Statistical Mechanics · Problem 8.5

Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions — R. K. Pathria

Physics — Statistical Mechanics · fermi-gas-second-derivatives-thermodynamic-relations

Problem

Evaluate

(\partial^2 P/\partial T^2)_v

(\partial^2\mu/\partial T^2)_v

and

(\partial^2\mu/\partial T^2)_P

of an ideal Fermi gas and check that your results satisfy the thermodynamic relations

C_V = VT\left(\frac{\partial^2 P}{\partial T^2}\right)_v - NT\left(\frac{\partial^2\mu}{\partial T^2}\right)_v

and

C_P = -NT\left(\frac{\partial^2\mu}{\partial T^2}\right)_P.

Examine the low-temperature behavior of these quantities.