Collision Theory

Max Trautz and William Cudmore McCullgh Lewis gave collision theory to interpret the exponential factor in the Arrhenius equation. Their theory was based on kinetic theory of gases. They assumed that the molecules are hard spheres and when they collide, the product is formed. It means that the rate depends on total number of collisions per unit volume, which is termed as collision number Z.

You can understand it by taking the example of Pool or Billiards. To sink a pool ball into the pocket, it is necessary to hit it by cue ball with desired kinetic energy. For a successful shot, the orientation of cue ball is as important as the kinetic energy.

Let’s take an example of a real reaction, the formation of methanol from methyl bromide.

CH₃Br + OH^- ⟶ CH₃OH +  Br ^–

As you know, the hybridization of methylbromide is sp³ and its shape is tetrahedral, but it gains inverted umbrella shape for this reaction. Formation of methanol depends on the orientation of methylbromide when OH^-collides with it.   If OH^- attacks from the methyl side, it leads to the formation of methanol. If OH^-attacks from the bromide side, it bounces back and no product is formed.

Right now I am trying to explain the importance of orientation in any reaction. We will discuss the detailed reaction mechanisms in our future posts of organic reaction mechanisms.

Trautz and Lewis first took a simple reaction between similar molecules to calculate the collision factor Z.

2A(g) ⟶ Product

Collision between A-A is given by equation:

Z_A-A  = ½ √2π  σ²ū N²_A              .......(1)

where π= 3.14,

σ is the distance,

N_Ais the number of molecules per unit volume,

ū is molecular mean velocity = (8k_BT/πm)^1/2

where ,

k_Bis the Boltzmann constant,

T is the temperature in Kelvin,

m is the molecular mass,

On putting the value of ū in equation 1:

Z_A-A  = ½ √2 πσ²(8k_BT/π m)^1/2 N²_A

Z_A-A  = 2 σ² N²_A(π k_B T/m)^1/2                  .......(2)

This quantity is known as collision number Z and its unit is m^-3 s^-1. This equation is restricted to the reactions between same molecules. They modified the equation of collision number Z for the reactions which involve two different molecules like A and B.

For the collision between two different molecules like:

A + B ⟶ Product

Z_A-B  = d ² N_A N_B (8π k_BT)^1/2                 .......(3)

                                    µ

Where d is the distance between the centres of A and B when collision occurs or it is the sum of the radii of A and B.

µ is the reduced mass =  m_A m_B

                                       m_A + m_B

On putting the value of µ in equation 3, we get:

Z_A-B  = d ² N_A N_B (8π k_BT m_A + m_B)^1/2    

                                               m_A m_B

Trautz and Lewis suggested that multiplying collision number with Arrhenius Factor gives the rate of formation of product (ν) in terms of number of molecules formed per unit volume and per unit time.

ν  =  d ² N_A N_B (8π k_BT)^1/2  e^{-Ea / RT}         

                               µ

And to get the rate constant, they divided the above equation by N_A N_B and multiplied it by Avogadro constant L,

k = L d ² (8π k_BT)^1/2  e^{-Ea / RT}      

                     µ

k = Z_AB e^{-Ea / RT}

The pre-exponential factor in this equation is called the collision frequency factor and is denoted by symbols Z_AA (for same molecules) or Z_AB(for different molecules). They applied this theory to the following reaction:

2HI ⟶ H₂ + I₂

And the calculated value of Z was quite closer to the experimental value. This theory gained confidence but when it was applied to other reactions, the difference between experimental and calculated values was quite considerable. By a lucky coincidence,  Lewis and Trautz picked the reaction which gave them accurate results. In order to resolve the discrepancies, they introduced another pre-exponential factor known as the steric factor P, which represents the fraction of the total number of collisions that are effective and from correct orientation.

 k = PZ_ABe^{-Ea / RT}

The introduction of P did improve the collision theory, but it is quite complicated to estimate the value of P as it involves the orientation.

The hard sphere collision theory may give satisfactory explanation to the reactions which involve gaseous reactants and products but it fails to explain the reactions between ions or in solutions. In our next post of chemical kinetics, we will discuss yet another theory to decipher the chemical reactions, the Transition State Theory.

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