 Solution

ID: past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2014/paper-1/1/c/solution

For

N

identical classical noninteracting particles, use

Z_{N} = Z_{1}^{N} / N!

. At fixed

N

ln Z_{N} = N ln V + \frac{3 N}{2} ln T + constant,

p = k_{B} T (\frac{\partial l n Z _{N}}{\partial V})_{T} = \frac{N k _{B} T}{V}, U = k_{B} T^{2} (\frac{\partial l n Z _{N}}{\partial T})_{V} = \frac{3}{2} N k_{B} T .

Thus, for one mole,

p V = RT, C_{V, m} = \frac{3}{2} R .

The supplied

T^{3/2}

dependence includes only three translational degrees of freedom; additional thermally active internal modes would change the heat capacity.

 New to topics? Read the docs here!