Partial molar properties relations among

This result, known as the Gibbs-Duhem equation, imposes a constraint on how the partial molar properties of any phase may vary with temperature, pressure, and composition. In particular, at constant T and P it represents a simple relation among the Af/ to which measured values of partial properties must conform. 

The partial derivative is a linear operator therefore, the partial molar derivative (3.4.5) may be applied to all those expressions given in 3.2, producing partial molar versions of the fundamental equations. In particular, when we apply the partial molar derivative to the integrated forms (3.2.29)-(3.2.31) of the fundamental equations, we obtain the following important relations among partial molar properties ... 

In a similar fashion a large collection of relations among the partial molar quantities can be developed. For example, since dCj cT)p M — —S- for a pure fluid, one can easily show that (8G /dT)pj j. — —5j for a mixture. In fact, by extending this argument to other mixture properties, one finds that for each relationship among the thermodynamic variables in a pure fluid, there exists an identical relationship for the partial molar thermodynamic properties in a mixture ... 

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