Mole fraction

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

$x_i = \frac{n_i}{n_{tot}}$

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

$\sum_{i=1}^{N} n_i = n_{tot} ; \; \sum_{i=1}^{N} x_i = 1$

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 $N_i$ 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 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.

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

Related quantities

Mass fraction

The mass fraction $w_i$ can be calculated using the formula

$w_i = x_i \cdot \frac {M_i}{M}$

where $M_i$ is the molar mass of the component $i$ and $M$ is the average molar mass of the mixture.

Replacing the expression of the molar mass:

$w_i = x_i \cdot \frac {M_i}{\sum_i x_i M_i}$

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 $\rho_i$ is given by:

$x_i = \frac{\rho_i}{\rho} \cdot \frac{M}{M_i}$

where $M$ is the average molar mass of the mixture.

$\rho_i = x_i \rho \cdot \frac{M_i}{M}$

Molar concentration

The conversion to molar concentration $c_i$ is given by:

$c_i = \frac{{x_i \cdot \rho}}{{M}} = x_i c$

or

$c_i = \frac{{x_i \cdot \rho}}{{\sum_i x_i M_i}}$

where $M$ is the average molar mass of the solution, c total molar concentration and $\rho$ 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:

$x_i= \frac{{\frac{{m_i}}{{M_i}}}}{{\sum_i \frac{{m_i}}{{M_i}}}}$

Spatial variation and gradient

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

References

1. 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.). 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.). 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". http://physics.nist.gov/Pubs/SP811/sec08.html. National Institute of Standards and Technology. Retrieved 5 July 2014. External link in |website= (help)