Algebraic determination of partial molar quantities

As stated in the introduction to this chapter, the methods used and the equations developed in the previous sections are formally applicable to all extensive properties. However, there are two classes of properties those 

The partial molar properties are not measured directly per se, but are readily derivable from experimental measurements. For example, the volumes or heat capacities of definite quantities of solution of known composition are measured. These data are then expressed in terms of an intensive quantity—such as the specific volume or heat capacity, or the molar volume or heat capacity—as a function of some composition variable. The problem then arises of determining the partial molar quantity from these functions. The intensive quantity must first be converted to an extensive quantity, then the differentiation must be performed. Two general methods are possible (1) the composition variables may be expressed in terms of the mole numbers before the differentiation and reintroduced after the differentiation or (2) expressions for the partial molar quantities may be obtained in terms of the derivatives of the intensive quantity with respect to the composition variables. In the remainder of this section several examples are given with emphasis on the second method. Multicomponent systems are used throughout the section in order to obtain general relations. 

We must multiply the molar volume by the total number of moles in order to obtain the volume of the solution containing a total of i=1 n, moles thus 

The partial molar volume of the kth component is then given by 

According to the first method the indicated multiplication would be performed, the mole fractions expressed in terms of the mole numbers and the differentiation carried out. According to the second method the indicated 

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