Bulter-Volmer方程与Nernst方程的关系

Bulter-Volmer equation

$$i=i_c-i_a=FAk^0 \left [c_O(0,t) \exp \dfrac{-\beta F(E-E^{0\prime})}{RT}- c_R(0,t) \exp \dfrac{(1-\beta)F(E-E^{0\prime})}{RT}\right] \tag {1}$$

Nernst equation

Bulter-Volmer方程是动力学方程，那么从它是否能推出热力学的关系？在平衡状态下净电流$i=0$，且反应物的本体浓度bulk concentration等于表面浓度surface concentration，即：

$$c_O=c_O^* ; c_R=c_R^*$$ $$FAk^0 \left [c_O^* \exp \dfrac{-\beta F(E-E^{0\prime})}{RT}- c_R^* \exp \dfrac{(1-\beta)F(E-E^{0\prime})}{RT}\right]=0 \tag {2}$$

整理得：

$$c_O^*\exp \dfrac{-\beta F(E-E^{0\prime})}{RT}- c_R^* \exp \dfrac{(1-\beta)F(E-E^{0\prime})}{RT}=0 \tag {3}$$

再整理得：

$$\dfrac {\exp \dfrac{(1-\beta)F(E-E^{0\prime})}{RT}}{ \exp \dfrac{-\beta F(E-E^{0\prime})}{RT}}=\exp \dfrac{F(E-E^{0\prime})}{RT}=\dfrac{c_O^*}{c_R^*} \tag {4}$$

此时为平衡状态则$E=E_{eq}$，根据上式可以得到Nernst方程：

$$E_{eq}=E^{0\prime}+ \dfrac {RT}{F} \ln \dfrac{c_O^*}{c_R^*} \tag{5}$$

---

Reference

[1] Allen J. Bard, Larry R. Faulkner; Electrochemical methods-Fundamentals and applications.