Transcript 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, k
r
A B P
v
k r
encounter rate [A][B] minimum energy requirement steric requirement.
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 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]
A 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
a d
2 reactions 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
( 2
d
) 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 P AB A B
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
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 ‡ P C ‡
K
‡
p C
‡
p
θ
p A p B
[ 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 ‡
K
‡
kT h
‡
K
‡ where
q
C ‡ denotes the partition function for all the other modes of the complex.
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
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
θ m 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
V
m θ 3 C ‡
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
(proportional to -Δ ‡
G
/
RT
).
(=-Δ r
G
θ /
RT
) is plotted against ln
k
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
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 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 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 B -H C H A -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.
H + Cl 2
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