As mentioned before, it is reasonable to consider electron hole concentration constant at the both membrane surfaces, thus the reverse reaction rate of Eq. (7a) and the forward reaction rate of Eq. (7b) are zero-order. On the other hand, it is fair to consider a non-elementary reaction order respect to oxygen partial pressure and oxygen vacancy concentration. Therefore, Eqs. (8a) and (8b) can be rewritten in the following forms