Blundell Thermal Physics · Problem 21.4

Statistical mechanics of an ideal gas

Physics — Statistical Mechanics · two-level-atom-partition-function-heat-capacity

Problem

An atom in a solid has two energy levels: a ground state of degeneracy

g_1

and an excited state of degeneracy

g_2

at an energy

\Delta

above the ground state. Show that the partition function

Z_{\rm atom}

Z_{\rm atom} = g_1 + g_2 e^{-\beta\Delta}. \tag{21.46}

Show that the heat capacity of the atom is given by

C = \frac{g_1 g_2 \Delta^2 e^{-\beta\Delta}}{k_{\rm B}T^2(g_1 + g_2 e^{-\beta\Delta})^2}. \tag{21.47}

A monatomic gas of such atoms has a partition function given by

Z = Z_{\rm atom} Z_N, \tag{21.48}

where

Z_N

is the partition function due to the translational motion of the gas atoms and is given by

Z_N = (1/N!)[V/\lambda_{\rm th}^3]^N

. Show that the heat capacity of such a gas is

C = N\left[\frac{3}{2}k_{\rm B} + \frac{g_1 g_2 \Delta^2 e^{-\beta\Delta}}{k_{\rm B}T^2(g_1 + g_2 e^{-\beta\Delta})^2}\right]. \tag{21.49}