构象布居（玻尔兹曼） Conformer populations (Boltzmann) — 公式推导、符号与局限

\[\frac{n_2}{n_1} = \exp\!\left(-\frac{\Delta G^\circ}{RT}\right),\qquad x_i = \frac{e^{-G_i/RT}}{\sum_j e^{-G_j/RT}}\]

环己烷取代基的 A 值cyclohexane A-values

\[\Delta G^\circ = -RT\ln K = -RT\ln\frac{[\text{平伏键}]}{[\text{直立键}]}\]\[\Delta G^\circ = -RT\ln K = -RT\ln\frac{[\text{equatorial}]}{[\text{axial}]}\]

含义Meaning

快速互变的构象之间达到平衡，各构象分数由自由能差决定。环己烷取代基处于直立键与平伏键的自由能差称为 A 值。

Rapidly interconverting conformers equilibrate; their fractions follow the free-energy differences. For cyclohexane substituents the axial–equatorial difference is the A-value.

符号说明Symbols

符号Symbol	含义 Meaning	单位 Unit
\(\Delta G^\circ\)	两构象自由能差free-energy difference	kJ·mol⁻¹
\(x_i\)	构象 i 的摩尔分数mole fraction of conformer i	1

推导Derivation

构象互变是普通化学平衡：\(K = n_2/n_1\)，\(\Delta G^\circ = -RT\ln K\)。
Conformational interconversion is an ordinary equilibrium.
多个构象时用玻尔兹曼加权并归一化。
For many conformers weight by Boltzmann factors and normalise.
可观测量（如 NMR 偶合常数）为各构象值的布居加权平均：\(\langle J\rangle = \sum x_iJ_i\)。
Observables (e.g. NMR J) are population-weighted averages.

适用条件与局限Assumptions & limitations

需要构象互变快于实验时间尺度；低温下互变变慢可分别观测。
Requires fast interconversion; at low T conformers may be seen separately.
要用自由能而非仅用能量（简并度和熵贡献，如 gauche 构象有两个）。
Use free energies, not just energies (degeneracy — two gauche forms — and entropy).

例题Worked example

甲基环己烷 A = 7.3 kJ/mol：25 °C 时 K = e^(7300/2478) = 19，平伏键构象占 95%。叔丁基 A ≈ 20 kJ/mol，几乎全部平伏。

Methylcyclohexane A = 7.3 kJ/mol: K = 19, 95 % equatorial at 25 °C. tert-Butyl (A ≈ 20 kJ/mol) is essentially all equatorial.