 Solution

A non-ideal gas departs from $p=n{k}_{B}T$ because interactions and finite particle size affect its pressure. At sufficiently low number density $n=N/V$, the virial expansion separates these corrections:The second virial coefficient has dimensions of volume per particle, while $B_3$ has dimensions of volume squared. These are density-series coefficients, not the coefficients of a pressure-series expansion with the same names.

For classical particles interacting through a pair potential $u(r)$,Repulsion contributes positively and attraction negatively. A hard core of diameter $d$ gives $B_2=2\pi d^3/3$, four times the geometric volume of one sphere. The factor reflects the volume excluded to the centre of a second particle together with pair counting; it is not simply the occupied geometric volume.

Expanding the Van der Waals equation at $nb\ll 1$ givesThe virial coefficients of the van der Waals gas are therefore $B_2=b-a/(k_{B}T)$ and $B_3=b^2$. The Boyle temperature, $T_B=a/(k_{B}b)$, makes the leading correction vanish, but does not remove the higher coefficients or turn the gas exactly ideal. More realistic potentials give temperature-dependent higher coefficients that need not equal the van der Waals values.

A useful measurement is a sequence of dilute isothermal pressure–volume observations. Plotting $[p/(n{k}_{B}T)-1]/n$ against $n$ exposes $B_2$ as the zero-density intercept and $B_3$ as its initial slope. Control temperature and density calibration, check equilibration and adsorption on the container walls, and vary the fitted density interval to test truncation error. One measured virial coefficient cannot uniquely reconstruct the pair potential. Near coexistence or a critical point a short density series may fail, and at quantum-degenerate densities exchange statistics add corrections even without interactions.