General Case Potential and Concentration Gradient

In the general case, the potential and concentration gradients appear in pores simultaneously [433, 436 41, 445]. It is assumed that the concentrations of the redox species outside the pores are Cq and and the current may be expressed in terms of the overpotential as in Eqs. (9.79) and (9.82). To solve for the concentration and potential gradient, Eqs. (9.1) and (9.2) must be solved simultaneously  

Taking the second derivative of the overpotential and substitution of the current gradient gives 

This equation must be solved together with Eq. (9.57). They both represent second-order differential equations for rj(x) and a(x) with the following conditions  

This solution means that there is a linear relation between the potential and concentration in the pores. There are two limiting cases  

The value of the parameter v for typical experimental conditions, n= 1, D = 10 cm s ps = 10 2 cm, and concentrations, Cq, 1 and 10 mM, equals p 10 and 10 V, respectively. This means that for typical concentratirMis the process is limited by the concentration gradient in the pores. Only in the case of very large concentrations of electroactive species or solvent reduction/oxidation (water electrolysis, chlorine evolution in concentrated solutions) might the potential gradient be important. 

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