Module 6 :  Reaction Kinetics and Dynamics
Lecture 29 : Temperature Dependence of Reaction Rates
  29.2

The Maxwell Boltzmann Distribution of Molecular Speeds

We have already commented that molecules have to collide with sufficient speeds so that the reaction occurs. Where does this sufficient energy come from? At a given temperature, not all molecules have the same speeds. During intermolecular collisions, these speeds change too. If we represent a molecular velocity by , it has three independent components x, y and z in the three directions x, y and z. Let us consider monoatomic gas of molecular mass m. The probability F ( x, y, z) that a given molecule will have velocity components lying between x and x+ dx, y and y + dy and z and z + dz is written as

 

F (x, y, z ) d z d y d z = f (x ) f (y) f (z) dx dy dz

 (29.16)
F is written as a product of three fs because x, y and z are independent and since nature does not distinguish between x, y and z (unless directional fields like gravitational or electromagnetic are present), the form of f is the same in the three directions. Again, since there is no distinction between positive and negative x, f depends on | x| or x2. We can rephrase (29.16) as


F(x2 , y2 , z 2 ) = f (x2 ) f (y2 ) f (z2 ) (29.17)
The only function that satisfies an equation like eq (29.17) is an exponential function since e x * x + y * y + z * z = e x *x e y * y e z * z and so we conclude that f (x2 ) may be written as
 
f (x2 ) = C e C e (29.18)
We take only the negative exponent (C and d are positive) because a positive exponent implies that very large velocities have very high probabilities. This scenario is highly unlikely. To evaluate C, We invoke the physical argument that the velocity has to lie somewhere between - to + and that the total probability is one, i.e.,
 
(29.19)
The above integral is a standard integral = 1/ 2 ( / a )1/2. Thus eq (29.19) becomes
 
dx = C ( / b )1/2. (29.20)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1