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You state Then you can assume equilibrium, thus the ionic flux $j_i\propto\nabla\mu$ should be zero So it seems to me that the authors you read are not defining $$ RT \ln(c_i(\mathbf r)/c_0)+z_i e \phi(\mathbf r)\equiv0 $$ but instead defining $$ \mu_i(\mathbf r)=\mu_i^0. $$ That is, the chemical potential doesn't change.

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