Chemical Kinetics : rate of a chemical reaction
Before a chemical reaction can take place the molecules involved
must be raised to a state of higher potential energy. They are then
said to be activated or to form an activated complex.
 Ea

In 1889 Arrhenius said:
1) van’t Hoff eq. for temperature coefficient of equilibrium
constant is
2)
d ln K c E

2
dT
RT
mass-action law relates equilibrium constant to
the ratio of rate constants
k
Kc 

f
kb
Hence a reasonable eq. for the variation of rate
constant with temperature is
d lnk E a

2
dT
RT
Where Ea is the
activation energy of
the reaction
If Ea does not depend on temperature, we can
integrate this last eq. to obtain
Ea
ln k  
 ln A
RT
where ln A is the constant of integration. Hence
k  Ae
E a / RT

This is the famous Arrhenius eq. for the rate constant.
According to Arrhenius, molecules must acquire a certain
critical energy Ea before they can react. The Boltzmann
factor eE a / RT is the fraction of molecules that manages to
obtain the necessary energy. This interpretation is still
held to be essentially correct.

Henry Eyring (1901-1981)
The rate of any chemical reaction can be formulated in terms of
its activated complex.
The rate of reaction is the number of activated complexes
passing per second over the top of the potential energy barrier.
This rate is equal to the concentration of activated complexes
times the average velocity with which a complex moves across to
the product side.
Calculation of conc. of activated complexes is greatly
simplified if we assume that they are in equil. with the
reactants.
This equil. can then be treated by means of
thermodynamics or statistical mechanics.
Transition State Theory
Consider this equilibrium:
A  B  AB  products

equil. constant for the formation of the complex is

[AB]
K 
[A][B]

the conc. of complexes
is thus [AB]  K  [A][B]
according to transition state theory, the rate of reaction is

d[A]/dt  AB 
(rate of passage over barrier)

The rate of passage over the barrier is equal to the
frequency with which the complex flies apart into the
products.
The complex flies apart when one of its vibrations becomes
a translation, and what was formerly one of the bonds
holding the complex together becomes simply the line of
centers between separating fragments.
The frequency  is equal to  /h where  is the
average energy of the vibration leading to
decomposition. Since by hypothesis this is a
thoroughly excited vibration at temperature T, it has its
classical energy   kT and hence frequency   kT/h


The reaction rate is therefore

d[A]
kT


 k2[A][B]  K [A][B]
dt
h

with rate constant
kT 
k2 
K
h
This is the general expression given by transition state
theory for the rate constant of any elementary reaction. To
be precise, the expression for k2 should be multiplied by a
factor  called the transmission coefficient, which is the
probability that the complex will not recross the transition
state and dissociate back into products. In basic TST,   1
The activated complex is similar to a normal stable
molecule in every respect save one. The sole
difference is that one of its vibrational degrees
of

freedom is missing, having been transformed into the
translation along the reaction coordinate. Instead of
3N-6 vibrational modes, it has 3N-7 modes (non-linear
case).
We can formulate k2 in thermodynamic terms by
introducing the standard free energy change
G  RT ln K
o

This is the difference between the free energy of the
activated complex and that of the reactants, when all
are in their standard states.
kT G o / RT kT S o / R H o / RT
k2 
e

e
e
h
h
The quantities G o H o S oare called the free energy of
activation, the heat of activation, and the entropy of activation.
The heat of activation is almost equivalent to the experimental
energy of activation Ea, except for a PV term which is negligible
for solid or liquid systems.

Chemical
Before
can correct for a wrong choice of
transition state this way as well
challenges for computational modeling
1) where/what is the transition state?
2) what is the reaction coordinate?
Schematic representation of
the free energy landscape
with two stable, attractive
wells separated by a transition
state ridge, which connects
the highest free energy points
of all possible paths
connecting the reactant and
product states. The dotted line
represents a new trajectory
that was branched of at point
p from an old trajectory (bold
line) and surpasses the TS
ridge at a lower point.
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Example of a complicated reaction coordinate:
aqueous proton transfer reaction


AH (aq)  B (aq)  A(aq)  BH (aq)
what is the reaction coordinate?
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Transition path sampling
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