Problem S10.4. Consider a membrane of thickness Ax which separates two regions containing dilute aqueous solutions of ions. One particular ion, i, with charge z; can diffuse through the membrane. It has diffusion coefficient, D, inside the membrane. A constant electric field, E, directed perpendicular to the membrane surface exists inside the membrane. Assume that a steady current, J, of ions diffuses across the membrane. The phenomenological equation relating the current to the gradient of the electrochemical potential of the ion inside the membrane is J = L(au/ax), where L is the phenomenological coefficient, L = Dc/RT, and c is the concentration of ion i. For dilute solutions, the gradient of the electrochemical potential of the ion is H = (RT/c)(dc;/dx) +z; F(do/dx), where is the electric potential, F is Faraday's constant, and E=-(do/dx). Show that the ion current inside the membrane can be written

where cand care the ion concentrations on the inside left and right surfaces of the membrane, respectively, and A = OR - OL is the difference in electric potential on the inside left and right surfaces of the membrane.

Transcribed Image Text:

Dz, FA (c-exp(z,FA/RT)) J = RTAX (1-exp(zFA6/RT))