Thermodynamic Derivation of the Distribution Law

The thermodynamic derivation of the distribution law is based upon he principle that if there are two phases in equilibrium (i.e. two immiscible solvents containing the same solute dissolved in them), the chemical potential* of a substance present in them must be same in both the phases.

From thermodynamics, we know that the chemical potential of a substance is a solution given by
                                 
Whereis the standard chemical potential and ‘a’ is the activity** of the substance (solute) in the solution.

Thus for the solute in liquid A, we have
                             
Similarly for the solute in liquid B we have
                             
But as already stated, since the liquids A and B are in equilibrium,
 
Further at a given temperature, and are constant for given substance in the particular solvents. Hence at constant temperature, we have from equation (3.3.4).
                   
and therefore     
                     
This is the exact expression of the distribution law. However, if the solutions are dilute, the activates are equal o the concentrations so that the expression (3.3.6) is modified to
                        
Which is the original form of the distribution law.

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