# Mole fraction

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:

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

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 (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:

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

or

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

These can be used for solving PDEs like:

or

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

or

Mole amounts can be eliminated by forming ratios:

Thus the ratio of chemical potentials becomes:

Similarly the ratio for the multicomponents system becomes

## Related quantities

### Mass fraction

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

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 . Then the mole fractions of the components will be:

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

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.

### 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:

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

### Molar concentration

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

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}## Spatial variation and gradient

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

## References

- 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. Retrieved 5 July 2014.