University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2015

Transcription

1 Lecture : Transition State Teory. tkins & DePaula: University o Wasinton Departent o Ceistry Ceistry 453 Winter Quarter 05. ctivated Kinetics Kinetic rate uations are overned by several principles. Te irst is te Law o Mass ction. It state tat or a ceical reaction k P (.) te rate o oration o product i.e. d[p]/dt is dependent o te reaction concentrations: dp k (.) dt However, te explicit dependence o stoicioetric coeicients is not eneral and riorously only applies to eleentary reactions. Deviations ay occur or ultiple steps involvin reaction interediates. Te Principle o Detailed alance states te relationsip o kinetic constants k at uilibriu. For a reversible reaction: at uilibriu te kr orward and reverse reactions are ual: rate rate orward r k k k K k r reverse (.3) Equation.3 eans te kinetic constants k and k r are not independent. Teir ratio is te uilibriu constant. In uilibriu terodynaics te van t Ho uation ives te teperature dependence o te entalpy: dln K H (.4) dt RT rrenius proposed an analoous relationsip or te teperature dependence o te kinetic constant dln k Ea (.5) dt RT Te assuption o te rrenius uation.5 is tat te reaction rate is deterined by te activation barrier E a at wic point on te reaction coordinate olecules are in a i enery activated state or transition-state. Te diara below sceatizes te activation barrier E a. ceical or pysical process wose rate ollows uation.5 is called an activated process.

2 . Transition State Teory Transition State Teory (TST) or ctivated Coplex Teory (CT) is a reaction ecanis oriinally developed to describe as pase collision reactions. It as te eneral reaction scee: K k (.6) were te reactants and collide to irst or a transition state or activated coplex wic is desinated (). Te idea beind TST is tat te transition state is an unstable, sort-lived coplex. In te case o a siple diatoic collision te transition state () consists o te pair joined by a very weak bond. Te collidin olecules and are assued in uilibriu wit te transition state were Fiure.: arrier o ctivation o Heit E a. Te Transition State Coplex ored between reactants is at te top o te arrier K (.7) Tis uilibriu notation owever involves te transition state wic actually exists at a enery axiu and is tus transitory in nature, as its nae iplies. For te diatoic reaction ecanis te rate is iven by rate k k K k (.8) To evaluate te rate constant k, we can apply statistical etods to K q qn K N (.9) q q N N were te partition unctions o,, and are q, q, q, respectively. Te treatent o q ruires soe explanation about te properties o te transition coplex. It is assued tat wen and or te transition coplex te coplex acquires translation, rotational, vibrational and electronic otions, wic all ust be relected in te partition unction, i.e.

3 q q q q q trans rot vib elec 3/ / De / 3 e / k T 8 I e e 3/ 8 I / 3 e / k T e were in.5 we ave used te relationsip De (.0) In.5 te ass o te transition state in te translational partition unction is just. In te rotational partition unction were te reduced ass, I R is te oent o inertia o te transition coplex and R is te lent o te bond tat is in te process o orin between and. Te ter is te deeneracy o te round electronic state o te transition coplex. Te vibrational otion is treated in te ollowin way. We assue as and coe toeter to or te transition coplex tat translational otion alon te reaction coordinate is eventually converted in part to a vibrational otion o te bond also directed alon te reaction coordinate. We use te notation q, transq, rotq, vib / K Ne (.) were is a reduced vibrational partition q, vib / e k T unction. Now te vibrational o te bond in te transition coplex is assued to occur at a very low ruency suc tat q, vib (.) / k T e / In oter words te bond vibration is calculated in te i teperature liit were. Now we also assue tat te transition coplex is converted to product witin a vibrational period so tat k. Usin tis expression and uation.7 we obtain q, transq, rot / k k K N e (.3) q, transq, rot N / e

4 Tis is called te Eyrin uation. To uation.3 is added ad oc a constant. Tis is called te transission coeicient and expresses te act tat not all collisions result in te oration o te transition coplex. Norally 0.5, but can be quite sall or atoic collisions. Te inal or or te Eyrin uation is k K N (.4) q, transq, rot / were K e is absent te deree o vibrational reedo treated in te i teperature liit and ivin rise to te actor o. Reaction rates are norally easured as per ole quantities so in uation.9 N=N : RT k, N K K (.5). Exaples o Diatoic Transition State Calculations For te collision o two atos we ave 3/ V 8 R q, transq, rot 3 3/ V q 3 (.6) 3/ V q 3 Ten we obtain or te rate constant 3/ 8 R k N e I / / 3/ k T k T 8 k T N R e is reported in units o Joules per ole we ust write: 8 / RT / (.7) k N R e (.8) C. More Coplicated Reactions: Ternary Coplexes Te siplest exaple o a ternary transition coplex is te isotope excane H DHD H (.9) Te transition coplex as linear or HH D were te bonds lie alon te reaction coordinate. Te rate constant as te or:

5 q HHD / k e K (.0) qdqh H were. Te partition unctions or q D and q H are obtained as usual or atoic and diatoic species. Te transition coplex partition unction q HHD is treated as ollows. Te transition coplex is linear so it as a sinle oent o inertia and rotations are calculated by te sae procedure used or a linear triatoic like CO except tat = or HHD. Te vibrational partition unction is treated as ollows. Note or a linear triatoic tere are our vibrational odes in te partition unction correspondin to syetric and asyetric stretces and two uivalent bendin odes HHD HHD HHD asyetric syetric Doubly de enerate stretc stretc bend Te asyetric stretc contributes to te reaction coordinate, is tereore treated in te i teperature liit and yields te k T/ ter in te Eyrin uation. Te oter tree vibrational odes reain in te transition coplex partition unction 3/ 3 H D IHHD qhhd 3 j/ j e (.).C. Gibbs Enery o ctivation Returnin to te eneral expression or te olar reaction rate RT k, Nk K, we deine orally te Gibbs enery o activation G RTln K (.) Usin uation 8.5 we now deine te kinetic constant as: RT G / RT k, e (.3) Usin te correspondin relationsip between te Gibbs enery, entalpy and entropy... G H T S we urter obtain: RT G / RT RT S / R H / RT k, e e e (.4) Note uation.4 sets an upper liit or te activated reaction rate. For G 0 and at T=300K: 3 ax.380 JK 300K k 6.50 s (.5) Js