Download H + H–H H∙∙∙∙∙∙∙∙∙ H∙∙∙∙∙∙H H∙∙∙∙∙∙H∙∙∙∙∙∙H

§9.7 Transition state theory (TST)
Theory of Absolute reaction Rates
Theory of activated complex theory
A + B-C  A-B + C
During reaction, energies are being redistributed among bonds:
old bonds are being ripped apart and new bonds formed.
H + H–H  H∙∙∙∙∙∙∙∙∙ H∙∙∙∙∙∙H
 H∙∙∙∙∙∙H∙∙∙∙∙∙H (activated state)
 H∙∙∙∙∙∙H∙∙∙∙∙∙∙∙∙∙∙∙H  H–H + H
This process can be generalized as:
A + B-C  [ABC]  A-B + C
Activated complex
Transition state
The transition state theory (TST), attempting to explain
reaction rates on the basis of thermodynamics, was developed
by H. Eyring and M. Polanyi during 1930-1935.
TST treated the reaction rate from a quantum mechanical
viewpoint involves the consideration of intramolecular forces
and intermolecular forces at the same time.
Basic consideration
According to TST, before undergoing reaction, reactant molecules
form an activated complex which is in thermodynamic equilibrium
with the molecules of the reactants. The activated complexes, the
energy of which is higher than both reactants and products, is treated as
an ordinary molecule except that it has transient existence and
decomposes at a definite rate to form the product.
r    cAB
7.1 Potential energy surfaces
According to the quantum mechanics, the nature of the chemical
interaction (chemical bond) is a potential energy which is the function
of interatomic distance (r):
V  V (r )
The function can be obtained by solving Schrödinger equation for a
fixed nuclear configuration, i.e., Born-Oppenheimer approximation.
The other way is to use empirical equation. The empirical equation
usually used for system of two atoms is the Morse equation:
Morse equation:
V (r )  De{exp[ 2a(r  r0 )]  2 exp[ a(r  r0 )]}
decomposition asymptote
When r = r0, Vr (r = r0) = -De
r, Vr (r) = 0
where De is the depth of the
wall of potential, or the
dissociation energy of the
bond. r0 is the equilibrium
interatomic distance, i.e.,
bond length, a is a parameter
with the unit of cm-1 which
can be determined from
spectroscopy.
r > r0, interatomic attraction,
Zero point energy: E0 = De-D0
r < r0, interatomic repulsion.
J. Comp. Chem., 2011, 32, 5: 797-809
For triatomic system A + BC  AB + C
C
C
rA

rBC
B
A
A
rAC
rAB
rBC
B
B
V = V(rAB, rBC, rAC ) = V(rAB, rBC , )
For triatomic system, the potential is a four-dimension function.
In 1930, Eyring and Polanyi make  = 180 o, i.e., collinear collision
and the potential energy surface can be plotted in a three dimensions /
coordination system.
 = 180 o
A
rAB
B
rBC
C
V = V(rAB, rBC)
Eyring et al. calculated the energy of the triatomic system:
HA + HBHC  HAHB+ HC
using the method proposed by London.
Schematic of LEP Potential
energy surface
Contour diagram of the
potential energy surface
Projection of LEP potential surface
peak
Which way should the reaction follows?
peak
reaction path or reaction coordinate.
valley
Saddle point
INTERMOLECULAR POTENTIAL
ENERGY SURFACE FOR CS2 DIMER
Journal of Computational Chemistry
Volume 32, Issue 5, pages 797-809, 12 OCT 2010 DOI: 10.1002/jcc.21658
http://onlinelibrary.wiley.com/doi/10.1002/jcc.21658/full#fig10
Activated complex has no recovery force. On any special vibration
(asymmetric stretching), it will undergo decomposition.
Whenever the system attain saddle point, it will convert to product
with no return.
7.2 Kinetic treatment of the rate constant of TST
For reaction:
The rate of the reaction depends on two factors:
1) the concentration of the activated complex (c)
2) the rate at which the activated complex dissociates into
products()
r    cAB
According to equilibrium assumption
cAB
K 
cA cB
r    K  cA cB
k    K
According to statistical thermodynamics, K can be expressed
using the molecular partition function.
cAB
q
f
 E0 
K 


exp  

cA cB qA qB f A f B
 RT 
E0 is the difference between the zero point energy of activated
complex and reactants. q is the partition function, f is the partition
function without E0 stem and volume stem.
For activated complex with three atoms, f can be written as a
product of partition function for three translational, two rotational,
and four vibrational degrees of freedom.
f   f * f '
Only the asymmetric stretching can lead to decomposition of the
activated complex and the formation of product.
For one-dimension
vibrator:
For asymmetric
stretching
1
f 
  h  
1  exp 

k
T
 B 
*
h  kBT
f* 
kBT
h 
kBT 

f 
f '
h 
kBT f  '
 E0 
k    K   
exp  

h  f A f B
RT


kBT f  '
 E0 
k
exp  

h fA fB
RT


statistical expression for the
rate constant of TST
For a general elementary reaction
kBT f  '
 E0 
k
exp  

h  fi
RT


In which f’ can be obtained from partition equation and E0 can be
obtained from potential surface. Therefore, k of TST can be
theoretically calculated. Absolute rate theory
For example:
For elementary equation:
H2+ F  HHF  H + HF
Theoretical:
k = 1.17  1011 exp(-790/T)
Experimental:
k = 2  1011 exp(-800/T)
7.3 Thermodynamic treatment of TST
For nonideal systems, the intermolecular interaction makes
the partition function complex. For these cases, the kinetic
treatment becomes impossible.
In 1933, LaMer tried to treat TST thermodynamically.
kBT f  '
 E0 
k
exp  

h fA fB
RT


kBT 
k
K
h
f'
 E 
K
exp   0 
fA fB
 RT 
G   RT ln K 
G  H   T S
Standard molar entropy of activation, standard molar enthalpy of
activation
G   RT ln K 
 G 
K   exp  

RT


kBT 
k
K
h
 G 
kBT
k
exp  

h
RT


G  H   T S
 S 
 H  
kBT

exp 
 exp  

h
 R 
 RT 
The thermodynamic expression of the rate of TST is different from
Arrhenius equation
kBT 
k
K
h
According to GibbsHolmholtz equation
kBT
ln k  ln
 ln K 
h
d ln k
1 d ln K 
 
dT
T
dT
d ln K U 

dT
RT 2
H   U   PV
d ln K H   PV

dT
RT 2
d ln k RT  H   PV

dT
RT 2
 d ln k 
Ea  RT 

 dT 
Ea  RT  H   PV
2
Ea  RT  H   PV
For liquid reaction: PV = 0
Ea  RT  H 
PV  nRT  (1  n) RT
For gaseous reaction:
n is the number of reactant molecules
Ea  H   nRT
k
 S 
 H  
k BT
exp    exp  

h
 R 
 RT 
thermodynamic expression
of the rate of TST.
 S  n
k BT
 Ea 
k
exp 
 e exp  

h
R
RT




 S  n
kBT
 Ea 
k
exp 
 e exp  

h
RT
RT




 E 
k  A exp   a 
 RT 
 S 
kBT n
A
e exp 

h
RT


k BT
is a general constant with unit of s-1
Z'
of the magnitude of 1013.
h
k SCT
 Ea 
 PZ ' exp  

 RT 
 S 
P  exp 

 RT 
The pre-exponential factor depends on the standard entropy of
activation and related to the structure of activated complex.
 S 
P  exp 

RT


suggests that the steric factor can be estimated
from the activation entropy of the activated
complex.
Example:
reactions
P
exp(S/R)
(CH3)2PhN + CH3I
0.5  10-7 0.9  10-8
1986 Noble Prize
Hydrolysis of ethyl acetate
2.0  10-5 5.0  10-4
Canada
Decomposition of HI
John C. Polanyi
1929/01/23 ~
Decomposition of N2O
0.5
0.15
1
1