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Mole fraction

In chemistry, the mole fraction or molar fraction (<math>x_i</math>) is defined as the amount of a constituent (expressed in moles), <math>n_i</math>, divided by the total amount of all constituents in a mixture, <math>n_{tot}</math>:[1]

<math>x_i = \frac{n_i}{n_{tot}}</math>

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

<math>\sum_{i=1}^{N} n_i = n_{tot} ; \; \sum_{i=1}^{N} x_i = 1</math>

The same concept expressed with a denominator of 100 is the mole percent or 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 <math>N_i</math> divided by the total number of all molecules <math>N_{tot}</math>. The mole fraction is sometimes denoted by the lowercase Greek letter <math alt="χ">\chi</math> (chi) instead of a Roman <math>x</math>.[2][3] For mixtures of gases, IUPAC recommends the letter <math>y</math>.[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 ratio 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.


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

Related quantities

Mass fraction

The mass fraction <math>w_i</math> can be calculated using the formula

<math>w_i = x_i \cdot \frac {M_i}{M}</math>

where <math>M_i</math> is the molar mass of the component <math>i</math> and <math>M</math> is the average molar mass of the mixture.

Replacing the expression of the molar mass:

<math>w_i = x_i \cdot \frac {M_i}{\sum_i x_i M_i}</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 <math>\rho_i</math> is given by:

<math>x_i = \frac{\rho_i}{\rho} \cdot \frac{M}{M_i}</math>

where <math>M</math> is the average molar mass of the mixture.

<math>\rho_i = x_i \rho \cdot \frac{M_i}{M}</math>

Molar concentration

The conversion to molar concentration <math>c_i</math> is given by:

<math>c_i = \fracTemplate:X i \cdot \rhoṃ = x_i c </math>


<math>c_i = \fracTemplate:X i \cdot \rhoTemplate:\sum i x i M i </math>

where <math>M</math> is the average molar mass of the solution, c total molar concentration and <math>\rho</math> is the density of the solution .

Mass and molar mass

The mole fraction can be calculated from the masses <math>m_i</math> and molar masses <math>M_i</math> of the components:

<math> x_i= \frac{{\fracTemplate:M iTemplate:M i}}{{\sum_i \fracTemplate:M iTemplate:M i}}</math>

Spatial variation and gradient

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


  1. ^ a b c IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "amount fraction".
  2. ^ Zumdahl, Steven S. (2008). Chemistry (8th ed. ed.). Cengage Learning. p. 201. ISBN 0-547-12532-1. 
  3. ^ Rickard, James N. Spencer, George M. Bodner, Lyman H. (2010). Chemistry : structure and dynamics. (5th ed. ed.). Hoboken, N.J.: Wiley. p. 357. ISBN 978-0-470-58711-9. 
  4. ^ Thompson, A.; Taylor, B. N. "The NIST Guide for the use of the International System of Units". National Institute of Standards and Technology. Retrieved 5 July 2014.