### Chapter 22: Reaction Dynamics

```Atkins & de Paula:
Atkins’ Physical Chemistry 9e
Chapter 22: Reaction Dynamics
Chapter 22: Reaction Dynamics
REACTIVE ENCOUNTERS
22.1 Collision theory
 rate constant, kr  encounter rate  minimum energy requirement  steric requirement.
A BP
v  k r [A][B]
1/ 2
 8 RT 
1/ 2
1/ 2
1/ 2
c 
  c  (T / M )  v   (T / M ) N AN B   (T / M ) [A][B]
 M 
k r   (T / M )1/ 2 e  Ea / RT
k r  P (T / M )1/ 2 e  Ea / 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:
1
Z AB
 8kT 
 N A2 [A][B],
  
  
Z AA
 4kT 
 N A2 [A]2
  
 m A 
2
1
2
  d 2 d  12 (d A  d B ),  
m A mB
m A  mB
Chapter 22: Reaction Dynamics
collisionfrequency, z  crel N A
crel
 8kT 

 
  
1
2
collisiondensity,Z AA  12 zN A  12 crel N 2A
Z AB  crel N AN B
1
 Z AB
collision cross-section
 8kT 
 N A2 [A][B]
  
  
2
volume of tube  
N  1 /   1 / crel t
z  1 / t  crel N
22.1(b) The energy requirement
dN A
  ( )vrel N AN B
 ( )  0 when ε  ε a
dt
d[A]
  ( )vrel N A [A][B]
dt

d[A]
     ( )vrel f ( )d  N A [A][B]
f ( ); Boltzmanndistribution of energy
 0

dt

k r  N A   ( )vrel f ( )d
0
Chapter 22: Reaction Dynamics

k r  N A   ( )vrel f ( )d
0
  12 v
2
rel
vrel , A B
 d 2  a2 

 vrel cos  vrel 
2
 d

1
2
2
  d 2  a2  2 
d 2  a2
2
1
1
    A B  
 
2   (vrel , A B )  2   vrel 
2
d2
  d
 
amax , above which reactionsdo not occur
1
a  amax   A B   a
2
  
  
(  ) amax
d 2
2
 amax
 1  a d 2 
 ,
  ( )  1  a 
 
 


   a   ( )  0 &    a   ( )  
vrel , A B
Chapter 22: Reaction Dynamics

k r  N A   ( )vrel f ( )d
0
  
f (v)dv  4 

 2kT 
3/ 2
v 2 e  v
2
/ 2 kT
  
      f (v)dv  4 

 2kT 
  12 v 2 , dv  d /( 2  )1 / 2
3/ 2
 2   / kT d
 e
(2 )1/ 2
 
3/ 2
 1 
1 / 2  / kT
f (v)dv  2 
d  f ( )d
  e
 kT 






0
0
0
 1 
 ( )vrel f ( )d  2 

 kT 
 ( )e  / kT d   

a
3/ 2


0
ax
ax
e ax
2
1/ 2
k r  N A   ( )vrel f ( )d  N Acrel e  Ea / RT
0
e ax
1/ 2
 8   1  
     ( )e  / kT d

 kT   kT  0
xeax
 e dx   a C ,  xe dx   a  a C
  a   / kT
 1  e
d           (kT ) 2 e 
 

 8kT   a / kT
 e
 ( )vrel f ( )d   




1/ 2
 2 
 ( )   1/ 2 e  / kT d
 
a
/ kT
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.
1/ 2
 8kT 
 rate constant from collision theory, k r  P 
 N Ae  E / RT
  
a
 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.
 E 

P  1 
E


s 1
s 1
 E 
 kb
 kb ( E )  1 
E


for E  E 

s; the # of modes of motion,E ; energyrequiredfor the bondbreakage,E ; energyavailablein thecollision
Lindemann-Hinshelwood
mechanism
Exp. data for
unimolecular
isomerization of
trans-CHD=CHD
RRK model
s
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 v  k d [A][B] AB : encounterpair,d : diffusion
AB  A  B v  k d [AB]
AB  P
v  k a [AB]
a : activatedprocess
k d [A][B]
d [AB]

 k d [A][B] k d [AB] k a [AB]  0  [AB] 
dt
k a  k d
kk
d [P]
 k a [AB]  k r [A][B], k r  a d
dt
k a  k d
When k d  k a  k r  k d
: diffusion - controlledlimit
When k a  k d  k r 
ka kd
 ka K
k d
: activation- controlledlimit
Chapter 22: Reaction Dynamics
22.2(b) Diffusion and reaction
A  B  AB in solution!
c
 2 c 3dimension
[B]
 D 2    DB  2 [B] 
; diffusion equation (Fick' s second law of diffusion)
t
x
t
[B]
 0   2 [B]r  0; r signifies a quantity that varies with the distance r
t
d 2 [B]r 2 d [B]r
b
spherically symmetry system
2
 [B]r      



0

General
solution
:
[
B
]

a

r
r 2
r r
r

[ B]r  [B] ([B] is bulk value) as r  , [ B]r  0 at r  R (the distance where reaction occurs)
 R 
[B]r  1  [B]
r 

Rate of reaction  4R 2 J ( J : molar flux of B toward A)
D [B]
 d [B]r 
From Fick' s first law J  DB 
 B

R
 dr  r  R
A
Rate of reaction  4R  DB [B] for
all
 4R  DB [B]N A  4R  DB N A [A][B]
DB  DA  DB  D  A is not stationary
d [P]
 k d [A][B]  4R  DN A [A][B]  k d  4R  DN A
dt
Chapter 22: Reaction Dynamics
By using Stokes- Einsteinequation;
kT
kT
DA 
DB 
( RA , RB ; hydrodynamic radius,  ; viscosityof medium)
6RA
6RB
RA  RB  12 R   k d  4R  DN A 
8RT
3
22.3 The material balance equation
Generalized diffusion equation: thediffusion equationincluding convection
[J]
 2 [J]
[J]
 D 2 v
t
x
x
[J]
 2 [J]
[J]
Includingchemicalreaction
 D 2 v
 k r [J]; materialbalanceequation
t
x
x
 No convection; [J]  [J]e kr t [J] : for no reaction
n0
 x 2 / 4 Dt
e
A(Dt)1/ 2
 For generalcases, we can solve thematerialbalanceequation numerically!!
 No reaction;[J] 
Chapter 22: Reaction Dynamics
TRANSITION STATE THEORY
 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 J  RT [ J ]

 [C‡ ] 
C‡  P
v  k r [A][B]
pC ‡ p θ
p A pB
RT ‡
K [A][B]
θ
p
v  k ‡ [C ‡ ]
RT
kr  θ k ‡ K ‡
p
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
θ
  q θ  J 
N
q
‡
A
K    J,m  e   r E0 / RT  K ‡  θ Cθ e   r E0 / RT where p  1 bar &  r E0  E0 (C‡ )  E0 (A)  E0 (B)
qA qB
 J  N A  
1
Partition function for specific vibration which leads to product formation; q 
‡
1  e  h / kT
1
kT
h ‡  kT  q 

 h ‡
 h ‡
1  1 
 
kT


kT
q where qC‡ denotes the partition function for all the other modes of the complex.
‡ C‡
h
θ
N
q
kT
A
‡
‡
C‡   r E0 / RT
K‡ 
K
K

e
( K ‡ ; K ‡ with one vibrational mode of C‡ discarded)
‡
θ θ
h
qA qB
qC ‡ 
Chapter 22: Reaction Dynamics
22.4(c) The rate constant
kr  k ‡
RT ‡
kT ‡
‡ kT RT
‡
K


K


K ‡ ; Eyringequation
pθ
h ‡ p θ
h C
For qCθ‡ , we have to know thesize, shape,and st ructureof act ivatedcomplex verydifficult !
22.4(d) The collision of structureless particles
Vmθ
q  3
J
J 
θ
J
h
(2mJ kT )1/ 2
Vmθ 
RT
pθ
~ 2
hcB 
2I
θ
 2 IkT  Vm
A  B  C (A B), q corresponds to rotatioalmode   q   2  3
    C‡
‡
I  r 2 ,

θ
C‡
θ
C‡
m A mB
, mC‡  m A  mB
m A  mB
 AB 
kT
kT RT  N A 3A 3B  2 IkT    r E0 / RT


N


e
kr  


A
2
θ 
3
θ 

h
h p   C‡Vm   
  C‡ 
1/ 2

k r  




  8 kT
3
 2 IkT    r E0 / RT
 2 e
  
N Ae a
 8kT  2   r E0 / RT
         r 2 , Ea   r E0
 N A  2 r e
  
 E / RT
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 + CO2[HOCO] ‡ HO+CO
Chapter 22: Reaction Dynamics
22.5 Thermodynamic aspects
22.5(a) Activation parameters
Gibbs energyof activation, ‡G   RT ln K ‡
kT RT ‡G / RT ‡G  ‡ H T‡S
kT RT
 is absorbed into S term
‡ S / R  ‡ H / RT
kr  
e














k

Be
e
B

r
h pθ
h pθ
  ln k r 
‡
k r  Ae Ea / RT  Ea  RT 2 
  Ea   H  2 RT
 T 
‡ S / R  Ea / RT
 k r  e Be
2
‡ S / R
 A  e Be
2
e
Pe
‡ S steric / R
 correlation analysis, a procedure in which ln K (=-ΔrGθ/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.
‡
θ
a
‡
d [P]
[
C
]
c
 k ‡ [C ‡ ] K  C  K 
dt
aA aB
[A][B]
C
K 
 A B
‡
k ‡ K k r0  k ‡ K when  1
k r0
kr 
   
 k r 
K
K
d [P]
 k r [A][B]
dt
From Debye- Huckel limit inglaw (log    z  z- AI 1/ 2 , A  0.509for aq. at 250 C),
log A   AzA2 I 1/ 2
log B   AzB2 I 1/ 2
log C‡   A( z A  z B ) 2 I 1/ 2


log k r  log k r0  A z A2  z B2  ( z A  z B ) 2 I 1/ 2
 log k r0  2 AzA z B I 1/ 2
Exercise Example 22.3!
Chapter 22: Reaction Dynamics
THE DYNAMICS OF MOLECULAR COLLISIONS
22.6 Reactive collisions
22.6(a) Experimental probes of reactive collisions
 infrared chemiluminescence, a process in which vibrationally excited molecules emit
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.
HA + HB-HC  HA-HB + HC
Chapter 22: Reaction Dynamics
 saddle point, the highest point on a potential energy surface encountered along the
reaction coordinate.
HA + HB-HC  HA-HB + HC
Chapter 22: Reaction Dynamics
 saddle point, the highest point on a potential energy surface encountered along the
reaction coordinate.
HA + HB-HC  HA-HB + HC
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
HA + HB-HC 
HA-HB + HC
Chapter 22: Reaction Dynamics
HA + HB-HC  HA-HB + HC
Chapter 22: Reaction Dynamics
22.8(a) The direction of attack and separation
300
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.
H + Cl2  HCl +Cl
attractive surface
repulsive surface
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.
direct mode process
complex mode process
```