Derivation of equation (9-40): Our model for calculating  is equation (9-26), whose denomi...

Derivation of equation (9-40): Our model for calculating  is equation (9-26), whose denominator is the total number of particles N and whose numerator is the total energy of the system, which we here call Utotal. Start with the denominator:

Insert the quantum gas density of states and an expression for the distribution, using ± to distinguish the Bose-Einstein from the Fermi-Dirac. Then change variables: E = y2, and factor  out of the denominator. In the integrand will be a factor

Using , a sum of two integrals results, each of Gaussian form. The integral thus becomes two terms in powers of 1/B. Repeal the process, but instead find an expression for Utotlal in terms of 1/B, using

Divide your expression for Utotlal by that for N, both in terms of 1/B. Now 1/B can safely be eliminated by using the lowest-order expression for N in terms of 1/B.