Development of other relations

Substituting this equation into Equation (4.55) and setting [dS] = 0, 

The coefficient must be zero because [dT] need not be zero. Therefore 

Because we are interested in the independent variables T and V, we may set E as a function of T and V at constant mole numbers of the components, from which 

Substitution of these three equations into Equation (4.22) and division by dnj yields 

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