{title2=$B_2$}
{wiki}

The <second virial coefficient> is the leading interaction correction to the <ideal gas law>: at fixed <temperature>, $B_2(T)=\lim_{n\to0}(Z-1)/n$, where $Z=p/(nk_BT)$ and $n$ is <number density>. For identical classical particles in a homogeneous three-dimensional <gas>, with an isotropic <pair potential> $u(r)$ and $r$ the centre-to-centre separation,
$$
B_2(T)=-\frac12\int_{\mathbb R^3} f(|\mathbf r|)\,d^3\mathbf r
=2\pi\int_0^\infty\left[1-e^{-u(r)/(k_BT)}\right]r^2\,dr,
\qquad f(r)=e^{-u(r)/(k_BT)}-1.
$$
The function $f$ is the <Mayer f-function>. The factor $4\pi r^2dr$ counts the <volume> of a thin spherical shell; the factor $1/2$ counts each unordered pair once. The result has units of <volume>. Repulsive regions, $u>0$, contribute positively, while attractive regions, $u<0$, contribute negatively. https://davidtong.org/teaching/statistical-physics/statmechhtml/S2#S2.S5.SS2[Tong, section 2.5.2, equations 2.75–2.76]{ref}

This is a striking bridge from a microscopic <pair potential> to a macroscopic <pressure> reading. At sufficiently small <number density>, $B_2>0$ raises <pressure> relative to an <ideal gas> at the same <temperature> and <number density>, while $B_2<0$ lowers it. A single value of $B_2$, however, cannot determine the entire <pair potential>: many different functions have the same weighted <integral>. Measurements over several <temperatures> supply more information.

The <integral> must exist. For example, a bounded short-range attraction outside a hard core has no difficulty. If instead $u(r)\sim A/r^s$ as $r\to\infty$, then $1-e^{-u/(k_BT)}\sim u/(k_BT)$ and the radial tail behaves like $\int r^{2-s}dr$: it converges only for $s>3$. A bare unscreened <Coulomb potential> therefore falls outside this formula's ordinary short-range setting. This condition on the tail does not by itself establish the stability or convergence of the full <virial expansion>.

For an exactly integrable toy attraction, choose the following square-well <pair potential>, with diameter $\sigma>0$, attraction depth $\epsilon>0$, and range multiplier $a>1$:
$$
u(r)=\begin{cases}
+\infty,&0\le r<\sigma,\\
-\epsilon,&\sigma\le r<a\sigma,\\
0,&r\ge a\sigma.
\end{cases}
$$
Set $T^*=k_BT/\epsilon$ and $b_{\mathrm{HS}}=2\pi\sigma^3/3$, the <second virial coefficient> of a <hard-sphere gas>. Integrating the core and attractive shell separately gives the exact toy result
$$
B_2(T)=b_{\mathrm{HS}}\left[1-(a^3-1)\left(e^{1/T^*}-1\right)\right].
$$
The core contributes $b_{\mathrm{HS}}$; the shell contributes $-b_{\mathrm{HS}}(a^3-1)(e^{1/T^*}-1)$. Heating weakens the relative <Boltzmann factor> enhancement of attractive configurations, so $B_2$ approaches the positive hard-core value as $T^*\to\infty$. For the illustrative choice $a=3/2$, the zero occurs at
$$
T^*_0=\frac{1}{\log\left(1+\frac{1}{a^3-1}\right)}
=\frac{1}{\log(27/19)}\approx2.846.
$$
At this <temperature>, $Z=1+B_3n^2+\cdots$: cancellation removes the linear correction, not every correction. At low $T^*$, the negative shell contribution becomes large, requiring a correspondingly smaller <number density> for a useful truncated <virial expansion>. The square well is a deliberately chosen model, not a fit to the <carbon dioxide> experiment.


 Second virial coefficient (source code)