7.2: Partial Molar Volume

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The partial molar volume of compound A in a mixture of A and B can be defined as

\[ V_A = \left (\dfrac{\partial V}{\partial n_A} \right)_{p,T,n_B} \nonumber \]

Using this definition, a change in volume for the mixture can be described using the total differential of \(V\):

\[ dV = \left( \dfrac{\partial V}{\partial n_A}\right)_{p,T,n_B} dn_A + \left( \dfrac{\partial V}{\partial n_B}\right)_{p,T,n_A} dn_B \nonumber \]

\[ dV = V_a \, dn_A + V_b\,dn_B \nonumber \]

and integration yields

\[V = \int _0^{n_A} V_a \, dn_A + \int _0^{n_B} V_b\,dn_B \nonumber \]

\[ V = V_a \, n_A + V_b\,n_B \nonumber \]

This result is important as it demonstrates an important quality of partial molar quantities. Specifically, if \(\xi_i\) represents the partial molar property \(X\) for component i of a mixture, The total property \(X \) for the mixture is given by

\[X = \sum_{i} \xi_in_i \nonumber \]

It should be noted that while the volume of a substance is never negative, the partial molar volume can be. An example of this appears in the dissolution of a strong electrolyte in water. Because the water molecules in the solvation sphere of the ions are physically closer together than they are in bulk pure water, there is a volume decrease when the electrolyte dissolves. This is easily observable at high concentrations where a larger fraction of the water in the sample is tied up in solvation of the ions.