含义Meaning
阻止溶剂通过半透膜进入溶液所需施加的额外压强。形式与理想气体方程相同（1887）。
The extra pressure needed to stop solvent flowing through a semipermeable membrane into the solution — formally like the ideal-gas law (1887).
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\Pi\) | 渗透压osmotic pressure | Pa |
| \(c_\mathrm{B}\) | 溶质物质的量浓度molar concentration of solute | mol·m⁻³ |
| \(A_2\) | 第二维里系数second virial coefficient | mol·m³·kg⁻² |
| \(M_n\) | 数均摩尔质量number-average molar mass | kg·mol⁻¹ |
推导Derivation
平衡时膜两侧溶剂化学势相等：\(\mu_\mathrm{A}^*(p) = \mu_\mathrm{A}(x_\mathrm{A}, p + \Pi)\)。
At equilibrium the solvent chemical potential is equal on both sides: \(\mu_\mathrm{A}^*(p) = \mu_\mathrm{A}(x_\mathrm{A}, p + \Pi)\).
右边 \(= \mu_\mathrm{A}^*(p) + RT\ln x_\mathrm{A} + \int_p^{p+\Pi} V_{m,\mathrm{A}}\,dp\)，得 \(\Pi V_{m,\mathrm{A}} = -RT\ln x_\mathrm{A}\)。
The right side equals \(\mu_\mathrm{A}^*(p) + RT\ln x_\mathrm{A} + \int V_{m,\mathrm{A}}\,dp\), giving \(\Pi V_{m,\mathrm{A}} = -RT\ln x_\mathrm{A}\).
稀溶液 \(-\ln x_\mathrm{A} \approx x_\mathrm{B} \approx n_\mathrm{B}/n_\mathrm{A}\)，且 \(n_\mathrm{A}V_{m,\mathrm{A}} \approx V\)，故 \(\Pi = n_\mathrm{B}RT/V = cRT\)。
Dilute: \(-\ln x_\mathrm{A} \approx n_\mathrm{B}/n_\mathrm{A}\) and \(n_\mathrm{A}V_{m,\mathrm{A}} \approx V\), so \(\Pi = cRT\).
适用条件与局限Assumptions & limitations
稀溶液、理想半透膜（只透过溶剂）。
Dilute solution and an ideal semipermeable membrane.
高分子溶液偏差大，需用维里形式外推 \(c \to 0\) 求 \(M_n\)。
Polymer solutions need the virial form extrapolated to \(c \to 0\) to obtain \(M_n\).
例题Worked example
人体血浆约 0.30 mol/L 溶质粒子，37 °C 时 Π = 0.30×1000×8.314×310 ≈ 7.7×10⁵ Pa（7.6 atm）；生理盐水 0.9% NaCl 与之等渗。
Blood plasma (~0.30 mol/L particles) at 37 °C: Π ≈ 7.7×10⁵ Pa (7.6 atm); 0.9 % saline is isotonic.