Extensivity and Partial Molar Quantities

In multicomponent systems, thermodynamic functions such as volume V, Gibbs free energy G, and many other thermodynamic functions that can be expressed as functions of p, Tand Nk are extensive functions of Nk. This extensivity gives us general thermodynamic relations, some of which we will discuss in this section. Consider the volume of a system as a function of p, T and Nk V = V p,T,Nk). At constant p and T, if all the mole numbers were increased by a factor X, the volume V would also increase by the same factor. This is the property of extensivity we have already discussed several times. In mathematical terms, we have 

At constant p and T using Euler s theorem, as in Section 4.4, we can arrive at the relation 

It is convenient to define partial molar volumes as the derivatives 

Partial molar volumes are intensive quantities. As in the case of the Gibbs-Duhem relation, we can derive a relation between Viak by noting that, at constant p and T, 

Comparing dVobtained from (5.5.4) and (5.5.5), we see ihdXYjk k dVrok)pj 0, in which we have explicitly noted that the change dV is at constant p and T. From this relation it follows that  

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