含义Meaning
速率常数随温度指数增长（1889）。活化能越大，对温度越敏感。经验规则“温度升高 10 K 速率约增 2–4 倍”对应 Ea 约 50–100 kJ/mol。
Rate constants rise exponentially with temperature (1889); larger activation energies mean greater sensitivity. The rule ‘2–4× per 10 K’ corresponds to Ea ≈ 50–100 kJ/mol.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(A\) | 指前因子（频率因子）pre-exponential factor | 同 k |
| \(E_a\) | 活化能activation energy | kJ·mol⁻¹ |
| \(T\) | 热力学温度temperature | K |
推导Derivation
范特霍夫方程 \(d\ln K/dT = \Delta H^\circ/RT^2\)，K = k₊/k₋，阿伦尼乌斯类比写出 \(d\ln k/dT = E_a/RT^2\)。
By analogy with van ’t Hoff, \(d\ln k/dT = E_a/RT^2\).
若 Ea 与温度无关，积分得 \(k = Ae^{-E_a/RT}\)。
Integrate with constant Ea.
物理意义：只有能量超过 Ea 的碰撞（玻尔兹曼因子 \(e^{-E_a/RT}\)）才能反应。
Physically only collisions above Ea (Boltzmann fraction) react.
ln k 对 1/T 作图（阿伦尼乌斯图）为直线，斜率 −Ea/R。
An Arrhenius plot of ln k vs 1/T has slope −Ea/R.
适用条件与局限Assumptions & limitations
宽温度范围内 A 与 Ea 会随温度变化（曲线弯曲）；隧穿、多通道反应、酶失活使其偏离。
Over wide ranges A and Ea vary (curvature); tunnelling, competing channels and enzyme denaturation cause deviations.
负活化能出现在含放热预平衡的复合反应中。
Negative apparent Ea arises from exothermic pre-equilibria.
例题Worked example
Ea = 50 kJ/mol：从 298 K 到 308 K，k₂/k₁ = exp[50000/8.314×(1/298 − 1/308)] = 1.93。
Ea = 50 kJ/mol: 298 → 308 K speeds the reaction 1.93×.