= Virial expansion
{wiki}
The <virial expansion> describes the first departures from the <ideal gas law> as a <power series> in <number density>. For a homogeneous, single-species <gas> in three dimensions at <thermal equilibrium>, write $n=N/V$, with $V$ the <volume (thermodynamics)>, and define the <dimensionless> compressibility factor $Z$ by
$$
Z=\frac{p}{nk_BT}=1+B_2(T)n+B_3(T)n^2+\cdots,
\qquad p=nk_BT\bigl[1+B_2(T)n+B_3(T)n^2+\cdots\bigr].
$$
Here $p$ is absolute <pressure>, $T$ is absolute <thermodynamic temperature>, and $k_B$ is the <Boltzmann constant>. The <second virial coefficient> determines the initial slope away from $Z=1$; higher coefficients describe further corrections. The symbol $Z$ here is an <equation of state> ratio, not a <canonical partition function>. https://davidtong.org/teaching/statistical-physics/statmechhtml/S2#S2.S5[Tong, Statistical Physics, section 2.5, equation 2.69]{ref}
\Image[/second-virial-pvt-and-slope.png]
{title=Controlled gas expansion and extraction of a dilute-gas slope: schematic apparatus and explicitly chosen toy curves}
The left panel sketches the measurement principle; the right panel uses deliberately chosen <polynomials>, $Z=1+bx+6x^2$ with $x=n\ell^3$ and $b=\pm2$, where $\ell$ is a reference length. These curves are not measured data. Their initial slopes have opposite signs, but both bend because of the quadratic term. Using a finite secant from $x=0$ to $x=h$ gives $(Z(h)-1)/h=b+6h$, rather than $b$: reducing the fitting range reduces this particular truncation bias. This algebra explains why measuring one nonideal <pressure> is insufficient to extract a reliable <second virial coefficient>.
Waxman and Hastings's 1971 Burnett experiment used sample and expansion chambers in a thermostatic bath, with a valve between them and a <pressure> transducer linked to a piston gauge. They measured successive <isothermal expansions>: evacuate the expansion chamber, open the valve, record the equilibrated <pressure>, isolate and evacuate that chamber again, then repeat. This avoids separately measuring the absolute gas amount and chamber <volumes>. For <carbon dioxide> at $50\,{}^\circ\mathrm C$, their measurements below $35$ bar gave a molar <second virial coefficient> of approximately $-102.2\,\mathrm{cm^3\,mol^{-1}}$. https://nvlpubs.nist.gov/nistpubs/jres/75C/jresv75Cn3-4p165_A1b.pdf[Waxman and Hastings, Journal of Research of the NBS 75C (1971), pages 165–176, abstract and apparatus]{ref}
An idealized reduction makes the signal explicit. Let $q=(V_A+V_B)/V_A$ be the calibrated chamber <volume> ratio, $p_i$ the pre-expansion <pressure>, and $p_f$ the post-expansion <pressure>. Particle conservation during one valve opening gives $n_f=n_i/q$. At the same <temperature>, division of the two <equations of state> yields
$$
\frac{p_i}{p_f}=q\frac{Z(n_i,T)}{Z(n_i/q,T)}
=q+(q-1)\frac{B_2(T)}{k_BT}p_i+O(p_i^2).
$$
Thus the intercept determines $q$, while the dilute-limit slope determines the <second virial coefficient>. In the actual experiment, <helium> runs calibrated the cell constant and chamber deformation corrections were included. Agreement of the <carbon dioxide> low-<pressure> intercept with the <helium> calibration, and the absence of an anomalous low-<pressure> upturn, checked for gas adsorption on chamber walls. https://nvlpubs.nist.gov/nistpubs/jres/75C/jresv75Cn3-4p165_A1b.pdf[Waxman and Hastings (1971), data reduction and adsorption checks, especially equation 13 and page 176]{ref}
For a limited dilute-gas fit, hold <temperature> fixed, allow <thermal equilibrium> after every expansion, calibrate absolute <pressure> and the chamber ratio, and check leaks and sample composition. Repeat the analysis with a smaller upper <pressure> cutoff and an added curvature term; compare the resulting <second virial coefficients> with their uncertainties. Too wide a range mixes in higher coefficients; too narrow a range makes the nonideal signal hard to distinguish from instrument error. The displayed chamber equation neglects deformation, dead spaces and adsorption, so it is a teaching reduction rather than the complete experimental analysis.
Units matter. With <number density> in $\mathrm{m^{-3}}$, $B_2$ has units $\mathrm{m^3}$ and $B_3$ has units $\mathrm{m^6}$. For molar density $c=n/N_{\mathrm A}$ in $\mathrm{mol\,m^{-3}}$, the same <equation of state> is
$$
Z=\frac{p}{cRT}=1+B_{2,\mathrm m}c+B_{3,\mathrm m}c^2+\cdots,
\qquad B_{2,\mathrm m}=N_{\mathrm A}B_2,\qquad
B_{3,\mathrm m}=N_{\mathrm A}^2B_3.
$$
Here $N_{\mathrm A}$ is the <Avogadro constant> and $R=N_{\mathrm A}k_B$ is the <molar gas constant>; the molar coefficients have units $\mathrm{m^3\,mol^{-1}}$ and $\mathrm{m^6\,mol^{-2}}$. A coefficient tabulated per <mole> must therefore multiply molar density, not <number density>.
For the classical formulas below, use a <nondegenerate gas> with <thermal de Broglie wavelength> $\lambda_{\mathrm{th}}$ satisfying $n\lambda_{\mathrm{th}}^3\ll1$, negligible quantum corrections to relative motion, and a stable, sufficiently short-ranged <pair potential>. Diluteness must also control the omitted interaction terms: $|B_2n|\ll1$ alone is not enough when positive and negative contributions cancel. A finite truncation is a local low-<number density> approximation, not an <equation of state> for a dense liquid or across a <phase transition>. https://davidtong.org/teaching/statistical-physics/statmechhtml/S2#S2.S2.SS4[Tong, section 2.2.4: classical-density condition]{ref} https://davidtong.org/teaching/statistical-physics/statmechhtml/S2#S2.S5.SS1[Tong, section 2.5.1: dilute interacting gas]{ref}
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