含义Meaning
溶有非挥发性溶质的溶液沸点高于纯溶剂，升高值与溶质粒子的质量摩尔浓度成正比。
A solution of a non-volatile solute boils higher than the pure solvent, by an amount proportional to solute-particle molality.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(K_b\) | 沸点升高常数（水 0.512）ebullioscopic constant (water 0.512) | K·kg·mol⁻¹ |
| \(i\) | 范特霍夫因子van ’t Hoff factor | 1 |
| \(T_b^*\) | 纯溶剂沸点boiling point of pure solvent | K |
| \(M_\mathrm{A}\) | 溶剂摩尔质量molar mass of solvent | kg·mol⁻¹ |
推导Derivation
沸腾时 \(\mu_\mathrm{A}(g) = \mu_\mathrm{A}(l) = \mu_\mathrm{A}^*(l) + RT\ln x_\mathrm{A}\)，即 \(\ln x_\mathrm{A} = \Delta_\mathrm{vap}G_m/RT\)。
At boiling \(\mu_\mathrm{A}(g) = \mu_\mathrm{A}^*(l) + RT\ln x_\mathrm{A}\), so \(\ln x_\mathrm{A} = \Delta_\mathrm{vap}G_m/RT\).
对 T 求导并用吉布斯–亥姆霍兹方程：\(d\ln x_\mathrm{A}/dT = -\Delta_\mathrm{vap}H_m/RT^2\)。
Differentiate with Gibbs–Helmholtz: \(d\ln x_\mathrm{A}/dT = -\Delta_\mathrm{vap}H_m/RT^2\).
从 \(x_\mathrm{A}=1\)（\(T_b^*\)）积分到 \(x_\mathrm{A}\)（\(T_b\)），稀溶液 \(\ln x_\mathrm{A} \approx -x_\mathrm{B}\)、\(T_bT_b^* \approx T_b^{*2}\)：
Integrate from \(x_\mathrm{A}=1\) at \(T_b^*\); for dilute solutions \(\ln x_\mathrm{A} \approx -x_\mathrm{B}\) and \(T_bT_b^* \approx T_b^{*2}\):
\[x_\mathrm{B} = \frac{\Delta_\mathrm{vap}H_m}{R}\cdot\frac{\Delta T_b}{T_b^{*2}}\]再用 \(x_\mathrm{B} \approx b_\mathrm{B}M_\mathrm{A}\) 得 \(\Delta T_b = K_b b_\mathrm{B}\)。电解质乘以 i。
With \(x_\mathrm{B} \approx b_\mathrm{B}M_\mathrm{A}\): \(\Delta T_b = K_b b_\mathrm{B}\); multiply by i for electrolytes.
适用条件与局限Assumptions & limitations
稀溶液（一般 b < 0.1 mol/kg 较准）、溶质不挥发、不进入气相。
Dilute (best below ~0.1 mol/kg), non-volatile solute.
电解质的实际 i 小于离子数（离子缔合），浓度越高越明显。
Real i for electrolytes is below the ion count (ion pairing), increasingly so at higher concentration.
例题Worked example
0.10 mol/kg 葡萄糖水溶液沸点升高 0.051 K；同浓度 NaCl 约 0.10 K（i ≈ 1.9）。
0.10 mol/kg glucose raises water’s boiling point by 0.051 K; NaCl at the same molality by ≈ 0.10 K (i ≈ 1.9).