# Volume fraction

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Volume fractions ${\displaystyle \phi _{i}}$ are useful alternatives to mole fractions ${\displaystyle x_{i}}$ when dealing with mixtures in which there is a large disparity between the sizes of the various kinds of molecules; e.g., polymer solutions. They provide a more appropriate way to express the relative amounts of the various components.

In any ideal mixture, the total volume is the sum of the individual volumes prior to mixing.

Caution: in non-ideal cases the additivity of volume is no longer guaranteed. Volumes can contract or expand upon mixing and molar volume becomes a function of both concentration and temperature. This is why mole fractions are a safer unit to use.

If ${\displaystyle v_{i}}$ is the volume of one molecule of component ${\displaystyle i}$, its volume fraction in the mixture is

${\displaystyle \phi _{i}\equiv {\frac {N_{i}v_{i}}{V}}}$

where the total volume of the system is the sum of the contributions from all the chemical species

${\displaystyle V=\sum _{j}N_{j}v_{j}\,}$

The volume fraction can also be expressed in terms of the numbers of moles by transferring Avogadro's number ${\displaystyle N_{A}}$ ≈ 6.023 x 1023 between the factors in the numerator.

${\displaystyle \phi _{i}\equiv {\frac {n_{i}V_{i}}{V}}}$

where ${\displaystyle n_{i}=N_{i}/N_{A}}$ is the number of moles of ${\displaystyle i}$ and ${\displaystyle V_{i}}$ is the molar volume, and

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

As with mole fractions, the dimensionless volume fractions sum to one by virtue of their definition.

${\displaystyle \sum _{i}\phi _{i}\equiv 1\,}$

Thermodynamic functions using volume fractions reduce to mole-fraction expressions for mixtures of rigid molecules of roughly equal size. For macromolecules, there is a question about whether they behave as flexible, random coils (see Flory-Huggins solution theory), or whether they have compact structures like globular proteins. In addition to entropic questions, there are others concerning energy.

For real mixtures, see Partial molar volume.

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