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Virial coefficients

Coefficients in the low-density expansion of the equation of state; encode interactions via integrals (Mayer -function) and connected-graph expansions.
Virial coefficients

Virial coefficients Bn(T)B_n(T) describe interaction corrections to the ideal-gas law in the low-density regime.

They are most naturally derived from the activity/cluster expansion (see ) and appear as coefficients in the density expansion of the pressure.

Virial expansion of the equation of state

Let ρ=N/V\rho=N/V be the number density and let β=1/(kBT)\beta = 1/(k_B T), with temperature TT as in .

The virial expansion is

βp  =  ρ  +  B2(T)ρ2  +  B3(T)ρ3  +  ⋯ , \beta p \;=\; \rho \;+\; B_2(T)\rho^2 \;+\; B_3(T)\rho^3 \;+\; \cdots,

where pp is the pressure (see ).

Equivalently, the compressibility factor satisfies

Z(ρ,T)  =  βpρ  =  1+B2(T)ρ+B3(T)ρ2+⋯ . Z(\rho,T) \;=\; \frac{\beta p}{\rho} \;=\; 1 + B_2(T)\rho + B_3(T)\rho^2 + \cdots.

Interpretation:

  • B2(T)B_2(T) captures the leading “two-body” correction (excluded volume and/or attraction).
  • Higher Bn(T)B_n(T) encode genuine nn-body correlation effects created by interactions.

Second virial coefficient from the Mayer ff-function

For a classical fluid with pair potential u(r)u(r) (translation-invariant), define the Mayer ff-function

f(r)  =  e−βu(r)−1. f(r) \;=\; e^{-\beta u(r)} - 1.

In dd dimensions, the second virial coefficient is

B2(T)  =  −12∫Rdf(r) ddr. B_2(T) \;=\; -\frac{1}{2}\int_{\mathbb{R}^d} f(r)\, d^dr.

Example: hard spheres (3D)

For hard spheres of diameter σ\sigma in d=3d=3,

  • u(r)=∞u(r)=\infty for r<σr<\sigma and u(r)=0u(r)=0 for r≥σr\ge \sigma,
  • hence f(r)=−1f(r)=-1 for r<σr<\sigma and f(r)=0f(r)=0 otherwise,

so

B2  =  −12∫∣r∣<σ(−1) d3r  =  12⋅4π3σ3  =  2π3σ3. B_2 \;=\; -\frac{1}{2}\int_{|r|<\sigma} (-1)\, d^3r \;=\; \frac{1}{2}\cdot \frac{4\pi}{3}\sigma^3 \;=\; \frac{2\pi}{3}\sigma^3.

Higher virial coefficients and Mayer graphs

For n≥3n\ge 3, Bn(T)B_n(T) can be written as integrals over connected graphs built from ff-bonds, most systematically via the cluster integrals in .

Concretely, one typically computes:

  1. the connected-graph coefficients bn(T)b_n(T) in the activity expansion of log⁡Ξ\log \Xi, then
  2. eliminates the activity to rewrite βp\beta p as a series in ρ\rho.

In the common convention where the ideal term is normalized so that b1=1b_1=1, the first relations are

B2=−b2,B3=4b22−2b3. B_2 = -b_2, \qquad B_3 = 4b_2^2 - 2b_3.

(Conventions vary by where thermal-wavelength factors are placed; the structure “virial coefficients are polynomials in cluster integrals” is robust.)

Convergence and regime of validity

The virial series is an asymptotic/analytic expansion in ρ\rho whose convergence depends on temperature and the interaction potential. Rigorous convergence criteria are typically proved using cluster expansion methods (see and ).

Prerequisites