chap 22. molecular reaction dynamics -...

Chap 22. Molecular reaction dynamics

(2015) Chemical Kinetics by M Lim 1

• Collision theory: reactions between simple species in the gas phase

• Reactions in solutions: diffusion-controlled, activation-controlled

• Activated complex theory: (transition state theory)k in terms of thermodynamic parameters

• Potential energy surfaces and the motion of molecules through these surfaces: the highest level of sophistication

• The transfer of electrons in homogeneous systems and at electrodes using transition state theory


Chapter 22 in the 9th edition

• Numerical Problems: 22.1, 22.3, 22.5, 22.7, 22.8, 22.10, 22.13

• Theoretical Problems: 22.15, 22.18, 22.19, 22.21, 22.22, 22.23

• Applications: 22.24, 22.25, 22.28, 22.29

Assigned Problems

Reactive encounters


Here we consider two elementary approaches to the calculation of reaction rates (gas-phase reactions and reactions in solutions). Both approaches are based on

1. reactant molecules must meet2. they should have a certain Emin.

22.1 Collision theory[ ][ ] rk

rA B P v k A B+ → =

[ ][ ]min

min

(steric requirement) (encounter rate)

(steric requirement) (encounter rate)r

v A BE

Ek

× ×

× ×

∝ ×

∴ ∝

22.1 (a) Collision rates in gases


The collision density, ZAB, is the number of (A, B) collisions in a region of the sample per V and t.

( ) [ ]

( ) [ ] [ ][ ]2

for a single molecule B with the A molecules present

8 where

12

rel rel AA

A Brel

A B

AB A rel AB

AA

zz c NumberDensity c N A

kT m mcm m

Z z NumberDensity zN B c N A B

Z z Numbe

σ σ

µπµ

σ

= × =

= =+

∴ = × = =

= × ( ) [ ]2212 rel AArDensity c N Aσ=

[ ][ ]28AB A

kTZ N A Bσπµ

=( )2 1,

2 A Bd d d dσ π= = +

[ ]224AA A

A

kTZ N Am

σπ

=

22.1 (a) Collision rates in gases


N2 at room T and p, with d=280 pm

[ ] [ ]

[ ]

( )

( )

2

2 22 2

2

2210 23 1 3

3

2 2 22 2 3 2

4 4

4 8.314 298 1 2.80 10 6.02 10 28 10 0.0821 298

AA A AA A

N

kT RTZ N A N Am M

pN

RT

m mol mol dm

kg m sm mol mol dm m mkg

σ σπ π

ππ

− − −−

−− −

= =

=

⋅ ⋅ = ⋅ × × ⋅ × ⋅

= ( )1 6 6 3 1

34 3 1

10

5.00 10

s dm m s

m s

− − − −

− −

=

= ×


22.1 (b) The energy requirementReaction occurs only when a collision occurs with sufficient energy→ incorporate this requirement by writing σ as a function of KE of

approach of the two colliding species (i.e., σ → σ(E))

[ ] ( ) [ ][ ] ( )

( ) ( ) ( )( ) [ ][ ]

( ) ( ) ( )0

0

σ ε ε

σ ε ε ε ε

σ ε ε ε ε

∞

∞

− = = =

=

∴ =

∫

∫

ABrel A rel rel

A

rel A

r A rel

d A Z v N A B v vdt N

v f d N A B

k N v f d( ) 1 for

0 for

aa

a

εσ ε σ ε εε

ε ε

= − ≥

= <

σ −= aE RTr A relk N c e


Reactive collision cross section( ) 1 for

0 for

aa

a

εσ ε σ ε εε

ε ε

= − ≥

= <

( )

2 2

, 2

2 22 2

, 2

2 2

2

22 2

2

2 2 2 2max max

cos

1 12 2

1 1

1 1

rel A B rel rel

rel A B rel

A B

A B A B

a a

d av v vd

d av vd

d ad

a a dd

a d a d

θ

µ µ

ε ε

ε εε ε

ε επ πε ε

−

−

−

− −

−= =

−=

−=

= − → = −

∴ = − → = −


( ) 1 for

0 for

aa

a

εσ ε σ ε εε

ε ε

= − ≥

= <

( )

( )

( ) ( ) ( )

2

2

3 2 3 22 2

3 2

3 2

0

1 2 2 , 2 2

24 42 2 2

1 2

1 22

rel rel rel

v kT kT

kT

kTr A rel A

v v d v dv dv dv

df v dv v e dv ekT kT

e d f dkT

k N v f d N e dkT

µ ε

ε

ε

ε εε µ ε µ µ µεµ µ

µ µ ε επ ππ π µ µε

π ε ε ε επ

εσ ε ε ε π σ ε επ µ

− −

−

∞ −

= → = = = =

= =

= =

= = ∫

( )

0

1 2

0

8 1 kTAN e d

kT kTε

ε

σ ε ε επµ

∞

∞ − =

∫

∫

( )

1 2

1 2

8 1 1

8 1

a

a

kTaA

kTA a

N e dkT kT

N e dkT kT

ε

ε

ε

ε

εσ ε επµ ε

σ ε ε επµ

∞ −

∞ −

= −

= −

∫

∫


( )

( )

( )

1 2

1

8 1

1

1

ε

ε

ε ε ε

ε ε ε

ε ε εε ε ε

ε εε ε

ε

ε

σ ε ε επµ

ε ε ε ε ε ε ε

ε ε

ε

∞ −

∞ ∞ ∞− − −

∞ ∞=∞ − −→− − −

=

∞ −

= −

− = −

= − = → =

=

∫

∫ ∫ ∫

∫ ∫

∫

a

a a a

a a

aa a

a

kTr A a

kT kT kTa a

kT akT akT kT a

a

k N e dkT kT

e d e d e d

e d kTe kTe e d ea

d de dda da a

( )

( ) ( ){ } ( ) ( )

( )

2

2

2 2

1 2

1

8 1 8

ε ε εε

ε

ε εε

ε

ε ε ε εε

ε

ε

ε

εε ε

ε ε ε

ε ε ε ε ε

σ ε ε ε σπµ πµ

∞− −−

∞ − −−

∞ − − − −−

∞ −

→ − = − −

∴ = +

− = + − =

= − =

∫

∫

∫

∫

a a a

a

a a

a

a a a a

a

a

a a aa a

kT kTkTa

kT kT kT kTkTa a a

kTr A a A

e e d e ea a

e d kT e kTe

e d kT e kTe kTe kT e

k N e d NkT kT k

( )2

2

2

2

1

1 2 1

1

8 8

ε

εσ σπ µ

σ

µ π

−

− −

−

= =

=

a

a a

a

kT

kT E RTA

E Rl

A

TA re

kT eT kT

kT kTN e N e

k N c e


22.1 (b) The energy requirement

Ea: the minimum KE along the line of approach that is needed for reaction

A: a measure of the rate at which collisions occur in the gas

2 222

: weak temperature dependence, ln 1 1

2 2

8

1

σπµ

− −= →

∝

= + = +

a aA

E RT E

a

Rr

T

a

k

A A TEd kRT RT E RT

dT T R

eAeT

T

kN


22.1 (c) The steric requirementExperimental σ may be different from theoretical σ obtained

for non-reactive collisions or tables of molecular radii: To compensate for the disagreement,

Steric factor, P. (ex, due to relative orientation of the colliding species)

A harpoon mechanism


* *Reac , : (target area for reaction)

tive cross-section Pσ σ σ=

Ex 22.1 Estimate steric factor for the reaction H2 + C2H4 → C2H6at 628 K when experimental A = 1.24×106 M−1s−1 at 298K.

2 2 4 11 1 1

6

8 8 7.37 102

1.7 1 the more complex the molecule, the smaller th0 e va( )lue of

H C Htheoretical A A

kT kTA N M

P P

N sσ σ

σπµ πµ

− −

−

+ = = = ×

∴ = ×

: local peoperties of the reaction (orientation, how close they must come to react)

: transport property (how often particles come together): energy criteri

arel A

E RTA

A

r B

B

k P Pe f

f

c N Z

Z

P

σ −∴ = =

on


Harpoon mechanism: when the two are close enough an electron flips across from K to Br2 → ions → Columbic interaction takes place

2for the reac4.5 tionP K Br KBr Br= + → +

2

2

2 20

4K Br

eK Br K Br E I EARπε

+ −+ → + = − −

Ex 22.2 Estimate P for the harpoon mechanism of K + Br2 → KBr + Br by calculating the distance at which it becomes energetically favorable for the electron to leap from K to Br2.

2

2* 2

max0 max

4

When 0, harpooning occurs K Bre I EAE R

Rσ π

πε=≤ ∴ ≥ −

( ) 2

2

2

2* 2 2

max2 2

0

1 19 1 19

1 400 4

420 7.0 10 , 250 4.2 10

4.2

K BrK Br

K Br

R eP d R R pmd dI EA

I kJ mol J EA kJ mol J

P

σσ πε

− − − −

= = = = + = −

= = × = = ×

=


(recall) 21.8 Unimolecular reactions-> RRK

*

* (slow, rds)

a

a

b

k

k

k

A A A A

A P−

→+ +←

→

21.8(a) Lindermann-Hinshelwood mechanism

[ ] [ ] [ ][ ] [ ] [ ] [ ][ ]

22*

0 * * * aa a b

b a

d A k Ak A k A A k A A

dt k k A−−

≈ = − − ∴ =+

[ ] [ ] [ ][ ] [ ] [ ]

[ ]

[ ] [ ] [ ]

2

*

If , and

b a b ab

b a b a

b aa b

a

d P k k A k k Ak A k A k

dt k k A k k A

d P k kk A k k A kdt k

− −

−−

= = = ≡ + +

>> ≈ =

[ ] [ ][ ] [ ]

1 1 1 1 vs straight line is expected

low : bimolecular high : unimolecular

−= + →a

b a a

kk k k k A k A

A A


22.1 (d) The RRK modelThe steric factor P for unimolecular gas phase reactions.

s: # of modes of motion over which E may be dissipated.E*: E required for the bond of interest to breakE: E available in the collision

( )

1*

*

1*

1

1

for

−

−

= −

= −

≥

s

b

s

bE kE

EPE

k E

E E


22.2 Diffusion-controlled reactionCollision in gasesEncounter in solution (cage effect): less frequent collision but longer stay

[ ] [ ][ ] [ ] [ ]

[ ] [ ][ ]

encounter pair

0

:d a

d

k k

k

d d a

d

a d

A B P

d ABk A B k AB k AB

dtkAB A

A A

B

B

k

B

k

−

−

−

→+ →←

= − − ≈

∴ =+

[ ] [ ] [ ][ ] [ ][ ]2 2 a d a da

a d a d

d P k k k kk AB A B k A B kdt k k k k− −

= = = ≡ + +

29 -1 -1

2

(i) , " "

10 (Enzyme catalyzed, the recombination of radicals)

(ii) ,

- limit

- " r

a d d

a da d a

d

k k k kM s

k kk

diffusion controlled

activationk k controlledKkk

−

−−

>> ≈

≈

<< ≈ = eaction"


22.2(b) Diffusion and reactionWhen the two reactants molecules react if they come within a distance R*.

*483

d A

d

k R DNRTk

π

η

=

≈*

**

*

*

6 6

1 16

2

46

44

8

46

8 33

A BA B

A BA B

A B

A

A

d A

kT kTD DR R

kTD D DR R

RR R

kTDR

kTR NR

kT N

k R DN

RT

πη πη

πη

πη

ππη

π

ηη

= =

= + = +

≈ ≈

≈

= =

= =

kd is independent of R*→ independent of the identities of the reactants

Abl. kd for recombination of I atoms in hexane (0.326 cP) at 298K.

4

7 3 1 1 10 1 1

8 8 8.314 2983 3 3.26 10

2.03 10 2.03 10

dRTk

m mol s M sη −

− − − −

⋅ ⋅= =

⋅ ×

= × = × 6kTD f af

πη= =


When the two reactants molecules react if they come within a distance R*.Let a A molecule is at the center (r = 0)

[ ] [ ]

[ ] [ ]

2 2

2

Recall that

When a system reached a steady state, 0 0

B

r

Bc D c D Bt t

BB

t

∂∂= ∇ → ∇ =

∂ ∂∂

= → ∇ =∂

B

R*

[ ] [ ] [ ][ ]

[ ] [ ][ ]

*

*

*

4

is not stationary:

4

4

A A B A

B B A

A

d A

d Pv A N R D N A B

dtA D D D D

d PR DN A B

d

k N

t

R D

π

π

π

=

= =

→ + ≡

=

∴

A

[ ] [ ] [ ]

2 22

2 2 2 2

2

2

2

2

2 1 1 1 Since sinsin sin

2 in spherically symmetric system

2 0 r rr

r r r r

r r rd B d B bB a

dr r dr r

θθ θ θ θ φ

∂ ∂ ∂ ∂ ∂∇ = + + + ∂ ∂ ∂ ∂ ∂

∂ ∂= +∂ ∂

+ = → = +

[ ] [ ][ ]

[ ] [ ] [ ] [ ]

[ ] [ ]

*

*

*

*

*

*

*

*

. . bulk value

0 react within a distance

1

r R

r

r R

rB Br R

r R

r

B C B B

B R

d B Bdr R

d B BJ D D

dr

Rr

R

B B

∞

=

=

==

=

=

∴ → =

= −

= − = −

[ ]**2 *4 4 for a moleculeA Br R

v R J R D B Aπ π=

= =


Review 22-1• Collision density:

•

[ ][ ] [ ]22 28 4, σ σπµ π

= =AB A AA AA

kT kTZ N A B Z N Am

( ) ( ) ( )2 0

σ ε ε ε ε σ

σ −

∞ −= =

→

∫ a

a

E RTA rel A r

r A

e

l

l

E Re

T

k

e

N v f d N c

P c N

e

The more complex the molecule, the smaller the value of PA harpoon mechanism

[ ] [ ][ ]

encounter pair

- limit

- reactio

:

n

( )

( )

−

−−

−

→+ →←

= >>+

<<

d a

d

k k

k

a da d

a d

a d

diffusion controlled

activation controlled

A B P

d P k k A B k kd

AB A

kk k

B

t k

* 843

πη

= ≈d ARTk R DN

chap 22. molecular reaction dynamics -...

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