# Chemistry:Mole fraction

__: Proportion of a constituent in a mixture__

**Short description**In chemistry, the **mole fraction** or **molar fraction** (* x_{i}* or

**χ**) is defined as unit of the amount of a constituent (expressed in moles),

_{i}*n*, divided by the total amount of all constituents in a mixture (also expressed in moles),

_{i}*n*

_{tot}.

^{[1]}This expression is given below:

- [math]\displaystyle{ x_i = \frac{n_i}{n_\mathrm{tot}} }[/math]

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

- [math]\displaystyle{ \sum_{i=1}^{N} n_i = n_\mathrm{tot} ; \ \sum_{i=1}^{N} x_i = 1. }[/math]

The same concept expressed with a denominator of 100 is the **mole percent**, **molar percentage** or **molar proportion** (**mol%**).

The mole fraction is also called the **amount fraction**.^{[1]} It is identical to the **number fraction**, which is defined as the number of molecules of a constituent *N _{i}* divided by the total number of all molecules

*N*

_{tot}. The mole fraction is sometimes denoted by the lowercase Greek letter χ (chi) instead of a Roman

*x*.

^{[2]}

^{[3]}For mixtures of gases, IUPAC recommends the letter

*y*.

^{[1]}

The National Institute of Standards and Technology of the United States prefers the term **amount-of-substance fraction** over mole fraction because it does not contain the name of the unit mole.^{[4]}

Whereas mole fraction is a ratio of moles to moles, molar concentration is a quotient of moles to volume.

The mole fraction is one way of expressing the composition of a mixture with a dimensionless quantity; mass fraction (percentage by weight, wt%) and volume fraction (percentage by volume, vol%) are others.

## Properties

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

- it is not temperature dependent (such as 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:
- [math]\displaystyle{ \begin{align} 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{align} }[/math]

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

- [math]\displaystyle{ \left(\frac{\partial x_1}{\partial x_2}\right)_{\frac{x_1}{x_3}} = -\frac{x_1}{1 - x_2} }[/math]

or

- [math]\displaystyle{ \left(\frac{\partial x_3}{\partial x_2}\right)_{\frac{x_1}{x_3}} = -\frac{x_3}{1 - x_2} }[/math]

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

- [math]\displaystyle{ \begin{align} 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{align} }[/math]

These can be used for solving PDEs like:

- [math]\displaystyle{ \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} }[/math]

or

- [math]\displaystyle{ \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} }[/math]

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

- [math]\displaystyle{ \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} }[/math]

or

- [math]\displaystyle{ \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} }[/math]

Mole amounts can be eliminated by forming ratios:

- [math]\displaystyle{ \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} }[/math]

Thus the ratio of chemical potentials becomes:

- [math]\displaystyle{ \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} }[/math]

Similarly the ratio for the multicomponents system becomes

- [math]\displaystyle{ \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}} }[/math]

## Related quantities

### Mass fraction

The mass fraction *w _{i}* can be calculated using the formula

- [math]\displaystyle{ w_i = x_i \frac{M_i}{\bar{M}} = x_i \frac {M_i}{\sum_j x_j M_j} }[/math]

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 [math]\displaystyle{ r_n = \frac{n_2}{n_1} }[/math]. Then the mole fractions of the components will be:

- [math]\displaystyle{ \begin{align} x_1 &= \frac{1}{1 + r_n} \\[2pt] x_2 &= \frac{r_n}{1 + r_n} \end{align} }[/math]

The amount ratio equals the ratio of mole fractions of components:

- [math]\displaystyle{ \frac{n_2}{n_1} = \frac{x_2}{x_1} }[/math]

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.

- [math]\displaystyle{ x_{1(123)} = \frac{n_{(12)} x_{1(12)} + n_{13} x_{1(13)}}{n_{(12)} + n_{(13)}} }[/math]

### Mole percentage

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

### Mass concentration

The conversion to and from mass concentration *ρ _{i}* is given by:

- [math]\displaystyle{ \begin{align} x_i &= \frac{\rho_i}{\rho} \frac{\bar{M}}{M_i} \\[3pt] \Leftrightarrow \rho_i &= x_i \rho \frac{M_i}{\bar{M}} \end{align} }[/math]

where *M̄* is the average molar mass of the mixture.

### Molar concentration

The conversion to molar concentration *c _{i}* is given by:

- [math]\displaystyle{ \begin{align} c_i &= x_i c \\[3pt] &= \frac{x_i\rho}{\bar{M}} = \frac{x_i\rho}{\sum_j x_j M_j} \end{align} }[/math]

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*of the components:

_{i}- [math]\displaystyle{ x_i = \frac{\frac{m_i}{M_i}}{\sum_j \frac{m_j}{M_j}} }[/math]

## Spatial variation and gradient

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

## References

- ↑
^{1.0}^{1.1}^{1.2}IUPAC,*Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "amount fraction". doi:10.1351/goldbook.A00296 - ↑ Zumdahl, Steven S. (2008).
*Chemistry*(8th ed.). Cengage Learning. p. 201. ISBN 978-0-547-12532-9. - ↑ Rickard, James N.; Spencer, George M.; Bodner, Lyman H. (2010).
*Chemistry: Structure and Dynamics*(5th ed.). Hoboken, N.J.: Wiley. p. 357. ISBN 978-0-470-58711-9. - ↑ Thompson, A.; Taylor, B. N. (2 July 2009). "The NIST Guide for the use of the International System of Units". National Institute of Standards and Technology. https://www.nist.gov/pml/pubs/sp811/index.cfm.

Original source: https://en.wikipedia.org/wiki/Mole fraction.
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