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7.2: Partial Molar Volume

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    84331
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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 \(V\):

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

    or

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

    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.


    This page titled 7.2: Partial Molar Volume is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Patrick Fleming.

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