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]

${\displaystyle x_{i}={\frac {n_{i}}{n_{\mathrm {tot} }}}}$

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

${\displaystyle \sum _{i=1}^{N}n_{i}=n_{\mathrm {tot} };\;\sum _{i=1}^{N}x_{i}=1}$

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 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

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:

$w_{i}=x_{i}\cdot {\frac {M_{i}}{\sum _{i}x_{i}M_{i}}}$

Molar mixing ratio

The mixing of two pure components can be expressed introducing the amount or molar mixing ratio of them ${\displaystyle r_{n}={\frac {n_{2}}{n_{1}}}}$. Then the mole fractions of the components will be:

${\displaystyle x_{1}={\frac {1}{1+r_{n}}}}$

${\displaystyle x_{2}={\frac {r_{n}}{1+r_{n}}}}$

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:

$x_{i}={\frac {\rho _{i}}{\rho }}\cdot {\frac {M}{M_{i}}}$

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:

${\displaystyle c_{i}={\frac {x_{i}\cdot \rho }{M}}=x_{i}c}$

or

${\displaystyle c_{i}={\frac {x_{i}\cdot \rho }{\sum _{i}x_{i}M_{i}}}}$

where M 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:

${\displaystyle x_{i}={\frac {\frac {m_{i}}{M_{i}}}{\sum _{i}{\frac {m_{i}}{M_{i}}}}}}$

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". National Institute of Standards and Technology. Retrieved 5 July 2014.