Exchange contribution to the second virial coefficient ($B_2^{\mathrm{ex}}=-\eta {\lambda }^3/2^{5/2}$)

For a single internal species of noninteracting particles with quadratic dispersion in three dimensions, expand the grand canonical partition function at small quantum gas fugacity. With $\eta =+1$ for bosons and $\eta =-1$ for fermions, $n{\lambda }^3=z+\eta z^2/2^{3/2}+O(z^3)$ and $\beta p{\lambda }^3=z+\eta z^2/2^{5/2}+O(z^3)$. Eliminating $z$ gives $p=n{k}_{B}T[1-\eta n{\lambda }^3/2^{5/2}+\cdots ]$. Bosonic bunching lowers the pressure at fixed density; fermionic exclusion raises it. This correction needs no interaction potential. For $g$ equally populated internal species, the coefficient in terms of total number density is reduced by $g$. In the dilute one-species limit the equal-time pair detection correlation is $g^{(2)}(r)=1+\eta e^{-2\pi r^2/{\lambda }^2}$. The exchange contribution to the two-particle cluster integral is therefore $-\frac{1}{2}\int [g^{(2)}(r)-1]d^{3}r=-\eta {\lambda }^3/2^{5/2}$, providing a direct pair-counting check of the coefficient.