Foundations of Chemical Kinetics. Lecture 9: Generalizing collision theory

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1 Foundations of Chemical Kinetics Lectue 9: Genealizing collision theoy Mac R. Roussel Depatment of Chemisty and Biochemisty

2 Spheical pola coodinates z θ φ y x

3 Angles and solid angles θ a A Ω θ=a/ unit: adians 2 Ω= A / unit: steadians dω = sinθ dθ dφ

4 Reactive scatteing: moving on fom simple collision theoy In ou pevious teatment, we concentated on the collision, and said nothing about what happens afte. Not all collisions ae the same: Molecules emege fom them at diffeent angles, with diffeent enegies, etc. To bing the elevant issues into focus, conside a molecula-beam expeiment to be descibed in detail late) in which the two eactants aive fom diffeent diections.

5 Geomety of the molecula-beam expeiment z dω eaction volume V θ B C φ A A + B C

6 Reactive scatteing continued) Count molecules of C emeging fom eaction volume in time dt in solid angle dω in diection θ, φ). dn C dt dω θ, φ) in mol s 1 ) By analogy to the simple collision theoy, we wite dn C θ, φ) = V [A][B] dt dω I R θ, φ, v )v Lf v ) dv IR θ, φ, v ) is called the diffeential coss-section. Instead of taking the mean elative speed, we ecognize that thee is a distibution of molecula speeds. v = v A v B f v ) dv is the Maxwell-Boltzmann) pobability that the elative speed is between v and v + dv.

7 Reactive scatteing continued) dn C dt θ, φ) = V [A][B]L = V [A][B]L v = v = I R θ, φ, v )v f v ) dv dω I R θ, φ, v )v f v ) dv sin θ dθ dφ We can integate ove the angula vaiables, since only IR depends on them. Define the total coss-section σ R v ) = d[c] dt 2π π = [A][B]L I R θ, φ, v ) sin θ dθ dφ σ R v )v f v ) dv

8 Reactive scatteing continued) d[c] dt = [A][B]L σ R v )v f v ) dv 1) The Maxwell-Boltmann distibution fo the elative speed is f v ) = 4πv 2 ) µ 3/2 exp µv 2 ) 2πk B T 2k B T Substitute the ate constant into equation 1) and note that the ate is in the mass-action fom to conclude ) µ 3/2 k = 4πL σ R v )v 3 exp µv 2 ) dv 2πk B T 2k B T

9 Reactive scatteing continued) k = 4πL σ R v )v 3 ) µ 3/2 exp µv 2 ) dv 2πk B T 2k B T Intoduce the elative kinetic enegy k = K = 1 2 µv 2 v 2 = 2K /µ dk = µv dv 4L k B T [2πµk BT ] 1/2 σ R K )K exp K ) dk k B T

10 What do these equations tell us? ) µ 3/2 k = 4πL 2πk B T = 4L k B T [2πµk BT ] 1/2 σ R v )v 3 exp µv 2 ) dv 2k B T σ R K )K exp K ) dk k B T 1. The coss-section depends on the elative speed o, equivalently, on the kinetic enegy of the collision. 2. The Ahenius dependence on tempeatue pesumably comes fom the Maxwell-Boltzmann dependence of the speed distibution. 3. The peexponential facto and activation enegy ae not eally sepaate quantities. 4. Classically, the coss-section would have a shap cut-off at enegies below the activation enegy.

11 Example: Piecewise constant coss-section Suppose that the coss-section has the following ideal gas/had collide) fom, with a cut-off to epesent the activation enegy: Then, o { fo K < ɛ σ R K ) = a σ fo K ɛ a k = 4Lσ k B T [2πµk BT ] 1/2 = 2Lσ 2k B T + ɛ a ) πµkb T k = 2Lσ 2RT + E a ) πµm RT ɛ a exp K exp K k B T ɛ ) a k B T exp E ) a RT ) dk

12 Example: Piecewise constant coss-section continued) k = 2Lσ 2RT + E a ) πµm RT exp E ) a RT The peexponential facto is A = σl 8RT + E a ) πµm RT Compae this to the simple collision-theoy esult: 8RT A ct = σl πµ m

13 Bette example: Coss-section goes continuously to zeo at K = ɛ a { fo K < ɛ σ R K ) = a σ 1 ɛ a /K ) fo K ɛ a k = 4L k B T [2πµk BT ] 1/2 ɛ a σ 1 ɛ a /K ) K exp 2k B T = 2Lσ πµ exp ɛ a = Lσ 8RT πµ m exp E a RT agees with simple collision theoy k B T ) ) K ) dk k B T

14 Genealizations So fa, we have seen the diffeential and total coss-sections. Diffeential coss-section: takes into account angula dependence of poduct distibution suitable fo analyzing esults of molecula-beam expeiments Total coss-section: aveaged ove all angles suitable fo undestanding bulk kinetics The coss-section can also depend on the initial and final quantum states of the eactants and poducts. This is called a state-to-state coss-section. State-to-state coss-sections ae useful fo advanced expeiments in which the eactants ae pepaed in a specified state and the states of the poducts ae discoveed by spectoscopy.