Chapter 22: Reaction Dynamics
22.1 Collision theory

rate constant, k r
 encounter rate
 minimum energy requirement
 steric requirement.
A

B

P v
 k r
[A][B] c
 k r
8 RT

M
1 / 2
 c

( T / M )
1 / 2
 
( T / M )
1 / 2 e

E a
/ RT
 v
 
( T / M )
1 / 2 N
A
N
B
 
( T / M )
1 / 2
[A][B] k r

P

( T / M )
1 / 2 e

E a
/ RT
22.1(a) Collision rates in gases
 collision density, the number of (A,B) collisions in a region of the sample in an interval of time divided by the volume of the region and the duration of the interval:
Z
AB
 


8 kT



1
2
N
2
A
[A][B] ,
Z
AA
 


4 kT
 m
A


1
2
N
2
A
[A]
2
   d
2 d

1
2
( d
A
 d
B
),
  m m
A
A
 m
B m
B

Chapter 22: Reaction Dynamics collision frequency, z
  c rel
N
A collision
Z
AB
 density, Z
AA
 c rel
N
A
N
B

1
2 z N
A

1
2
 c rel
N 2
A

Z
AB
 


8 kT



1
2
N
A
2
[A][B] c rel



8 kT



1
2 collision cross-section volume of tube
 
N z


1 /
1 /

 t


1 /
 c rel
 t
 c rel
N
22.1(b) The energy requirement d N
A dt
  
(

) v rel
N
A
N
B

(

)

0 when ε  ε a d [ A]
  
(

) v rel
N
A
[A][B] dt d [ A] k r dt

 



0


(

) v rel f (

) d



N
A
[A][B]
N
A
0
 

(

) v rel f (

) d
 f (

); Boltzmann distributi on of energy

Chapter 22: Reaction Dynamics k r

N
A
 
0

(

) v rel f (

) d

 
1
2
 v
2 rel v rel , A

B
 v rel cos
  v rel

 d
2 d

2 a
2


1
2
1
2
 
( v rel , A

B
)
2 
1
2
 


 v rel

 d
2 d

2 a
2


1
2



2
 
A

B
  d
2  a
2 d
2 a max a

, above a max

which

A

B

 a
2 max

1

 a d
2 reactions
 a do not occur
   )  a
2  ,
     
(

)
   a
 
(

)

0 &
   a
 
(

)
 

1

 a
 v rel , A

B

Chapter 22: Reaction Dynamics k r

N
A

0


(

) v rel f (

) d
 f ( v ) dv

4


2
 kT
3 / 2 v
2 e
  v
2
/ 2 kT    1
2
 v
2  ,   d  /( 2  )
1 /  2  f ( v ) dv

4


2
 kT
3 / 2 

2



 e
 
/ kT d

( 2

)
1 / 2 f ( v ) dv

2

1
 kT
3 / 2

1 / 2 e
 
/ kT d
  f (

) d


0


(

) v rel f (

) d


0


(

) e
 
/ kT d


2

 

  a

1
 kT
 1
3 / 2

0


(


)

2



 a

 e
 
/ kT d



1 / 2

1 / 2 e
 
/ kT d
 


8
 kT


1 / 2
1 kT

0


(

) e
 
/ kT d


  a x e
 dx
 
 e

 a x

C a

,
  xe
 a x dx


 e
 a x
2

 xe
 a x

C
 
( kT )
2
 e
  a
/ kT

0


(

) v rel f (

) d
  


8 kT



1 / 2 e
  a
/ kT k r

N
A

0


(

) v rel f (

) d
 
N
A
 c rel e

E a
/ RT

Chapter 22: Reaction Dynamics
22.1(c) The steric requirement

steric factor, P = σ */
σ
.

reactive cross-section, σ *, the area within which a molecule must approach another molecule for reaction to occur.
 rate constant from collision theory, k r

P



8 kT



1 / 2
N
A e

E a
/ RT
 harpoon mechanism, a process in which electron transfer precedes atom extraction.
(Exercise Example 22.2!)

Chapter 22: Reaction Dynamics
22.1(d) The RRK model

The Rice–Ramsperger–Kassel model (RRK model), a model that takes into account the distribution of energy over all the bonds in a molecule.
P


 1
 s ; the # of
E
E
 
 s

1 modes of
 k b
( E ) motion, E



1

E
E
 
 s

1 k b for E

E


; energy required for the bond breakage, E ; energy available in the collision
Lindemann-Hinshelwood mechanism
RRK model s
Exp. data for unimolecular isomerization of trans -CHD=CHD

Chapter 22: Reaction Dynamics
22.2 Diffusion-controlled reactions
 cage effect, the lingering of one molecule near another on account of the hindering presence of solvent molecules.

Chapter 22: Reaction Dynamics
22.2(a) Classes of reaction
 diffusion-controlled limit, a reaction in which the rate is controlled by the rate at which reactant molecules encounter each other in solution.
 activation-controlled limit, a reaction in solution in which the rate is controlled by the rate of accumulating sufficient energy to react.
A

B
AB
AB



AB
A

B
P v v

 k d k
 d
[A][B]
[AB] v
 k a
[AB]
AB a
: encounter
: activated pair, d process
: diffusion d [AB] dt
 k d
[A][B]
 k
 d
[AB]
 k a
[AB]

0

[ AB ]
 k d
[A][B] k a
 k
 d d [ P ]
 dt k a
[AB]
 k r
[A][B] ,
When k
 d
 k a
 k r
 k d k r
 k a k a
 k d k
 d
: diffusion controlled limit
When k a
 k
 d
 k r
 k a k
 d k d
 k a
K : activation controlled limit

Chapter 22: Reaction Dynamics
22.2(b) Diffusion and reaction
A

B

AB in solution!

 t c

D

 x
2 c
2
 3  
At steady state;

[ B ]
 t
 

0
D

B
 2
 2 [
[ B ]
B ] r



[ B ]
0

; t
; diffusion equation (Fick' s second law of diffusion) r signifies a quantity t hat varies with the distance r
 2
[ B ] r
[ B ] r
       

[B] ([B] is bulk value ) as r d
2
[
 r
B ] r
2
 
, [ B ] r

2 r d [ B ] r
 r

0 at r

0

General solution : [ B ] r
 a

R

(the distance where reaction occurs)


[ B ] r


 1

R
 r

 [ B ] b r
Rate of reaction

4

R

2
J ( J : molar flux of B toward A)
From Fick' s first law J
Rate of reaction


D
B d [ B ] r dr r

R


D
B
[
R

B ]
4

R

D
B
[ B ]
  
4

R

D
B
[ B ] N
D
B

D
A

D
B

D  A is not stationary
A

4

R

D
B
N
A
[ A ][ B ] d [ P ]
 dt k d
[ A ][ B ]

4

R

DN
A
[ A ][ B ]
 k d

4

R

DN
A

Chapter 22: Reaction Dynamics
By using Stokes Einstein equation;
D
A
 kT
6

R
A
D
B
 kT
6

R
B
( R
A
, R
B
; hydrodynam
R
A

R
B

1
2
R
  k d

4

R

DN
A

8 RT
3
 ic radius,

; viscosity of medium)
22.3 The material balance equation
Generalize d diffusion equation : the diffusion equation including convection

[ J]
 t

D
 2
[ J]
 x
2
 v

[ J]
 x
Including

No chemical convection ; [J]
 reaction


[ J]
 t

[ J ] e
 k r t
[ J] :
 for
D
 2
[ J]
 x
2
 no reaction v

[ J]
 x
 k r
[ J]; material balance equation

No reaction; [J]

A ( n
0

Dt )
1 / 2 e
 x
2
/ 4 Dt

For general cases, we can solve the material balance equation numericall y!
!

Chapter 22: Reaction Dynamics
 transition state theory (or activated complex theory, ACT), a theory of rate constants for elementary bimolecular reactions.
 transition state, the arrangement of atoms in an activated complex that must be achieved in order for the products to form.
22.4 The Eyring equation
A

B

C
‡
K
‡  p
C
‡ p
θ p
A p
B
 p   
C
‡ 
P
[ C
‡
]

RT p
θ
K v
 k
‡
[ C
‡
]
‡
[A][B]
Our task!!
v
 k r
[A][B] k r

RT p
θ k
‡
K
‡

Chapter 22: Reaction Dynamics
22.4(a) The rate of decay of the activated complex

transmission coefficient, κ , the constant of proportionality between the rate of passage of the complex ( k
‡
) through the transition state and the vibrational frequency along the reaction coordinate (
 ‡
); k
‡
=
κ  ‡
.
22.4(b) The concentration of the activated complex
K




J



 q
θ
J, m
N
A





J
 e
  r
E
0
/ RT 
K
‡ 
N q
θ
A
A q
θ
C
‡ q
θ
B e
  r
E
0
/ RT where p
 
1 bar &
 r
E
0

E
0
( C
‡
)

E
0
( A )

E
0
( B )
Partition function for specific vibration which leads to product formation; q

1
1
 e
 h
 ‡
/ kT h
 ‡  kT
 q

1




1

1 h
 ‡ kT
 



 kT h
 ‡ q
C
‡
 kT h
 ‡ q
C
‡ where q
C
‡ denotes the partition function for all the other modes of the complex.
K
‡  kT h
 ‡
K
‡
K
‡ 
N q
A
θ
A q q
θ
C
‡
θ
B e
  r
E
0
/ RT
( K
‡
; K
‡ with one vibration al mode of C
‡ discarded)

Chapter 22: Reaction Dynamics
22.4(c) The rate constant k r
 k
‡
RT p
θ
K
‡   ‡ kT h
 ‡
RT p
θ
K
‡   kT h
K
‡
C
‡
; Eyring equation
For q
θ
C
‡
, we have to know the size, shape, and structure of activated complex
 very difficult!
22.4(d) The collision of structureless particles q
J
θ 
V m
θ
 3
J

J
 h
( 2
 m
J kT )
1 / 2
V
θ m

RT p
θ
A

B

C
‡
(A  B), q
θ
C
‡ correspond s to rotatioal mode hc B
 2
   q
θ
C
‡




2 IkT
 2 


V m
θ
 3
C
‡
I
  r
2
,
  m m
A
A
 m
B m
B
, m
C
‡
 m
A
 m
B k r
  kT h
RT p
θ




N
A
 3
 3
A
C
‡

V m
θ
3
B







2 IkT
 2 

 e
  r
E
0
/ RT   kT h
N
A



 
A


C
‡
B



 3



2 IkT
 2 

 e
  r
E
0
/ RT
 
N
A 


8 kT

2 


 r
2 e
  r
E
0
/ RT  k r
 




8 kT


1 / 2

N
A e


E a
/ RT
     r
2
, E a
  r
E
0

Chapter 22: Reaction Dynamics
22.4(e) Observation and manipulation of the activated complex

Na + I decay
 Photoreaction of IH∙∙∙OCO van der Waals complex
IH∙∙∙OCO 
HOCO resembles the activated complex of H + CO
2

[HOCO]
‡ 
HO+CO

Chapter 22: Reaction Dynamics
22.5 Thermodynamic aspects
22.5(a) Activation parameters
Gibbs energy of activation ,
 ‡
G
 
RT ln K
‡ k r k r



Ae kT h
RT p
θ e
  ‡
G / RT      H           S term  k r

E a
/ RT 
E a

RT
2
 ln

T k r

E a
  ‡
H

2 RT
 k r
 e
2
Be
 ‡
S / R e

E a

A
 e
2
Be
 ‡
S / R
/ RT
P
 e
 ‡
S steric
/ R

Be
 ‡
S / R e
  ‡
H / RT
B
 kT h
RT p
θ
 correlation analysis, a procedure in which ln K (=-Δ r
G
θ
/ RT ) is plotted against ln k
(proportional to -Δ
‡
G / RT ).
 liner free energy relation (LFER), a linear relation obtained in correlation analysis; reaction becomes thermodynamically more favorable.

Chapter 22: Reaction Dynamics
22.5(b) Reactions between ions
 kinetic salt effect, the effect of a change in ionic strength on the rate constant of a reaction.
d [ P ] dt
 k
‡
[ C
‡
] K
 a
C
‡ a
A a
B

K

[ C
‡
] c
θ
[A][B]
K




A
C
‡

B d [ P ]
 k r
[A][B] dt k r
 k
‡
K
K

 k r
 k K     k r
 k r
0
K

From Debye H  u  ckel limiting law ( log

 log log log

A
 
Az
A
2
I
1 / 2

C
‡ k r

 
A ( z
A log k r
0 

A
 z
B z
A
2
)
2

I
1 / 2 z 2
B log

B

( z
A



Az
B
2
I
1 / 2 z
B
) 2

I 1 / 2
 log k r
0 
2 Az
A z
B
I
1 / 2
  z
 z
-
AI
1 / 2
, A

0.509
for aq.
at 25
0
C),
Exercise Example 22.3!

Chapter 22: Reaction Dynamics
22.6 Reactive collisions
22.6(a) Experimental probes of reactive collisions
 infrared chemiluminescence, a process in which vibrationally excited molecules emit infrared radiation as they return to their ground states.
IR chemiluminescence
O+CS

CO +S

Chapter 22: Reaction Dynamics
 laser-induced fluorescence (LIF), a technique in which a laser is used to excite a product molecule from a specific vibration–rotation level and then the intensity of fluorescence is monitored.

Chapter 22: Reaction Dynamics
 multiphoton ionization (MPI), a process in which the absorption of several photons by a molecule results in ionization.
 resonant multiphoton ionization (REMPI), a technique in which one or more photons promote a molecule to an electronically excited state and then additional photons are used to generate ions from the excited state.
A laser pulse excites electrons in a semiconductor surface (10 layers C
60 on a Cu(111) substrate) which in turn pass their energy to adsorbed molecules (NO).
REMPI measures the motion of the desorbed molecules.

Chapter 22: Reaction Dynamics
 reaction product imaging, a technique for the determination of the angular distribution of products.
Reaction products detected in the Streamer Chamber when a 1.1-GeV-per-nucleon beam of holmium-165 collided with a holmium-165 target at the Bevalac.

Chapter 22: Reaction Dynamics
22.7 Potential energy surfaces
 potential energy surface, the potential energy as a function of the relative positions of all the atoms taking part in the reaction.
H
A
+ H
B
-H
C

H
A
-H
B
+ H
C

Chapter 22: Reaction Dynamics
 saddle point, the highest point on a potential energy surface encountered along the reaction coordinate.
H
A
+ H
B
-H
C

H
A
-H
B
+ H
C

Chapter 22: Reaction Dynamics
 saddle point, the highest point on a potential energy surface encountered along the reaction coordinate.
H
A
+ H
B
-H
C

H
A
-H
B
+ H
C

Chapter 22: Reaction Dynamics

Example of potential energy surfaces.
Ultrafast reaction dynamics of the complete photo cycle of an indolylfulgimide studied by absorption, fluorescence and vibrational spectroscopy

Chapter 22: Reaction Dynamics
22.8 Some results from experiments and calculations
H
A
H
A
+ H
B
-H
C
-H
B
+ H
C


Chapter 22: Reaction Dynamics
H
A
+ H
B
-H
C

H
A
-H
B
+ H
C

Chapter 22: Reaction Dynamics
22.8(a) The direction of attack and separation
30 0

Chapter 22: Reaction Dynamics
22.8(b) Attractive and repulsive surfaces
 attractive surface, a potential energy surface in which the saddle point occurs early on the reaction coordinate.
 repulsive surface, a potential energy surface in which the saddle point occurs late on the reaction coordinate.
2


Chapter 22: Reaction Dynamics
22.8(c) Classical trajectories
 direct mode process, a bimolecular process in which the switch of partners takes place very rapidly.
 complex mode process, a bimolecular process in which the activated complex survives for an extended period.