# Partial molar volume

Partial molar volumes are applicable to real mixtures, including solutions, in which the volumes of the separate, initial components do not sum to the total. This is generally the case, in distinction to the paradigm of ideal mixtures. For real mixtures, there is usually a contraction or expansion on mixing due to changes in interstitial packing and differing molecular interactions. Even so, the total volume is the sum over the partial molar volumes times the numbers of moles, because the volume is a homogeneous function of degree one in the amounts of the various chemical species present.

${\displaystyle V(\lambda n_{1},\lambda n_{2},...,\lambda n_{z})=\lambda V\,}$

For example, if the amount of everything in the system is doubled at constant temperature and pressure, the volume doubles because all the molecular circumstances remain the same throughout.

${\displaystyle V=\sum _{j}n_{j}{\overline {V}}_{j}\,}$

in which the partial molar volume is

${\displaystyle {\overline {V}}_{i}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\partial V}{\partial n_{i}}}}$

and ${\displaystyle n_{i}}$ is the number of moles of component ${\displaystyle i}$. As noted, ${\displaystyle T}$ and ${\displaystyle P}$ are held constant when taking these partial derivatives. These quantities can be measured experimentally, as well as being theoretically useful. They are not constants but vary with the composition of a system.

Partial molar quantities can be defined for all the extensive thermodynamic variables in a system.