# Mole fraction

In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, n$_i$ (expressed in unit of moles, symbol mol), and the total amount of all constituents in a mixture, n$_{tot}$ (also expressed in moles):
    $x_{i}={\frac {n_{i}}{n_{\mathrm {tot} }}}$
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 ${\mathsf {N}}/{\mathsf {N}}$ 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:
    $\sum _{i=1}^{N}n_{i}=n_{\mathrm {tot} };\ \sum _{i=1}^{N}x_{i}=1$
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:
    ${\begin{aligned}x_{1}&={\frac {1-x_{2}}{1+{\frac {x_{3}}{x_{1}}}}}\\[2pt]x_{3}&={\frac {1-x_{2}}{1+{\frac {x_{1}}{x_{3}}}}}\end{aligned}}$
Differential quotients can be formed at constant ratios like those above:
    $\left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{1}}{1-x_{2}}}$
or
    $\left({\frac {\partial x_{3}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{3}}{1-x_{2}}}$
The ratios X, Y, and Z of mole fractions can be written for ternary and multicomponent systems:
    ${\begin{aligned}X&={\frac {x_{3}}{x_{1}+x_{3}}}\\[2pt]Y&={\frac {x_{3}}{x_{2}+x_{3}}}\\[2pt]Z&={\frac {x_{2}}{x_{1}+x_{2}}}\end{aligned}}$
These can be used for solving PDEs like:
    $\left({\frac {\partial \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3}}$
or
    $\left({\frac {\partial \mu _{2}}{\partial n_{1}}}\right)_{n_{2},n_{3},n_{4},\ldots ,n_{i}}=\left({\frac {\partial \mu _{1}}{\partial n_{2}}}\right)_{n_{1},n_{3},n_{4},\ldots ,n_{i}}$
This equality can be rearranged to have differential quotient of mole amounts or fractions on one side.
    $\left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{n_{2},n_{3}}=-\left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{\mu _{1},n_{3}}=-\left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\mu _{1},n_{3}}$
or
    $\left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{n_{2},n_{3},n_{4},\ldots ,n_{i}}=-\left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{\mu _{1},n_{2},n_{4},\ldots ,n_{i}}$
Mole amounts can be eliminated by forming ratios:
    $\left({\frac {\partial n_{1}}{\partial n_{2}}}\right)_{n_{3}}=\left({\frac {\partial {\frac {n_{1}}{n_{3}}}}{\partial {\frac {n_{2}}{n_{3}}}}}\right)_{n_{3}}=\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{n_{3}}$
Thus the ratio of chemical potentials becomes:
    $\left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{\frac {n_{2}}{n_{3}}}=-\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{\mu _{1}}$
Similarly the ratio for the multicomponents system becomes
    $\left({\frac {\partial \mu _{2}}{\partial \mu _{1}}}\right)_{{\frac {n_{2}}{n_{3}}},{\frac {n_{3}}{n_{4}}},\ldots ,{\frac {n_{i-1}}{n_{i}}}}=-\left({\frac {\partial {\frac {x_{1}}{x_{3}}}}{\partial {\frac {x_{2}}{x_{3}}}}}\right)_{\mu _{1},{\frac {n_{3}}{n_{4}}},\ldots ,{\frac {n_{i-1}}{n_{i}}}}$

## Related quantities
### Mass fraction

The mass fraction w$_i$ can be calculated using the formula
    $w_{i}=x_{i}{\frac {M_{i}}{\bar {M}}}=x_{i}{\frac {M_{i}}{\sum _{j}x_{j}M_{j}}}$
where M$_i$ is the molar mass of the component i and M̄ is the average molar mass of the mixture.

### Molar mixing ratio

The mixing of two pure components can be expressed introducing the amount or molar mixing ratio of them $r_{n}={\frac {n_{2}}{n_{1}}}$. Then the mole fractions of the components will be:
    ${\begin{aligned}x_{1}&={\frac {1}{1+r_{n}}}\\[2pt]x_{2}&={\frac {r_{n}}{1+r_{n}}}\end{aligned}}$
The amount ratio equals the ratio of mole fractions of components:
    ${\frac {n_{2}}{n_{1}}}={\frac {x_{2}}{x_{1}}}$
due to division of both numerator and denominator by the sum of molar amounts of components. This property has consequences for representations of phase diagrams using, for instance, ternary plots.

#### Mixing binary mixtures with a common component to form ternary mixtures

Mixing binary mixtures with a common component gives a ternary mixture with certain mixing ratios between the three components. These mixing ratios from the ternary and the corresponding mole fractions of the ternary mixture x$_{1(123)}$, x$_{2(123)}$, x$_{3(123)}$ can be expressed as a function of several mixing ratios involved, the mixing ratios between the components of the binary mixtures and the mixing ratio of the binary mixtures to form the ternary one.
    $x_{1(123)}={\frac {n_{(12)}x_{1(12)}+n_{13}x_{1(13)}}{n_{(12)}+n_{(13)}}}$

### Mole percentage

Multiplying mole fraction by 100 gives the mole percentage, also referred as amount/amount percent [abbreviated as (n/n)% or mol %].

### Mass concentration

The conversion to and from mass concentration ρ$_i$ is given by:
    ${\begin{aligned}x_{i}&={\frac {\rho _{i}}{\rho }}{\frac {\bar {M}}{M_{i}}}\\[3pt]\Leftrightarrow \rho _{i}&=x_{i}\rho {\frac {M_{i}}{\bar {M}}}\end{aligned}}$
where M̄ is the average molar mass of the mixture.

### Molar concentration

The conversion to molar concentration c$_i$ is given by:
    ${\begin{aligned}c_{i}&=x_{i}c\\[3pt]&={\frac {x_{i}\rho }{\bar {M}}}={\frac {x_{i}\rho }{\sum _{j}x_{j}M_{j}}}\end{aligned}}$
where M̄ is the average molar mass of the solution, c is the total molar concentration and ρ is the density of the solution.

### Mass and molar mass

The mole fraction can be calculated from the masses m$_i$ and molar masses M$_i$ of the components:
    $x_{i}={\frac {\frac {m_{i}}{M_{i}}}{\sum _{j}{\frac {m_{j}}{M_{j}}}}}$

## Spatial variation and gradient

In a spatially non-uniform mixture, the mole fraction gradient triggers the phenomenon of diffusion.