5.62 Spring 2004 Lecture #34, Page 1. Transition-State Theory

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1 5.6 Spring 004 Lecture #34, Page 1 Transition-State Teory A good teory must tae into account te internal degrees of freedom of te reactants and teir angle of approac. An approac nown as transition state teory (activated complex teory)(absolute rate teory) does so in an approximate way. POTENTIAL ENERGY SURFACE A correct teory must consider te internal structure of molecules and te forces acting on atoms in te molecules because bonds are being broen and formed during a reaction. During a reactive collision, te force on an atom depends on bot intramolecular forces (forces between atoms in a molecule) and intermolecular forces (forces between molecules). Must treat te two colliding reactants as a single uantum mecanical system. Tis system exists only during collision process. Te system s potential energy is calculated te same way te potential energy for nuclear rational motion is calculated. Witin te orn- Oppeneimer approximation, solve Ψ E Ψ el el el el at fixed nuclear configuration. Te resulting E el is te potential energy at tat nuclear configuration. Systematically vary te nuclear configuration (grid of points) to get potential energy as a function of nuclear coordinates. Problem: too many nuclear coordinates! Can t plot potential energy as a function of more tan coordinates. A plot of potential energy versus coordinates is a 3D plot were te potential energy is a SURFACE. Potential energy for more tan coordinates is still called a surface even troug tere are more tan coordinates.

2 5.6 Spring 004 Lecture #34, Page EXAMPLE: +F F+ For collinear approac: tere are independent variables R F and R on wic te potential depends Te potential energy of tis system can be represented as a 3D surface as a function of R F and R. Te 3D surface an be represented as a D contour potential energy surface. Lines on a contour map represent EQUIPOTENTIALS. Valleys correspond to initial and final states.

3 5.6 Spring 004 Lecture #34, Page 3 (To go from one valley to anoter reuires crossing a saddle point) REACTION COORDINATE FOR F + REACTION COORDINATE minimum energy pat along deepest part of potential energy surface Note: reaction coordinate ere corresponds to antisymmetric F ration TRANSITION STATE transitory [F] complex wit a definite structure dissociates witin one alf antisymmetric ration TRANSITION STATE TEORY An approac to calculating a rate constant by reducing te dynamics of te reaction to an euilibrium between te reactants and te transition state along te reaction coordinate. A+ [A] products

4 5.6 Spring 004 Lecture #34, Page 4 Uses statistical mecanics to treat te euilibrium. Te reaction coordinate is te 1D antisymmetric ration of te transition state.

5 5.6 Spring 004 Lecture #34, Page 5 Transition State Teory Transition state teory Activated Complex Teory Absolute Rate Teory +F [F] F+ Assume euilibrium between reactants +F and te transition state. Treat te transition state as a molecule wit structure wic unimolecularly decays wit rate constant. d[f] dt [F] K [ ][F] [ F] K [ ][F] as units of sec -1 (unimolecularly decay). Te motion along te reaction coordinate loos lie an antisymmetric ration of of tis ration. Terefore can be approximated by te freuency of te antisymmetric ration ν[sec -1 ] [ F], one-alf cycle ν freuency of antisymmetric ration (bond formation and cleavage loos lie antisymmetric ration) d[f] ν K dt [ ][F] * d[f] ( / N) ν * e F dt ( / N)( / N) K E / T [ ][F] * ( trans / N) rot g 0 F * F e ( trans / N)( trans / N) rot g0 g0 E / T [ F]. Tis Reaction coordinate is antisymmetric rational mode of ration is fully excited because it leads to te cleavage of te - bond and te formation of te -F bond. For a fully excited ration,

6 5.6 Spring 004 Lecture #34, Page 6 ν << T Te rational partition function for te antisymmetric mode is 1 1 e *, asym ν / T T ν, since e ν / T 1 ν T K *' ( trans / N) rot g 0 F * F e ( trans / N)( trans / N) rot g0 g0 E / T were 1 3n 5 1 *' i/ T i 1 1 e ν 3n 6 1 *' 1 i/ T i 1 1 e ν if te transition state is linear if te transition state is linear (n # atoms in transition state) *' is te partition function from wic te antisymmetric rational mode is excluded; it because te reaction coordinate So K T K ν ' ' K is te special K excluding te partition function for te reaction coordinate Wat is E?

7 5.6 Spring 004 Lecture #34, Page 7 Since a molecule cannot ave a rational energy lower tan its zero point energy, te effective barrier along te reaction coordinate is E V0 + ( ZPE) TS ( ZPE) R For linear F, n3, so 3n rational modes, tus E V + ν + ν ν 0 sym. st. bend difference in ZPE FORMULATION OF TST TST T K ' but not all reactant molecules mae it all te way to products some are reflected bac to separated reactants. Tus, TST T K ' κ, were κ transmission coefficient EVALUATION OF TST POTENTIAL ENERGY SURFACE KNOWN: E - directly from potential energy surface I - calculate from geometric structure of transition state ν - analyze sape of potential in saddle point region κ - trajectory calculations (consider κ 1 for now) +F F + T300K, m amu, m F 19 amu Translational part

8 5.6 Spring 004 Lecture #34, Page 8 3 ( trans / N) N m 3/ ( / )( / ) ( T ) m m F N F N π 3/ ( J s) 3 ( π J / K 300K) / mol 3/ m / Rotational part mol g σ I σ 1 mg 48 I rot 8π IT σ mg 46 (assume linear TS) I σ 54.4 rot rot I σ Vibrational part [ F] is a linear transition state 3n rational degrees of freedom (one ration is reaction coordinate) ν s 5771 K( stretc) ν 633K ν b 573 K( x degen bend)

9 5.6 Spring 004 Lecture #34, Page 9 *' * 1 ν s / T ν b/ T ( 1 e ) ( 1 e ) ( ) ν / T 1 e Electronic part g 0 4 F g 0 4 g 0 1 g g g 0 F Calculate E V0 3.8 J/ mol ν s 14 1 ν s s 14 1 ν b s (ow do we guess values for ν and 13 1 s ν b?) 1 E V0 + N νs + νb ν 6.1 J/ mol Calculate T/ 3 T J/ K 300K J s All togeter now s 1 1 TST T ' κ K 1 1( s )(.5 10 m mol )( 54.4)( 1.38) e TST e / RT 7 6.1/ RT m mol s at 300K

10 5.6 Spring 004 Lecture #34, Page 10 EXP m mol s - NOT AD AGREEMENT! Experimental value is smaller because κ is probably not 1. Sometimes TST will be smaller tan EXP because of tunneling. Tis model for TST does not tae te uantum mecanical penomenon of tunneling into account. Tunneling can mae te reaction rate become faster tan te TST prediction. If TST < EXP, it may mean tat tere is some tunneling contribution. Additional Material on TST Want to get TST into Arrenius form TST T κ K ' but K e RTln K G ' G / RT so T T κ e κ e e TST G / RT S / R / RT because G TS NOW: E + nrt n # molecules in TS molecularity of reaction Were (molecularity: eg: unimolecular, bimolecular, etc) e.g.: So: n T κ e e e TST S / R 1 E / RT T e TST m E / RT

11 5.6 Spring 004 Lecture #34, Page 11 (were m1) Again, teory predicts a temperature dependence to te pre-exponential factor wic is difficult to observe experimentally unless te rate constant is measured over a wide temperature range (At least a factor of 5). Now: d ln e e lnt e TST dln κ + + dt dt 1 E + T RT dln Eact Arrenius : dt RT TST dln dln dt dt Eact 1 E + RT T RT E RT + E act S / R 1 E / RT Again, experimental E act is larger tan E because E act is a difference between te average energy of molecules in te ensemble and te average energy of molecules tat react, wile E is a microscopic uantity, a tresold energy along PES. Notice E act is not a barrier along PES. COMPARISON OF TRANSITION STATE TEORY AND COLLISION TEORY Calculate TST in te limit of te assumptions of collision teory (ie: simplified TST): 1) Collisions of ard speres ) Only translational degrees of freedom

12 5.6 Spring 004 Lecture #34, Page 1 Treat as an atom a ard spere of mass (no rotation, no ration) Treat F as a diatomic molecule (One unnown parameter, d -F ) + F F F + Wit tese assumptions T ( trans / N) F ( trans / N)( trans / N) e E / RT rot Note: no rational partition function for F is included because te one rational mode for te pseudo diatomic molecule transition state as become te reaction coordinate. ( + ) 3/ π m m T T π m T [ π mt F ] 3 3 N N F 3 N 8π 3/ 3/ IT σ e E / RT Te reason tere is no rotational or rational partition function for is not tat we are assuming te ig-t limit, but rater is treated as if it were an atom. Now I µ d F, µ m m m F + m F 1/ m + m π d N e 8T F F E / RT π m m F σ

13 5.6 Spring 004 Lecture #34, Page 13 Tis loos identical to te collision teory result. Collision teory is not based on termodynamics. 1/ CT 8T E0 / RT π dae πµ Calculate value for TST in limit of collision teory assumptions (ie: wwat fraction of collisions are effective because tey ave sufficient translational energy along te line of centers?): σ 1 π d F m 8 E / RT / e m mol s Compare to TST TST 7 E / RT / e m mol s TST is smaller because it reflects te more restrictive collinear steric reuirement. K CT is an upper bound because collision teory treats reactants as speres wit no favored direction of approac (but wit explicit reuirement of te effective collision energy).