The Two Equations for Partial Molar Properties

If we have experimental data, or an equation that is believed to represent such experimental data for some extensive property as a function of concentration, we can compute the partial molar values by the method of tangent slopes or the method of tangent intercepts. In either case we could plot the data and make the geometric constructions. However, the mathematical procedure, which our computers can do for us is much more useful. 

If we have the data in the form of an equation for the molar value of some property (e.g., the molar volume) as a function of mol fraction, then from the geometry of Figiure 6.6, and from Eq. 6.18, we can see that, for any value of Xa 

The partial molar volumes for ethanol computed here and in Example 6.1 are the same. The method used in Example 6.1 does not give a value for the partial molar volume of water, which this method does. See also Problem 6.4.  

This equation has no common name a good name for it would be the tangent intercepts calculating equation. It is shown here for volume, but equally applicable for any property whose molar values can be expressed as a function of mol fraction. It is not restricted to binary mixtures it can be applied to species a (or b or c) in mixtures of any number of species. Example 6.4 shows the application of this equation. 

The corresponding equation for the method of tangent slopes (Eq. 4.19) requires that we have the property equation in the form of an extensive property, stated as a function of the number of mols present of each species. If we have, for example, an equation for v as a function of Xj we can multiply both sides of that equation by the total number of mols present, j-. This makes the following changes 

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