Second virial coefficient ($B_2$)

The second virial coefficient is the leading interaction correction to the ideal gas law: at fixed temperature, $B_2(T)={\lim }_{n\to 0}(Z-1)/n$, where $Z=p/(n{k}_{B}T)$ 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,The function $f$ is the Mayer f-function. The factor $4\pi r^{2}dr$ 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. ^[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 $1-e^{-u/(k_{B}T)}\sim u/(k_{B}T)$ 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 $ϵ>0$, and range multiplier $a>1$:Set $T^{\ast }=k_{B}T/ϵ$ 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 resultThe core contributes $b_{\mathrm{HS}}$; the shell contributes $-b_{\mathrm{HS}}(a^3-1)(e^{1/T^{\ast }}-1)$. Heating weakens the relative Boltzmann factor enhancement of attractive configurations, so $B_2$ approaches the positive hard-core value as $T^{\ast }\to \infty$. For the illustrative choice $a=3/2$, the zero occurs atAt this temperature, $Z=1+B_3{n}^2+\cdots$: cancellation removes the linear correction, not every correction. At low $T^{\ast }$, 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.

For a steep soft repulsion, the effective excluded radius decreases with temperature. An inverse-twelfth-power core gives a positive second-virial contribution proportional to $T^{-1/4}$. A hard core instead has a fixed positive excluded-volume limit. These asymptotics explain why different inert-gas model sketches have different high-temperature tails.

In the number-density virial expansion, $p/(n{k}_{B}T)=1+B_{2}n+\cdots$, so $B_2$ has units of volume. With molar concentration instead, the coefficient is $B_{2,\mathrm{mol}}=N_A{B}_2$, where $N_A$ is the Avogadro constant.

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.

For a classical dilute gas with pair potential $u(r)=ϵ[1-(r/a{)}^3]$ for $r<a$ and zero otherwise, put $x=ϵ/(k_{B}T)>0$. Substitution $t=(r/a{)}^3$ in the Mayer f-function formula givesIt is positive for finite positive temperature, tends to $2\pi a^3/3$ as $T\to 0$ in the classical model, and behaves as $\pi a^3ϵ/(3k_{B}T)$ at high temperature. The low-temperature mathematical limit need not remain in the dilute classical regime for a real molecular gas.