Dialysis reaction in dialysate

In dialysis (see Section 4.3.1), the rate at which a solute is removed from a feed to the dialyzate through the dialysis membrane is proportional to the concentration difference on two sides of the membrane. If a chemical reaction can be carried out in the dialyzate side between the solute and an added agent such that neither the products nor the agent can diffuse to the feed side, then the solute 

Consider species 1 (e.g. phenol), species 2 (e.g. NaOH) and species 3 (e.g. sodium phenolate) in a batch dialysis system having a feed solution volume Vf and dialyzate solution volume V. If the membrane area separating the two solutions is A , the molar rate of change of solute 1 in the feed chamber is 

Usually, species 2 is present in the dialyzate in large excess one can assume reaction (5.4.30) to be in equilibrium  

The total amount of phenol transferred from the feed to the dialysate chamber at any time t from t = 0, with reaction, is 

Consider the value of the same quantity, without reaction, Vf [Cy - Ci/f)]. The ratio of the values with and without reaction, for a very long time, where exp(-Mt) 0 is given by 

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