Chapter 6: Adiabatic reaction dynamics and Transition State Theory Bimolecular reactions Transition State theory... 72

Transcription

1 Winter 204 Capter 6: Adiabatic reaction dynamics and Transition State Teory imolecular reactions... 7 Transition State teory Derivation of universal factor T Kinetic Isotope effect Capter 6: Adiabatic reaction dynamics and Transition State Teory General reaction: A + + C... X + Y + Z... Rate at wic euilibrium is establised: Eg. d A rate( t) φ ( t) A a b c C Y y z Z X [ w ]: concentration of species w. (as molarity or partial pressure) ( t) : penomenological rate constant (migt depend on time) φ In general, euation migt only be valid for limited amounts of time/concentration regimes. General description can be very complicated as many elementary steps migt be involved in te reaction. To mae progress, we need to reduce a reaction to elementary steps of unimolecular or bimolecular reactions. Ten we can apply principles of statistical mecanics. We will consider reactions on a single potential energy surface (adiabatic, orn- Oppeneimer approimation). Simplest case: Unimolecular reaction A Rate euation: d A A( t) t ( ) d A Steady State: 0 At ( ) A e A e e e Ke A e We ave seen before ow to calculate euilibrium constants in stat mec. Eperimental determination. Use ecess A, neglect [ ] Capter 6: Adiabatic Reaction Dynamics and Transition State Teory 70

2 Winter 204 [ ] [ A] At ( ) t [ Ao ] t ( ) [ A] e d A ln At Eponential decay of [ A ] under conditions of ecess can be fitted to get. o Of course one could also start wit ecess of [ ], and measure. Tis would allow a cec of relations. Oter types of reactions: e K A e A! + C d[ A] [ A] [ ][ C] e C e Ke A e At Ao e t If [ ][ C ] is negligible ( ) [ ] e imolecular reactions A +! C d[ A] [ A][ ] [ C] e Steady State: C Ke A e e * t In presence of ecess, suc tat [ ] is constant At ( ) [ A] e, [ ] o * o imolecular reaction is te limit. Elementary reaction steps never involves more tan 2 colliding molecules. Negligible probability to ave 3 molecules colliding simultaneously. If tree molecules are needed, ten one as to first create a comple of two molecules and collide tis wit a tird molecule needed for reaction. Capter 6: Adiabatic Reaction Dynamics and Transition State Teory 7

3 Winter 204 In our discussion of inetics (later on) you will learn ow to tin about building complicated reactions from elementary reaction steps. Here we focus on elementary reactions and focus on ow we can calculate rate constants from first principles using information from uantum cemistry and statistical mecanics calculations Transition State teory To develop euations for te rate of reactions we will assume termal euilibrium for all species A,,, were refers to te (fleeting) transition state or activated comple. Overall reaction: A a model: A! d a : activation, K e [ ] [ A] rn a K A d d : deactivation, rn : rate to go from to Crucial idea: consider forward reaction only ( rate to products). A rn N rn A rn N A (Note: In cemical inetics one usually uses concentrations to derive tings. In statmec we are used to epressing tings in partial pressures. I am doing te derivation ere, based on partial pressure as a measure of concentration. ) rn : rate of moving to product for molecules at transition state. K A / N A / N A " " A Capter 6: Adiabatic Reaction Dynamics and Transition State Teory 72

4 Winter 204 Use same formula as for cemical euilibrium! Same euilibrium principles. Te factor N is included in te translational partition function as usual. It is a bit of a nuisance to write it correctly. Calculating A : calculate molecular structure, vibrational freuencies, energy at minimum. A A A A A A e zp t R v For transition state, we can proceed in analogous way : calculate structure of transition state; calculate vibrational freuencies, transition state electronic energy. Someting special about (non- linear TS) We ave 3N 7 armonic, real vibrational freuencies, and one imaginary freuency, wic needs to be treated in a special way. Te normal mode of te imaginary freuency is along te reaction coordinate. rn ( ) rn e zp t R v rn : component of partition function along reaction coordinate! : partition function of transition state, for all motions ecept vibration along reaction coordinate It ten follows tat te forward reaction rate is given by rn! rn rn rn A A We will see tat rn rn Or: T T T A 3 ~0 s A if bimolecular ( A + C ) Te determining factor in A is K e K zp as it is for cemical euilibria. Te teory is very similar. e ( ΔEe+ΔEzp )/ T A t R v t A R A v A Capter 6: Adiabatic Reaction Dynamics and Transition State Teory 73

5 Winter 204 We get an eponential temperature dependence from Δ Ee +Δ Ezp, power of T dependence from te remaining fraction, often just a constant if same of molecules for reactants and T transition state. Oter contribution Δ E zp : te difference in zeropoint energy between reactant and transition state 3N 7 3N 6 Δ E ω ω ( ) ( ) zp i TS i R 2 i 2 i Please note te number of normal modes in eac sum. If te molecular structure is linear, te number of normal modes increases by, as usual. Derivation of universal factor T a) Partition function rn δ δ Potential is flat in transition state region o, o Assume particle in te bo types of partition functions rn 2πm T δ 2 Notes: ) 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 now m! 2) We do not now δ eiter!! Capter 6: Adiabatic Reaction Dynamics and Transition State Teory 74

6 Winter 204 Second assumption: all molecules in te transition state region tat ave a positive velocity will V react. Te rate depends on ow fast tey traverse te transition state region: δ We can calculate ow fast molecules traverse te TST region: rn V δ P( V )dv 0 PV ( ): probability to ave V Once again, use euilibrium arguments. V follows Mawell- oltzmann distribution P( V ) e mv 2 /2 T e mv 2 /2 T dv Here we use a oltzmann distribution for inetic energy Integrate (Gaussian integrals) rn δ T 2πm Assumption: Even at te transition state geometry, te velocity distribution V is given by Mawell- oltzmann (were potential energy is very ig). Tis seems a bit counter intuitive. I do not ave a good rationale for tis. A great virtue of te above derivation is: 2πm rn rn T δ 2 δ Unnown m, δ cancel. How convenient! T T 2πm It will be clear tat tis factor is an estimate. Transition state teory is not as robust as oter aspects of statistical mecanics. Tere are variations on te teme in te scientific literature. Kinetic Isotope effect Isotope effects are strongest (or even observable only) if protons are replaced by Deuterium ) Primary isotope effect: H or D is involved in reaction pat, or te main atom tat is moving a. Isotope substitution does not affect electronic structure calculations same PES, same stationary points However mass enters mass- weigted Hessian, and tis affects te freuencies ω ~ g m (For diatomics. Analogous for polyatomics) pure proton freuency is 2 times as large as D freuency. A good eample would be to compare te vibrational freuencies in HCl and DCl. Capter 6: Adiabatic Reaction Dynamics and Transition State Teory 75

7 Winter 204 If proton movements defines reaction coordinate, ten in te reactant tis vibrational freuency 3N 6 contributes to Ezp ωi. 2 i ut it is missing from transition state 3N 7 Ezp ω 2 i i, ligt eavy e ( )/ T > ω eavy ~ 2 ω ligt 2! ω R R eavy ω ligt (for D vs. H) ligt particle faces less of a barrier tan eavy particle [ ] [ ] > [ A] [ A] ligt eavy Also note tat m did cancel out in te derivation of T. Te cause of te effect is not tat ligt particles move faster. Te effect is due to zeropoint motion (purely uantum!) rn rn is te same for any particle (or collective coordinate). b) In secondary isotope effect, te isotope of interest is not involved in te reaction coordinate zeropoint effects more or less cancel, minor residual effect on /. One can epect te effect of isotopic substitution to be muc smaller ten. So te effect of isotopic substitution can tell you someting about te nature of te transition state. ligt eavy A good eample of a primary inetic isotope effect would be te HCN to CNH reaction rate vs. DCN to CND. Tis is clearly a primary isotope effect and easy to calculate using Gaussian calculations plus matlab processing. It will also illustrate te concept of calculations on transition states and reaction rates witout bogging us down in computational compleities. We will eplore in te computer lab! Capter 6: Adiabatic Reaction Dynamics and Transition State Teory 76