Since , in the low density limit, the fugacity can be expanded in the form
Writing out the first few terms in the pressure and density equations, we have
Substituting the expansion for into the density equation gives
This equation can now be solved perturbatively, equating like powers of on both sides. For example, working only to first order in , yields:
When this is substituted into the pressure equation, and only first order terms in the density are kept, we find
which is just the classical ideal gas equation. Working, now, to second order in , we have, from the density equation
and the equation of state becomes
From this, we can read off the second virial coefficient
It is particularly interesting to note that there is a nonzero second virial coefficient in spite of the fact that there are no interactions among the particles. The implication is that there is an ``effective'' interaction among the particles as a result of the fermionic spin statistics. Moreover, this effective interaction is such that is tends to increase the pressure above the classical ideal gas result ( ). Thus, the effective interaction is repulsive in nature. This is a consequence of the Pauli exclusion principle: The particle energies must be distributed among the available levels in such a way that no two particles can occupy the same quantum state, thus giving rise to an ``effective'' repulsion between them.
If we look at the third order correction to the pressure, we find that
so that . Thus, one must go out to third order in the density expansion to find a contribution that tends to decrease the pressure.