Volume fraction

Volume fractions $phi\_i$ are useful alternatives to mole fractions $x\_i$ 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 $v\_i$ is the volume of one molecule of component $i$, its volume fraction in the mixture is

:$phi\_i\; equiv\; frac\{N\_iv\_i\}\{V\}$

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

:$V\; =\; sum\_j\; N\_jv\_j\; ,$

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

:$phi\_i\; equiv\; frac\{n\_iV\_i\}\{V\}$

where $n\_i\; =\; N\_i\; /\; N\_A$ is the number of moles of $i$ and $V\_i$ is the molar volume, and

:$V\; =\; sum\_j\; n\_jV\_j\; ,$

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

:$sum\_i\; phi\_i\; equiv\; 1\; ,$

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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