Question

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 atom is Z_atom is Z_atom = g_1 + g_2 e^-beta Delta. (a) Show that the heat capacity of the atom is given by C = g_1 g_2 Delta^2 e^-beta Delta/k_B T^2 (g_1 + g_2 e^-beta Delta)^2. A monatomic gas of such atoms has a partition function given by Z = X_atom^N Z_t, where Z_t is the partition function due to the translational motion of the gas atoms and is given by Z_t = [V/lambda_th^3]^N/N!. (b) Show that the heat capacity of such a gas is C = N [3/2 k_b + g_1 g_2 Delta^2 e^-beta Delta/k_b T^2(g_1 + g_2 e^-beta Delta)^2].

Answer 1

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