Again,

(5.159)

Entropy is given by

(5.160)

The Helmholtz function is given by

(5.161)

Once the partition function Z has been evaluated, all the thermodynamic properties can thus be estimated.

Monatomic Ideal Gas

The atoms of a monatomic ideal gas possess only translational kinetic energy. For an atom in cell i,

(5.162)

For all the cells in the space occupied by the gas, the partition function is:

(5.163)

where the degeneracy g_i is given by

(5.163)         (5.164)

Let I =

Assuming

The integrand I is

(5.166)   (5.167)

Substituting in the equation,

(5.168)

Substituting N_i by d⁶N,

(5.169)

Integration of this equation over all values of v_x, v_y and v_z gives

(5.170)

The number of atoms per unit volume of ordinary space is thus a constant, confirming that the atoms are uniformly distributed in the gas volume.
The Eq. (5.169), when integrated over all the values of x, y and z, gives the distribution of atoms in velocity space, as given below:

(5.171)

This equation is precisely the same as Maxwell-Boltzman velocity distribution as derived from the kinetic theory of gases and given by Eq. (21.49), provided K is recognized as the Boltzman constant.

Now,

(5.172)   (5.173)

The internal energy of the gas from Eq. (5.159),

Thus,

(5.174)

The results agree with the kinetic theory and the equipartition principle.
Considering the Helmholtz function

(5.175)