含义Meaning
对理想气体方程做两项修正：b 扣除分子自身体积，\(a/V_m^2\) 计入分子间吸引使压强降低。第一个能描述气液转变的状态方程（1873）。
Two corrections to the ideal-gas law: b removes the molecules’ own volume and \(a/V_m^2\) accounts for attraction lowering the pressure. The first equation of state to describe the gas–liquid transition (1873).
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(a\) | 吸引常数attraction parameter | Pa·m⁶·mol⁻² |
| \(b\) | 排斥体积（约为分子体积的 4 倍）excluded volume (≈4 × molecular volume) | m³·mol⁻¹ |
| \(V_m\) | 摩尔体积molar volume | m³·mol⁻¹ |
推导Derivation
体积修正：分子可自由运动的空间是 \(V_m - b\)；硬球模型中 \(b = 4N_A\cdot\tfrac43\pi(d/2)^3\)。
Volume correction: free space is \(V_m - b\); for hard spheres \(b = 4N_A\cdot\tfrac43\pi(d/2)^3\).
压力修正：靠近器壁的分子被内部分子向内拉，撞壁力减小，减少量正比于撞壁分子数和拉它的分子数，即 \(\propto (n/V)^2\)。
Pressure correction: wall-bound molecules are pulled back by the bulk, reducing pressure in proportion to \((n/V)^2\).
临界点处等温线有拐点：\((\partial p/\partial V)_T = 0\) 且 \((\partial^2 p/\partial V^2)_T = 0\)，联立解得 \(T_c, p_c, V_{m,c}\)。
At the critical point the isotherm has an inflection: \((\partial p/\partial V)_T = (\partial^2 p/\partial V^2)_T = 0\), giving \(T_c, p_c, V_{m,c}\).
低于 \(T_c\) 时等温线出现 S 形“范德华环”，用麦克斯韦等面积规则画水平线得到气液共存压强。
Below \(T_c\) isotherms develop a van der Waals loop; Maxwell’s equal-area rule replaces it with the coexistence line.
适用条件与局限Assumptions & limitations
定量精度有限：预测 \(Z_c = 0.375\)，而真实气体约 0.23–0.31。
Only semi-quantitative: predicts \(Z_c = 0.375\) versus 0.23–0.31 for real gases.
a、b 实际随温度变化；工程上用 Redlich–Kwong、Peng–Robinson 等改进方程。
a and b vary with temperature; engineering uses Redlich–Kwong, Peng–Robinson, etc.
S 形环中 \(\partial p/\partial V > 0\) 的部分不稳定，无物理意义。
The part of the loop with \(\partial p/\partial V > 0\) is unstable and unphysical.
例题Worked example
CO₂：a = 0.364 Pa·m⁶/mol²，b = 4.27×10⁻⁵ m³/mol，预测 T_c = 8a/(27Rb) = 304 K（实测 304.1 K）。
CO₂ (a = 0.364, b = 4.27×10⁻⁵): predicted T_c = 304 K (measured 304.1 K).