Activated Complex Theory or Transition State Theory | Physical Chemistry PDF Download

Activated complex Theory of Bimolecular Reaction or Transition state Theory or Eyring Equation

The activated complex forms between reactants as they collide. The difference between the energy of the activated complex and the energy of the reactants is the activation energy, E_a.

(a) Exothermic react ion (b) Endothermic reaction According to Eyring the equilibrium is between reactants and the activated complex.
Consider A and B react to form an activated complex that undergoes decay, resulting in product formation.
The activated complex represents the system at the transition state.
This complex is stable.

where (AB)^# is the act ivated complex and k₁ is the equilibrium constant between reactants and activated complex.
If (AB)^# one of the vibrat ional degrees of freedo m has become a translat ional degree of freedo m.
From the classical mechanics,

k_B = Boltzmann constant

from the quantum mechanics,
energy = hv

 than  
 

The vibrat ional frequency v is the rate at which the activated complex mo ve across the energ y barrier i.e. the rate constant k₂ is ident ified by v.
Then the reaction is

 
⇒ 
then 

 for conventional rate

 
∴

i.e. k₂ = v × k₁
k₂ = frequency × k₁                                            …(1)

where k₁ = equilibrium constant = k_eq

Relation between k₁ and ΔG^#:
k₁ = equilibrium constant =
&  ΔG^# = ΔH^# - TΔS^#
where ΔG#, ΔH# & ΔS# are the standard free energy o f act ivat ion, enthalpy o f act ivat ion and entropy of activation. 

k₂ = keq × frequency = k₁× frequency

                                    …(2)

                   …(3)
∵ 
∴ …(4)

and  Δn_g =  difference is number of mo les between transit ion state and reactant

Relation between ΔE^# and E_a

We know that
 

On taking log

On differentiate above equation

or

and                          (from arrhanius equation)
then

 E_a = ΔE^# + RT                                                     …(5)

Relation between E₀ & ΔE^#

E₀ = Collision energy

Value of A, using above equations From Eyring theory and Arrhenius theory we have

rate constant =  

                              …(6)
 

Problem.  Consider the decomposition of 

NOCl, 2NOCl(g) → 2NO(g) + Cl₂(g) 

The Arrhenius parameters for this reaction are A = 1.00 × 10¹³ M^–1 s^–1 and E_a = 10⁴ kJ mol^–1. Calculate ΔH^# and ΔS^# for this reaction with T = 300 K.
 Sol. We know that

                     ΔH^# = ΔE^# + Δn_gRT

where Δng = difference in number of moles between act ivated complex and reactant.

                      ΔH^# = ΔE^# + (-1) RT
                       ΔH^# = ΔE^# - RT

&                    E_a = ΔE^# + RT

then               ΔE^# = E_a - RT
∴                    ΔH^# = ΔE^# + RT = E_a - RT - RT = E_a - RT
                      ΔH^# = 104 kJ mol^-1 - 2(8.314 J mol^-1 K^-1) (300 K)
                       = 99.0 kJ mol^-1
We know that

                           [Δn_g = -1]
Taking log     
                      

ΔS# = -12.7 J mol^-1 K^-1

Note: for unimolecular      Δn_g = 0

for bimolecular Δn_g = -1
for trimolecular            Δn_g = -2
&                                 ΔH^# = ΔE^# + Δn_gRT                                [∵ Ea = ΔE^# + RT]4
                                    = E_a - RT + Δn_gRT
⇒                                 E_a = ΔH^# + RT - Δn_gRT
for unimolecular            E_a = ΔH^# + RT
for bimolecular              E_a = ΔH^# + 2RT
for trimolecular              E_a = ΔH^# + 3RT

The Pre-equilibrium Approximation: Consider the following reaction

(i) First, equilibrium between the reactants and the intermediate is maintained during the course of the reaction. (ii)  The intermediate undergoes decay to form product.
Then the rate law expression is

∵ I is in equilibrium wit h the reactant then

   = equilibrium constant

[I] = k_C [A][B]

∴ 

The Lindemann Mechanism 

Lindemann mechanism for unimolecular reactions involves two steps. First reactants acquire sufficient energy to undergo reaction through a bimolecular collision.

In this,  A^* is the act ivated reactant and undergoes one of two reactions.

Then, rate of product formation is

& rate of formation of A^*

Applying the steady-state approximation

Then 

It state that the observed order dependence on [A] depends on the relative magnitude of k_–1[A] versus k₂. At high reactant concentration, k_–1[A] > k₂ and

i.e. the product formation is first order at high pressure. At low reactant concentration ^k₂ > k_–1[A] and

i.e. at low pressure, the rate of formation product is second order in [A].
Then

k_uni is the apparent rate constant for the reaction defined as

and 

when k_–1[A] >> k₂ i.e. at high concentration k_uni

when k_–1[A] << k₂ i.e. at low concentration k_uni = k₁[A]

 k_uni = k₁[A]