Mole fraction

${\displaystyle x_i=\frac{n_i}{n_{\mathrm{t}\mathrm{o}\mathrm{t}}}}$

It is denoted x_i (lowercase Roman letter x), sometimes χ_i (lowercase Greek letter chi). (For mixtures of gases, the letter y is recommended.)

It is a dimensionless quantity with dimension of ${\displaystyle \mathsf{N}/\mathsf{N}}$ and dimensionless unit of moles per mole (mol/mol or mol⋅mol⁻¹) or simply 1; metric prefixes may also be used (e.g., nmol/mol for 10⁻⁹). When expressed in percent, it is known as the mole percent or molar percentage (unit symbol %, sometimes "mol%", equivalent to cmol/mol for 10⁻²). The mole fraction is called amount fraction by the International Union of Pure and Applied Chemistry (IUPAC) and amount-of-substance fraction by the U.S. National Institute of Standards and Technology (NIST). This nomenclature is part of the International System of Quantities (ISQ), as standardized in ISO 80000-9, which deprecates "mole fraction" based on the unacceptability of mixing information with units when expressing the values of quantities.

The sum of all the mole fractions in a mixture is equal to 1:

${\displaystyle \sum _{i=1}^N{n}_i=n_{\mathrm{t}\mathrm{o}\mathrm{t}};\sum _{i=1}^N{x}_i=1}$

Mole fraction is numerically identical to the number fraction, which is defined as the number of particles (molecules) of a constituent N_i divided by the total number of all molecules N_tot. Whereas mole fraction is a ratio of amounts to amounts (in units of moles per moles), molar concentration is a quotient of amount to volume (in units of moles per litre). Other ways of expressing the composition of a mixture as a dimensionless quantity are mass fraction and volume fraction.

Properties

Mole fraction is used very frequently in the construction of phase diagrams. It has a number of advantages:

• it is not temperature dependent (as is molar concentration) and does not require knowledge of the densities of the phase(s) involved
• a mixture of known mole fraction can be prepared by weighing off the appropriate masses of the constituents
• the measure is symmetric: in the mole fractions x = 0.1 and x = 0.9, the roles of 'solvent' and 'solute' are reversed.
• In a mixture of ideal gases, the mole fraction can be expressed as the ratio of partial pressure to total pressure of the mixture
• In a ternary mixture one can express mole fractions of a component as functions of other components mole fraction and binary mole ratios:

  ${\displaystyle \begin{array}{cc}{x}_1& =\frac{1-x_2}{1+\frac{x_3}{x_1}}\\ x_3& =\frac{1-x_2}{1+\frac{x_1}{x_3}}\end{array}}$

Differential quotients can be formed at constant ratios like those above:

${\displaystyle {\left(\frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{x}_2}\right)}_{\frac{x_1}{x_3}}=-\frac{x_1}{1-x_2}}$

or

${\displaystyle {\left(\frac{\mathrm{\partial }{x}_3}{\mathrm{\partial }{x}_2}\right)}_{\frac{x_1}{x_3}}=-\frac{x_3}{1-x_2}}$

The ratios X, Y, and Z of mole fractions can be written for ternary and multicomponent systems:

${\displaystyle \begin{array}{cc}X& =\frac{x_3}{x_1+x_3}\\ Y& =\frac{x_3}{x_2+x_3}\\ Z& =\frac{x_2}{x_1+x_2}\end{array}}$

These can be used for solving PDEs like:

${\displaystyle {\left(\frac{\mathrm{\partial }{\mu }_2}{\mathrm{\partial }{n}_1}\right)}_{n_2,n_3}={\left(\frac{\mathrm{\partial }{\mu }_1}{\mathrm{\partial }{n}_2}\right)}_{n_1,n_3}}$

or

${\displaystyle {\left(\frac{\mathrm{\partial }{\mu }_2}{\mathrm{\partial }{n}_1}\right)}_{n_2,n_3,n_4,\dots ,n_i}={\left(\frac{\mathrm{\partial }{\mu }_1}{\mathrm{\partial }{n}_2}\right)}_{n_1,n_3,n_4,\dots ,n_i}}$

This equality can be rearranged to have differential quotient of mole amounts or fractions on one side.

${\displaystyle {\left(\frac{\mathrm{\partial }{\mu }_2}{\mathrm{\partial }{\mu }_1}\right)}_{n_2,n_3}=-{\left(\frac{\mathrm{\partial }{n}_1}{\mathrm{\partial }{n}_2}\right)}_{{\mu }_1,n_3}=-{\left(\frac{\mathrm{\partial }{x}_1}{\mathrm{\partial }{x}_2}\right)}_{{\mu }_1,n_3}}$

or

${\displaystyle {\left(\frac{\mathrm{\partial }{\mu }_2}{\mathrm{\partial }{\mu }_1}\right)}_{n_2,n_3,n_4,\dots ,n_i}=-{\left(\frac{\mathrm{\partial }{n}_1}{\mathrm{\partial }{n}_2}\right)}_{{\mu }_1,n_2,n_4,\dots ,n_i}}$

Mole amounts can be eliminated by forming ratios:

${\displaystyle {\left(\frac{\mathrm{\partial }{n}_1}{\mathrm{\partial }{n}_2}\right)}_{n_3}={\left(\frac{\mathrm{\partial }\frac{n_1}{n_3}}{\mathrm{\partial }\frac{n_2}{n_3}}\right)}_{n_3}={\left(\frac{\mathrm{\partial }\frac{x_1}{x_3}}{\mathrm{\partial }\frac{x_2}{x_3}}\right)}_{n_3}}$

Thus the ratio of chemical potentials becomes:

${\displaystyle {\left(\frac{\mathrm{\partial }{\mu }_2}{\mathrm{\partial }{\mu }_1}\right)}_{\frac{n_2}{n_3}}=-{\left(\frac{\mathrm{\partial }\frac{x_1}{x_3}}{\mathrm{\partial }\frac{x_2}{x_3}}\right)}_{{\mu }_1}}$

Similarly the ratio for the multicomponents system becomes

${\displaystyle {\left(\frac{\mathrm{\partial }{\mu }_2}{\mathrm{\partial }{\mu }_1}\right)}_{\frac{n_2}{n_3},\frac{n_3}{n_4},\dots ,\frac{n_{i-1}}{n_i}}=-{\left(\frac{\mathrm{\partial }\frac{x_1}{x_3}}{\mathrm{\partial }\frac{x_2}{x_3}}\right)}_{{\mu }_1,\frac{n_3}{n_4},\dots ,\frac{n_{i-1}}{n_i}}}$