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.