# 草稿:巴特勒-褔尔默方程

${\displaystyle j=j_{0}\cdot \left\{\exp \left[{\frac {\alpha _{a}zF}{RT}}(E-E_{eq})\right]-\exp \left[-{\frac {\alpha _{c}zF}{RT}}(E-E_{eq})\right]\right\}}$

${\displaystyle j=j_{0}\cdot \left\{\exp \left[{\frac {\alpha _{a}zF\eta }{RT}}\right]-\exp \left[-{\frac {\alpha _{c}zF\eta }{RT}}\right]\right\}}$

• ${\displaystyle j}$：电极的电流密度，A/m2 (defined as i = I/A)
• ${\displaystyle j_{o}}$：exchange current density，A/m2
• ${\displaystyle E}$：electrode potential，V
• ${\displaystyle E_{eq}}$：equilibrium potential，V
• ${\displaystyle T}$absolute temperature，K
• ${\displaystyle z}$：该电极反应中涉及的电子数目
• ${\displaystyle F}$Faraday constant
• ${\displaystyle R}$universal gas constant
• ${\displaystyle \alpha _{c}}$：so-called cathodic charge transfer coefficient，无量纲
• ${\displaystyle \alpha _{a}}$：so-called anodic charge transfer coefficient，无量纲
• ${\displaystyle \eta }$：activation 過電位 (defined as ${\displaystyle \eta =(E-E_{eq})}$)。

## 质量传递的控制

${\displaystyle i_{\text{limiting}}={\frac {zFD}{\delta }}C^{*}}$

• D是的 扩散的系数;
• δ是的扩散层的厚度；
• C* 是的浓度活性的(限制性)的种类在大量的电解质。
• C(0,t)是依赖于时间的浓度在零距离从表面。

## 极限情况

• 低过电势的区域(称为"偏性"，即，当E≈电子当量)，巴特勒–沃尔公式简化了：
${\displaystyle i=i_{0}{\frac {zF}{RT}}(E-E_{eq})}$;
• 高过电势地区，那里的管家–沃尔公式简化了的 塔菲尔公式中：
${\displaystyle E-E_{eq}=a_{c}-b_{c}\log(i)}$ 一阴极反应，当E<<E均衡，或
${\displaystyle E-E_{eq}=a+b\log(i)}$ 为阳极反应，当E>>Eeq

## 参考文献

1. ^ Mayneord, W. V. John Alfred Valentine Butler. 14 February 1899-16 July 1977. Biographical Memoirs of Fellows of the Royal Society. 1979, 25: 144–178. doi:10.1098/rsbm.1979.0004.
2. ^ J. O'M. Bockris, A.K.N.Reddy, and M. Gamboa-Aldeco, "Modern Electrochemistry 2A. Fundamentals of Electrodics.", Second Edition, Kluwer Academic/Plenum Publishers, p.1083, 2000.