Blundell Thermal Physics · Problem 21.3

Statistical mechanics of an ideal gas

Physics — Statistical Mechanics · number-of-states-below-energy

Problem

Show that the number of states in a gas with energies below

E_{\max}

\int_0^{\sqrt{2mE_{\max}/\hbar^2}} g(k)\,\mathrm{d}k = \frac{V}{6\pi^2}\left(\frac{2mE_{\max}}{\hbar^2}\right)^{3/2}. \tag{21.45}

Putting

E_{\max} = \frac{3}{2}k_{\rm B}T

, show that the number of states is

\Xi V n_{\rm Q}

where

\Xi

is a numerical constant of order unity.