(b) Analytical solutions of the material
balance equation
[ J ]
t
 [J ]
2
 D
x
t
[ J ]  k  [ J ]e
*
2
 kt
0
 k[J ]
dt  [ J ]e
 kt
2
[J ] 
n0e
 x / 4 Dt
A ( Dt )
1/2
Using numerical method: Euler method or 4th order Runge-Kutta method to
Integrate the differential equation set.
Fig. 1 Snapshots of stationary 2D spots (A) and stripes (B) in a thin layer and 2D images of
the corresponding 3D structures (A→B, C; B→E to G) in a capillary.
T Bánsági et al. Science 2011;331:1309-1312
Published by AAAS
Fig. 3 Stationary structures in numerical simulations.
Stationary structures in numerical simulations. Spots (A),
hexagonal close-packing (B), labyrinthine (C), tube (D),
half-pipe (E), and lamellar (F) emerging from asymmetric
[(A), (B), and (E)], symmetric [(D) and (F)], and random (C)
initial conditions in a cylindrical domain. Numerical results
are obtained from the model:
dx/dτ = (1/ε)[fz(q – x)/(q + x) + x(1 – mz)/(ε1
+ 1 – mz) – x2] + ∇2x;
dz/dτ = x(1 – mz)/(ε1 + 1 – mz) – z + dz∇2z,
where x and z denote the activator, HBrO2 and the oxidized
form of the catalyst, respectively; dz is the ratio of diffusion
coefficients Dz/Dx; and τ is the dimensionless time.
Parameters (dimensionless units): q = 0.0002; m = 0.0007;
ε1 = 0.02; ε = 2.2; f = (A) 1.1, (B) 0.93, [(C) to (F)] 0.88;
and dz = 10. Size of domains (dimensionless): diameter = 20
[(A) to (C) and (F)]; 14 [(D) and (E)]; height = 40.
T Bánsági et al. Science 2011;331:1309-1312
Published by AAAS
Transition State Theory
(Activated complex theory)
• Using the concepts of
statistical
thermodynamics.
• Steric factor appears
automatically in the
expression of rate
constants.
24.4 The Eyring equation
•
The transition state theory pictures a reaction between A and B as proceeding
through the formation of an activated complex in a pre-equilibrium:
A + B ↔ C‡

K 
pC  p

( `‡` is represented by `±` in the math style)
pA pB
•
•
The partial pressure and the molar concentration have the following relationship:
pJ = RT[J]
thus

[C ]  K

RT
p
•
[ A ][ B ]
The activated complex falls apart by unimolecular decay into products, P,
C‡
•

→
vk
So
v = k‡[C‡]
P

RT
p


K [ A ][ B ]
Define k 2  k  RT K 

p
v = k2[A][B]
(a) The rate of decay of the activated
complex
k‡ = κv
where κ is the transmission
coefficient. κ is assumed to be about 1 in
the absence of information to the contrary.
v is the frequency of the vibration-like
motion along the reaction-coordinate.
(b) The concentration of the activated complex
Based on Equation 17.54 (or 20.54 in 7th edition), we have
K



N Aq
C



e
  E 0 / RT
with ∆E0 = E0(C‡) - E0(A) - E0(B)
q AqB

are the standard molar partition functions.
qJ
q 
1
1e
 hv / kT
provided hv/kT << 1, the above partition function can
be simplified to
1
q
1  (1 
hv

kT
hv
 )
kT
Therefore we can write
kT 
qC‡ ≈
qc

hv

where q C  denotes the partition function for all the other modes of the
complex.
K‡ =
kT
hv

K



K


N
A
qC 


q AqB
e
  E 0 / RT
(c) The rate constant
combine all the parts together, one gets
k2  k

RT
p
k 2  v

K
kT RT
hv p



K

then we get
k2  
kT
h


K
(Eyring equation)
C
To calculate the equilibrium constant in the Eyring equation, one
needs to know the partition function of reactants and the activated
complexes.
Obtaining info about the activated complex is a challeging task.
(d) The collisions of structureless particles
A + B → AB
Because A and B are structureless atoms, the only
contribution to their partition functions are the
translational terms:

qJ 

Vm
3
J
q
h

( 2  m J kT )
Vm 
1/ 2
RT
p



C
J 

 2 IkT  V m

 
2
 
 C
3
3
kT RT  N A  A  B

k2  
 
3

h p   C Vm
  2 IkT    E / RT

e 0
2
  

 8 kT 
k 2  N A 

 u 
1/ 2
r e
2
  E 0 / RT
24.5 Thermodynamic aspects
Kinetics Salt Effect
Ionic reaction A + B ↔ C‡
C‡
→ P
d[P]/dt = k‡[C‡]
the thermodynamic equilibrium constant
K 
a
C


a Aa B

C
0

 K
[C ]
[ A ][ B ]

d[P]/dt = k2[A][B]
k2
[C ]
 A  B [ A ][ B ]
Then
Assuming


k2 
k K
K
is the rate constant when the activity coefficients are 1 ( k 20  k  K )
0
k2 
Debye-Huckle limiting law
log(k2) = log(
0
k2
k2
K
log(  J )  Az J I
2
1/2
) + 2AZAZBI1/2
with A = 0.509
(Analyze this equation)
Experimental tests of the kinetic
salt effect
•
Example: The rate constant for the base hydrolysis of [CoBr(NH3)5]2+ varies
with ionic strength as tabulated below. What can be deduced about the
charge of the activated complex in the rate-determining stage?
I
0.0050 0.0100 0.0150 0.0200 0.0250 0.0300
k/ko
0.718
0.631
0.562
0.515
0.475
0.447
Solution:
I1/2
Log(k/ko)
0.071
-0.14
0.100
-0.20
0.122
-0.25
0.141
-0.29
0.158
-0.32
0.173
-0.35
24.6 Reactive Collisions
• Properties of incoming molecules
can be controlled:
1. Translational energy.
2. Vibration energy.
3. Different orientations.
• The detection of product molecules:
1. Angular distribution of products.
2. Energy distribution in the product.
Detection techniques
 infrared chemiluminescence, a process in which
vibrationally excited molecules emit infrared radiation as
they return to their ground states.
 laser-induced fluorescence, 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.
 multiphoton ionization (MPI), a process in which the
absorption of several photons by a molecule results in
ionization.
24.6 Reactive collisions (cont..)
 reaction product imaging, a technique for the
determination of the angular distribution of products.
 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.
 state-to-state cross-section, σnn', the cross section for
reaction in which a specified initial state changes into a
specified final state.
 state-to-state rate constant, the rate constant for a
specified state-to-state reaction; knn = σnnvrelNA.
24.7 Potential energy surface
• Can be constructed from experimental measurements or from
Molecular Orbital calculations, semi-empirical methods,……
Contour plot
Potential energy is a function of the relative positions of all the atoms taking
part in the reaction.
Potential energy surfaces, pt. 2.
Various trajectories through the
potential energy surface
24.8 Results from experiments and
calculations
(a) The direction of the attack and separation
Attractive and repulsive surfaces
Crossing crowded dance floors.
S Bradforth Science 2011;331:1398-1399
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Classical trajectories