能斯特分配定律 Nernst distribution law — 公式推导、符号与局限

\[K_D = \frac{c_\mathrm{B}(\text{有机相})}{c_\mathrm{B}(\text{水相})}\]\[K_D = \frac{c_\mathrm{B}(\text{organic})}{c_\mathrm{B}(\text{aqueous})}\]

n 次萃取后水相剩余比例fraction left in water after n extractions

\[\frac{m_n}{m_0} = \left(\frac{V_\mathrm{w}}{K_DV_\mathrm{o} + V_\mathrm{w}}\right)^n\]

含义Meaning

一定温度下，溶质在两种互不相溶溶剂中达到平衡时浓度之比为常数。萃取的理论基础。

At equilibrium the ratio of a solute’s concentrations in two immiscible solvents is constant — the basis of extraction.

符号Symbol	含义 Meaning	单位 Unit
\(K_D\)	分配系数partition coefficient	1
\(V_\mathrm{o}, V_\mathrm{w}\)	每次所用有机相、水相体积organic and aqueous volumes per extraction	L
\(n\)	萃取次数number of extractions	1

平衡时两相中溶质化学势相等，稀溶液中得 \(c_\mathrm{o}/c_\mathrm{w} = \exp[-(\mu_\mathrm{o}^\circ - \mu_\mathrm{w}^\circ)/RT]\) = 常数。
Equal chemical potentials in both phases give \(c_\mathrm{o}/c_\mathrm{w}\) = const.
一次萃取后水相剩余 \(m_1 = m_0 V_\mathrm{w}/(K_DV_\mathrm{o} + V_\mathrm{w})\)；重复 n 次连乘。
One extraction leaves \(m_1 = m_0 V_\mathrm{w}/(K_DV_\mathrm{o} + V_\mathrm{w})\); repeat n times.
同样总体积的萃取剂，分多次萃取比一次萃取效率高（“少量多次”）。
With the same total solvent, several small extractions beat one large one.

溶质在两相中形态相同（不缔合、不电离）；否则用分配比 D。
Same species in both phases; otherwise use the distribution ratio D.
辛醇–水分配系数 \(\lg P\)（logP）是药物亲脂性的指标。
The octanol–water lg P (logP) is the standard lipophilicity index.