凝固点降低 Freezing-point depression — 公式推导、符号与局限

\[\Delta T_f = i\,K_f\, b_\mathrm{B}, \qquad K_f = \frac{R\,T_f^{*2} M_\mathrm{A}}{\Delta_\mathrm{fus}H_m}\]

凝固点法测摩尔质量molar mass by cryoscopy

\[M_\mathrm{B} = \frac{K_f\, m_\mathrm{B}}{\Delta T_f\, m_\mathrm{A}}\]

含义Meaning

溶液的凝固点低于纯溶剂。常用于测定摩尔质量、制防冻液和融雪剂。

A solution freezes below the pure solvent — used for molar-mass determination, antifreeze and road salt.

符号Symbol	含义 Meaning	单位 Unit
\(K_f\)	凝固点降低常数（水 1.86，苯 5.12，樟脑 ≈ 40）cryoscopic constant (water 1.86, benzene 5.12, camphor ≈ 40)	K·kg·mol⁻¹
\(\Delta_\mathrm{fus}H_m\)	溶剂的摩尔熔化焓molar enthalpy of fusion of solvent	J·mol⁻¹

凝固时纯固体溶剂与溶液中的溶剂平衡：\(\mu_\mathrm{A}^*(s) = \mu_\mathrm{A}^*(l) + RT\ln x_\mathrm{A}\)。
At freezing, pure solid solvent equilibrates with solvent in solution: \(\mu_\mathrm{A}^*(s) = \mu_\mathrm{A}^*(l) + RT\ln x_\mathrm{A}\).
与沸点升高同样的推导，把 \(\Delta_\mathrm{vap}H\) 换成 \(\Delta_\mathrm{fus}H\)，符号相反。
Same derivation as boiling-point elevation with \(\Delta_\mathrm{fus}H\) and opposite sign.
因 \(\Delta_\mathrm{fus}H < \Delta_\mathrm{vap}H\)，同一溶剂 \(K_f > K_b\)，凝固点法更灵敏。
Since \(\Delta_\mathrm{fus}H < \Delta_\mathrm{vap}H\), \(K_f > K_b\) — cryoscopy is more sensitive.

假设析出的是纯溶剂固体（不形成固溶体）。
Assumes the solid that separates is pure solvent (no solid solution).
稀溶液近似；高浓度下（如防冻液 50% 乙二醇）只能定性。
Dilute-solution approximation; only qualitative for concentrated antifreeze.

1.00 g 未知物溶于 50.0 g 苯，凝固点降低 0.640 K：M = 5.12×1.00/(0.640×0.0500) = 160 g/mol。

1.00 g in 50.0 g benzene lowers the f.p. by 0.640 K: M = 160 g/mol.