Collision theory of unimolecular reaction, transition state theory of absolute reaction rate and characteristics of chain reaction

Collision theory of unimolecular reaction

Lindemann tried to explain the formation of unimolecular reaction on the basis of bimolecular collision. According to this theory the activated molecule that has formed due to bimolecular collision does not convert immediately into product. But there exist certain time lag between activation and product formation. Because of this time lag the activated molecule may be deactivated by collision with other molecule. Consequently the rate of reaction is not dependent to all molecules that are activate but only to those that remain active for sometime and converted into product.

Let us consider a reaction:

A+A\underset{k_2 (deactivation) }{\overset{k_1 (activation)}{\rightleftarrows}} A^*+A

activated molecule

A^*$$\rightarrow{k_3}$$ products

Therefore the rate of reaction,

-$$\frac{d[A]}{dt}$$ = k₃[A^*].............(i)

Since A^*is a short lived intermediate so steady state principle can be applied

Rate of formation of A^*= k₁[A]²

Rate of decomposition of A^o= k₂[A^*][A] + k₃[A^*]

At steady state,

k₁[A]²= k₂[A^*][A] + k₃[A^*]

= [A^*] {k₂[A] + k₃}

or, [A^*] = $$\frac{k_1[A]^2}{k_2[A] + k_3}$$..........(ii)

from equation (i) and (ii), we get,

-$$\frac{d[A]}{dt}$$ = $$\frac{k_3 k_1 [A]^2}{k_2[A] + k_3}$$

i.e, Rate of reaction-$$\frac{d[A]}{dt}$$ =$$\frac{k_3 k_1 [A]^2}{k_2[A] + k_3}$$...........(iii)

A+A\underset{k_2}{\overset{k_1}{\rightleftarrows}}A^*+A

Here two cases arise from equation (iii)

Case I : When k₂[A]>>k₃, then k₃can be neglected in the denominator. So equation (iii) becomes

Rate of reaction (-$$\frac{d[A]}{dt}$$) =$$\frac{k_3 k_1 [A]^2}{k_2[A] }$$

i.e Rate = k[A]......(iv)

where, k = $$\frac{k_3 k_1}{k_2}$$ = a new constant

The equation (iv) indicates that the reaction is first order. Thus these conditions appears only when the concentraion of normal molecule is higher as compared to the activated molecule and hence rate of deactivation becomes high as compared to rate of activation. In this way unimolecular reaction is explained on the basis of collision theory.

Case II: When k₃>>k₂[A] then k₂[A] + 3≈ k₃

So equation (iii) becomes, Rate of reaction (-$$\frac{d[A]}{dt}$$) =$$\frac{k_3 k_1 [A]^2}{k_3}$$

i.e Rate = k₁[A]².........(v)

This equation (v) indicates that the reaction is of second order. This condition appears only when rate of activation is higher as compared to rate fo deactivation.

Transition state theory (theory of absolute reaction rate)

According to this theory reactant molecule donot convert directly into product but convert via formation of short lived state called transition state. This state remains in equilibrium with reactant molecules. Transition state has higher energy as compared to reactant and product and its decomposition results into product formation.

Let us consider a general reaction

A + B $$\longrightarrow$$ products

The postulated steps can be represented as,

A + B $$\longrightarrow$$ [AB^*] $$\longrightarrow$$ products

The specific reaction rate of this recation is given by,

k = k^*$$\frac{RT}{Nh}$$......(i)

where, k^*= equilibrium constant for formation of transition state or activated complex

k^*= $$\frac{[AB]^*}{[A][B]}$$

R= univerasl gas constant

T= temperature

N= Avogadro's number

h = plank's constant

Now, from thermodynamics

ΔG^*= -RT lnk^*

or, lnk^*= -$$\frac{\Delta G^*}{RT}$$

or, k^*= e^{-$$\frac{\Delta G^*}{RT}$$}..........(iii)

From equation (i) and (iii) we get

k = e^{-$$\frac{\Delta G^*}{RT}$$} . $$\frac{RT}{Nh}$$.........(III)

The equation (iii) consists of two term, the first term $$\frac{RT}{Nh}$$ is constant at constant temperature, Thus the value of specific reactant route depends only on second term,e^{-$$\frac{\Delta G^*}{RT}$$}this term involvesΔG^*, smaller is the value of k and hence slower will be the reaction rate i.e greater the free energy of activation, slower will be the reaction rate, we know that,

ΔG =ΔH - TΔS

or,ΔG^* =ΔH^*- TΔS^*

Equation (iii) becomes,

k = $$\frac{RT}{Nh}$$× e^{-($$\frac{\Delta H^*-T \Delta S^*}{RT}$$}

k = $$\frac{RT}{Nh}$$ e^{-$$\frac{\Delta H^*}{RT}$$}e^{$$\frac{\Delta S^*}{R}$$}..............(iv)

From equation(iv) we can determine the entropy of activation ($$\Delta$$ S^*). If the entropy of activation along with specific reaction rate are known as enthalpy of activation ($$\Delta$$ H^*) can be easily determined from experimentally determined value of Eq

$$\Delta$$ H^*= Eq - RT (For first order reaction)

$$\Delta$$ H^*= Eq - 2RT (For second order reaction)

$$\Delta$$ H^*= Eq - 3RT (for third order reaction)

Advantages of transition state theory over collision theory

• Collision theories applied for gaseous reaction but transition state theory applied for both gaseous phase and solution phase reaction.
• The concept of formation of activated complex seems to be logical, more superior their molecules collide and converts into product.
• In collision theory, probablisitic factor has been introduced but in transition state theory free energy change of activation has been introduced.

Chain reaction

A reaction that proceeds in series of successive steps initiated by primary process is known as chain reaction. For example: photochemical combination between H₂and Cl₂and between H₂and Br_2.

Characteristics of chain reaction

1. Generally a chain reaction consist of three steps:

a. Chain initiation

b. Chain propagation

c. Chain termination

2. The probabilistic factor for non chain reaction lies between 1-10^-9where as it is greater than 1 for chain reaction.

In non chain reaction, highest rate is observed at the beginning and rate decreases with time but chain reaction starts at almost zero rate, rate rises immediately to peak value and then decreases.

As chain start almost at zero rate so it takes for its initiation. This is known as induction period. Induciton period is a characteristics of chain reaction. Chain reaction are more sensitive than non-chain reaction. So even change in shape of vessel may change their rate.

References