关于Arrhenius公式的几点讨论

doi:10.3866/PKU.DXHX202204035

Abstract

Abstract:

The Arrhenius equation is one of the most important equations in chemical kinetics and has been discussed by scientists for more than a century since its establishment. The historical process of the equation and further explanations of its relationship with the van't Hoff equation are traced. The physical meaning of the Tolman activation energy is discussed with some suggested modifications. The variation of the activation energy with temperature is discussed, and the temperature range for applying the Arrhenius equation is quantitatively analyzed. It is suggested that the three-constant equation be used even in the normal temperature range for reactants with a complex molecular structure. The similarity between some thermodynamic equations and the Arrhenius equation in physical chemistry textbooks are explained. These discussions will help with correctly understanding and applying the Arrhenius equation.

Key words: Arrhenius equation, Tolman activation energy, Temperature validity range, Thermodynamic relation

MSC2000:

Chenxi Zhang, Han Su, Shuyong Zhang. Discussions on the Arrhenius Equation[J].University Chemistry, 2022, 37(7): 2204035.

Figures/Tables 4

公式	微分式	积分式	物理意义	使用条件
Arrhenius公式	$ \frac{{{\text{dln}}k}}{{{\text{d}}T}} = \frac{{{E_{\text{a}}}}}{{R{T^{\text{2}}}}} $	$ \ln k=-E_{\mathrm{a}} / R T+B $	k–T	满足$ r = k{c^n} $或$ r = k{p^n} $、小温度范围
van't Hoff方程	$ {(\frac{\text{dln}{K}_{p}}{\text{d}T})}_{p}=\frac{\Delta {H}^{\ominus}}{R{T}^{2}} $ $ {(\frac{\text{dln}{K}_{c}}{\text{d}T})}_{p}=\frac{\Delta {U}^{\ominus}}{R{T}^{2}} $	$ \text{ln}{K}_{p}^{\ominus}=-\Delta {H}^{\ominus}\text{/}RT+B $$ \text{ln}{K}_{c}^{\ominus}=-\Delta {U}^{\ominus}\text{/}RT+B $	K–T	化学平衡、恒压或恒容
化学势(任意组分)	纯组分或溶剂：$ \frac{\mathrm{d} \ln \left(f / p^{\ominus}\right)}{\mathrm{d} T}=\frac{\mu^{\ominus}-\mu}{R T^{2}}$	纯组分或溶剂： $ \mathrm{ln}(f\text{/}{p}^{\ominus})=-\frac{{\mu }^{\ominus}-\mu }{RT} $	标准态→计算状态，a–T	化学势平衡
	液态混合物或溶液组分：$ \frac{\mathrm{d} \ln a_{\mathrm{B}}}{\mathrm{d} T}=\frac{\mu^{\ominus}-\mu}{R T^{2}} $	液态混合物或溶液组分：$ \text{ln}{a}_{\text{B}}=-\frac{{\mu }^{\ominus}-\mu }{RT} $
	液滴表面：$ \frac{\mathrm{d} \ln \left(p_{\mathrm{r}} / p\right)}{\mathrm{d} T}=\frac{\mu-\mu_{\mathrm{r}}}{R T^{2}}$	表面：$ {\text{ln}}\frac{{{p_{\text{r}}}}}{p} = - \frac{{\mu - {\mu _{\text{r}}}}}{{RT}} $
依数性公式	凝固点降低公式：$ \left(\frac{\mathrm{d} \ln a_{\mathrm{A}}}{\mathrm{d} T}\right)_{p}=-\frac{\Delta_{\mathrm{fus}} H_{\mathrm{m}}^{\ominus}}{R T^{2}} $ 沸点升高公式：$ \left(\frac{\mathrm{d} \ln a_{\mathrm{A}}}{\mathrm{d} T}\right)_{p}=-\frac{\Delta_{\text {vap }} H_{\mathrm{m}}^{\ominus}}{R T^{2}}$	凝固点降低公式：$ \text{ln}{a}_{\text{A}}=\frac{{\Delta }_{\text{fus}}{H}_{\text{m}}^{\ominus}}{R}(\frac{1}{{T}_{\text{f}}^{}}-\frac{1}{{T}_{\text{f}}}) $ 沸点升高公式：$ \ln a_{\mathrm{A}}=\frac{\Delta_{\mathrm{vap}} H_{\mathrm{m}}^{\ominus}}{R}\left(\frac{1}{T_{\mathrm{b}}}-\frac{1}{T_{\mathrm{b}}^{}}\right)$	非挥发性溶质，$ {\Delta }_{\text{fus}}{H}_{\text{m}}^{\ominus}-{T}_{\text{f}} $或$ {\Delta }_{\text{vap}}{H}_{\text{m}}^{\ominus}-{T}_{\text{b}} $	相平衡，含不挥发性溶质的稀溶液
Clausius-Clapeyron方程	$ \frac{\mathrm{d} \ln p}{\mathrm{~d} T}=\frac{\Delta_{\text {vap }} H_{\mathrm{m}}}{R T^{2}} $	$ {\text{ln}}\frac{{{p_2}}}{{{p_1}}} = \frac{{{\Delta _{{\text{vap}}}}{H_{\text{m}}}}}{R}(\frac{1}{{{T_1}}} - \frac{1}{{{T_2}}}) $	单组分纯物质相平衡时，p–T	气-液或气-固相平衡
吸附热方程	$ \left(\frac{\mathrm{d} \ln \|p\|}{\mathrm{d} T}\right)_{a}=-\frac{\Delta_{\mathrm{ads}} H_{\mathrm{m}}}{R T^{2}}$	$ {\text{ln}}\frac{{{p_2}}}{{{p_1}}} = \frac{{{\Delta _{{\text{ads}}}}{H_{\text{m}}}}}{R}(\frac{1}{{{T_2}}} - \frac{1}{{{T_1}}}) $	吸附量一定时，p_质，平衡–T	吸附平衡，吸附等量线

References 13

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方程	lnk ? C = k/T
m	0.0	1.0	2.0	3.0	4.0	5.0	6.0
斜率k	?9622.32	?9939.54	?10256.76	10573.97	?10891.19	?11208.41	?11525.62
线性相关度R²	1	1	1	1	0.99999	0.99999	0.99999
25 ℃ E_a (二常数)/(J?mol^?1)	79999.968	82637.336	85274.703	87911.987	90549.35	93186.721	95824.005
25 ℃ E_a (三常数)/(J?mol^?1)	80000	82478.819	84957.638	87436.457	89915.276	92394.096	94872.915
25 ℃相对误差E_r	0.00%	?0.19%	?0.37%	?0.54%	?0.71%	?0.86%	?1.00%

Discussions on the Arrhenius Equation

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