The ideal fermion gas 417At this point, let us recall the procedure for calculating the equation of state in
the grand canonical ensemble. The free energy in this ensemble is−PV, and
PV
kT= lnZ(ζ,V,T). (11.4.12)Moreover, the average particle number is the thermodynamic derivative with respect
to the fugacityζ:
〈N〉=ζ∂
∂ζlnZ(ζ,V,T). (11.4.13)Next, the fugacityζmust be eliminated in favor of〈N〉by solving forζin terms
of〈N〉and substituting into eqn. (11.4.12). Thus, in order to obtain the equation
of state in the grand canonical ensemble, we must carry out the products in eqn.
(11.4.10) and then apply the above procedure. Although we saw in Section 6.5 that
this is straightforward for the classical ideal gas, the procedurecannot be performed
exactly analytically for the quantum ideal gases. For an ideal gas ofidentical fermions,
the equations we must solve are
PV
kT= lnZ(ζ,V,T) = ln[
∏
n(
1 +ζe−βεn)
]g
=g∑
nln(
1 +ζe−βεn)
〈N〉=ζ∂
∂ζlnZ=g∑
nζe−βεn
1 +ζe−βεn, (11.4.14)
and for bosons, they become
PV
kT= lnZ(ζ,V,T) = ln[
∏
n1
1 −ζe−βεn]g
=−g∑
nln(
1 −ζe−βεn)
〈N〉=ζ∂
∂ζlnZ=g∑
nζe−βεn
1 −ζe−βεn. (11.4.15)
It is not difficult to see that the problem of solving forζin terms of〈N〉is nontrivial
for both particle types. In the next two sections, we will analyze the ideal fermion and
boson gases individually and investigate the limits and approximations that can be
applied to compute their thermodynamic properties.
11.5 The ideal fermion gas
As we did for the ideal Boltzmann gas in Section 11.3, we will consider the thermody-
namic limitL→∞of the ideal fermion gas, so that the spacing between energy levels
becomes small. Then the sums in eqns. (11.4.14) can be replaced by integrals over a
continuous variable denotedn. For the pressure, this replacement leads to