氧化还原滴定的计量点电位 Equivalence-point potential of a redox titration — 公式推导、符号与局限

\[E_\text{sp} = \frac{n_1E_1^{\circ\prime} + n_2E_2^{\circ\prime}}{n_1 + n_2}\]

可准确滴定的条件（25 °C，对称电对）feasibility criterion (25 °C, symmetric couples)

\[E^{\circ\prime}_1 - E^{\circ\prime}_2 \ge \frac{0.059 \times 3(n_1 + n_2)}{n_1 n_2}\ \mathrm V\ (\text{误差} \le 0.1\%)\]\[E^{\circ\prime}_1 - E^{\circ\prime}_2 \ge \frac{0.059 \times 3(n_1 + n_2)}{n_1 n_2}\ \mathrm V\ (\text{error} \le 0.1\%)\]

含义Meaning

对称电对的氧化还原滴定，计量点电位是两电对条件电位按电子数加权的平均。滴定突跃大小取决于条件电位差。

For symmetric couples, the equivalence potential is the electron-weighted average of the two formal potentials; the size of the jump depends on their difference.

符号说明Symbols

符号Symbol	含义 Meaning	单位 Unit
\(E^{\circ\prime}\)	条件电位（式量电位）formal potential	V
\(n_1, n_2\)	两电对转移电子数electrons transferred by each couple	1

推导Derivation

计量点时两电对处于平衡，溶液电位对两电对都满足能斯特方程。
At equivalence both couples share the same potential.
写出 \(E = E_1^{\circ\prime} + \frac{0.059}{n_1}\lg\frac{c_{\mathrm{Ox}_1}}{c_{\mathrm{Red}_1}}\) 和 \(E = E_2^{\circ\prime} + \frac{0.059}{n_2}\lg\frac{c_{\mathrm{Ox}_2}}{c_{\mathrm{Red}_2}}\)。
Write both Nernst equations.
分别乘 n₁、n₂ 相加，计量点处对数项相消（\(c_{\mathrm{Ox}_1}/c_{\mathrm{Red}_1} = c_{\mathrm{Red}_2}/c_{\mathrm{Ox}_2}\) 的计量关系），得加权平均。
Multiply by n₁, n₂ and add; the log terms cancel at equivalence.

适用条件与局限Assumptions & limitations

只适用于对称电对；Cr₂O₇²⁻/Cr³⁺ 这类不对称电对需另行推导，且计量点电位与浓度有关。
Symmetric couples only; asymmetric ones (Cr₂O₇²⁻/Cr³⁺) differ and depend on concentration.
条件电位依赖介质（酸度、配位剂）。
Formal potentials depend on the medium.

例题Worked example

Ce⁴⁺ 滴定 Fe²⁺（1 mol/L H₂SO₄）：E°′ = 1.44 与 0.68 V，E_sp = (1.44 + 0.68)/2 = 1.06 V，用邻二氮菲亚铁（1.06 V）作指示剂。

Ce⁴⁺ vs Fe²⁺ in 1 M H₂SO₄: E_sp = 1.06 V; ferroin (1.06 V) indicates it.