\[Z = \frac{pV_m}{RT} = 1 + \frac{B(T)}{V_m} + \frac{C(T)}{V_m^2} + \cdots\]
第二维里系数与分子间势能的关系second virial coefficient from the pair potential
\[B(T) = -2\pi N_A\int_0^\infty \left(e^{-u(r)/k_\mathrm{B}T} - 1\right) r^2\,dr\]含义Meaning
把 Z 展开为 \(1/V_m\) 的幂级数。系数可由统计力学严格地与分子间势能联系起来，是连接实验与分子间作用的桥梁。
A power series of Z in \(1/V_m\). Statistical mechanics links each coefficient rigorously to intermolecular potentials.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(B, C\) | 第二、第三维里系数second and third virial coefficients | m³·mol⁻¹, m⁶·mol⁻² |
| \(u(r)\) | 分子对势能pair potential | J |
推导Derivation
低密度时 Z → 1；把偏差按密度展开（迈耶集团展开）。
Z → 1 at low density; expand the deviation in powers of density (Mayer cluster expansion).
B 只与两分子相互作用有关，C 与三分子作用有关。
B reflects pair interactions, C triplets.
范德华气体：\(Z = \frac{1}{1-b/V_m} - \frac{a}{RTV_m} \approx 1 + \frac{b - a/RT}{V_m}\)，即 \(B = b - a/RT\)。
For van der Waals: \(B = b - a/RT\).
\[Z \approx 1 + \left(b - \frac{a}{RT}\right)\frac{1}{V_m} + \frac{b^2}{V_m^2} + \cdots\]
适用条件与局限Assumptions & limitations
级数只在中低密度收敛；接近液体密度时失效。
The series converges only at low-to-moderate density.
高阶系数难以测定，一般截断到 B 或 C。
Higher coefficients are hard to measure; usually truncated after B or C.