Question

Derive Boltzmann’s formula for the probability of atoms in thermal equilibrium occupying a state E at absolute temperature T. 

Answer 1

Consider a collection of N molecules of a large number of energy states, E₁, E₂, E₃ etc such that there are N₁ molecules in state E₁, N₂ in E₂ and so on. The nature of energy is immaterial. The number of ways in which N molecules can be accommodated in various states is given by

The underlying idea is that the state of the system would be state if W is a maximum. 

Taking logs on both sides and applying Stirling’s approximation ln W =

If the system is in a state of maximum thermodynamic probability, the variation of W with respect to change in N_i is zero, that is

We now use the Lagrange method of undetermined multipliers. Multiplying (5) by α and (6) by β and adding to (7), we get