The partition function for a gas is given by

Q(N, V, T) =  (v - Nb)^N 

The internal energy of the gas is

Concept:

• The Maxwell–Boltzmann distribution concerns the distribution of an amount of energy between identical but distinguishable particles.
• It represents the probability for the distribution of the states in a system having different energies. A special case is the so-called Maxwell distribution law of molecular velocities.
• The final expression for the internal energy U of a system following the Boltzmann distribution of energy is:

U = .......(1)

Explanation:-

• The partition function for a gas is given by

Q(N, V, T) =  (v - Nb)Ne .........(2)

• By taking ln both the sides of equation (2) we got,

or, 

or, 

or, 

or, 

or,

• Now, putting the value of  in equation (1) we get,

U = 

or, U = k_BT² ×  -  kBT2 × 

or, U = 

Conclusion:-

Hence, the internal energy of the gas is