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# Volume fraction

'''Volume fractions''' $phi_i$ are a useful alternative 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 present; eg polymer solutions (See Flory-Huggins solution theory) They provide a more appropriate way to express the relative amounts of the various component|chemical species In any ''ideal'' mixture the total volume is the summation|sum of the individual volumes prior to mixing If $v_i$ is the volume of one molecule of component $i$ its volume fraction in the mixture is
$phi_i equiv frac\left\{N_iv_i\right\}\left\{V\right\}$

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 $N_A$ ≈ 6023 x 1023 between the factors in the numerator
$phi_i equiv frac\left\{n_iV_i\right\}\left\{V\right\}$

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
$sum_i phi_i equiv 1$

For ideal mixtures thermodynamic functions expressed in terms of volume fractions asymptotically approach the mole fraction representation as the various molecular volumes become equal
For real mixtures there is usually a contraction or expansion on mixing due to interstitial packing and different molecular interactions so the volumes of the separate initial components do not sum to the total Even in that case the total volume is the sum of the partial molar volumes because the volume is a homogeneous function of degree one in the amounts of components present
$V\left(lambda n_1 lambda n_2 lambda n_z\right) = lambda V$

For example if the amount of everything in the system is doubled the volume doubles regardless of the molecular interactions
$V = sum_j n_joverline V_j$

in which
$overline V_i equiv frac\left\{partial V\right\}\left\{partial n_i\right\}$