 12 10 A C b.Oxlo" 1C. = l.OxlO Ox.O org _:) 11 v° - -1...20 a-b=-c»r- 2 c - 4.0xl0 c ' 8 org 10.0 ml n g R !.0xl0 K„ „ - 1.0x10 K 3 I623xl0 Ex, D Ex.H • n.iV c..n.W qA inZ Mol of A inZ MM of B Mol of C org R org % "c in" qR inZ extracted »»(rart»d extracted 8 8 2.0 4.0xl0~ 8.00 3.2O3xl0" 0.86 O.903xl0"9 0.09 ó.957xlo"'° < 0.01 8 4.0 8.0x10 15.76 6. 304x1o" 1.83 1.469x10 0.19 1.494xlo"9 < 0.01 8 6.0 !.2xlo" 23.21 9.284xlo" 2.93 2.347xl0~8 0.30 2.4!lxlo"9 < 0.01 7 8.0 1.6xI0~ 30.33 I.2I3xl0~ 4.17 3.338xl0"8 0.43 3.468xl0~9 Ü.01 7 10.0 2.0x10 37.08 I.483xi0" 5.56 4.453xl0"8 0.59 4.687xlo"9 0.02 7 12.0 2.4x10 43.42 l.737xI0~ 7.12 5.702xl0"8 0.76 6.093xl0*9 0.02 7 14.0 2.8x10 49.32 1.973xl0" 8.86 7.095xlo"8 0.96 7.7I0xlo"9 0.03 16.0 3.2xlO~ 54.76 2.190xlO~ 10.80 8.640xl0"8 1.19 9.568xlo"9 0.04 7 18.0 3.6x10* 59.72 2.389xl0" 12.91 I.033xl0~' 1.46 1.169xlo"8 0."5 7 20.0 4.0x10 64.20 2.568xl0" 15.21 1.217xl0"7 1.76 l.4IOxlo"8 0.06 7 22.0 4.4xl0~ 68.22 2.729xl0" 17.67 1.4l4xlo"7 2.10 l.b82xlo"8 0.07 7 24.0 4.8x10 71.81 2.872xl0" 20.30 1.624xl0"7 2.48 1.986xI0"8 0.08 7 26.0 5.2xlO~ 74.98 2.999 0" 23.06 1.845xl0"7 2.90 2.327xlo"8 0.09 28.0 5.6x10 77.78 3.11lxl0~' 25.93 2.074xl0"7 3.38 2.706xl0~8 0.11 7 30.0 6.0x10 80.25 3.210xlo" 28.89 2.311xl0"7 3.90 3.123xlo"3 0.13 7 32.0 6.4x10 82.42 3.297xi0" 31.92 2.554xl0~7 4.47 3.582xl0"8 0.15 7 34.0 6.8x10 84.33 3.373xlo" 34.99 2.799xl0"7 5.10 4.036xlo"8 0.17 7 36.0 7.2xt0~ 86.01 3.440xI0" 38.08 3.046xl0~ 5.79 4.635xlo"8 0.19 7 38.0 7.6xl"" 87.50 3.5OOxl0"7 41.18 3.294xl0"7 6.54 5.234xl0"9 0.22 7 8 40.0 8.0xI0" 88.81 3-.552xH/"7 4.4.26 3.54lxin~7 7.35 5.88ixl0" 0.25 Figure 12 ml of lit rant i.q* 2.qB 3.qc - 71 - CHAPTER 5. MULTIELEMENT ISOTOPE DILUTION ANALYSIS BY MEANS OF RADIOMETRIC TITRATION SUMMARY A theoretical concept is derived for multielement isotope dilution analysis in a liquid-liquid extraction system. The practical execution is based on a radiometric titration. To that purpose a prototype of a titration/extraction vessel with continuous flow has been constructed. For calculations of the initial concentrations of the elements of interest from the experimental data, a computer program has been developed. - 72 - 5.1 INTRODUCTION Substoichiometric isotope dilution analysis is a useful tool in the determination of trace constituents. In general a fraction of the com ponent of interest is isolated by adding a substoichiometric amount of a specific reagent. The sample is diluted with a known amount of a ra dioactive standard solution and the isolated activity is compared to that of the undiluted standard. This method is based on the assumption of a linear relation between the amounts of added reagent and isolated radioactivity [1,2]. This simple correlation is not valid any more if the reagent is non-specific. This puts a severe limit to tie applicability of isotope dilution analysis. This text presents a theoretical concept and a practical procedure for the use of non-specific reagents in liquid-liquid extractions. The basic idea is that the concentration of the participating species may be derived from a radiometric titration in which the number of succes sive additions of the titrant equals or surpasses that of the consti tuents of interest. The activity is added as a spike or - in model experiments - by irradiating one of the components. The fractional ex traction into the organic phase is observed by measuring the decrease of activity in the aqueous phase. For the calculations, all conditional extraction constants must be known. Computer programs have been developed, one to predict the titration curve for a given set of initial concentrations (QFORW (*)T3]) and one to obtain the initial concentrations from the results of the titration (QBACK (*)). The second program may also be used to obtain values of the conditional extraction constants from experiments with complex stan dard solutions. A titration vessel with continuous flow has been constructed which enables an uninterrupted observation of the extraction process. The applicability of the technique is demonstrated by means of the ti- -4 -6 tration of aqueous solutions containing 3x10 M Cu(II) or 3x10 Cu(II) and 10 M Pb(II) respectively with Zn(DDC)„ dissolved in chloroform. 64 In this experiments use is made of Cu (Tj = 12.8 h ; E =511 keV) as the radiotracer. (*) Both computer programs are given in Appendix II and IV respectively. - 73 - 5.2 PRINCIPLE 5.2.1 Definitions Y(y) - metal Y with valency y, where Y = A,B,..,N,R X - monovalent chelating agent, used as re .gent * c , c„ - initial concentrations of the metal ions in the aqueous y R phase c , c - concentrations of R (as RX ) and X respectively in the K A IT organic solution of reagent to he added V - volume of the aqueous phase Vor g - initial volume of the organic phase AV org - volume of the separate additions of the reagent solution n - number of additions CY'],[R'] - concentrations of all species of Y and R respectively in the aqueous phase [4]. 5.2.2 Theoretical considerations Titrating an aqueous solution of metal ions with a chelating agent, X, dissolved as the complex RX in an organic solvent, after each addition the following displacement reaction will reach equilibrium. r Yy+ + y (RX ) t r(YX ) + y Rr+ (1) ' r org y org The equilibrium constant K for this reaction can be given in terms of an< the conditional extraction constants Kgx y * Kgx R : (K >r LX YY Kg = ^ ' (2) (KÈx,R) where [YX ] .[H]y fcX,ï [Y'].CHX]y org and - 74 - r [RXrJnr.[H] ^x R = ?r (*> bX'K [R'].CHX]r org Valency indications will be omitter* in this text. As an acceptable simplification, it can be assumed that the undissocia- ted organic acid HX as well as the complexes YX and RX are extracted quantitatively, whereas the ion^c species are kept quantitatively in the aqueous phase. The mass balances are: [Y'].V + [YX ] .(V° + n.AV ) = cv . V (5) y org org org' Y o * [R'].V + [RX ] .(V + n.AV ) = c .V + c .n.AV (6) r org org org nR nR org |[HX] + y[YX 1 + r[RX J j.(V° + n.AV ) = r.n.AV (7) [ org J y org r org] org org' X org Combining equations (3) with (5) and (4) with (6; respectively and sub stituting the results into equation (7), one obtains: * n.AV N f org \ n.AV r-(cR + CR'——*J rg X V r [HX]° = c . V° + n.A23V- V° + n.AV [Hj org org org org + ^x,R-[HX]oor g Y(y) = N(n) y.c Y £I (8) V° + n.AV [H]y Y(y) = A(a) -2IÊ 2I£ + y V KI v.[HX] tx,Y org The extraction efficiency,q, for Y is defined by the ratio of the amount of Y in the organic phase tot the total amount of Y present in the sys tem. [YX ] .(V° + n.AV ) q y org org org (9) Y cY.V - 75 - Using a derivation similar to that of equation (8), it can be shown that equation (9) can be written as: (V° - n .AV )/V org org (10) v° + n.AV TH]y org org + V .[HX]y "Ex ,vY org After each addition of titrant RX , the technique of isotope dilution enables one to observe a number of activities for Y. Since the activity of Y is closely related to the qv and the q-values of all participating metals are correlated, measurement of the activity of one isotope only is sufficient. As the theoretically predicted value for the activity of Y after the n addition of reagent is derived from the pre-estimation of the concentrations, the least-squares method can be applied to determine the unknown values of the initial concentrations. Equation (10) is used repeatedly to calculate required [HX] values. The computer program (QBACK) by the aid of which the calculations can be carried out, was run on a CYBER 175 computer and required 39.000 words of central memory. 5.2.3 Optimization of the K* - values An optional feature of the computer program QBACK can be used to calcu late the Kl -values from the experimental data obtained by titrating standard solutions. To start the iterative calculations, a first approx imation of these values is needed. For this purpose literature data are used. Next, the individual Kl -values are optimized. - 76 - 5.3^EXPERIMENTAL -4 Titrations of 3x10 M Cu(II) solutions were carried out to check the apparatus. Experiments with a mixture of 3x10 M Cu(II) and 10 M Pb(II) were performed to control the procedure. The pH was kept at 3.4 + 0.3 which is within the pH range in which the formation of HX is negligible and the metals are present as ions. The concentration of Zn(DDC)? (reagent) in the organic solvent to be added was 10 Ti in the first experiments and 2.5x10 M in the second one. 5.3.1 Chemicals and equipment - Double distilled water; chloroform 'pro analysi'; concentrated HNO- 'purissimum'; concentrated NaOH 'pro analysi' - Cu-foil and Pb(N0_)2 'pro analysi'. - Zn(DDC)„, prepared according to ref.[5]. - A 250-yl syringe "3aton Rouge'. - A titration vessel (see section 4.2.). - A 5-cm liad shield. 3 - A 43 cm coaxial Ge(Li) detector connected to a 400-channel analyzer with automatic readout. 64 The apparatus applies the net photopeak of Cu at 511 keV. The usual counting time is 300 - 600 s. 5.3.2 The titration vessel The array is shown diagrammatically in Figure 5.1. Figure 5.2 of ZN(DDC)2 in chloroform. It is transported to the organic phase by the flowing of the aqueous solution. The counting vessel has a volume of about 10 ml. - 77 - The time needed to reach equilibrium after each addition of reagent is approximately 40 minutes (cf. Figure 5.4), FLOW DIRECTION *AQ 1 GLASS FILTER TO SEPARATE THE AQUEOUS PHASE FROM POSSIBLE ORGANIC SOLVENT ..MIST" v0RG< XJÈrPJ t INJECTION SYSTEM FOR ADDITION OF REAGENT FIG. 1: DIAGRAM OF THE EXTRACTION / TITRATION VESSEL. Figure 5.2 Total extraction/titration apparatus with counting facility - 78 - Figure 5.3 Detailed picture of titration vessel 5.3.3 Procedure A Cu-foil of 20 mg is irradiated for 3 minutes at a thermal neutron flux of 5x10 cm s resulting in an activity of 1.3 mCi Cu. The metal is dissolved in 100 pi cone. HNO , the solution is diluted to 100 ml and the pH is adjusted to 3.4 + 0.3 by addition of cone. NaOH. For the experiments carried out to check the apparatus, this solution is iiiade up to 1000 ml. In het case of experiments with a mixture of Cu(II) and Pb(II) a 250-ml -1 . -4 aliquot of a 4x10 M Pb(N0 )„ solution is added to 10 ml of the 3x10 M Cu(II) solution and the mixture is made up to 1000 ml with double distilled water. -2 A 10 M stock solution of Zn(DDC)„ is prepared by dissolving 361 mg of the compound prepared according to Wyttenbach et al in 100 ml chloroform. It is kept in the dark to avoid decomposition by the influence of u.v. light [5]. Twnty five ml of the aqueous solution to be titrated are introduced in the lower part of the vessel and 50 ml in the upper part. Then 25 ml chloroform are introduced in the lower part and the system is closed. The aqueous phase is pumped arcund to remove all air from the tubings. Measurement of the count rate per 5-10 minutes is repeated until the net peak, corrected for decay has become constant. - /y - Titration comprises 4-5 additions of 200 or 400 yl reagent solution respectively. After each step the system is allowed to reach equilibrium and for statistical reasons, the countrate is measured 5-8 times over a 5~ 10-minutes period. For each observation the time elapsed since the beginning of the titration, the counting result as well as the number of the addition are recorded. After the titration is finished, the following data - needed for calcu lation of the initial concentrations by computer - are recorded on punch cards. Card 1 : V ;[H ];AV ; c ; number of elements, R included. Card 2 : estimation of the concentrations : CA' CB V CR ' Card 3 : valencies of A, B, R : *» J D j • * • • 9 r• Card 4 : the conditional extraction constants of the element with the uncertainties : ^x.A' AKÉx,A ' ^x.R' AKÊx,R * Card 5 : values of the A's of the radioisotopes used. E.g. if only the activity of element C is added to the aqueous phase, this is recorded as follows: 0.0 0.0 0.693/Tj(C) 0.0 0.0 Card 6 - card z : on each card the essential data of one observation, i.e.: the number of the addition ; time elapsed since the titra tion was started ; the element of which the activity is measured (1,2, ... or n) ; the net peak. 5.4 RESULTS 5.4,1 Experiments to check the apparatus Figure 5.4 gives the variation of the observed count—rate with time after additions of reagent in a radiometric titration of Cu(II) with Zn(DDC)_. It can be seen that the time meeded to reach equilibrium is about 40 min. t 1x10 m 9xKf 5 8x 10 er i •- z S 7x10* 6x10 5x10 60 120 180 240 300 360 TIME IN MINUTES FIG. i VARIATION OF THE OBSERVED COUNT-RATE WITH TIME IN A RADIOMETRIC TITRATION OF Cu (B) WITH ZN (DDCI2 The results of a titration of a 3.21x10 "* M Cu(II) solution with a -2 1.0x10 M Zn(DDC)_ solution in chloroform are summarized in Table 5.1, The q-values predicted by the program 'QF0RW' as well as the experimen tal q-values are given. It follows that there is a good agreement be tween theory and experiments carried out in the titration apparatus. After each addition of reagent, n measurements of the radioactivity of the aqueous phase in equilibrium state have been done. TABLE 5.1 Theoretical t ind experimental q-valiLie s (in %) for the 4 2 titration of 3. 21xl0~ M Cu(II) wi th l.OxlO" M Zn(DDC)2 dissolved in chloroform. The vilue of the pH is 3.2 . L 8 = 12 .81 + 0.07 3 ; log 2 3 ° Vcu «Ex.Zn " ' t 0-2 [3]. Meq Zn(DDC)2 q (theoretical) q (experimental) n added in % in % 2.0xlO"3 8.31 8.07 + 0.21 6 4.0xlO-3 16.61 16.63 + 0.13 5 6.0xlO-3 24.92 24.96 +0.11 5 3 8.0xl(f 33.22 33.24 + 0.05 5 - 81 - 5»4»2 Radiometric titrations of some solutions of metal ions A few titrations of aqueous solutions of metal ions with ZN(DDC). dis solved in chloroform have been carried out to obtain data that could serve as input for the computerprogram 'QBACK*. Since the titration apparatus gives reliable results, with those data the exactness of the calculations by computer can be hecked out. 5.4»2.1 Titration of a 3»2xl0~4 M_Cu(II) solution with a 1.0xlO~2_M Zn (DDC)? solution in chloroform. With the ions of only one metal in the aqueous phase, in principle one addition of reagent suffices to obtain the data required for the calcu lation of the initial concentration of that metal in the aqueous phase. In this experiment the influence of the number of additions of Zn(DDC) on the accuracy of the calculated initial concentration of Cu(II) is investigated. The results of this experiment are summarized in Table 5.II, 5.6,2.2 Titration of a mixed solutioTi_cf_3^15xlO"^_M_Cu(III_and_lJ,Q8lQlL M_Pb(ïïl_wi£h_2i5xig^_M^niD^22-^i^^lYS^-i!:-£!!i2£2£°I5- In the case of a mixture of the ions of two metals dissolved in the aqueous phase, at least three additions of reagent have to be done (cf. section 5,1), In view of both the slow adjustment of equilibrium and the relatively long counting times, after these three additions the titration was finished. The results are given in Table 5,III. - 82 - Table 5.11 4 2 Titration of 3.21xl0~ M Cu(II) with a 1.0xl0~ M Zn(DDC)2 solution in chloroform. V = 75.0 ml ; V = 25.0 ml ;AV = 0.2 ml ; pH = 3.2 aq org org Kl „ - 12.81 + 0.07 ; Ex,Cu «Ex.Zn = 2'3 " °'2 4 •1 >Cu = 9.02344xl0~ (min)~ • number of time elapsed net peak in [Cu(II)] calculated by computer addition since T in equilibrium o min. state 0 0 82725 after after after after 0 6 82464 1 add. 2 add. 3 add. 4 add. 0 12 81720 0 18 81605 0 24 81261 3.30 x 3.21 x 3.20x 3.20x -4 -4 0 30 81085 10~4 M 10 M 10 M 10_4M 81 70940 + 7.9 x + 3.7x + 2.2x + 1.5x 87 70343 JO"6 M 10"6 M 10_6M 10"6M 93 70323 99 69760 105 69155 111 68838 2 162 59706 2 168 55423 2 174 59079 2 180 58610 2 186 58553 3 231 50385 3 237 50224 3 243 50046 3 249 49682 3 255 49463 4 300 42224 4 306 42040 4 312 41728 4 318 41564 4 324 41289 - S3 - Table 5.Ill Titration of a mixed solution of 3.15x)0~ M Cu(II) and 1.0xl0~ M -4 Pb(II) with 2.5x10 M Zn(DDC)2 dissolved in chloroform. V = 75.0 ml ; V = 25.0 ml ; V = 0.4 ml ; pH = 3.1 aq org org Kl _ = 12.81 + 0.07 ; Kl _. = 7.0 + 0.1 ; K' „ = 2.3 + 0.2 Ex.Cu - Ex,Pb - Ex.Zn A, - 9.02344xl0~4(min)-1. Cu • number of time elapsed sine? net peak in equi CCu and CPb addition T = 0 (in min) . librium state calculated 0 0 11763 0 15 11485 0 30 11558 0 45 11254 0 60 11209 0 75 10933 cCu = 3.11x 130 6402 10~6 M + 145 6211 6.8xl0"8 M 160 6190 175 6036 cp, = 1.07x 190 6001 10 M + 2 250 2697 2 265 2603 3.3xl0~3 M 2 280 ?520 2 295 2497 2 310 2433 2 325 2486 - 84 - 5.5.REFERENCES [1] Tolgyessy,J., Braun,T., Kyrs.M., Isotope Dilution Analysis, Pergamon, 1972, Oxford. [2] Ruzicka,J., Stary.J., Substoichiometry in Radiochemical Analysis, Pergamon, 1968, Oxford. [3] Ooms, P.C.A., Leendertse,G.P., Brinkman,U.A.Th., Das.H.A. unpublished results. [4] Ringbom,A. Complexation in Analytical Chemistry, Interscience, 1963, New York. [5] Wyttenbach,A., Bajo.S., Analytical Chemistry, 47 (1) 1975 - 85 - CHAPTER 6. ISOTOPE DILUTION ELECTRQANALYSIS SUMMARY Isotope dilution electroanalysis presents an analogon to isotope dilution analysis by liquid-liquid extraction. As a first tentative, two different methods for the determination of cadmium by isotope dilution electroanalysis at controlled potential have been investi gate. The first method consists in comparing the remaim...^ .jdioactivity of two 5-ml counting aliquots taken from two Cd(II) containing solu tions after simultaneous electrodeposition of equal amounts 01 cadmium at the mercury pool cathodes of two serially connected identical electrolytic vessels. The second method is based on the comparison of the radioactivities remaining in the solutions after subsequent electrodeposition of equal amouits of cadmium from both a sample and a standard solution. In this case, both electrolyses are carried out in the same cell. To ensure electrodeposition of equal amounts of cadmium, use is made of a coulo- m(ter. - 86 - 6.1. INTRODUCTION The application of substoichiometric radioisotopic dilution principles to controlled-j-otential coulometry was introduced by Ruzicka and Benes [1]. They developed a method for the determinat' >ilver, using two identical electrolytic cells which are connected in series. Landgrebe et al. [2] extended the method to other metals, using mercury pool cathodes instead of silver cathodes. In the present work the procedure with two serially connected electro lytic cells is compared to that with one single cell. Special attention was paid to the reproducibility and reliability of both techniques. Because of its favourable electrochemical properties, cadmium has been used as test element. 6.2. PRINCIPLE 6.2.1. Procedure I: Two electrolytic cells connected in series If two identical electrolytic cells are connected in series and a known amount (M ) of the element of interest is added to the control c cell and a sample containing an unknown amount (M ) is introduced into the sample cell, equal known amounts (Y) of a radioisotope of the element with activity A are added to each cell. c During the electrolysis equation (1) is valid. control cell = sample cell il) Assuming an equal current yield in both cells for the element to be analyzed, the result is a deposition of equal amounts (M) of the metal. After the electrolysis, the decrease of radioactivity of the solutions in each cell is determined and M is calculated as follows. s In the control cell: AA A C C A(M + Y) - M • Y (2) c c - 87 - In the sample cell AA A c (3) A(M .) M + Y s s Moreover, equation (4) holds: A(M + Y) = A(M + Y) = M (4) Combination of equations (2) -(4) gives: (5) In order to obtain the condition of equal-current yield in both cells, the following conditions must be fulfilled: a) Sample cell and control cell have to be identical, since the migra tion of the metal ions to the cathode is determined by the cell ge ometry (distance anode-cathode; area of both electrodes). Besides, stirring of the solution in both electrolytic cells must be done in an identical way. b) No side reactions may occur at the cathodes, unless these reactions occur in both cells to the same extent. As an example, the chemical reaction of 0„ with the electrolyzed metal may interfere the analysis. This interference can be prevented by a prolonged pre-electrolysis. c) The potential at which the electrolysis will b*- carried out must be chosen in such a way that no electrochemical side reactions may occur in one or both cells. Therefore the concentrations in sample and control cell must be of the same order of magnitude and the electrolysis potential should not be too negative. b) Temperature in both cells must be the same. 6i2i2i_Procedure_IIi_Electroly_sis_in_one_single As an alternative to simultaneous electrolysis in serially connected ceUs, electrolysis of sample and standard solution can be carried out separately, using a single electrolytic cell. Again, electrolysis takes - 88 - place at controlled potential; however, different from the procedure described in section 6.2.1., now a coulotneter has to be inserted in the circuit to register the amount of electric charge used up. 6.3. EXPERIMENTAL 6i3i2^._A2EêE§ÏH5 A life appearance of the serially connected electrolytic cells employed is shown in Figure 6.1. The circuit is represented diagrammatically in Figure 6.2. Figure 6.1: Life appearance of two serially conne-fd p ?-jctrolyt ic cells ifiEF "° 'pOTENTIOSW SAMPLE CELL CONTROL CELL FIG. 2: DIAGRAM OF APPARATUS FOR ELECTROLYSIS AT CONTROLLED POTENTIAL IN SERIALLY CONNECTED ELECTROLYTIC CELLS - 89 - The anodes consist of platinum wires, bent in the shape of a helix with total length of about 3 cm. They are separated from th solutions by a fritted glass filter. The reference electrodes used are identical sat urated calomel electrodes of a commercial type. 2 The cathodes are mercury pools of about 1.5 cm in area. The me" -iry is added to the cells by the aid of a glass shovel (Figures o.3 and 6.4) which permits reproducible results in dosing the mercury. Figure 6.3: Glass shovel for dosing mercury in a reproducible way Figure 6.4: Practical execution of dosing mercury with the aid of the glass shovel - 90 - The solutions in both cells are stirred by means of two identical glass stirring rods fitted to an axis driven by the same stirring apparatus. Separate provisions were constructed, permitting to take counting sam ples without removal of the caps, i.e. continuing stirring and main taining the nitrogen pressure inside the cells during those samplings. Figure 6.5 shows the electrolytic vessel used to pre-electrolyse solutions of supporting electrolyt for Ions periods of time. I«-N, MS S3 0 0 SC.E _JSa £ O MAGNETIC STIRRER FIG. 5: ARRANGEMENT FOR THE PRE- ELECTROLYSIS Of SUPPORTING ELECTROLYTE SOLUTION. E - POWER SUPPLY Pi -PLATINUM INDICATOR ELECTRODE Hq-MERCURY POOL WORKING ELECTRODE S C E. - SATURATED CALOMEL ELECTRODE V - DIGITAL VOLTMETER In all experiments use was made of the potentiostat, built according to Wenking. The current integration equipment was of the Wenking SS1-70 type. Two digital voltmeters, constructed at the Free University at Amsterdam were used for the cell voltage measurements. - 91 - §ili2i_Reagents A stock solution of the supporting electrolyte KC1 was prepared by dis solving 0.1 Mol KC1 'suprapur* in 1 1 double distilled and boiled water. Further purification of this solution was accomplished bv long time electrolysis at a controlled potential of -1.0 V .s. S.C.E. in the pre- - electrolytic vessel shown in Figure 6.5. The Cd solution to be analyzed was prepared !-\ dissolving about 85 mg Cd in 100 pi KN0_ cone, 'suprapur' after irradiation during 15 min at a thermal neutron flux of 5 x 10 ^m "s . Th* Cö solution was Jilted with a 0.1 N KC1 solution, prepared as described above, to 500 ml. 115 measurements of the radioactivity were carried out on "Cd (T{ = 53.5 hr ; E « 485 and 530 keV). 6.3.3^_Procedures Procedure I To each of the cells was added 25 ml of 0.1 M KC1 followed by 5.0 ml 0.1 M KCl, containing the Cd. Before the electrolysis was started, the solutions were deaerated by passing nitrogen. Both cells were pre- electrolyzed at -400 mV vs. S.C.E. until the current became constant. Then the electrolysis was' started, the control cell potential being held at -650 mV vs. S.C.E. The time of electrolysis was 20 - 30 min. Without interruption of the electrolysis, 5 ml aliquots of both standard and sample solution were simultaneously withdrawn from each cell. They were counted on a Philips PW4580 automatic gamma analyser with 3" x 3" perforated Nal detector. Procedure_II Procedure II does not differ essentially from procedure I, except for the fact, that both sample and standard solution were electrolyzed separately after each other in the same cell. - 92 - 6.4. RESULTS 6.4.1. Results from Procedure I Original Count-rate Count-rate count-rate in ref. cell after elec [Cd] in in both after elec trolysis [Cd] found % of both cells cells, in trolysis in sample cell in sample deviation before elec c(5min) c(5min) in c(5mL») cell in sample trolysis per total per total per total in 2.54 x 10~4 39519 34170 34332 2.62 x 10~4 + 3.15 2.54 x 10~4 37523 31631 32306 2.87 x 10"4 +13.00 2.91 x 10~4 47380 39231 40009 3.22 x 10~4 +10.65 2.91 x 10~4 44988 37655 38118 3.11 x J0~4 + 6.87 6.4.2. Results from Procedure II Original Trans Deposition of standard Deposition of sample concen mitted in % in % in % tration charge in M in M relative relative relative of Cd in in C (+. 0.3%) (+ 0.3%) to to to mox.l (+ 0.3%) trans trans standard mitted mitted charge charge 2.61xl0~4 19.8xl0-2 9.851xl0-7 99.5% 9.89xl0~7 99.9 100.4 2.57xl0~4 20.1xl0-2 10.08 xl0~7 100.3% 9.96xl0"7 99.1 98.8 2.83x10-4 20.2xl0~2 10.01 xl0"7 99.1% 10.03xl0"7 99.3 100.2 2.83xl0"4 2O.lxl0"2 10.02 xio"7 99.7 10.05xlO~7 100.0 100.3 - 93 - 6.5. DISCUSSION The two procedures for substoichiometric isotope dilution electroanalysis do not differ essentially. Two advantages of the single cell procedure should be mentioned. - since electrolysis of both sample and standard solution is carried out in the same cell, no difficulties will arise due to the cell geometry. - different frota the set-up with the serial connected cells, electro lysis of both sample and standard solution will take place at con trolled potential. For the procedure with two electrolytic cells connected in series, the highest demands have been made on the similarity of the electrolytic vessels. In section 6.2.1. the conditions are listed that must be ful filled for equal current yield in both cells. Nevertheless, in the results from procedure I a systematic positive deviation from the real values of [Cd] is found. Most probably this is due to non-controllable electrochemical sidereactions in the sample cell. The systematic positive deviation as well as the poor reproducibility of the results from procedure I made it superfluous to develop a theore tical model for multielement isotope dilution electroanalysis at con trolled non-selective potential. Comparing the results obtained by applying the respective procedures, it is obvious that procedure II provides more reproducible data. The discrepancy between the results of the two procedures is due to the fact that in procedure I the potential of the sample eel? cannot be kept constant. 6.6. REFERENCES [1] Ruzicka, J. and Benes, P., Collection Czechoslov. Commun. 26 (1961) 1784. [2J Landgrebe, A.R., Mc Clendon, L.7., Devoe, J.R., Pella, P.A., Purdy, W., Anal. Chimica Acta 3< (1967) 151. - 95 - APPENDIX I DESCRIPTION OF THE COMPUTER PROGRAMMES QFORW AND QPLOT - 96 - I.I. DESIGN OF THE SOFTWARE The software for the calculations of the various q values plus associated errors (due to uncertainties in specified K„ values) as a function of the volume of the organic phase and providing an option to have these functions visualized in plots, consists of two FORTRAN main programmes and a number of subprogramnes. 1.1.1. QFORW This programme processes the inputdata that define the system of chemical elements, calculates and prints the q values at given volumes of organic phase, i.e. at given amounts of reagents RX added to the original organic phase. QFORW was intended to be of moderate size, thus permitting short turn-around times for its execution. If requests for plots are made to QFORW this only results in the creation of an appropriate plotdatafile (PLDATA) in addition to QFORW's normal calculations, whereas the space consuming plotpart is left to the second programme. 1.1.2 QPLOT This programme performs the actual plottask. Data defining the system of chemical elements, corresponding default values for the number of plots, for titles etc. are read from the file PLDATA (created in a previous QFORW run). Some additional information concerning the number of plots, the titles or the state(s) of the chemical system may still be given as input in order to replace the default values from the PLDATA file. Since some 28 plot subprogrammes are actually involved, the execution of QPLOT requires a significantly larger amount of central memory than that needed for QFORW. 1. 1.3. QMIMAPA_|XTEFFi_XZEROa_RMAIN The central part of the computations is the determination of a (HX) value for each state of the systt-j». This value is ob- org tained as the root of a nonlinear reduced balance equation (the main equation). Values of q are easily derived from the (HX) value. - 97 - If (HX) is to he computed for a sequence of states, a (HX) from a previous state will be used for the calcu lations at a next state. The computations are performed by a library routine -ZBRENT- and the subprogrammes QMIMAP, EXTEFF, XZERO and FXMAIN: QMIMA? - to compute the q value, with upper and lower error bound and (optionnaly) the p value of a given element in the system. EXTEFF - to compute the q value of a given element. XZERO - to find the root of the 'Win equation". FXMAIN - to evaluate the "main equations" lefthandside for arbitrary (HX) Both QFORW and QPLOT call the most general sub- programme QMIMAP. EXTEFF is called from QMIMAP and so on. Remark: If QFORW and QPLOT are to be used in different jobs - and the software was designed to enable this - arrangements must be made to save the PLDATA file. In the jobs available at ECN, to be run on a CYBER 175 computer under the NOS/BE operating system, the PLDATA files are cataloged as permanent files. 1.2. SUMMARY OF INPUT VARIABLES I.2.I. ISEH£.YÏliS^lÊS_i2_2E25H 1.2.1.1. Variables which describe the initial state of the system (first case) Nl - number of elements in the system, including the reagent metal (at most 11). HA - concentration of H in the aqueous phase. VAQ - volume of the aqueous phase. V0RG1 - volume of the organic phase at the initial state. C(l) - concentration of remaining elements at initial state (first case). C(2)...C(NJ) - concentration of remaining elements at initial state MU(1)...MU(NJ) - valencies of the elements TLG(1)...TLG(N1)- log 1L values for the elements - 98 - TLER(1)...TLER(N1) - errors in log IC values. 1.2.1.2. Variables defining the NDV states of the system which arise from the additions. DVORG - Volume of the organic phase in one addition. NDV - Total n-anber of additions. CRADD - Concentration of the reagent in the additions. 1.2.1.3. Variables defining NCA cases. Each case differs from a former one in its C(J) value. NCA - number of initial states, also referred to as: number of cases. FCA - factor, used to C(l)(case K)={C(1)(First case)}. (l.+FCA) 1.2.1.4. Some additional variables IPLOT - piotindicator : only if IPLOT = 1, data for QPLOT will be written to PLDATA. INUM - iteration counts indicator : only if INUM + 0, iteration counts from the numerical process are printed. The task of QFOP'J can be summarized as: Print q - and p - values of the NDV states for each of the NCA cases. In the output of QFORW the elements are denoted by A,B, ...etc. R is always used for the reagent metal. 1.2.2. JSEH£_YSEi2ÏilÊ5_ÏS_Q?t9ï NELP - number of elements for which the q-curve has to be plotted (may not exceed Nl) NIN1PL - number of q-curves in one plot, except possibly for the last plot. KEL(J)...KEL(NELP) - order in which the q-curves for the various elements have to be distributed over the plots. More precisely: the q-curves for elements numbered as KEL(J)...KEL(NINIPL) appear in plot 1, those for eleirants KEL(NIN1PL + 1)... KEL(2*NIN1PL) arpear in plot 2, etc. C(J) - initial concentration of first element (first plot case). NCAP - number of plot cases, i.e. number of initial plot states. FCAP - factor, used to modify C(l) from one plot case to another Note: QPLOT may handle several plotcases, a - 99 - plot case being defined similarly to a case in QFORW. VPMTN,VPMAX - lover and upper limit for the volume of added organic phase. Hence, VPMAX - VPMIN is the range of the independent variable in all plots. PTEXT(l)...... PTEXT(NELP) - Text strings of two characters to serve as the names for the elements. Element I is named PTEXT(l), etc. A name appears as a subscript to q in the legend of the plot. IERR - inclusion-of-error indicator. If IERR equals the string 'NO ERRORS', no errors in the log K_ values are assumed. No error bounds are drawn in the plots. Summarizing, for each of the NCAP cases QPLOT draws the q-curves (NINIPL per plot) of the NELP specified elements and shades the area between the error bounds. 1.3. INPUT SPECIFICATION 1.3.1. l5Eïï£_5E££Üi£5£i2S_£2L.2I251? QFORW expects its input - free formatted - as follows. on card 1 : VAQ HA V0RG1 DVORG CRAD Nl on card 2 : C(l) C(2) C(N1) on card 3 : MU(!) MU(2) MU(N1) on card 4 : TLG(l) TLER(l) TLG(2) TLER(2) TLG(Nl) TLER(NI) on card 5 : NOV NCA FCA IPLOT on card 6 : INUM Variables having a name starting with I,K,M or N require integer values to be specified on input. If Nl is large, more than one card may be needed for the specification of C(l). C(N1) and/or TLG(l) TLER(l) TLG(Nl) TLER(Nl). Further details on the usage of free formatted input can be found in the section 'Deviations from ANSI - FORTRAN'. Note: Deleting card 6 has the same effect as including card 6 with a zero for INUM. - 100 - Input specification for QPLOT The values of all variables needed by QPLOT to plot the q-curves are obtained fro» the PLDATA file. There «ay be situations, however, where other values are «ore appropriate. A number of variables - the QPLOT input variables described in this section - may therefore receive prescribed (Modified PLDATA -) v lues. Moreover, since the desired modifications not necessarily apply to all of the input variables, five types will be distinguished. Itmut to QPLOT then may consist of data for some or all of the types. If input is given, the first card oust be a type indi cator card. The five types, each type with its defining variables in the order as given, are listed below, followed by the (default) values they receive if this type card is missing. type 1: NCAP C(l) FCAP Nl C(l) FCA defaults type 2: VPMIN VPMAX 0 NDV*DV0RG defaults type 3: NIN1PL NELP KEL(1)..KEL(NELP) minumum(Nl,5) Nl 1... NELP defaults type 4: IERR 'ERRORS' defaults type 5: PTEXT(l) PTEXT(2) ...PTEXT(NELP) A B ... default? If input is submitted to QPLOT, it must be a card 1 with: KTYPE(l) KTYPE(5) which will be read according to F0RMAT(5I1) and the type cards specified by KTYPE(l)..,KTYPE(5). E.g. : card 1 contains : 15 This means card 2 will be interpreted as a type 1 and card 3 as a type 5 card. Data on cards of type 1 through 3 are free formatted. The same remarks hold as made for the QFORW input. A type 4 card is read using FORMAT(A9). If IERR - 'NO ERRORS', errors in K_ values will not be taken into account. The text strings PTEXT(l)...... PTEXT(NELP), to occur on the type 5 card, are read with FORMAT(11(A2,2X)). - 101 - 1.4. DESCRIPTION OF SUBPROGRAf4ES LOADED FROM LIBRARIES 1.4.1. DISSPLA routines [1] ANGLE (ANGM) This routine rotates text in messages ANGM-Angie of Messages in degrees (counterclockwise from horizontal) BARSHD (XCARAY, Y1ARAY, Y2ARAY, NBRS, BARWTH, ANGS, GATRAY, NGAPS, IWORK, NWORK) This routine draws shaded bars having arbitrary shade line angle and spacing. XCARAY - array containing X-values at center of bars. Y1ASAY - array containing lower or upper Y-values. Y2ARAY - array containing upper or lower Y-values. NBRS - number of bars to be drawn. BARWTH - dimensions of bar width > 0, bar width in inches < 0, ratio of bar width to distance between first two values, ANGS - angle of shading in degrees (counter clockwise from horizontal) GAPRAY - array or scalar of distance(s) between shading lines, in inches. NGAPS - such that NGAPS is length of GAPRAY. > 0, both shading and perimeter (1, if scalar) • 0, no shading, only perimeter. < 0, only shading, no perimeter. IW0RK - workspace array of at least 1.5 times the number of intersections of shading lines with perimeter of the tallest bar. NWORK - dimension of IWORK BGNPL (IPLOT) This routine resets all DISSPLA parameters to default values. IPLOT - serial number of the plot, if 0 no summary on printer. C0NNPT (XT0,YT0) This routine draws a straight line from the current position to a given point. - 102 - XTO, YTO- the x,y coordinates to the endpoint of the line, measured in inches from the physical origin. CURVE (XARAY, YARAY.NPNTS.IMARK) This routine draws the curve or plots the data points. XARAY - array containing the x-values. YARAY - array containing the y-values. NPNTS - number of points to be plotted. IMARK - number of points between each marker. 0 means no markers, points connected. DONEPL This routine closes the plot file. ENDPL (IPLOT) This routine ends the plot and creates a new physical page. IPLOT - serial number of the plot, if < 0, no summary on printer. FRAME This routine draws a frame around the subplot area. GRACE (GRMAR) This routine sets the margin around the subplot area, beyond which curves will be scissored, to an arbitrary value. GRMAR - width of grace margin around subplot area, in inches. Default: 0.5 inches. GRAF (XORIG, XSTP, XMAX, YORIG, YSTP, YMAX) This routine sets up the basic linear axes. XORIG } - values at the axis origin. YORIG XSTP } - step interval in user units. YSTP XMAX } - values at the end of the axis. YMAX HEIGHT (HITE) This routine sets the height of strings and labels. KITE - height in inches. Default: 0.14 inches. INTNO (INUM, XPOS, YPOS) This routine plots an integer in inches from the physical origin. INUM - integer written as a string of characters. XPOS } - distance from physical origin in inches. YPOS - 103 - Remark : If XPOS or YPOS has a value the string "ABUT" the position of the end of the last message will be assigned to the start of the current message. MESSAG (LMESS, IMESS, XPOS, YPOS) This routine plots a text. LMESS - characters to be written. IMESS - number of characters in LMESS. XPOS } - distance from physical origin in YPOS . . inches. Remark : If the string count IMESS is equal to 100, and the string LMESS is terminated by a dollar sign, then DISSPLA will calculate the length of the string internally. Strings are always written between quotes. MXiALF (ALPHA, LSHIFT) This routine relates one of DISSPLA's (i * 1,2,....6) eight alphabets to a so-called escape. character. The escape character, say '*', when encountered in a plottext (i.e. input textstring of routines like TITLE and MESSAGE) causes the characters in this string following '*' and upto a new escape character to be drawn in the corresponding alphabet. Among the 8 alphabets there is an instruc tion set to perform such operations as sub- and superscripting, underlining, etc. Following the escape character related to this instruction alphabet special com mand strings have to appear in the plottext. ALPHA - name of the alphabet, one of the string 'STANDARD' for Upper Case Roman 'L/C5T0' for Lower Case Roman 'GREEK' for Upper Case Greek 'L/CGREEK* for Lower Case Greek 'RUSSIAN' for Upper Case Russian 'L/CRUSSN' for Lower Case Russian 'HEBREW' for Hebrew or 'INSTRUCTION' for the Instruction alphabet. LSHIFT - escape character. PAGE (PAGEX, PAGEY) This routine defines page border, PAGEX - horizontal dimension of page in inches PAGEY - vertical dimension of page in inches. PHYSOR (XPHYS, YPHYS) This routine causes routine TITLE to set the physical origin at a given point. XPHYS }- X,Y coordinates of the physical origin XPHYS measured in inches from the lower lefthand page corner. RESET ("NAME") This routine resets the parameter set ting subroutine NAME to its default value. SHADE (XARAY, YARAY, NPNTS, ANGS, GAPRAY, NGAPS, IWORK, NWORK) This routine shades any closed contour (without drawing a curve around the perimeter of the contour). XARAY - array of x-values defining the contour YARAY - array of y-values defining the contour NPNTS - number of points in the contour definition. ANGS - angle of shading in degrees (counter clock' wise from horizontal) GAPRAY - array or scalar of distance(s) between shading lines, in inches. NGAPS - length of GAPRAY. IWORK - workspace array of at least 1.5 times the number of intersections of shading lines with the contour. The actual workspace used by SHADE is returned in IWORK (1). NWORK dimension of IWORK. Remark If the first and last points in the arrays XARAY and YARAY are not the same - 105 - (i.e., the contour is not closed), they will be assumed to be connected by a straight line. STRTPT (XP, YP) This routine positions the pen at a given point. XP, YP - X, Y coordinates of the point, measured in inches from the physical origin. TITLE (LTITLE, ITITLE, LXNAME, LYNAME, IYNAME, XAXIS, YAXIS) This routine specifies the subplot area LTITLE - title of the plot (not used). ITITLE - number of characters in title. LXNAME - x-axis label. IXNAME - number of characters in x-axis label. LYNAME - y-axis label. IYNAME - number of characters in y-axis label. XAXIS - x-dimension of subplot area in inches. YAXIS - y-dimension of subplot area in inches. TRIPLX This routine sets the character style serif, triple stroke, 21 units high, using a variable letter width. XPOSN (XÜNX, YUNY) This routine calculates the x-distance in inches (from the physical origin) of a point given in units of the axes XUNX } - X,Y coordinates of point in units of YUNY x-axis and y-axis respectively. YAXANG (ANG) This routine causes angled labels on y-axis. ANG - angle from horizontal, in degrees. YPOSN (XUNX, YUNY) This routine calculates the y-distance in inches (from the physical origin) of a point given in units of the axrs. XUNX } - X,Y coordinates of point in units of YUNY x-axis and y-axis respectively. A further explanation of terms used in the description of the DISSPLA subprogrammes can be found in [1] - 106 - CALCOMP^ROUTINE [2] NEWPEN (KPEN) This routine causes a new penholder to be selected KPEN - number of the penholder to be selected KPEN may be 1,2 or 3. . IMSL ROUTINES [3] VSRTPM (A,N,IPERM) This routine sorts an array by absolute value into ascending order and returns the permutation vector. VSORTP Entry of VSRTM, to sort arrays by algebraic value. A - On input: array to be sorted On output: sorted array (of absolute values if VSORP is used) N - number of elements to be sorted IPERM - Integer array of length N On input: IPERM should contain the integers 1,2 N (i.e., IPERM(I) - I) on output: IPERM contains a record of the permutations made on A. VSORTZ (Z, IDIMZ, NROWZ, NCOLZ, IND, IPERM, SCR) This routine interchanges the rows or columns of a matrix using a given permutation vector, (e.g., as obtained from VSRTPM/VSORTP) Z - on input: matrix of dimension NROWZ by NCOLZ of which the ^-ows/columns have to be interchanged, on input: Z contains the resulting matrix. IDIMZ - Rowdimensisn of Z as specified in the calling programme (> NROWZ) NROWZ - Number of rows in the matrix. NCOLZ - number of columns in the matrix. IND - input parameter. If > 0, rows have to be interchanged. - 107 - Otherwise, columns will be interchanged. IPERM - integar permutation vector of length NROWZ if IND > 0 and of length NCOLZ otherwise;IPERM is destroyed on output. SCR - scratch array of length NROWZ if IND > 0 and of length NCOLZ otherwise. ZBRENT (FUNC, EPS, NSIG, A, B, ITMAX, IER) This routine determines the zero of a function which changes sign in a given interval. FUNG - user supplied external FUNCTION sub- programme to evaluate the function at any point in the interval. EPS - first convergence criterion. An approximation, B, is accepted as a zero if |FUNC(B)| £ EPS. NSIG - second convergence criterion. An approximation is accepted as a zero if it agrees with the former approxima tion to within NSIG decimals. A, B - on input : two points specified by the user such that FUNC(A) and FUNC(B) are opposite in sign. on output : A and B are altered. B will contain the best approximation of the zero of the function FUNC. ITMAX - on input: maximum number of iterations required to find the zero, on output: the actual number of iterations required to find the zero. IER - error return parameter. IER • 0 normal return (no errors) IER » 129 no convergence in ITMAX iterations IER - 130 input error ; FUNC(A) and FUNC(B) have the same sign. Remark : An approximation is accepted as a zero if either the first or the second con vergence criterion is satisfied. - 108 - NUMRCN (a local library at E.C.N. Petten) ROUTINE VIP (A, IA, B, IB, N, S) This routine, written in assembler, determines the innerproduct of two vectors a * (a,, a.,...,a ) and • - n (b,, b„,...,b ) defined as: l I n n a.b. and 0 for n = 0. Z l l i-1 The vectors are assumed to be located in the arrays A and B respectively, a. is stored in A(I), a. in A(1+IA), a. in A(I+2IA) and so on. A similar convention holds with res pect to b. A - Array containing vector a. IA - Storage increment between two succes sive components of a. B - Array containing vector b. IB - Storage increment for two successive b-elements. N - Length (n) of the vectors a and b. S - Innerproduct of a and b. VIPA Entry of VIP: the innerproduct is added to S. VIPS Entry of VIP: The inner product is subtracted from S. VIPD, VIPDA, VIPDS Entries of VIP, doing the inner pro duct accumulation in double precision. Have the same purposes as VIP, VIPA and VIPS respectively. - 109 - DEVIATIONS FROM ANSI-FORTRAN [4] In the QPLOT programme use is made of the following NON-ANSI FORTRAN input/output statements , which are Control Data extensions to ANSI FORTRAN. DECODE and ENCODE : to reformat data in memory, i.e.: to transfer information from one area of memory to another under FORMAT specifications. DECODE (c,fn,v) iolist This is similar to a formatted READ ENCODE (c,fn,v) iolist Is similar to a formatted WRITE c - unsigned integer constant or simple in teger variable specifying the length of each record, fn - FORMAT designator. Similar as its usage with READ/WRITE statements, v - variable or array name supplying the storage location of the record to be decoded/encoded. iolist - list of variables to be transmitted to/ list to receive variables from the lo cation specified by v. The first record starts with the leftmost character of the location specified by v and continues for c characters. In its usage in QPLOT - where c = 10 - each new record begins with a new computer word. In both QPLOT and QFORW use is made of: READ (u,*) iolist : To read data in a free format, i.e.: data s transmitted without using a FORMAT specification. This list directed input can be applied for coded records, u - input unit designator, used to determine the logical file name of the input file. In QPLOT and QFORW u is 5, which causes TAPE5 to be selected as input file name. In the PROGRAM statements TAPE5 has been equivalence^to the file INPUT, hence data is actually read from cards. - 110 - iolist - list "of variables defining the storage locations for the data. Input data consists of a string of values separated by one or more blanks» or by a comma or slash (preceeded or followed by any number of blanks). Both programmes call the Input/output status checking subprogram», included in the FORTRAN system library: EOF (u) to test for an end-of-file condition. u - input unit designator, similar as in READ (u,*) statement. The EOF-FUNCTION returns with a zero if no end-of-file is encountered, with a non-zero otherwise. 1.6. REFERENCES [1] DISSPLA - Beginners and Intermediate Manual, Vol. 1, Integrated Software Systems Corporation, San Diego, California. [2] CALCOMP - Software Reference Manual, No 1005, California Computer Products Inc., Anaheim, California. [3] IMSL - Library 3 Reference Manual, edition 6 , International Mathematical and Statistical Libraries, Houston, Texas. [4] FORTRAN EXTENDED version 4 Reference Manual, Pub. 60497800, Control Data Corporation, Sunnyvale, California. APPENDIX II TEXT OF THE COMPUTER PROGRAMMES QFORW AND QPLOT - 112 - PROGRAM QFORW(INPUT»OUTPUT»PLnATA,TAPE5*INPllT»TAPE6=Ol.lTPUT» * TAPE7=PLDATA) C QFORW .PREDICT Q-VALUES AND ASSOCIATED ERRvRS DUE TO C • UNCERTAINTIES IN TME K-EX VALUES C C C QFORW READS FROM INPUT ALL VARIABLES DEFINING C THE INITIAL STATE OF THE SYSTEM OF CHEMICAL C ELEMENTS C THE NUMBER OF SUBSEQUENT STATES ARISING TROM C AUDING ORGANIC PHASE C THE TOTAL NUMBER OF INITIAL STATES WITH DIFFERENT C CONCENTRATIONS OF THE FIRST ELEMENT C AND PRINTS P- AND Q-VALUES (PLUS ERROR BOUNDS FOR C 0) FOR ALL EXAMINED STATES C A SPECIAL FILE»PLDATA. WITH DATA FOR THE OPLOT-PROGRAM C IS CREATED IF A PLOTREQUEST IS MADE ON INPUT C C C C DIMENSION TLG(11).TLER(11)»TKMM(11»3)»QMM(3)»ITER(3) INTEGER TEXT(11) C ARRAYS IN QFORW ARE PROPERLY DIMENSIONED FOR 11 ELEMENTS C (INCLUDINGtR) AT MOST C LOGICAL KERROR LOGICAL NUMINF C NUMINF INDICATES WHETHER OR NOT NUMERICAL INFORMATION C (ITERATION COUNTS) HAS TO BE PRINTED C ITS ACTUAL VALUE DEPENDS ON THE INPUT VARIABLE INUM C DIMENSION PLT0AT(86) C TOTAL NUMBER OF DATA TO BE TRANSFERRED TO THE PLOTPROGRAM: QPLOT C (IF REQUESTED SO BY THE IPLOT VARIABLE) C EQUIVALENCE (PLTDAT(1)»VAQ)• (PLTDAT(SO)»TKMM(1*1)) EQUIVALENCE (PLTDAT (83) »VAMAX> f (PLTDAT (8<») »NCA> EQUIVALENCE (PL1ÜAT(85).FCA) • (PLTDAT(86)*KERROR) C FIRST 49 ENTRIES OF PLTDAT CORRESPOND WITH THE VARIABLtS IN C COMMON/PARM/ C NEXT 31 ENTRIES CORRESPOND WITH THE tNTRIES OF TKMM(11»3> C LAST 4 ENTRIES CORRESPOND WITH VAMAX ( * NDV*DVORG )»NCA» FCA C AND KERPOR IN THAT ORDER C NCA = NUMBER OF "MODIFIED CA " CASES C FCA = FACTOR USED TO COMPUTE THt INCREMENT IN TWO C SUCCESSIVE INITIAL CA-VALUES C ACTUALLY THIS INCREMENT IS t FCA»CA(FIRST CASE) C KERROR INDICATES WHETHER OR NOT ERRORS IN THE K-EX VALUES OCCUR C EQUIVALENCE ( ITER(l)•ITEhX(l) ) C ITER ARRAY IS INTRODUCED TO SIMPLIFY THE PRINTING STATEMENTS C FOP ITERX(J) J=lf?«3 C - 113 - COMMON/PARM/VAG»V0RG1.DVOPG.HA.CRADO.Nl«C(11)»MU( 11)tTKEX(11)» • VOkC>.CRADJ,«ADD.CS»DLA,XWNOM»lTERX(4) C C C VARIABLES IN CONMON/PARM/ C C VAO VOLUME OF AQUEOUS PHASE C VORG1 VOLUME OF INITIALLY PRESENT ORGANIC PHASE C OVORG VOLUME OF ORGANIC PHASE (WITH REAGENT) c IN ONE AOOITION c HA CONCENTRATION OF »»• IN THE AQUEOUS PHASE c CR ADD CONCENTRATION OF R IN ADDITIONS OVORG r* NI f y-ui;-> OF ELEMENTS*INCLUDING R c Nl = N • 1 c N ELEMENTS:A*B*C. c R WILL BE REFERRED TO AS THE (N»1)-TH c ELEMENT c C*MU«- c - TKEX ARRAYS OF 11 ELEMENTS EACH c Nl ELEMENTS ARE ACTUALLY USED c MEANING OF : c C(K) - INITIAL CONCENTRATION OF K-TH ELEMENT c IN AQUEOUS PHASE c MU(K) - VALENCY OF K-TH ELEMENT c TKEX(K) - CONDITIONAL EXTRACTION CONSTANT FOR c K-TH ELEMENT c c ( K = 1*?«... Nl ) c c THE FOLLOWING VARIABLES ARE AUXILIARY ONES ANO HAVE BEEN c PLACEO IN COMMON ONLY FOR COMMUNICATION BtTWEEN SUBPROGRAMS c WHICH OTHERWISE WOULD DUPLICATE SOME COMPUTATIONAL STEPS c THEY ARE SET (COMPUTED) IN XZERO «WHEREAS THE FORMER c VARIABLES ARE SET IN THE XZEKO CALLING (SUB) PROGRAMS c VOPG - TOTAL VOLUME OF CURRENT ORGANIC PHASE c VORG = VORG1 • VADÜ c CRADJ - ADJUSTED CONCENTRATION OF R IN THE AQUEOUS PHASE f c = CONCENTRATION OF R WHICH WOULD OCCUR IF ALL c ADDITIONS CONTRIBUTED TO THE AQUEOUS PHASE ONLY c CRADJs C(N1)« RADO/VAQ c RADD - TOTAL AMOUNT OF R ADDED c CS - SCALED CX-VALUE : (VADD/VORG)» MU(N1)*CRADD c DLA - RATIO OF VOLUMES (LAMBDA) c DLA s VORG / VAQ c XRNOM - AUXILIARY VARIABLE FOU ZEROFINDING c ITERX - ARRAY OF 4 ENTRIES CONTAINING INFORMATION ON THE c CONVERGENCE OF THE ROOT-FINDING PROCESS c ( TO OBTAIN A ZERO OF THE MAIN EQUATION ) c ITEKX(1),ITERX(2)#ITERX(3) GIVE THF WMBER OF c ITERATIONS REQUIRED TO FIND THE ZERO (HxORG) DURING c COMPUTATION OF OMlNfQ AND QMAX RESP. c ITERX<<•> i SCRATCH c c - 114 - DATA PLTDAT/86»0./ C INITIALIZATION OF DATA FOR PLDATA FILE C DATA NlMAX/11/ C MAXIMUM NR. OF ELEMENTS PERMITTED (INCLUDIN6:R) C OATA KERROR/.FALSE./ DATA NUMINF/.FALSE./ DATA TEXT/lHA«lHB.lHC*lHD.lHE«lHF«lHb*lHH*lHI*lHJ.lH»/ C IDENTIFIERS FOR CHEMICAL ELEMENTS C C C C «RITE(6*200) 200 FORMAT(1H1) C PAGE EJECT C READ(5**) VAQ*HA*VORGI«OVORG«CRADD»N1 IF(Nl.LE.NlMAX) GOTO 10 WR1TE(6.210) NltNlMAX 210 FORMAT(//.M THE SPECIFIED NUMBER OF ELEMENTS.INCLUDING R» IS TOO L •ARGE (MAXIMUM IS "•I2»M >"»/•" QFORW STOPS") GOTO 999 C 10 IF(N1.GE.2> GOTO 15 C ONLY ONE ELEMENT NOT ALLOWED C WRITE(6.215) 215 FORMAT (//*•• INPUT ERROR'S/»" NUMBER OF ELEMENTS (INCLUDING*. R) IS» • »I2.//) GOTO 999 C 15 READ(5««) (C(K)»K=1.N1) C INITIAL CONCENTRATIONS C CA«C(1) C AS CI]) MIGHT bE CHANGED» KEEP FIRST VALUE IN CA C «EAD(5.»> (MU(K)*Ksl»Nl) C VALENCIES C READ(5»»> (TLG(K)»TLER(K)»Ksl»Nl) C 10L0O OF K-EX VALUES WITH THEIR ERRORS C READ(5»») NDV*NCA»FCAfIPLOT C NDV - NUMBER OF ADDITIONS C NCA - NUMBER OF CASES WITH A MODIFIED C(l) C ( CO) = INITIAL CONTRATION OF THE FIRST ELEMENT ) C FCA - FACTOR USED TO UPDATE C(l> FROM ONE CASE TO THE NEXT C MORE PRECISELY: C C(l> (CASE) =(1*(CASE-1)»FCA) • CA C teHERE CA e C(l) (FIRST CASE ) »AS SPECIFIED ON C INPUT - 115 - C IPLOT - PLOT INDICATOR C IF IPLOT = 1 DATA FOR THE PLOT-PROGRAM Qf-LOT WILL BE C WRITTEN TO FILE : PLDATA C WRITING IS SKIPPED IF IPLOT.NE.1 C IF(NCA.GT.O.ANO.NOV.GT.O) GOTO 20 C NCA AND/OR NDV HAVE/HAS ILLEGAL VALUE(S) C WRITE<6.2?0) NCA.NDV 220 FORMAT(//*•• INPUT ERROR"»/»" NUMBER OF CA-CASES IS : "»I4*/* * •• NUMBER OF ADDITIONS IS i 'SI5»//) GOTO 999 C C C NOW COMPUTE: "MEAN",LOWER AND UPPER BOUND FOR K-EX 20 DO 30 K=1»N1 TKMM(K«2)=TM=10.»*TLG(K) C "MEAN" C FAC=10.»»TLER(K) IF(TLER(K).NE.O.) KERRv'Q=.TRUt. TKMM(K»1)=TM/FAC C LOWER BOUND C TKMM(K»3)=rM»FAC C UPPER BOUND C 30 CONTINUE C READ(5»*) INUM C SEE IF ITERATION COUNTS ARE REQUESTED C IF(EOF(5).NE.O.) GOTO 35 C NO ITERATION OUTPUT WANTED IF THIS INPUTCARD IS MISSING C NUMINF=INUM.NE.O C ONLY A NONZERO INPUT VARl„dLE INUM CAUSES THE ITERATION NUMBERS C TO BE WRITTEN ON OUTPUT C 35 WRITE(6.225) 225 F0RMATU8(/)»35X*46("-»>) WRITE(6*230) Nl*HA»VAO*VORG1*DVORG.CRADD*NCA»NDV 230 FORMAT(/»35X»»SVSTEM 0F"»I4*" ELEMENTS (INCLUDING : W )•••/• •35X»»C0NCENTRATI0N H*"*E12.5»/*35X»"V0LUME AQUEOUS PHASEM«FI0.3»/* •35X»»V0LUME ORGANIC PHASE (INITIALLY)"*F10.3*/*35X*»V0LUME OF ONE •ADDITION"*F10.3*/*35X*«CONCENTRATION OF R IN ONE AD0ITI0N"»EI2.5»/ •*35X*"NUMBER OF •MODIFIED CA» CASES"*I5*/»35X»"NUMBER OF ADDITIONS •»*I6*/*35X*46("-">•////) WRITE(6.240> 240 FORMAT(/»» ELEMENTS»*l0X»"VALENCY"»15X*"i0LOG(KEX> »*18X*»INITIAL C •ONCENTRATION"»/) TEXT(Nl)=TEXT(ll> WRITE(6*250) TEXT(l)*MU(1)*TLG(1)*TLER(1)*CA»FCA 250 F0RMAT(4X*A1*18X*I2.11X*F9.2*"("*F9.3»»>"* 10X,El?.5t" • (1* (CASE •-1>»"*F9.2*">"> - 116 - 00 40 K=2,N1 WRITE(6*260) TEXT(K)*MU(K)*TLG(K)«7LER(K)*C(K) 260 F0RMAT(4X*A1*1BX»I2.11X*F9.2*M("*F9.3«M>M*10X*E12.5) 40 CONTINUE C X=-l. C WHEN USED AS AN INITIAL GUESS IN THE X (AND Q*P) COMPUTATION C NO REASONABLE INITIAL GUESS IS ASSUMED TO BE KNOWN C VAMAXsNDV*DVORG C TOTAL (MAXIMUM) VOLUME ADDED C C(1)=CA C C(l)- CONCENTRATION AT THE FIRST CASE C IF(IPLOT.EO.l) WRITE(7) PLTDAT C WRITE PLOT-DATA IF REQUESTED C C DO 70 KA=1»NCA WRITE(6*200> i C PAGE EJECT C C(1)=(1.*(KA-1)»FCA)»CA C AUGMENT INITIAL CONCENTRATION FOR NEXT CASE C WRITE(6*265) C(l) 265 FORMAT(///•30X*»INITIAL CONCENTRATION : CA =»*E12.5»//) V0L=V0RG1 C RE-INITIALIZE VOLUME FOR EACH CA-CASE C DO 60 L=1*NDV VADD=L»DVORG V0L=VOL*DV0RG C AUGMENT VOLUMES TO TAKE NEXT ADDITION INTO ACCOUNT C COMPUTE HXORG BY COMPUTING P*Q FOR ELEMENT A C CALL QMIMAP(1*VADD*2»TKMM*X*0MM»P) HXORGsHA/X WRITE(6*270) VAOO*VOL*HXORG 270 FORMAT(//•" VOLUME AUDED = "*F10.2*" VOLUME ORGANIC" * » PHASE = "*F10.2*" HXORG « "*E12.5) IF(.NOT.NUMINF) WRITE(6*2B0) IF(NUMINF) WRITE(6*285) 280 FORMAT (//»•» ELEMENT»»* 10X»"P»»»7X*MQMjN"*8X*»Q"*8X*"QMAX,,*/> 2ft5 FORMAT(//•" ELEMENT"* 10X*"P"* 7X*"QMIN"*8X*»Q"*BX*"QMAX"* • 35X*»ITERATI0NS"*/) IF(.NOT.NUMINF) WRITE(6*290) TEXT(1),P*QMM IF(NUMINF) W»ITE(6»295) TEXT(1)*P*QMM*ITER 290 FORMAT<5X*A1*4X*F10.3»3E11.4) 295 FORMATC5X*A1*4X*F10.3*3E11.4*28X»3(3X*13)) - 117 - 00 SO K=2«N1 C REMAINING ELEMENTS C CALL QMlMAP(K»VADO«-l*TKMM*X*QMMtP> IF(.NOT.NUMINF) «RITE(6.290) TEXT(K)«P«QMM I*-(NUMINF> WRITE(6«?95) TEXT (K) .P»QMM»ITER 50 CONTINUE C 60 CONTINUE C 70 CONTINUE C 999 STOP END - 118 - PROGRAM QPLOT( INPUT.OUTPUT.PLDAT A.TAPES=INPUT.TAPE6=0'jTPUT. • TAPE7=PLnATA.TAPEl=65.TAPE2«65) C CPLOT .PLOT THE O-VALUES AND THEIR CONFIDENCE REGIONS C C C OPLOT READS ALL VARIABLES WHICH ARE REQUIRED TO DETERMINE C THE STATE OF A SYSTEM OF Nl ELEMENTS AS A FUNCTION OF THE C ADOCD VOLUME ORGANIC PHASE FROM THE FILE PLDATA C < Nl - NUMBER READ FROM PLÜATA AS WELL ) C THIS FILE IS ASSUMED TO BE WRITTEN BY THE OFORW PROGRAM C IF A USER INPUT RECORD OCCURS.SOMt OF THE VARIABLES FROM C THE PLDATA FILE WILL RE RF-SET C ADDITIONAL INPUT VARIABLES MAY DEFINE RELEVANT PARAMETERS C FOR THE VARIOUS PLOTS. C C C THE CONFIDENCE REGION IS THE REGION dETWEEN Q-MINCV) AND C Q-MAX(V) WITH V ( - VOLUME ADDED ORGANIC PHASE ) RUNNING C BETWEEN SPECIFIED LIMITS C U-MIN(V) AND Q-MAX(V) ARE FUNCTIONS OF THE VOLUME ADDED '. V C C CONFIDENCE REGIONS FOP DIFFERENT ELEMENTS WILL BE SHADED C USING DIFFERENT SHAOING ANGLES C C C LOGICAL KERROR DIMENSION TKMMU1.3) ,OMM(3) .KIND(5).KEL(11>»KELl (11) C DIMENSION'PLTDAT(86) C TOTAL NUMBER OF DATA TO BE TRANSFERRED FROM TH£ PROGRAM : QFORW C C EQUIVALENCE (PLTDAT(1)*VAO)* (PLTDAT(50)tTKMMO«1)) EQUIVALENCE (PLTDAT(03).VAMAX) . (PLTDAT(84)»NCA) EQUIVALENCE (PLTDAT(85)»FCA) , (PLTDAT(86).KERROR) C FIRST 49 ENTRItS OF PLTDAT CORRESPOND WITH THE VARlAdLES IN C COMMON/PARM/ C NEXT 33 ENTRIES CORRESPOND WITH THE ENTRIES OF TKMM(11»3) C LAST 4 ENTRIES CORRESPOND WITH VAMAX ( * NOV#OVORG )»NCA. FCA C AND KERROR IN THAT ORDER C NCA = NUMBER OF "MODIFIFO CA " CASES C CCA s INCREMENT FOR CONCENTRATION C(l) C KERROR INDICATES WHETHER OR NOT ERRORS IN THE K-ES VALUES OCCUR C C c DIMENSION POPL(600t5)»PQPLX(400> »ANGAR(5)»LEGARR(?).K0DE(5) DIMENSION OSRT(5)*IPRMQ(5)*IWORK(1000) INTEGER TEXT(11)tUTEXT(11).PTEXT(11) C C COMMON/PARM/VAÜ.V0RG1»DV0RG.HA»CRADD»N1.C(11)»MU(11)»TKEX(11), • V0RG.CRADJ.KADD»CS>DLA.XWNOM»ITERX(4> - I!9 - C C C VARIABLES IN COMMON/PARM/ C C VAQ - VOLUME. OF AQUEOUS PHASE C VOPG1 - VOLUME OF INITIALLY PRESENT ORGANIC PHASE C DVORG - VOLUME OF ORGANIC PHASE (WITH REAGENT) C IN ONE ADDITION C HA - CONCENTRATION OF H* IN THfc AQUEOUS PHASE C CNADD - CONCENTRATION OF R IN ADDITIONS DVORÖ C Nl - NUMbER OF ELEMENTS*INCLUDING H C NI = N • 1 C N ELEMENTS:A.B.C. C R WILL BE REFERStD TO AS THE C SYMBOLIC NAMES FOf* THE CHEMICAL ELEMENTS (DEFAULT) C DATA UTEXT/11MH / OATA IHEAD/PHO-/ DATA NPPL/200/ C NUMBER OF POINTS PER Q-PLOT C DATA PLT0AT/86»0./ DATA K0DE/1H(.1H *1H>,1H ,IH$/ DATA LEGARft(l)/lOHO»L0.5M0.7/ C THE • IN lOHU^ IS THE ESCAPE CHARACTER FOR THE DISSPLA C INSTRUCTION ALPHABET (SO DEFINED IN THE CALL TO MX3ALF) C L0.5ANÜ HO.7 ARE COMMANDS FROM THE INSTRUCTION ALPHABET C TO LOWER THE CHARACTER POSITION AND TO MODIFY THE C CHARACTER HEIGHTS C BOTH THE LEGARR- AND THE KODE-ARRAY ARE USED TO «RITE TEXTS C LIKES 0 PH IN THE LEGEND OF THE PLOT C HERE PB IS TO bE A SUBSCRIPT FOR 0 AND P MUST BE UPPER CASE C WHEREAS B SHOULD BE LOWER CASE C PB IS THE 2 CHARACTERS ELFMENT NAME SPECIFIED BY THE USER C ONLY A PART OF THE TEXT CAN THEREFORt BE FORMED IN ADVANCE C (ACTUALLY: LEGARR(1)«KODE(1) AND K0DE(3> ) C THE 2 CHARACTERS OF THE ELEMENTS NAME ARE ASSIGNED TO K0DE(2) C AND KODEU) FURTHER ON IN THE PROGRAM»AFTER WHICH LEGARR(2) C CAN BE SET C THEN»FINALLY»A COMPLETE TEXTSTRING WILL OCCUR IN ARRAY LEGARR C CONTAINING THE ELEMENTS NAME.ESCAPE CHARACTERS FOR THE C APPROPRIATE ALPHABETS FOR ITS 2 CHARACTERStAND INSTRUCTIONS TO C USE THE NAME AS A SUBSCRIPT FOR Q C DATA XASL.YASL/17..8.5/ DATA ALFkIN.ALFMAX/50..170./ C MINIMUM AND MAXIMUM SHADING ANGLES C C WEWINO 7 R£AD(7) PLTDAT C VARIABLFS IN COMMON/PARM/, ARRAY TKMM. VARIABLES VAMAX»NCA AND C FCA ARE REAO FROM FlLErPLDATA C TEXT(N1)=TEXT(11) C Nl-TH ELEMENT IS R C NCAP=NCA C NUMBER OF INITIAL CONCENTRATIONS C(l) TO CONSIDER C FCAP=FCA C FACTOR TO OETERMINE INCREMENT IN C(1)-CONCENTRATION C VPMlNsO. C MINIMUM VOLUME AOUF.O (VAUD) C VPMAXsVAMAX C MAXIMUM VADD r - 121 - NELP=N1 C NUMBER OF ELEMENTS TO BE PLOTTED C NIN1PL=S C NUMBER OF ELEMENTS FOR ONE PLOT C DO 10 K=1,N1 PTEXT(K)=UTEXT(K)=TEXT(K) 10 CONTINUE C TEXT : A»H»C» R C DO 12 K=1,N1 KEL(K)«K 12 CONTINUE C DEFAULT ELEMENT INDICES C C FCAP,CCAP.VPMlN«VPMAX.NELPfNlNlPL AND THE ARRAYS KEL AND UTEXT C RECEIVED DEFAULT VALUES BASED ON DATA FROM FILE PLDATA C HOWEVER : THEY MAY BE RESET BY USER INPUT OCCURRING ON C FILE INPUT C C C C C READ(S.lOn) KIND 100 F0RMAT(5I1) IF(EOF(5).NE.O.) GOTO 65 C AT MOST 5 CARDS MAY FOLLOW ON INPUT C DO 60 KRT=1»5 KK=KIND(KRT)*1 GOTO (60*?0*30t40*<»5f50> * KK C 20 KEAD(5»*> NCAPtC(l)»FCAP C LIST DIRECTED I/O GOTO 60 C 30 READ<5»«) VPMIN.VPMAX C LIST DIRECTED I/O GOTO 60 C 40 READ(5f») MMPL»NELPf (KEL C FOM REASONS OF PLOT-LAYOUT IT IS NOT ALLOWED TO EXCEED 5 C GOTO 60 C 45 PEAD(5tl35) IERR 135 FORMAT(AV) C SEE.IF ERRORS IN KEX-VALUES MUST bE TAKEN INTO ACCOUNT C KERROR=IERH.NE.<»HNO ERRORS C STRING "NO ERRORS" SPECIFIES NO ERRORS IN K-EX VALUES OCCUR C GOTO 60 C 50 READ(5»i-'rO> (PTEXT(J),J=l.NELP) 140 FORMATU1 (A2.2X)) C USE PTEXT-ARRAY AS SCRATCH ARRAY C C THE TEXT STRINGS ARE ASSUMED TO REFLECT THE ORDER OF THE C ELEMENTS.AS GIVEN IN THE QFORW PROGRAM C 60 CONTINUE C C C C KEL(J).J=1.NELP : INDICES OF ELEMENTS FOR WHICH THE C Q-VALUES HAVE TO HE PLOTTED C THE TEXT-STRINGS : UTEXT(J).J=1»NELP MUST BE REASSIGNED C TO THE PROPER ARRAY ELEMENTS ACCORDING TO KEL(J) .J=1.NELP C 65 DO 70 Ksl.NELP KELM=KEL(K) UTEXT(KELM)=PTEXT(K) KELKK)=KEL(K) 70 CONTINUE C CALL VSORTZ(PTEXT.ll,NELP.l»l»KELl.IWORK) C BRING TEXT STRINGS IN THE ORDER PRESCRIBED BY THE KfcL-ARRAY C WRITE(6.200> 200 FORMATC1H1) C PAGE EJECT C WRITE<6.225) 225 F0RMAT(18(/).35X.46("-")) WRITE(6*230) N1.HA.VAO.VORG1.CRADD.NCAP.VPMIN.VPMAX.NELP 230 F0RMAT(/.35X»"SYSTEM 0F".I4." ELEMENTS (INCLUDING : R )"•/• •35X."CONCENTRATION H»".E12.5./«35X."V0LUME AQUEOUS PHASE"»FI0.3./. »35X."V0LUME ORGANIC PHASE (INITIALLY)".F10.3./»35X."CONCENTRATION •OF R IN ONE ADDITION".E12.5»/.35X*»NUMBER OF «MODIFIED CA» CASES". •I5./.35X."V ADDED MlNlMUM".F10.2./»3bX."V ADDED MAXIMUM".F10.2. »/.35X."NUM«ER OF ELEMENTS FOR Gl-PLOTS". I4./.35X»46("-") •///> WRITE(6.240) 240 FORMAT(/•» ELfcMENT".lOX."VALENCY". 8X."KEX-MlN"*10Xt"KEX"t14X» » "KEX-MAX".12X."INITIAL CONCENTRATION"./) - 123 - WRITE(6.250> UTEXT(1).MU<1>.(TKMM(UJ).J=1,3> 250 FORMAT(3X»A9,10X.I2,3(5X.E12.5)»12X," CA"> DC 72 K=2.N1 WRITE(6,260) UTEXT(K).MU(K)»(TKMM(K.J).J=l»3)»C(K) 260 F0RMATC3X.A9»10X»I2.3(5X.E12.5)»10X»E12.5> 72 CONTINUE C DO 75 K=1,N£'LP KELM=KEL(M ENCODE(10.270»PTEXT(K)) IHEAU.UTEXT(KELM) 270 F0RMATCA2.A8) C ELEMENTS NAME IS PRECEDED bY 0 TO OBTAIN PLOT-TEXT C 75 CONTINUE C CALL CALC0M(4HPL0T.?MSW»2Hl3»2HBZ»2HN0) C DISSPLA IS TO USE THE CALCOMP-PLOTTER C NPLTS=1*(NELP-1)/MN1PL C NUMBER OF PLOTS PER CA-CASE C DELV=(VPMAX-VPMIN)/FLOAT(NPPL-1) C PLOT INCREMENT (VADD) C IPL = 0 Xs-1. CAP=C(1) C INITIAL CONCENTRATION FIRST ELEMENT AT FIRST CASE C DO 97 KA=WNCAP C(l)=(l.* NOW PERFORM INITIALIZATION PER PLOT CALL BGNPL(-IPL) CALL PAGE(21..11.) CALL MX1ALF(»L/CS".">") ALPHABET: LO*ER CASE ROMAN,ESCAPE CHARACTER:) CALL MX2ALF(»STAN"»"(") ALPHABET: UPPER CASE ROMAN.ESCAPE CHARACTER:( CALL MX3ALF(»INST"."*") INSTRUCTION ALPHABET.ESCAPE CHARACTER:» CALL PHYS0R(3..2.) CALL HEIGHT(.28) CALL TITLE(O.O."ML OF TITRANTS'MOO." ".1.XASL.YASL) CALL ANGLE(90.) CALL STRTPT(-2..-2.) CALL C0NNPT(-2..9.) CALL MESSAGC'PLOTNO. t".100.-2.3*0.) CALL INTNO(IPL."ARUT"."ABUT") CALL RESET("ANGLE") CALL FRAME CALL TWIPLX CALL YAXANG(0.) CALL MESSAG(»Q».l.-1.2.<».2> CALL GRAF(VPMlN."SCALE".VPMAX.O.."SCALE".1.) Ü0 87 L=1.NPPL VADO=VPMIN*(L-l)»DELV M0OEK=-2 ÜO 85 KELM=NEL1.NEL2 K*KEL(KELM) KKELM=KELM-NEL1*1 INDEX ELEMENT DO 78 10=1.3 OMM(IQ)=0. CONTINUE OVALUES FOR VAOO*0. IF(VAOD.EQ.O.) GOTO 80 CALL OMIMAP(K*VADD.MODEK*TKMM.X*OMM.P) EVALUATE Q-VALUES FOR VADD.NE.O. MODEKs-1 AVOID PECOMPUTATION OF X IN NEXT CALL CONTINUE POPL(Ml.KKELM)xQMM(1) POPL(M2.KKELM)=OMM(2) PUP» (M3.KKELM)=0MM(3) - 125 - CONTINUE M2=M2»1 M3=M3-1 LRML=LR-L PQPLX(L)=PQPLX(LRML)=VAOO VOLUME (PLOT) INCREMENT CONTINUE NOW PLOT ARRAY PUPL HAS BtEN FILLED NPLA=NEL2-NELl*l ACTUAL NUMBER OF ELEMENTS IN PLOT KPL ANGAR(1)=(ALFMIN*ALFMAX)/2. IPRMQ<1)=1 ASSIGNMENTS FOR ONE ELEMENT PER PLOT CASE IF(NPLA.EQ.l) GOTO 92 ÜELALF=(ALFMAX-ALFMIN)/FLOAT(NPLA-1) DO 90 IEL=1.NPLA ANGAR(IEL)=ALFMAX-(1EL-1)*DELALF QSRT(IEL)=-PQPL(IPi »IEL) Q-VALUES HALFWAY BETWEtN VPMIN AND VPMAX ARE USED TO DETERMINE THE SHADING-ANGLES ONLY THE ORDER OF THE O-VALUES IS RELEVANT IPRMO(IED=IEL INITIALIZATION OF THE PERMUTATION VECTOR CONTINUE CALL VS0RTP(OSRT»NPLA»IPRMO) SORT TME O-VALUES IN ASCENDING ORDER AND RECORD THE PERMUTATIONS DO 94 KELM=NEL1«NFL2 KKELM=KELM-NEL1*1 X8=VPMIN*(0.2S*(KKELM-1)»1.5)*(VPMAX-VPMIN)/XASL Y1B=-I.6/YASL Y2H=-U4/VASL CALL GRACE(?.0) RANG' *IPRMQ(KKELM> SHADiiMG ANGLE INDEX CALL CURVE(POPLXtPOPL(LR»KANGL)»NPPL»0) DRAW O-CURVt IF(.NOT.KERPOR) GOTO 93 - 126 - CALL NEWPEN(2) CALL SHADE CPOPLX»PQPL(1»KANGL>»LR-1»ANGAR(KKELM)• • O.ltltltfORKtlOOO) C SHADE AREA BETWEEN LOWER AND UPPER ERROR BOUNDS C CALL CURVE(PQPLX.PQPL SUBROUTINE OMIMAP(K.VADD»MODEK»TKMM.X.QMM»P) QMIMAP .COMPUTE Q-VALUES (»MEANM»MINIMUM»MAXIMUM) AND (OPTIONALLY) 1 THE P-VALUE FOR THE K-TH ELEMENT IT IS ASSUMED THAT EACH OF THE CONDITIONAL EXTRACTION CONSTANTS ( REX-VALUES ) IS ATTENOEO WITH AN UPPER AND A LOMER BOUND THEN QMAX FOR ELEMENT K IS THE Q-VALUE OBTAINED 6Y SUBSTITUTING THE UPPER KEX-VALUE FOR THE K-TH ELEMENT ANO LOWER BOUNDS FOR ALL OTHER ONES SIMILARLY : FOR QMIN INTERCHANGE UPPER ANO LOWER.... C PARAMETERS C C K INDEX CF ELEMENT (INPUT) C VADO TOTAL VOLUME OF ORGANIC PHASE ADDED AT THIS (INPUT) C STAGE C MODEK MODt IN WHICH QMIMAP IS CALLEU (INPUT) C O.LE.MODEK.LE.l SINCE THE PREVIOUS CALL ONLY C K HAS BEEN CHANGED (HENCE: C DO NOT WtCOMPUTE X ) C COMPUTE P C -l.LE.MODEK.LT.O SAME AS ABOVE«BUT DO NOT COMPUTE C A P-VALUE C AbS(MODEK).GT.l RECOMPUTE X AND COMPUTE P FOR C POSITIVE MODEK c TKMM ARRAY DIMENSIONED AS TKMM(13»3) (INPUT) C APPROPRIATE FOR N=10 ELEMENTS (EXCL. R) AT c MOST c ONLY THE UPPER Nl*3 PART IS USED c ( Nl = N • 1 ) c TKMM(I.l) LOWER BOUND OF KEX OF ELEMENT I c TKMM(It?) "MEAN" OF KEX FOR ELEMENT I c TKMM(It3) UPPER BOUND c VALUE OF HA/HXORG WHICH MAKES THE MAIN (INPUT/ c EQUATION ZERO»TRANSIENT PARAMETER OUTPUT) c ON INPUT : ESTIMATE c ON OUTPUT : FINAL VALUE c X(OUTPUT) = THE ZERO FOR "MEAN"-KEX CASE c ALSO»FOR INPUT» A "MEAN" X-GUESS WIIL DO c QMM ARRAY OF SIZE 3 (OUTPUT) c 0(1) MINIMUM O-VALUE c 0(?> "MEAN" O-VALUE c 0(3) MAXIMUM O-VALUE c P-VALUE OF K-TH ELEMENT»DtFINtD AS* (OUTPUT) c MU(N1)«AMOUNT OF R ADDED / c MU(K)«AMOUNT OF ELEMENT K PRESENT c COMPUTED ONLY IF MODEK IS POSITIVE c A NEGATIVE VALUE OF P IS RETURNED IF C(K>* 0 c c c - 128 - C C c VARIAHLES IN COMMON/PARM/ c c WAO - VOLUME OF AQUEOUS PHASE c VORG1 - VOLUME OF INITIALLY PRESENT ORGANIC PHASE c UVORG - VOLUME OF ORGANIC PHASE (WITH REAGENT) c IN ONE ADDITION c HA - CONCENTRATION OF H+ IN THE AQUEOUS PHASE c CPADO - CONCENTRATION OF R IN ADDITIONS DVORG c Nl - NUMBER OF ELEMENTS»INCLUDING R c Nl = N • 1 c N ELEMENTSIA.B.C. c R WILL BE REFERRED TO AS THE (NM)-TH c ELEMENT c C»MUf- c - TKEX - ARRAYS OF 11 ELFMENTS EACH c Nl ELEMENTS ARE ACTUALLY USED c MEANING OF : c C(K) - INITIAL CONCENTRATION OF K-TH ELtMtNT c IN AQUEOUS PHASE c MU(K) - VALENCY OF K-TH ELEMENT c TKEX(K) - CONDITIONAL EXTRACTION CONSTANT FOR c K-TH ELEMENT c ( K = It?,... Nl ) c c THE FOLLOWING VARIARLES ARE AUXILIARY ONES AND HAVE BEEN c PLACED IN COMMON ONLY FOR COMMUNICATION BETWEEN SUBPROGRAMS c WHICH OTHERWISE WOULD DUPLICATE SOME COMPUTATIONAL STEPS c THEY ARE SET (COMPUTED) IN XZERO tWHEREAS THE FORMER c VARIABLES ARE SET IN THE XZERO CALLING (SUB) PROGRAMS c c VORG - TOTAL VOLUME OF CURRENT ORGANIC PHASE c VORG = VORG1 • VADD c CRADJ - ADJUSTED CONCENTRATION OF R IN THE AQUEOUS PHASE c = CONCENTRATION OF R WHICH WOULD OCCUR IF ALL c AUDITIONS CONTRIBUTED TO THt AQUEOUS PHASE ONLY c CRADJs C(N1>* RADD/VAQ c RADD - TOTAL AMOUNT OF R ADDED c CS - SCALED CX-VALU£ : (VADD/VORG)* MU(Nl)»CRADU c ÜLA - RATIO OF VOLUMES (LAMBDA) c DLA = VORG / VAQ c XWNOM - AUXILIARY VARIABLE FOR ZEROFlNDING c ITERX - ARRAY OF 4 ENTRIES CONTAINING INFORMATION ON THE c CONVERGENCE OF THE ROOT-FINDING PROCESS c ( TO OBTAIN A ZERO OF THE MAIN EQUATION ) c ITERX(l)tITERX(2)»IT£RX(3) GIVE THE NUMBER OF c ITERATIONS REQUIRED TO FIND THE 7ER0 (HXORG) DURING c COMPUTATION OF QMlNtQ AND OMAX RESP. c ITERX(4) : SCRATCH c c c COMMON/PAPM/VAÜ»V0RG1•DVORG»HA»CRADD»Nl»C(11)tMU(11)»TKEX(11)* • VORG.CRADJ»RADD*CS»DLA»XWNOM.ITERX(<») - 129 - DIMENSION TKMM(I1,3)«0MM(3) C ARRAY TKMM APPROPRIATE FOP Nl = II AT MOST C IMOO=IABS(MOOEK) M00EO=3 C INITIAL MODE C IFUMOD.LE.l) MOOEO=0 C DO NOT RECOMPUTE X C DO 20 JJ=4»6 J=M0D(JJ*3)M C X FOR THE "MEAN" CASE MUST BE COMPUTED FIRS C DO 10 1=1.Nl TKEX(I)=TKMM(I«J) 10 CONTINUE C TKEX(K)=TKMM(K»<»-J) CALL EXTEFF(K.VAOD.MODEO»X»QMMU-J)> ITERX(4-J)=ITERX(4) IFU.EQ.2) XKEEP=X C X ONLY RELEVANT FOR "MEAN" CASE C M0OEQ=2 C VADD DOES NOT CHANGE WHEN J BECOMES 2*3 C 20 CONTINUE C IF(IMOD.LT.O) GOTO 999 C P IS NOT REQUESTED C P=-1.0 IF(C(K).NE.O.) P=DLA»CS/(MU(K)»C(K)> X=XKEEP C X FROM "MEAN" CASE C 999 RETURN END - 130 - SUBPOUTINE EXTEFF(K,VADD»MODEQ.X.Q> C EXTEFF .COMPUTE EXTRACTION EFFICIENCY FOR K-TH ELEMENT C C [ME Ü-VALUE IS COMPUTED BY FIRST CALCULATING THE ZERO OF THE C MAIN EQUATION (A VALUE OF HXORG FOB WHICH THE MAIN EQUATION C IS ZERO ) c MOST OF THE VAF«ABLES OCCURRING IN THE MAIN EQUATION (HENCE c INFLUENCING THE ZERO AND 0(K) ) ARE TRANSFERRED BY COMMON c c c c PARAMETERS c K INDEX OF ELEMENT c VADD TOTAL VOLUME OF ORGANIC PHASE AODEO TO THE (INPUT) c ORIGINAL SYSTEM (TO THE INITIAL STATE) (INPUT) c MOOEQ MODE IN WHICH SUBROUTINE EXTEFF IS CALLED (INPUT) c MODEQ = 0 SINCE THE PREVIOUS CALL TO tXTEFF ONLY c K HAS BEEN CHANGED c APPROPRIATE MODt WHEN THE Q-VALUES FOR c SOME OR ALL OF THE ELEMENTS ARE TO BE c COMPUTED UNUER THE SAME CONDITIONS c (LIKE: VAOD»C(K)tTKEX(K)«K=l•..Nl) c c THE COMPUTATION OF THE ZERO IS SKIPPED c AND THE PARAMETER X IS ASSUMED TO BE c THE ZERO c MODtQ = 1 SINCE THE PREVIOUS CALL : c VAPO HAS BEEN CHANGED c C(K)»TKEX(K)»K=1...N1 REMAINED c CONSTANT c CHANGING OF K IRRELEVANT c MODEQ = 2 SINCE THE PREVIOUS CALL c VADD REMAINED UNCHANGED c AT LEAST ONE OF THE C-* OR TKEX- c VALUES HAS BEEN CHANGED c CHANGING OF K IRRELEVANT c MODEQ "0»1«2 VADD AND ONE OF THE C-».OR TKEX- c VALUES HAVE BEEN CHANGED c CHANGING OF K IRRELEVANT c VALUE OF (H»)/HXORG WHICH MAKES THE MAIN (INPUT/ c EQUATION ZERO OUTPUT) c X IS A TRANSIENT PARAMETER c ON INPUT : ESTIMATE FOR (HO/HXORG c ON OUTPUT : FINAL VALUE AS COMPUTED BY ZBRENT c FOR MODEQ = 0 THE ESTIMATE IS TAKEN c AS THE FINAL VALUE c COMPUTED EXTRACTION EFFICIENCY (OUTPUT) c A NEGATIVE VALUE INDICATIES A FAILURE OF c THE ZERO FINDER ZBRENT c c c c c VARIABLES IN COMMON/PARM/ - 131 - C C VAQ - VOLUME OF AQUEOUS PHASE C VORG1 - VOLUME OF INITIALLY PRESENT ORGANIC PHASE C DVORG - VOLUME OF ORGANIC PHASE «WITH REAGENT) C IN ONE ADDITION C HA - CONCENTRATION OF H-» IN THE AQUEOUS PHASE c CRADD - CONCENTRATION OF R IN AUDITIONS DVORG c Nl - NUMBER OF ELEMENTS»INCLUDING R c Nl = N • 1 c N ELEMENTS:A.8,C. c R WILL HE REFERRED TO AS THE (N*1)-TH c ELEMENT c C»MU«- c - TKEX - ARRAYS OF 11 ELEMENTS EACH c Nl ELEMENTS ARE ACTUALLY USED c MEANING OF : c C(K) - INITIAL CONCENTRATION OF K-TH ELEMfcNT c IN AQUEOUS PHASE c MU(K) - VALENCY OF K-TH ELEMENT c TKEX(K) - CONDITIONAL EXTRACTION CONSTANT FOR c K-TH ELEMENT c ( K = 1*2,... Nl ) c c THE FOLLOWING VARIABLES ARE AUXILIARY ONES AND HAVE BEEN c PLACED IN COMMON ONLY FOR COMMUNICATION BETWEEN SUBPROGRAMS c WHICH OTHERWISE WOULD DUPLICATE SOME COMPUTATIONAL STEPS c c THEY ARE SET (COMPU' D) IN XZERO «WHEREAS THE FORMER c VARIABLES ARE SET IN THE XZERO CALLING (SUB) PROGRAMS c c VORG - TOTAL VOLUME OF CURRENT ORGANIC PHASE c VORG = VORG1 • VADD c CRADJ - ADJUSTED CONCENTRATION OF R IN THE AQUEOUS PHASE c = CONCENTRATION OF R WHICH WOULD OCCUR IF ALL c ADDITIONS CONTRIBUTED TO THE AQUEOUS PHASE ONLY c CRADJs C(N1>* RADD/VAQ c RADD - TOTAL AMOUNT OF R ADDED c CS - SCALED CX-VALUE : (VAÜÜ/VORG)» MU(N1)*CRADD c DLA - RATIO OF VOLUMES (LAMBOA) c DLA = VORG / VAQ c XRNOM - AUXILIARY VARIABLE FOR ZEROFINOING c ITERX - ARRAY OF <• ENTRIES CONTAINING INFORMATION ON THE c CONVERGENCE OF THE ROOT-FINDING PROCESS c ( TO OBTAIN A ZERO OF THE MAIN EQUATION ) c ITERX(1).ITERX(2)tITERX(3) GIVE THE NUMBER OF c ITERATIONS REQUIRED TO FIND THE ZERO (HXORG) DURING c COMPUTATION OF QMIN»Q AND QMAX RESP. c ITERX(4) : SCRATCH c c c COMMON/PARM/VAU»VORG1•DVORG»HA»CRADD»Nl*C(11)*MU(11)»TKEX(11), • VOHG»CRADJ»RADD»CS«DLA»XRNOM»ITERX(4) C DATA PINF/1.0E200/ - 132 - C PLUS INFINITY (FOR THIS PROGRAM) C ITF»X«»>=0 C INITIALIZATION NUMBER OF ITERATIONS C IMVADD.NE.O.) GOTO 5 Q-O. C NO EXTRACTION IN INITIAL STATE C IF(M0DEQ.EQ.2> GOTO 999 C OLA AND CS HAVE BEEN SET IN A PREVIOUS CALL C X=PINF OLA=VORGl/VAQ CS=0. C OLA WILL NOT RE SET IN VADU = O CASE BY ROUTINE XZERO C THE SAME HOLDS FOR CS C THIS CAN 8E SKIPPED IF MODEQ = 2 C GOTO 999 C 5 IF(MODEO.EQ.O) GOTO 10 XNEW=XZERO(VADD*MODEQ*X) C X IS USED AS XGUESS C X=XNE«* 10 OREV=X IFCX.GT.O.) QREV=1.0»X»»MU(K)/(DLA*TKEX(K)) U=l./OREV 999 RETURN END - 133 - FUNCTION XZERO(VAf)D.MODE»XGUESS> XZEBO .FIND THE ZERO OF THE MAIN EQUATION AT THE L-TH ADDITION XZERO = THAT VALUE OF HXORG/HA «FOR WHICH THE MAIN EQUATION IS ZERO XZEPO SHOULO HE POSITIVE IF HOMEVER A NEGATIVE VALUE IS RETURNEO THIS INDICATES A FAILURE OF THE ROOTFINOING PROCESS IN THIS CASE : XZERO = -IER FOR THE ERROR RETURN CODE IER OF Z8RENT.SEE t IMSL- DOCUMENTATION OF ZWRENT C PARAMETERS C C VADD TOTAL VOLUME OF ORGANIC PHASE ADDED TO THE ORIGINAL C SYSTEM WITH AN AQUEOUS PHASE (VOLUMEtVAQ) AND AN C ORGANIC PHASE (VOLUMEtVORGl) C IN THE INITIAL STATE (VAQtVOSGl) NO REAGENT OCCURS* C BUT THE REAGENT METAL MAY OCCUR IN THE AQUEOUS C PHASE WITH CONCENTRATION DENOTED AS: C(N1) C FOR AN EXPLANATION OF C(1)••..C(Nl) SEE: C •*» VARIABLES IN COMMON ••• C NOTICE THAT THE VOLUME VADD MAY BE ADDED IN A C NUMbER OF SUCCESSIVE ADDITIONS WHERE THE C CONCENTRATION OF R IS SUPPOSED TO BE THE SAME C FOP ALL THESE ADDITI0NS»NAMELY:CRAOD C C HXORG IN THE INITIAL STATE IS ZEROfYIELDING AN C XZERO-VALUE OF INFINITY C THE FUNCTION XZERO MUST HOWEVER NOT BE CALLED TO C PRODUCE THIS VALUE»THAT IS: C XZERO MUST NOT BE CALLED FOR VADD « O C ( INITIAL STATE ) C MODE MODE IN WHICH XZERO IS CALLED C MODE = 1 COMPARED WITH THE FORMER CALL TO XZERO c ONLY VADD HAS BtEN CHANGED c MODE = 2 VADD IS THE SAME AS IN THE PREVIOUS c CALL BUT AT LEAST ONE C(K) OR c TKEX(K)«K=l*2t..Nl HAS BEEN CHANGED c MODE "1«? VADD AND SOME C(K)*TKEX(K) HAVE REEN c CHANGED c XGUESS - INITIAL GUESS FOR THE ZERO c WILL BE USED ONLY IF GREATER THAN 0.0 c c c c THE MAIN EQUATION IS CONSIDERED AS A FUNCTION OF ONE VARIABLE c * X - IN WHICH MANY OTHER PARAMETERS OCCUR c E.G. VAQ*VORGl*HAf.ETC. c THE MAIN EQUATIONS ZERO (JXZERO) THEREFORE DEPENDS ON THOSE c PARAMETERS c THE DESIGN OF THE FORTRAN FUNCTION XZERO IS SUCH THAT ONLY c THE DEPENDENCY ON THE VOLUME OF ADDED ORGANIC PHASE IS - 134 - C GOVERNEO BY THE FUNCTIONS PARAMETER LIST (I.E. :VADD) C ALL OTHER PARAMETERS ARE TRANSFERRED VIA COMMON C C BESIDES THE NATURAL PARAMETERS VAQ.VORGl»HA,.ETC. SOME C AUXILIARY VARIABLES HAVE ALSO BEEN PLACED IN COMMON C ThfSE VARIABLE ARE SET IN XZERO AND WILL bE USED IN C OTHER SUBPROGRAMS C C C C VARIABLES IN COMMON/PARM/ c c VAO - VOLUME OF AQUEOUS PHASE c VORG1 - VOLUME OF INITIALLY PRESENT ORGANIC PHASE c DVORG - VOLUME OF ORGANIC PHASE (WITH REAGENT) c IN ONE ADDITION c HA - CONCENTRATION OF H* IN THE AQUEOUS PHASE c CRADD - COMCENTRATION OF R IN ADDITIONS DVORG c Nl - NUMHER OF ELEMENTS*INCLUDING R c Nl = N • 1 c N ELEMENTS:A.B.C. c R WILL BE REFERRED TO AS THE (N»1)-TH c ELEMENT c C.MU»- c - TKEX - ARRAYS OF 11 ELEMENTS EACH c Nl ELEMENTS ARE ACTUALLY USED c MEANING OF : c C(M - INITIAL CONCENTRATION OF K-TH ELEMENT c c IN AQUEOUS PHASE c MU(K> - VALENCY OF K-TH ELEMENT c TKEX(K) - CONDITIONAL EXTRACTION CONSTANT FOR c K-TH ELEMENT c ( K = 1, C ITERATIONS WEQUIRED TO FINO THE ZERO (HXORG) DURING C COMPUTATION OF QMlN.Q AND OMAX RESP. C ITEkXU) S SCRATCH C c c c C0MM0N/PARM/VAQ.V0RG1.DVORG.HA.CRADD.N1.C(11)*MU(11).TKEX(11), » VORG»CRAOJ.RADD»CS»DLA.XRNOM.ITERX(4) C EXTERNAL FXMAIN DATA EPSM/1.0F-13/. NSIG/13/ C MACHINE PRECISION CONSTANTS C DATA FAC/S./ C FAC IS USED TO PREDICT XLEFT/XRIGHT «HEN A GOOD XGUESS IS KNOWN C IF(MODE.EO.l) GOTO 20 C IF ARRIVED HERE ONE OF C(1)»...C(Nl).TKEX(1).... TKEX(Nl) C (AT LEAST) HAS BEEN CHANGED C CTMU=0. C DO 10 K=1.N1 CTMU-AMAX1(CTMU.MU(K)*C(K)»TKCX(K)) 10 CONTINUE XWNOM=HA*CTMU C NOMINATOR OF XRIGHT C IF(MODE.EG.2) GOTO 30 C 20 VORG=VORGl*VADD DLA=VORG/VAQ RADD=VADD»CRADD CS=MU(N1)»RADP/V0RG 30 CRADJ=C(N1)»RADD/VAQ XLFFT=HA/CS C FXMAIN POSITIVE FOR XLFFT C IF(XLEFT.LT.l) GOTO 40 XRIGHT=xRNOM/CS GOTO hO C UO Fl=FXMAINU.O) IF(F1.LT. 0.) GOTO 50 XLEFT=1.0 XRIGHT=XRNOM/CS GOTO 60 C 50 XRIGHT =1. C C A SAFE INTERVAL HAS «EEN DETERMINED C MAYBE XGUESS IS CLOSER TO THE ZERO C - 136 - 60 IF(XGUESS.GT.XkIGHT.OR. XGUESS.LT. XLEFT) GOTO 70 C XliUESS INSIDE INTERVAL XLEFT-XRIGHT FGUESS=FXMAIN(XGUESS> IFfFGUESS.LT.O.) GOTO 65 XLEFT=XGUESS XRPOT=FAC*XGUESS C POTENTIAL "BETTER" XRIGHT C IF (XRIGHT.LE.XRPOT) GOTO 70 C -HECK IF XRPOT IS SMALLER THAN CURRENT XRIGHT C ( CURRENT XRIGMT MIGHT HE FAR TOO LARGE) C FXRP=FXMAIN(XRPOT) IFCFXRP.LT.O.) XRIGHT=XRPOT C ACCEPT XRPOT AS NEW XRIGHT ONLY IF FXMAIN IS NEGATIVE AT XRPOT C GOTO 70 C C 65 XPlGHTsXGUESS XRPOT=XGUESS/FAC IF(XLEFT.GE.XRPOT) GOTO 70 C XGUESS/FAC MIGHT BE A BETTER XLEFT-VALUE C CHECK IF XGUESS/FAC LARGER THAN CURRENT XLEFT C FXRP=FXMAIN(XRPOT) IF(FXRP.GT.O.) XLEFT=XRPOT C ACCEPT XRPOT AS A NEW XLEFT ONLY IF FXMAIN IS POSITIVE AT XRPOT C 70 EPS=10.*EPSM» CS C ATTAINABLE ACCURACY C MAXIT=10000 C MAXIMUM NO. OF ITERATIONS C CALL ZBRENT(FXMAIN«EPS»NSIG.XLEFT»XRIGHT»MAXIT«IER) XZERO= XRIGHT ITERX(4)=MAXIT C NUMBER OF ITERATIONS REQUIRED C IF(IER.NE.O) XZERO=-FL0AT(IER) RETURN END - 137 - FUNCTION FXMAIN(X) FXMAIN .COMPUTE LEFThANDSIOE OF MAIN EQUATION FOR A GIVEN 1 VALUE OF X PAPAMETERS : X - STANDS FOK HA/HXORG»HXORG BEING THE CONCENTRATION (INPUT) OF H* IN THE ORGANIC PHASE FX = FXMAIN(X) : LEFTHANDSIOE OF MAIN EQUATION VARIABLES IN COMMON/PARM/ VAQ - VOLUME OF AQUEOUS PHASE VORG1 - VOLUME OF INITIALLY PRESENT ORGANIC PHASE DVORG - VOLUME OF ORGANIC PHASE (WITH REAGENT) IN ONE ADDITION HA - CONCENTRATION OF H* IN THE AQUEOUS PHASE CRADD - CONCENTRATION OF R IN ADDITIONS DVORG Nl - NUMbER OF ELEMENTS»INCLUDING R Nl = N • 1 N ELEMENTS:A»B»C. H WILL BE REFERRED TO AS THE (NM)-TH ELEMENT C»MU«- - TKEX - ARRAYS OF 11 ELEMENTS EACH Nl ELEMENTS ARE ACTUALLY USED MEANING OF : C(K) - INITIAL CONCENTRATION OF K-TH ELEMENT IN AQUEOUS PHASE MU(K) - VALENCY OF K-TH ELEMENT TKEX(K) - CONDITIONAL EXTRACTION CONSTANT FOR K-TH ELEMENT ( K = 1*2*... Nl ) THE FOLLOWING VARIABLES ARE AUXILIARY ONES AND HAVE BEEN PLACED IN COMMON ONLY FOR COMMUNICATION BETWEEN SUBPROGRAMS WHICH OTHERWISE WOULD DUPLICATE SOME COMPUTATIONAL STEPS THEY ARE SET (COMPUTED) IN XZERO .WHEREAS THE FORMER VARIABLES ARE SET IN THE XZERO CALLING (SUB) PROGRAMS VORG - TOTAL VOLUME OF CURRENT ORGANIC PHASE VORG = VORG1 • VADD CRADJ - ADJUSTED CONCENTRATION OF R IN THE AQUEOUS PHASE = CONCENTRATION OF R WHICH WOULD OCCUR IF ALL ADDITIONS CONTRIBUTED TO THE AQUEOUS PHASE ONLY CRADJ= C(ND* RADD/VAO RADD - TOTAL AMOUNT OF R ADDED CS - SCALED CX-VALUE : (VADD/VORG)* MU(N1)»CRADD ÜLA - RATIO OF VOLUMES (LAMBDA) - 138 - C OLA = VORG / VAO C XRNOM - AUXILIARY VARIABLE F0« ZEHOFINDING C ITFRX - ARRAY OF U ENTRIES CONTAINING INFORMATION ON THE C CONVERGENCE OF THE ROOT-FINDING PROCESS C ( TO OBTAIN A ZERO OF THE MAIN EQUATION ) C ITERX(1)«ITERX'2),ITESX<3> GIVE THE NUMBER OF C ITERATIONS REQUIRED TO FIND THE ZERO (HXORG) DURING C COMPUTATION OF QMlNtQ AND QMAX RESP. C ITERX(4) : SCRATCH c C c c c COMMON/PARM/VAU,VORG1.DVORG»HA»CRADD»Nl*C(11)*MU(11)«TKEX{11)• • VOKG«CRADJ*RADD*CStDLA«XRNOMtITERX(4) C DIMENSION T(13) C LOCAL SCRATCH-ARRAY APPROPRIATE FOR N«10 ELEMENTS (EXCL. R) C AT MOST C DATA ONE/l.O/ DATA PINF/1.0E2OO/ C PLUS INFINITY (FOR THIS PROGRAM) C C S=PINF IF(X.EQ.O.) GOTO 900 C FXMAIN TENDS TO INFINITY AS X TENDS TO ZERO C N2=N1*1 N3=N2*1 C CKEEP=C(N1) C(N1)=CRADJ C ASSIGN CWADJ TO C(N1) TEMPORARILY TO FACILITATE THE EXECUTION C OF THE FOLLOWING DO-LOOP C DO 10 K=1*N1 A1=MU(K)*C(K)/DLA A2»X*»MU END - 139 - APPENDIX III DESCRIPTION OF THE COMPUTER PROGRAMME QBACK - 140 - III.1. DESIGN OF THE SOFTWARE The software for the calculations of initial concentrations of the several elements present in an aqueous solution, from data obtained by activity measurements during a radiometric titration of that solution, consists of one FORTRAN main programme and a number of subprogrammes. III.l.l. gBACK This programme estimates initial concentrations of a number of elements dissolved in an aqueous phase from activity measurements performed during the radiometric titration of that solution as follows. Each activity measurement is represented by a set of four numbers: 1 = state index (1 = 0,1,?,,...), implying that a volume of IxAV of reagent has been added since t = 0. This index is also used as a subscript, t = time, j = element index (j • 1,2...,N). The elements A,B,...,R are numbered as I,2,..,N1. Nl = N + 1. This corresponds to the element R, originally bound to the added reagent. The index j is also used as a subscript, ij = measured activity. Hence the set of NM measurements consists of the NM quadru ples tl. »t.,j ,ij } where the subscript k denotes the index of measurement (k • 1,2...,NM). From the NM measurements the unknown initial concentrations are calculated by minimising the function NM E - i (R. r k-1 w (c)) = (c) where ^ = k x(jk) • tk 1Jk w, = weight of the k-th measurement: -A(jk) . tk e wk a) a) V')-S(jk) . ^ ! ^ -n -,.(ik.c)] . .. V . c. for j = 1,2,...,N aq j a. V . c + 1 . AV . c for j * NI J aq NMI1 r J A. • given decay constant of j-th element q. • extraction efficiency for the j-th element c. = initial concentration of the j-th element c * concentration of the reagent in the additions g. • unknown proportionality constant, computed from: NM * /ÏI • Cr - U) j m * (a) 2 k=l where Z treans: summation over those measurements for which jk = j' III.2. INPUT SPECIFICATION FOR QBACK The first four cards are identical to QFORW: on card 1: VAQ HA VORGI DVORG CRAD Nl on card 2: C(l) C(2) C(NI) on card 3: MU(1) MU(2) MU(N1) on card 4: TLG(l) TLER(l) TLG(2) TLER(2) TLG(NI) TLER(NI), However: on card 2 C(J) now denote estimates of the unknown concentrations. on card 5: LAMBDA(J), J - J,...,NI where LAMBDA(J) is the decay constant of the several elements. - 142 - Finally the input data of the NM activity measurements follow, one card for each measurement containing: V V V ijk" Like for QFORW, the input for QBACK is free formatted. QBACK is restricted to at most It elements (Nl < 11) and 500 activity measurements (NM < 500). HI.3. ROUTINES USED QBACK calls in a routine RNL (non-linear regression) to minimise the non-linear function of the unknown concentra tions. For subroutine RNL itself calls a routine RESIDU, which for an estimate of the parametervector p, might calculate the residual vactor R. The subroutine EXTEFF is called in RESIDU to compute q-values for a given volume of reagent added. EXTEFF and its required subprogrammes XZER0 and FXMAIN have been described in Appendix I. III.4. SUMMARY OF VARIABLES Variables in QRACK other than those used in QFORW are listed below. LAMBDA(J) - decay constant (A.) of the j-th element. LK(K) - state index 1, of the k-th measurement. TK(K) - time index t, of the k-th measurement. JK(K) - elementindex j of the k-th measurement. Y(K) - activity measured (ij,) of the k-th measurement. SY(K) - /!jk. W(K) - weight (w ) of the k-th measurement. NBETA - number of different elements of which the activity is measured. SIGMA(J) - estimate of error (a.) in j-th parameter. NM - number of activity measurements. NW - number of observations. NDF - number of degrees of freedom. SSQ - residual sum of squares (E). - 143 - R(K) - the k-th component of residual vector (R, )• JACOB - Jacobian matrix. CORREL - correlation matrix III.5. DESCRIPTION OF SUBPROGRAMMES LOADED FROM LIBRARIES 111.5.1. QPRO^RAMJ-i 1°£*1_ nbrai:y_§ t_EjC._N._Pe 11 en ) .ROUTINE RNL (RESIDU, NW, NP, NDF, NSIGP, P, R, SSQ, JACOB, SIGMA, CORREL, PROB, SCR, IER) This subroutine 'etermines the the parameter values and associated correlations for nonlinear regression problems, more precisely: Let R(K) = Y(K)-PHI(K,P) where the NW observations Y(K) have variance 1 and expectation PHI(K,P) where PHI is a nonlinear function of the unknown parameters P(I), 1=1 to NP. Given an external subroutine residu (P, NW, NP, R) to compute the residual vector R for an assumed parameter - setting P, RNL tries to minimise SSQ = euclidean norm of R. RESIDU (P,NW,NP,R) - external subroutine NW - nr. of observations = length of vector R. NP - nr. of parameters = length of vector P. NDF - nr. of degrees of freedom. NSIGP - nr. of significant digits of P required. P - entry: estimate of parameters exit : value of parameters. R - residual vector. SSQ - euclidean norm of R. JACOB - array of dimensions NW, NP containing Jacobian » DR/DP. SIGMA - standard deviations of parameters. If on entry, any a(I) is positive, it is assumed to be an estimate of a(I). If a(I) < 0, RNL will take as estimate 0.1 x P(I). - 144 - In either case, the estimate will be revised in the iteration process if either on entry or in the iteration process the estimate of a(I) exceeds 10 , it is set to 0 and this parameter is kept fixed. CORREL - correlation coefficients of parameters. PROB - probability that chisquare (NDF) less than SSQ. SCR - scratch array. IER - error indicator. The dimensions of the various arrays follow from the array declaration statement: iloal P(NP), R(NW), JACOB(NW.NP) SIGMA(NP), CORREL(NP,NP) , SCR(5.NP + 2.NW + NP.(NP + 1)). 111.5.2. NUMRCN_ (loca 1_1 ibr ary__a t _EiCiNi_Pe t ten ) .ROUTINE PA (IDIMA.NROWA.NCOLA,A,NFILE,HEADING,NRDIGITS SCALE) This routine prints a rectangular real array with a specified number of digits. IDIMA - row dimension of A as declared in the calling programme. NROWA - number of rows of A to be printed. NCOLA - number of columns of A to be printed, A - array containing real numbers. NFILE - filenumbers of outputdevice. HEADING - an alphanumeric used as a header for each block. E.G. HEADING = I0HTITLE.MXX will result in a new page for each block and heading: TITLE.MXX. SCALE - output parameter, computed by PA as a power of 10. Scale is a common multiplication factor. Scale is 100.0 means that all entries printed are to be multiplied by 100.0. Scale is selected such t i the largest entry is between 1.0 and 10.0. - 145 - NRDIGITS - If NRDIGITS = K + 1, then all entries are printed in the format SX.XXXX with the digits behind the decimal point. (S is sign) 1 < NRDIGITS < 9. III.6. DEVIATIONS FROM ANSI-FORTRAN Apart from the routines mentioned above and the subprogrammes EXTEFF, XZERO and FXMAIN (described in Appendix I) a random number generator routine from the FORTRAN system library is used. See reference |4| of Appendix I. RANF(n) - FORTRAN Intrinsic Function. Returns values uniformly distributed over the range (0,1); the values 0 and 1 are excluded, 'n' is a dummy argument which is ignored. The result is type real. - 147 - APPENDIX IV TEXT OF THE COMPUTER PROGRAMME QBACK - 148 - PROGRAM QBACK(INPUT.OUTPUT»TAPES=lNPUT,TAPE*=OUTPUT> C OBACK . PROGRAM TO COMPUTE INITIAL CONCENTRATIONS FROM ACTIVITY C 1 MEASUREMENTS COMMON/PARM/VAO*VORG1•DVORG.HA.CRAOD.N1»C (11>.MU( 1I).TKEX(11), • V0RG*CRA0J*RAD0*CS*0LA«XRN0M«tTrRX(4) C C C VARIABLES IN COMMON/PARM/ c c VAO - VOLUME OF AQUEOUS PHASE c VOPG1 - VOLUME OF INITIALLY PRESENT ORGANIC PHASE c OVORG - VOLUME OF ORGANIC PHASE (WITH REAGENT) c IN ONE AOOITION c HA - CONCENTRATION OF H* IN THE AQUEOUS PHASE c CRADO - CONCENTRATION OF R IN ADDITIONS QVORG c Nl - NUMBER OF ELEMENTS*INCLUDING R c NI = N • 1 c N ELEMENTS:A,8tC. c R WILL BE REFERRED TO AS THF (N*1)-TH c ELEMENT c C.MU»- c - TKEX - ARRAYS OF 11 ELEMENTS EACH c Nl ELEMENTS ARE ACTUALLY USED c MEANING OF : c C(K> - INITIAL CONCENTRATION OF K-TH ELEMENT c c IN AQUEOUS PHASE c MU(K) - VALENCY OF K-TH ELEMENT c TKEX(K) - CONDITIONAL EXTRACTION CONSTANT FOR c K-TH ELEMENT c ( K * lt?«... Nl ) c c THE FOLLOWING VARIABLES APE AUXILIARY ONES ANO HAVE BFEN c PLACED IN COMMON ONLY FOR COMMUNICATION BETWEEN SUBPROGRAMS c WHICH OTHERWISE WOULD DUPLICATE SOME COMPUTATIONAL STEPS c THEY ARE SET (COMPUTED» IN XZERO «WHEREAS THE FORMER c VARIABLES ARE SET IN THE XZERO CALLING (SUB) PROGRAMS c c VORG - TOTAL VOLUME OF CURRENT ORGANIC PHASE c VOPG = V'»RG1 • VADD c CRADJ - ADJUSTED CONCENTRATION OF R IN THE AQUEOUS PHASE c = CONCENTRATION OF R WHICH WOULD OCCUR IF «LL c ADDITIONS CONTRIBUTED TO THE AQUEOUS PHASE ONLY c CRADJ* C(Nl>* RADD/VAQ c RADD - TOTAL AMOUNT OF R ADDED c CS - SCALED CX-VALUE : (VADD/VORG)» MU(N1)*CRA0D c DLA - RATIO OF VOLUMES (LAMBDA) c DLA » VORG / VAO c XRNOM - AUXILIARY VARIABLE FOR ZEROFINDING c ITFRX - ARRAY OF * ENTRIES CONTAINING INFORMATION ON THE c CONVERGENCE OF THE ROOT-FINDING PROCESS c ( TO OBTAIN A ZERO OF THE MAIN EQUATION ) c ITERX(l),ITERX(?)tITERX(3) GIVE THE NUMBER OF ITERATIONS REQUIRED TO FIND THE 7ER0 (HXORfO DURING - 149 - C COMPUTATION OF QMIN.G AND QMAX RFSP. C ITERXU) : SCRATCH C c c c C COMMUNICATION BETWEEN OBACK AND RESIDU COMMON/RESPAR/ALFA•NM• • LK(500)»TK(500)»JK(500)»SY(5on)*W(500) • • G(ll)tTLGUl) C K-TH MEASUREMENT IS LK(K).TK(K)»JK(K)»Y(K> C <5Y(K)*SQRT(Y(K)) C W(K> = EXP(-LAMBDA(JK(K))«TK(K) ) / SY(K) C J-TH ADDITIONAL OBSERVATION t C TLG(J)= GIVEN LOG(KEX) OF J-TH ELEMENT C WITH MEAN ERROR TLER(J> C G(J)x 1/ TLER(J) C C ALFA x Otl DENOTES ABSENCE/PRESENCE OF ADDITIONAL OBSERVATIONS C NM s NR.OF MEASURFMENTS(EXCLUDING ADDITIONAL OBSERVATIONS) C REAL TLER(11)«LAMBDA(11)»P(22).R(511) REAL Y(500)«JACOB(511.22).SIGMA(22)»C0RREL(?2»22> EXTERNAL RESIDU REAL SCR(2000) INTEGFR KBETA(ll) INTEGER OUTPUT c COHESION OF VARIABLES c VARIABLE LOCATION INITIALISED DEFINED USEO c VAO PARM OBACK(I) REST"»» c V0RG1 PARM OBACK(I) c DVORG PARM QBACK(I) RESIo: c HA PARM OBACK(I) c CRADD PARM OBACK(I) RESIDU c Nl PARM OBACK(I) c C(J) PARM OBACK(I) RESIDU c MU(J) PARM OBACK(I) c TKEX(J) PARM OBACK RESIOU c ALFA RESPAR OBACK RESIDU c NM RESPAR OBACK RESIDU c LK(K) RESPAR OBACK(I) RESIDU c TMK) RESPAR OBACK(I) RESIDU c JK(K) RESPAR OBACK(I) RESIDU c SV(K) RESPAR OBACK RESIOU c W(K) RESPAR OBACK RESIDU c G(J> RESPAR OBACK RESIDU c TuG(J) RESPAR OBACK(I) RESIDU c - 150 - c VAPIABLE LOCATION INITIALISED DEEINEn USED c PSCALE(I) PSCALE RNL RNL.RESTDU c Pd) (OBACK) OBACK RNL RESIDU.OBACK c R(I> (OBACK) RESIDU ZXSSQ.QBACK c BETA(J) (RESIDU) RESIDU RESIDU c AKA(K> (RESIDU) RESIDU RESIDU c TBETA(J) (RESIDU) RESIDU RESIDU c SBETAU» (RESIDU) RESIDU RESIDU c TLER(J) (OBACK) OBACK'I) c LAMBOA(J) (OBACK) OBACK(I) c Y(K) (OBACK) OBACK(I) c JACOB(K,I) (OBACK) RNL OBACK c SIGMA(I) (OBACK) RNL QBACK c CORREL(l.J) (OBACK) RNL OBACK c SCR (QBACK) RNL c COMMUNICATION BETHEEN 1RN L AND RESIDU COMMON /PSCALE/ XINIT.I?F . PSCALE(4*11 c P(I>* PSCALE(I»NP) • PSCALE(I)»X(I) c P IS PARAMETERVECTOR c X IS SCALED PARAMETERVECTOR DATA XINIT /1.0E-5/ DATA RF/1.0E-10/ DATA PSCALE/2PM.0/ DATA NlMAX/11/t NMMAX/500/ C AT MOST 11 ACTIVE ELEMENTS (INCLUDING R) C AT MOST 500 MEASUREMENTS DATA TEN/10.0/ DATA OUTPUT /6/ 800 CONTINUE FOLLOWING INPUT IDENTICAL TO QFORW WRITE(6,200) 200 FORMAT(1H1) PAGE EJECT READ(5t«) VAQ.HA.V0RG1.DV0RG.CRADD.N1 PRINT «.VAQ.HA•V0RG1.DV0RG.CRA0D.N1 IF(Nl.LE.NlMAX) GOTO 10 WPITE(6«210) N1« C 15 READ(5.»> (C(K)«K=1,N1) PRINT •• (C(K)*K«1«N1) C INITIAL CONCENTRATIONS READ(5.»> (MU«K).K=1,N1) PRINT •* (MU(K)»K=1»N1) C VALENCIES READ(5**> (TLG(K)*TLER(K)«K*1*N1) PRINT •* (TLG(K)»TLER(K)«K=1.N1) C 10LOG OF K-EX VALUES WITH THEIR ERRORS C C SET KEX AND G(J) 00 30 J=1.N1 TKEX(J)=TEN»*TLG(J) G(J)xl.O/TLER(J) 30 CONTINUE C FOLLOWING INPUT SPECIFIC TO BACK READ (5«*l (LAMBDA(J)*J=1*N1> C DECAY CONSTANTS PRINT 90f (LAMBDA(J)*Jsl*Nl) 90 FORMAT ( 1H0» »OECAV CONSTANTS •/ UF10.8 J C READ ACTIVITY MEASUREMENTS PRINT 99 99 FORMAT(lH0«RX*lHKt9X*lHL»9X*lHT*9X*lHJt9X*lHY*9X*7HS0RT(Y)*3X«lHH • /) KsO 50 KxK»l READ <5t»> LK(K)»TK(K),JK(K)»Y(K) IF (EOF(S).NE.O.O) GOTO 60 PRINT 91, K, LK(K),TK(K),JK(K),Y(K) 91 FORMAT(lX«2I10tF10.5«I10»F10.0) IF (JK(K).LT.l.OR.JKCK).GT.Nl) GOTO 55 SY(K)*SQRT(Y(KI) W(K) = EXP(-L*MBDA(JK(K))*TK(K) ) / SY(K) PRINT 93*SY(K)«W(K) 93 FORMAT(lH»»50X»F10.5»F10.fl) IF (K.LT.NMMAX) GOTO 50 GOTO 60 C ERROR i 55 PRINT 92 92 FORMAT(45H0ERROR IN DATA? THIS MEASUREMENT IS IGNORED. ) K«K-1 GOTO 50 C ENO OF DATA 60 CONTINUE NM»K-1 C COMPUTE NR.OF ACTIVE PARAMETERS BETA DO 6? J«1,N1 6i? KBETA(J)»0 - 152 - 00 64 K=1.NM 64 KBETA(JK(K))*KBETA(JK(K>)*1 NBETA«0 00 65 J=l.Nl IF (KRETA(J).NE.0) NBETA=NBETAM 65 CONTINUE C TEST NM IF (NM.GT.N1*NBETA) 60T0 61 PRINT 991,NM 991 FORMAT(1H0* • NUMBEP OF MFASUREMENTS=».I3. • IS TOO SMALL.») STOP C INITIALISE PAPAMETEPVECTOR 61 DO 70 Jsl.Nl P(J)=C(J) SIGMA(J)=O.0 SIGMA(J»N1)«TLER(J> 70 P(J*N1)=TLG(J> PF*0 C CALCULATION THREE TIMES FOR SAME SET OF OAT»: C FIPSTtCONC» SECONO :CONC»KEX» THIRD:CONC*KFX BUT CONC.REAGENT=0 75 CONTINUE C FIRST COMPUTE WITH ALFA*0 ALFA«0.0 NW=NM*1 C R(NW) IS PENALTY FOR NEGATIVE C(J) NPsNl BO N0F«NW-NP-NBETA-1 NSIGP*6 CALL RNL(RESIDU.NW.NP.NDF.NSÏGP.P. • R.SSO.JACOB.SIGMA.CORREL.PROB.SCR.IER) PRINT 901.ALFA.NW.NP.NDF.IER.SSQ.PROB 901 FORMATdHl» «RESULTS OF RNL'.FOR ALFA= • «Flo.5/ • • NW.NP.NDF s • * 13.1H. I3.1H. 13 / • • ERROR INDICATOR» • * 13 / • • RESIDUAL SUM OF SQUARES* • * F10.5 / • • PROBABILITY OF CHISOUARE LESS THAN SSQ= •• F10.5 / ) PRINT 902«(I«P(I>«SIGMA(I)«I*1*NP) 90? F0RMATUH0.1HI.9X.1HP.9X.5HSIGMA /( 1X.I02.F20.10.E20.10 ) ) PRINT 903. (R(K).K»1.NW> 903 FORMATdHO.» RESIOUAL VECTOR:» / (10F10.5) ) CALL PA(NW.NW.NP.JACOB.OUTPUT.6H0JAC0B.NSIRP.ASCALE) CALL PA(NP.NP.NP.CORREL.OUTPUT.7H0CORREL.NSTGP.ASCALE» 85 CONTINUE IF (ALFA.LT.0.0) GOTO 800 IF (ALFA.EO.1.0) GOTO 850 C REPEAT WITH EXTENOEO MODEL ALFA«1.0 NM*Nt*~Nl«l NP«2*N1 GOTO 80 REPEAT «ITH CONCENTRATION REAGENS FIXED AT ZERO 850 ALFAs-1.0 P(N1>*0 SIGMA(N1)*1.1E100 GOTO 80 999 STOP END - 154 - SUBROUTINE RESIDU(P.NW.NP»R) C RESIDU .SUBROUTINE OF ORACK. CALLED BY RNL C 1 TO CALCULATE RESIDUAL R FROM PARAMETERS P C REAL P(NP)»R(NW> C NP^NR.OF PARAMETERS C NW=NR.OF OBSERVATIONS C COMMON : PARM*RESPAR*PSCALE REAL P(NP)tR(NW) COMMON/PARM/VAO*VORG1«nVOPG»HA.CRADO«Nl.C(11)«MU(11>«TKEX(11)* • VOR6«CRADJ«RADD»CS«DLA»XRNOM« ITFRU <<»> C C C VARIABLES IN COMMON/PARM/ C C VAO - VOLUME OF AQUEOUS PHASE C V0R61 - VOLUME OF INITIALLY PRESENT ORGANIC PHASE C DVORG - VOLUME OF ORGANIC PHASE (WITH REAGENT) C IN ONE ADDITION C HA - CONCENTRATION OF H* IN THE AQUEOUS PHASE c CRADO - CONCENTRATION OF R IN ADDITIONS OVORG c Nl - NUMBER OF ELEMENTStINCLUDING R c Nl * N • 1 c N ELEMENTS*. A»B»C. c R WILL BE REFERRED TO AS THF (N*1)-TH c ELEMENT c C«MU»- c - TKEX - ARRAYS OF 11 ELFMENTS EACH c Nl ELEMENTS ARE ACTUALLY UStD c MEANING OF : c C(K) - INITIAL CONCENTRATION CF K-TH ELEMENT c IN AQUEOUS PHASE c MU(K> - VALENCY OF K-TH ELEMENT c TKEX(K) - CONDITIONAL EXTRACTION CONSTANT FOR c K-TH ELEMENT c ( K = 1.2,... Nl ) c c THE FOLLOWING VARIABLES ARE AUXILIARY ONES AND HAVE BFEN c PLACED IN COMMON ONLY rOR COMMUNICATION BETWEEN SUBPROGRAMS c WHICH OTHERWISE WOULD DUPLICATE SOME COMPUTATIONAL STFPS c THEY ARE SET (COMPUTED) IN XZERO «WHEREAS THE FORMER c VARIABLES APE SET IN THE XZERO CALLING (SUB) PROGRAMS c c VORG - TOTAL VOLUME OF CURRENT ORGANIC PHASE c VORG = VORG1 * VADD c CRAOJ - ADJUSTED CONCENTRATION OF R IN THE AQUEOUS PHASE c » CONCENTRATION OF R WHICH WOULD OCCUR IF ALL c ADDITIONS CONTRIBUTED TO THE AOUEOUS PHASE ONLY c CRAOJ» C C XRNOM AUXILIARY VARIABLE FOR ZEROFINDING C ITERX ARRAY OF 4 ENTRIES CONTAINING INFORMATION ON THE C CONVERGENCE OF THE ROOT-FINDING PROCESS c ( TO OBTAIN A 7ER0 OF THE MAIN EQUATION ) c !TERX(1).ITERX(?).ITERX(3) GIVE THE NUMBER OF c ITERATIONS REQUIRED TO FIND THE 7ER0 (HXORft) DURING c COMPUTATION OF OMlNtO AND OMAX RFSP. c ITERX(4) : SCRATCH c c r C COMMUNICATION BETWEEN QBACK AND RESIDU COMMON/RESPAR/ALFA«NM« • LK(500)*TK15001•JK(500)*SY(500)*W(500) • • GUl).TLG(ll) C K-TH MEASUREMENT IS LK(K)«TK(K),JK(K)*Y(K) C SV(K)*SQRT(Y(K)) C W(K) » EXP(-LAMRDA(JK(K)>*TK(K) ) / <;V(K) C J-TH ADDITIONAL OBSERVATION : C TL6(J)= GIVEN LOG(KEX) OF J-TH ELEMENT C WITH MEAN ERROR TLER(J) C G(J)= 1/ TLER(J) C C ALFA x 0*1 DENOTES ABSENCE/PRESENCE OF ADDITIONAL OBSERVATIONS C NM> NR.OF MEASUREMENTS(EXCLUDING ADDITIONAL OBSERVATIONS) C C COMMUNICATION BETWEEN RNL AND RESIDU COMMON /PSCALE/ XINITtRFt PSCALE U*> C P(I)= PSCALE(I*NP) • PSCALE(I)*X(I) C P IS PARAMETERVECTOR C X IS SCALED PARAMETERVECTOR REAL RETA(ll)*AKA(500)tTBETA(ll)tSBETA(ll) DATA TEN/ 10.0/ C PRINT 9?t(P(I)*I*l*NP) C 92 FORMATUHO»«PARAMETERS RESIDU»/ UX.6E20.13)) C R(NW) IS PENALTY FOR NEGATIVE C(J) DATA PENALTY /O.O/ R(NW>*0 C SET SCALED INTERNAL PARAMFTERS DO 10 J*1«N1 C(J)«PSCALE(J^NP)*PSCALE(J)»P(J) IF (C(J).LT.O.O) R(NW)aR(NW)-PENALTV»C(J) IF (C(J).LT.O.O) C(J)»-C(J) C REVERSE SIGN OF NEGATIVE CONCENTRATIONS 10 CONTINUE - 156 - IF (ALFA.EO.0.0) GOTO 21 C SET ALSO R(ADDITIONAL) DO 30 J=1.N1 SJ*PSCALE(J*HP*H\> UPSCALE CALL EXTEFF(JK(K)tVAOD*MODEOtX*0) C TEST 0 IF (O.LT.0.0) GOTO 11 J=JK(K> AJsVAO»C(J) IF (J.EQ.N1I AJ=AJ»VADD*CPADD AKA(K)s WCK)*(1.0- 0) *AJ 30 CONTINUF GOTO 32 31 PPINT <>11 0«K 91 FORMAT(?5H0»*»ERR0R»»*r NEGATIVE 0= »F10.5,17H FOR MEASUREMENT» * • 13/ • RESULTS UNRELIABLE • ) RETURN 32 CONTINUF C CALCULATE PROPORTIONALITY BETA 00 40 J*1«N1 40 TPETA(J)=SBETAU)*0.0 00 50 K=1,NM J=JK(K) TBETA(J)sTBETA(J)^SV(K)»AKA(K) SBETA(J)sSBETA(J)*AKA(*)••? 50 CONTINUE 00 60 J=1.N1 IF (SBETA(J).EO.O.O) GOTO 60 C IF SBETA(J)30 • THEN BETA(J) IS UNDEFINED BETA(J)«TBETA(J)/SBETA(JJ 60 CONTINUE - 157 - C RESIDUAL VECTOR 00 70 K=1»NM 70 R(K)= SV(K) - BETA(JK(K>)* AKA(K) • •RF»(RANF(QQ>-0.5> C RANDOMISED RESIDUAL VECTOR C FOR TESTING PURPOSES C RNORM= VIP( R*ltR«l*NW.RNORM) C PRINT 97 »RN0RM,(P(I),I=1»NP) C 97 FORMAT(7H0RESIDU* AE?ft.l3> C PRINT 9n,(J,C(J).TKEX(J)»TBETA(J)*S8ETA(J).RETA(J)»J=l.Nl) C 99 FORMAT(1HO» 1HJ.9X«1HC,9X»4HTEX»6X.5HTBETA,5X»5HSBETA.5X»«HBETA/ C 1 (IX,110» 5E10.3 ) ) C PRINT 9ft«RN0RM C 98 FORMAT< 5H0SSO*» E20.13) RETURN EK3 - 159 - APPENDIX V DESCRIPTION OF APPARATUS FOR SEMI-AUTOMATIC LIQUID-LIQUID EXTRACTION - 160- V.l. INTRODUCTION In radio analysis liquid-liquid extraction is an useful sepa ration technique. In activation analysis the radio isotope(s) of the element(s) of interest can be obtained separated from the activity of the matrix, whereas in isotope dilution ana lysis the species to be analyzed can be concentrated in an easy way. To minimize the time of handling of the radioactive solution by a radiological analyst, the use of a semi automatic appa ratus for liquid-liquid extraction is preferable. V.2. THE 'ANALYTISCHE SYSTEM APPARATE (A.S.A) -MODULAR SYSTEM' OF ISMATEC At the radiochemical laboratory of E.C.N. Petten, a semi automatic liquid-liquid extraction apparatus of Ismatec has been tested for the determination of copper by means of ex traction with an organic solution containing neocuproine. The apparatus is built up with the following units (Figures V. I-V.4). V.2.I This first dispenser (2 x 5.0 ml adjustable) adds a buffer so lution (sodium citrate) to the acid-destructed sample to fa cilitate the adjustment of the pH by the titrator. V.2.2. Titrator The pH of the sample solution is adjusted by a 'Metrohm' ti tration unii E 526 with dosimat E 535. As was described in chapter 2, the yield of an extraction with a complexing agent depends on the pH of the aqueous solution. After the pH is adjusted, the second dispenser of the first unit adds an amount of a solution of hydroxylamine HC1 to re duce all Cu(II) to Cu(I). V.2.3. ^iSE£D5ÏD8-§D-_!SïïiDS_yDi£ The second dispenser is used to add both the solution of the reagent and the organic solvent to the sample solution. Simultaneously the aqueous and organic phases are mixed vigo- - 161 - rously by means of a stirrer, for a predetermined time. V.2.4. Dilcomo This part of the apparatus can be used as a diluting unit; in the determination of copper it served as a sampling unit. Figure v.1: General imnression of the semi automatic liquid-liquid extraction apparatus. Figure v ,2: Control unit. - ie>2 - Figure v.3: Dispensing and mixing unit, Figure '' Oil conn used as a sampling unit,