Sunday, October 16, 2022

Derivation of the mixed activation and transport-limited governing equations for first order electrochemical reactions

It took me awhile to work out all the steps to deriving the Koutecky-Levich equation so I wanted to save this derivation where it would be easily accessible. The general form of the Butler-Volmer equation for outer-sphere electron transfer reactions is given in the following: \begin{equation}\label{eq:eq1} i=i_{0} \left\{ \frac{c_{a}(0,t)}{c_(a,b)} e^{\frac{\alpha_{a}zF}{RT}}- \frac{c_c(0,t)}{c_(c,b)}e^{\frac{-\alpha_c zF}{RT}}\right\} ^{(a)} \end{equation} where \begin{equation} c_i(0,t) \end{equation} indicates the concentration of the reactant at the electrolyte-electrode interface and \begin{equation} c_{(i,b)} \end{equation} indicates the bulk concentration of the reactant. For concentration-gradient mediated transport to the electrolyte-electrode interface, the flux, j, of the reacting species in the electrolyte is given by, \begin{equation} \label{eq:eq2} j=-\frac{i}{zF} \end{equation} Then, from Fick’s first law, \begin{equation} \label{eq:eq3} |i|=zFD \frac{\partial c}{\partial x} \end{equation} And, using the first two terms in the Taylor series expansion for the concentration, we can rewrite Fick’s first law as, \begin{equation} \label{eq:eq4} |i|=zFD \frac{c_{c,b}- c_c(0,t)}{x_b-x_0}^{(b)} \end{equation} where we assign the distance, \begin{equation} x_b-x_0 = \delta \end{equation}. In the limit that, \begin{equation} c_c (0,t)=0 \end{equation}, * becomes, \begin{equation} \label{eq:eq5} |i_L | = zFD \frac{c_{c,b}}{\delta} ^{(c)} \end{equation} And, dividing equation b by c results in, \begin{equation} \label{eq:eq6} \frac{c_c (0,t)}{c_b} = 1-\frac{|i|}{|i_L |} ^{(d)} \end{equation} Focusing on just the cathodic term from a, and substituting d, results in the following, \begin{equation} \label{eq:eq7} i= -i_0\left\{\left ( 1-\frac{|i|}{|i_L | } \right ) e^{\frac{-\alpha _c zF}{RT}} \right\} \end{equation} This equation is transcendental in i, but an analytical solution can be obtained, as follows: \begin{equation} \label{eq:eq8} i = \left\{\left ( 1-\frac{|i|}{|i_L|} \right ) i_{act}\right\} \end{equation} \begin{equation} \label{eq:eq9} i_{act} = -i_0 e^{\frac{-\alpha_c zF}{RT}} (e) \end{equation} \begin{equation} \label{eq:eq10} 0 = i_{act} -\frac{|i|i_{act}}{|i_L |} -|i| \end{equation} \begin{equation} \label{eq:eq11} 0=|i|\left [\frac{i_{act}}{|i|} - \frac{i_{act}}{|i_L|}-1 \right ] \end{equation} \begin{equation} \label{eq:eq12} 0=\frac{i_{act}}{|i|} - \frac{i_{act}}{|i_L|}-1 \end{equation} \begin{equation} \label{eq:eq13} 1=\frac{i_{act}}{|i|} - \frac{i_{act}}{|i_L|} \end{equation} \begin{equation} \label{eq:eq14} \frac{1}{i_{act}} = \frac{1}{|i|} - \frac{1}{|i_L|} \end{equation} \begin{equation} \label{eq:eq15} \frac{1}{i_{act}} + \frac{1}{|i_L|} = \frac{1}{|i|} \end{equation} \begin{equation} \label{eq:eq16} \left [ \frac{1}{i_{act}} + \frac{1}{|i_L|} \right ]^{-1}=|i| \end{equation} \begin{equation} \label{eq:eq17} |i|=\frac{|i_{act} ||i_L |}{|i_{act} |+|i_L |} ^{(f)} \end{equation} Where f is the Koutecky-Levich equation and b, e, and f provide expressions that can be used to calculate and model transport-limited kinetics, activation kinetics, and mixed kinetics, respectively.

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