The Cluster Expansion and Virial Coefficients

Every result in the preceding modules assumed the gas particles ignore one another. The ideal-gas law, the Sackur–Tetrode entropy, and the quantum distributions all followed from a Hamiltonian with no interaction term. Real molecules attract at long range and repel at short range, and the equation of state acquires corrections that grow with density. The systematic way to organize those corrections is the virial expansion, a power series in the number density $n=N/V$,

whose coefficients $B_{\ell }(T)$ are determined by the interactions among $\ell$ molecules at a time. This lesson derives the expansion from the configuration integral, identifies $B_2(T)$ with a single integral over the pair potential, and evaluates it for the hard-sphere, square-well, and Lennard-Jones models.

The configuration integral

For $N$ identical classical particles with pairwise interactions, the Hamiltonian separates into kinetic and potential parts,

with $r_{ij}=|{\vec{r}}_i-{\vec{r}}_j|$ the separation of particles $i$ and $j$ and $u(r)$ the pair potential. The canonical partition function factors because the momentum integrals are unaffected by $U$,

where $\lambda$ is the thermal de Broglie wavelength from the momentum Gaussians and $Q_N$ is the configuration integral. All deviation from ideality lives in $Q_N$: for a non-interacting gas $U=0$ gives $Q_N=V^N$ and the ideal partition function $Z_N=V^N/N!{\lambda }^{3N}$ returns. The free energy is

so the interaction correction is the logarithm of $Q_N/V^N$, a dimensionless ratio that equals $1$ for the ideal gas. Computing it exactly is impossible for any realistic $u(r)$; the cluster expansion computes it order by order in density.

The Mayer function

The integrand $e^{-\beta U}={\prod }_{i<j}{e}^{-\beta u(r_{ij})}$ is a product of factors, each close to $1$ wherever the corresponding pair is far apart, since $u(r)\to 0$ at large separation. Subtracting that background defines the Mayer function

which vanishes wherever the pair does not interact and is appreciable only when the two particles are within range of one another. In terms of $f$ the Boltzmann weight is an exact product,

Expanding the product generates a sum over all sets of pairs: the term $1$, the single-bond terms ${\sum }_{i<j}{f}_{ij}$, the two-bond terms $\sum f_{ij}{f}_{k\ell }$, and so on. Because each $f$ is short-ranged, a term with several bonds contributes only when the bonded particles are simultaneously close, an event whose probability falls with density. Ordering the expansion by the number of bonds is therefore an ordering in powers of $n$.

The Mayer function encodes the shape of the potential. Inside the repulsive core $u\to +\infty$ forces $f\to -1$; in the attractive well $u<0$ makes $e^{-\beta u}>1$ so $f>0$; far outside, $f\to 0$. Its sign changes where $u$ changes sign, and its magnitude is set by the competition between $|u|$ and $k_{B}T$.

The second virial coefficient

Keep only the single-bond terms, which dominate at low density. Their contribution to the configuration integral is

Each pair term integrates the $N-2$ spectator coordinates freely, giving $V^{N-2}$, and the two bonded coordinates over $f_{ij}$. Shifting to the relative coordinate $\vec{r}={\vec{r}}_i-{\vec{r}}_j$ leaves one free center-of-mass integration worth $V$,

There are $\left(\genfrac{}{}{0ex}{}{N}{2}\right)=\frac{1}{2}N(N-1)$ pairs, so

Taking the logarithm with $\ln (1+x)\approx x$ and forming the pressure $P=-\partial F/\partial V$ from $F=F_{\text{ideal}}-k_{B}T\ln (Q_N/V^N)$ gives

which identifies the second virial coefficient

the angular integration having produced the $4\pi r^2$ shell factor. The sign of $B_2$ follows the sign of $f$: a repulsive potential makes $f<0$ and $B_2>0$ (pressure above ideal), while an attractive potential makes $f>0$ and $B_2<0$ (pressure below ideal). The connection to the virial theorem of the previous module is direct — both express the leading interaction correction to $PV=N{k}_{B}T$ through an integral over the intermolecular force.

The cluster expansion and higher coefficients

The single-bond truncation is the first term of a systematic hierarchy. Grouping the terms of ${\prod }_{i<j}(1+f_{ij})$ by connected sets of bonds, the logarithm $\ln Q_N$ becomes a sum of cluster integrals: a diagram is a set of particles (points) joined by $f$-bonds, and each connected diagram contributes an integral over the positions of its points. The systematic bookkeeping is cleanest in the grand canonical ensemble, where $\ln \Xi$ equals the sum of all connected clusters and the pressure follows from $PV=k_{B}T\ln \Xi$.¹

Not every diagram contributes to a virial coefficient. The virial coefficients are given by the irreducible clusters alone — the diagrams that remain connected after any single point is removed. The three-particle chain $f_{12}{f}_{23}$ is reducible (deleting point $2$ disconnects it) and its contribution is cancelled by disconnected pieces, so the third virial coefficient comes only from the triangle,

with particle $1$ fixed at the origin. Each successive $B_{\ell }$ is an integral over the irreducible $\ell$-point clusters, and these coefficients assemble the density expansion of the pressure — the virial series. In practice $B_2$ carries most of the low-density physics and is the only coefficient with a simple closed form for realistic potentials.

Hard spheres and the excluded volume

The hard-sphere potential is the sharpest model of a repulsive core:

The integral is elementary,

independent of temperature. This is four times the volume of a single sphere: with molecular volume $v_0=\frac{4}{3}\pi (\sigma /2{)}^3=\frac{\pi }{6}{\sigma }^3$, one has $B_2=4v_0$. The factor of four is geometric — two impenetrable spheres of diameter $\sigma$ cannot approach closer than $\sigma$ center to center, so each pair excludes a sphere of radius $\sigma$, whose volume is eight times a molecule's and is shared between the two particles.

The square well and the Boyle temperature

A hard core alone gives a temperature-independent $B_2$; attraction introduces temperature dependence. The square-well potential adds a finite attractive shell to the hard core,

with well depth $\epsilon$ and range $R\sigma$. The Mayer function is $-1$ inside the core, $e^{\beta \epsilon }-1>0$ in the well, and $0$ outside, so

At high temperature $\beta \epsilon \to 0$ the bracket approaches $1$ and $B_2$ tends to the hard-sphere value $\frac{2\pi }{3}{\sigma }^3>0$: the fast molecules barely feel the shallow well and the repulsive core dominates. At low temperature the exponential grows, the bracket turns negative, and $B_2<0$: the attraction dominates and lowers the pressure below ideal. Between the two limits $B_2$ passes through zero at the Boyle temperature $T_B$, defined by $B_2(T_B)=0$.

The Lennard-Jones gas

Real molecules interact through a smooth potential, and the Lennard-Jones form combines a $r^{-12}$ repulsion with a $r^{-6}$ dispersion attraction,

with $u(\sigma )=0$, a minimum of depth $\epsilon$ at $r_{\min }=2^{1/6}\sigma$, and the $-r^{-6}$ tail fixed by the van der Waals dispersion force between induced dipoles. Its second virial coefficient has no elementary closed form, but the qualitative behavior matches the square well: positive and tending to a constant at high temperature, negative at low temperature, crossing zero at a Boyle temperature $k_B{T}_B\approx 3.42\epsilon$.² Written in reduced units $T^{\ast }=k_{B}T/\epsilon$ and $B_2^{\ast }=B_2/(\frac{2}{3}\pi {\sigma }^3)$, the curve is the same for every substance obeying a Lennard-Jones law, and measured $B_2(T)$ data for the noble gases collapse onto it once $\sigma$ and $\epsilon$ are fitted — an early instance of the corresponding-states idea developed in the next lesson.

The two competing pieces of $B_2$ carry distinct physical content. The positive, repulsive piece is the excluded volume $b\sim {\sigma }^3$, weakly dependent on temperature and set by molecular size. The negative, attractive piece scales as $-\epsilon /k_{B}T$ times a volume, growing as the gas cools. Isolating these two terms produces the van der Waals equation, whose constants $a$ and $b$ are the attractive and repulsive contributions to $B_2$.

Summary

• Interactions live entirely in the configuration integral $Q_N=\int {\prod }_i{\mathbf{d}}^3{r}_i{e}^{-\beta U}$; the free energy correction is $-k_{B}T\ln (Q_N/V^N)$, which equals $1$ inside the log for an ideal gas.
• The Mayer function $f_{ij}=e^{-\beta u_{ij}}-1$ is short-ranged and turns the Boltzmann weight into ${\prod }_{i<j}(1+f_{ij})$; expanding by number of bonds is an expansion in density.
• The virial expansion $PV/N{k}_{B}T=1+B_{2}n+B_3{n}^2+\cdots$ has coefficients fixed by irreducible clusters: $B_2=-\frac{1}{2}\int f{\mathbf{d}}^{3}r$ and $B_3=-\frac{1}{3}\int f_{12}{f}_{13}{f}_{23}{\mathbf{d}}^3{r}_2{\mathbf{d}}^3{r}_3$.
• Hard spheres give $B_2=\frac{2\pi }{3}{\sigma }^3=4v_0$, the temperature-independent excluded volume. The square well adds an attractive term $-(R^3-1)(e^{\beta \epsilon }-1)$ that makes $B_2$ negative at low $T$.
• $B_2(T)$ crosses zero at the Boyle temperature, where repulsion and attraction cancel and the gas is ideal to first order in density; for Lennard-Jones, $k_B{T}_B\approx 3.42\epsilon$.