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Transition State Theory Winter 2014 Chem 350: Statistical Mechanics and Chemical Kinetics Chapter 6: Adiabatic reaction dynamics and Transition State Theory ...................................................... 70 Bimolecular reactions ............................................................................................................................ 71 Transition State theory .......................................................................................................................... 72 kT Derivation of universal factor B ....................................................................................................... 74 h Kinetic Isotope effect ............................................................................................................................. 75 Chapter 6: Adiabatic reaction dynamics and Transition State Theory General reaction: A B C... X Y Z... + + + + Rate at which equilibrium is established: a b c d ⎡ A⎤ ⎡ A⎤ ⎡B⎤ ⎡C⎤ ⎣ ⎦ rate t k t ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ Eg. − = ( ) = φ ( ) x y z dt ⎡ X ⎤ ⎡Y ⎤ ⎡Z ⎤ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ [w]: concentration of species w . (as molarity or partial pressure) ktφ ( ) : phenomenological rate constant (might depend on time) In general, equation might only be valid for limited amounts of time/concentration regimes. General description can be very complicated as many “elementary” steps might be involved in the reaction. To make progress, we need to reduce a reaction to elementary steps of unimolecular or bimolecular reactions. Then we can apply principles of statistical mechanics. We will consider reactions on a single potential energy surface (adiabatic, Born-Oppenheimer approximation). k1 Simplest case: Unimolecular reaction A B k −1 d ⎡ A⎤ ⎣ ⎦ k ⎡ A t ⎤ k ⎡B t ⎤ Rate equation: − = 1 ( ) − −1 ( ) dt ⎣ ⎦ ⎣ ⎦ d ⎡ A⎤ ⎣ ⎦ 0 At A Steady State: = ⎣⎦⎡⎤( ) =⎡⎤eq dt ⎣⎦ kA⎡⎤ k ⎡⎤ B 11⎣⎦eq= − ⎣⎦ eq ⎡⎤Beq k ⎣⎦==K 1 A eq k ⎣⎦⎡⎤eq −1 We have seen before how to calculate equilibrium constants in stat mech. Experimental determination k1 . Use excess A , neglect kB−1 [ ] Chapter 6: Adiabatic Reaction Dynamics and Transition State Theory 70 Winter 2014 Chem 350: Statistical Mechanics and Chemical Kinetics dA[ ] −=kdt [A] 1 ⎣⎦⎡⎤At( ) ln =−kt1 [Ao ] At A e−kt1 ⎣⎦⎡⎤( ) =[ o ] Exponential decay of [A] under conditions of excess can be fitted to get k1 . Of course one could also start with excess of [B], and measure k−1 . This would allow a check of relations. k ⎡⎤Beq 1 ==⎣⎦K k A eq −1 ⎣⎦⎡⎤eq Other types of reactions: k1 A! B + C k −1 dA[ ] −=kA11[ ]− k− [ BC][ ] dt ⎡⎤⎡⎤BCeq eq k ⎣⎦⎣⎦==K 1 A eq k ⎣⎦⎡⎤eq −1 If is negligible At A e−kt1 [BC][ ] ⎣⎦⎡⎤( ) =[ o ] Bimolecular reactions k1 A+ B!C k −1 dA[ ] −=kAB11[ ][ ]− k− [ C] dt ⎡⎤Ceq k Steady State: ⎣⎦==K 1 AB eq k ⎣⎦⎣⎦⎡⎤⎡⎤eq eq −1 In presence of excess B , such that B is constant At A e−kt* , kkB* [ ] ⎣⎦⎡⎤( ) =[ o ] = 1 [ o ] Bimolecular reaction is the limit. Elementary reaction steps never involves more than 2 colliding molecules. Negligible probability to have 3 molecules colliding simultaneously. If three molecules are needed, then one has to first create a complex of two molecules and collide this with a third molecule needed for reaction. Chapter 6: Adiabatic Reaction Dynamics and Transition State Theory 71 Winter 2014 Chem 350: Statistical Mechanics and Chemical Kinetics In our discussion of kinetics (later on) you will learn how to think about building complicated reactions from elementary reaction steps. Here we focus on elementary reactions and focus on how we can calculate rate constants from first principles using information from quantum chemistry and statistical mechanics calculations Transition State theory To develop equations for the rate of reactions we will assume thermal equilibrium for all species AB,#, , where # refers to the (fleeting) transition state or “activated complex”. Overall reaction: k1 [B] k1 A B Keq == k−1 Ak [ ] −1 k k a rxn ⎡#⎤ ⎣ ⎦ ka model: A! # → B = = K# kd ⎡ A⎤ k ⎣ ⎦ d k : activation, k : deactivation, k : rate to go from # to B a d rxn Crucial idea: consider forward reaction only ( k# rate to products). k ⎡ A⎤ = k ⎡#⎤ 1 ⎣ ⎦ rxn ⎣ ⎦ ⎡#⎤ N k = k ⎣ ⎦ = k # 1 rxn ⎡ A⎤ rxn N ⎣ ⎦ A (Note: In chemical kinetics one usually uses concentrations to derive things. In statmech we are used to expressing things in partial pressures. I am doing the derivation here, based on partial pressure as a measure of concentration. ) k : rate of moving to product for molecules at transition state. rxn ⎡#⎤ ⎣ ⎦ ⎛ q# / N# ⎞ q# = K# = → " " ⎡ A⎤ ⎝⎜ q / N ⎠⎟ q ⎣ ⎦ A A A Chapter 6: Adiabatic Reaction Dynamics and Transition State Theory 72 Winter 2014 Chem 350: Statistical Mechanics and Chemical Kinetics Use same formula as for chemical equilibrium! Same equilibrium principles. The factor N is included in the translational partition function as usual. It is a bit of a nuisance to write it correctly. Calculating qA : calculate molecular structure, vibrational frequencies, energy at minimum. AA AAA qqqqqqAezptRv= For transition state, we can proceed in analogous way q# : calculate structure of transition state; calculate vibrational frequencies, transition state electronic energy. Something special about # (non-linear TS) We have 37N − harmonic, real vibrational frequencies, and one imaginary frequency, which needs to be treated in a special way. The normal mode of the imaginary frequency is along the reaction coordinate. q = q# ⋅ q#q# q#q# q# = q# ⋅q! # rxn ( e zp t R v ) rxn # q# : component of partition function along reaction coordinate rxn q : partition function of transition state, for all motions except ‘vibration’ along # reaction coordinate It then follows that the forward reaction rate is given by qrxn # ⎛ q!# ⎞ k1 = krxn = krxnqrxn q ⎝⎜ q ⎠⎟ A A # kBT 13− 1 We will see that krxnqrxn = ~10 s h k T ⎛ q ⎞ k B # 1 = ⎜ ⎟ h ⎝ q ⎠ A k T ⎛ q ⎞ k B # A B # C Or: 1 = ⎜ ⎟ if bimolecular ( + → ) h ⎝ q q ⎠ A B q# The determining factor in is Ke Kzp as it is for chemical equilibria. The theory is very similar. qA # # # q − ΔE +ΔE /k T q q q # = e ( e zp ) B ⋅ t R v q q Aq Aq A A t R v Chapter 6: Adiabatic Reaction Dynamics and Transition State Theory 73 Winter 2014 Chem 350: Statistical Mechanics and Chemical Kinetics We get an exponential temperature dependence from ΔEEezp+Δ, power of T dependence from the remaining fraction, often just a constant if same # of molecules for reactants and kT transition state. Other contribution B h ΔEzp : the difference in zeropoint energy between reactant and transition state 1137NN−− 36 ΔE =hhωω− zp∑∑( i)TS( i ) R 22ii==11 Please note the number of normal modes in each sum. If the molecular structure is linear, the number of normal modes increases by 1, as usual. kT Derivation of universal factor B h a) Partition function q# rxn ⎡⎤δδ Potential is flat in transition state region xx, ⎢⎥oo−+ ⎣⎦22 → Assume particle in the box types of partition functions # 2πmkBT qrxn = 2 ⋅δ h Notes: 1) if reaction coordinate involves a single atom, we can assign mass m. In general, reaction coordinate is some collective motion (normal mode of transition structure). We do not know m ! 2) We do not know δ either !! Chapter 6: Adiabatic Reaction Dynamics and Transition State Theory 74 Winter 2014 Chem 350: Statistical Mechanics and Chemical Kinetics Second assumption: all molecules in the transition state region that have a positive velocity will V react. The rate depends on how fast they traverse the transition state region: x δ We can calculate how fast molecules traverse the TST region: 1 ∞ k = V P V dV rxn ∫0 x ( x ) x δ PV( x ): probability to have Vx Once again, use equilibrium arguments. Vx follows Maxwell-Boltzmann distribution mV 2 /2k T e− x B P Vx = ( ) ∞ mV 2 /2k T e− x B dV ∫−∞ x Here we use a Boltzmann distribution for ‘kinetic energy’ 1 kBT Integrate (Gaussian integrals) → krxn = δ 2πm Assumption: Even at the transition state geometry, the velocity distribution Vx is given by Maxwell-Boltzmann (where potential energy is very high). This seems a bit counter intuitive. I do not have a good rationale for this. A great virtue of the above derivation is: ⎛ ⎞ ⎛ ⎞ # 2πmkBT 1 kBT kBT qrxn ⋅ krxn = ⎜ 2 ⋅δ ⎟ ⋅⎜ ⎟ = h δ 2πm h ⎝ ⎠ ⎝ ⎠ Unknown m, δ cancel. How convenient! It will be clear that this factor is an estimate. Transition state theory is not as robust as other aspects of statistical mechanics. There are variations on the theme in the scientific literature. Kinetic Isotope effect Isotope effects are strongest (or even observable only) if protons are replaced by Deuterium 1) Primary isotope effect: H or D is involved in reaction path, or the main atom that is moving a. Isotope substitution does not affect electronic structure calculations → same PES, same stationary points However mass enters mass-weighted Hessian, and this affects the frequencies g ω ~ (For diatomics. Analogous for polyatomics) m → pure proton frequency is 2 times as large as D frequency. A good example would be to compare the vibrational frequencies in HCl and DCl. Chapter 6: Adiabatic Reaction Dynamics and Transition State Theory 75 Winter 2014 Chem 350: Statistical Mechanics and Chemical Kinetics If proton movements defines reaction coordinate, then in the reactant this vibrational frequency 36N − 1 E h contributes to zp= ∑ ω i . i=1 2 But it is missing from transition state 37N − 1 E h zp= ∑ ω i,# i=1 2 1 k R R /k T light − !(ω heavy −ωlight ) B 1 ≈ e 2 > 1 ω ~ ω (for D vs. H) k heavy 2 light heavy → light particle faces less of a barrier than heavy particle ⎛⎞[##] ⎛⎞[ ] ⎜⎟> ⎜⎟ ⎜⎟AA ⎜⎟ ⎝⎠[ ] light ⎝⎠[ ] heavy k T Also note that m did cancel out in the derivation of B . The cause of the effect is not h that light particles move faster. The effect is due to zeropoint motion (purely quantum!) k ⋅q# is the same for any particle (or collective coordinate). rxn rxn b) In secondary isotope effect, the isotope of interest is not involved in the reaction coordinate → zeropoint effects more or less cancel, minor residual effect on kklight/ heavy . One can expect the effect of isotopic substitution to be much smaller then.
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