Mole fraction

 ( Dimensionless Quantities Of Chemistry ) 

It is a dimensionless quantity with dimension of $\backslash mathsf\{N\}/\backslash mathsf\{N\}$ and dimensionless unit of moles per mole ( mol/mol or molmol⁻¹) or simply 1; metric prefixes may also be used (e.g., nmol/mol for nano-). When expressed in percent, it is known as the mole percent or molar percentage (unit symbol %, sometimes "mol%", equivalent to cmol/mol for centi-). The mole fraction is called amount fraction by the International Union of Pure and Applied Chemistry (IUPAC) and amount-of-substance fraction by the U.S. National Institute of Standards and Technology (NIST). This nomenclature is part of the International System of Quantities (ISQ), as standardized in ISO 80000-9, which deprecates "mole fraction" based on the unacceptability of mixing information with units when expressing the values of quantities.

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

$\backslash sum\_\{i=1\}^\{N\}\; n\_i\; =\; n\_\backslash mathrm\{tot\}\; ;\; \backslash \; \backslash sum\_\{i=1\}^\{N\}\; x\_i\; =\; 1$

• 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 , 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:
• : $\backslash begin\{align\}$

 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}}
     

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

$\backslash left(\backslash frac\{\backslash partial\; x\_1\}\{\backslash partial\; x\_2\}\backslash right)\_\{\backslash frac\{x\_1\}\{x\_3\}\}\; =\; -\backslash frac\{x\_1\}\{1\; -\; x\_2\}$

or

$\backslash left(\backslash frac\{\backslash partial\; x\_3\}\{\backslash partial\; x\_2\}\backslash right)\_\{\backslash frac\{x\_1\}\{x\_3\}\}\; =\; -\backslash frac\{x\_3\}\{1\; -\; x\_2\}$

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

$\backslash begin\{align\}$

 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}
     

These can be used for solving PDEs like:

 \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}
     

or

 \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}
     

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

  \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}
     

or

  \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}
     

Mole amounts can be eliminated by forming ratios:

 \left(\frac{\partial n_1}\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}
     

Thus the ratio of chemical potentials becomes:

  \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}
     

Similarly the ratio for the multicomponents system becomes

  \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}}
     

$w\_i\; =\; x\_i\; \backslash frac\{M\_i\}\{\backslash bar\{M\}\}\; =\; x\_i\; \backslash frac\; \{M\_i\}\{\backslash sum\_j\; x\_j\; M\_j\}$

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

$\backslash begin\{align\}$

 x_1 &= \frac{1}{1 + r_n} \\[2pt]
 x_2 &= \frac{r_n}{1 + r_n}
     

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

$\backslash frac\{n\_2\}\{n\_1\}\; =\; \backslash frac\{x\_2\}\{x\_1\}$

due to division of both numerator and denominator by the sum of molar amounts of components. This property has consequences for representations of using, for instance, .

$x\_\{1(123)\}\; =\; \backslash frac\{n\_\{(12)\}\; x\_\{1(12)\}\; +\; n\_\{13\}\; x\_\{1(13)\}\}\{n\_\{(12)\}\; +\; n\_\{(13)\}\}$

$\backslash begin\{align\}$

                    x_i &= \frac{\rho_i}{\rho} \frac{\bar{M}}{M_i} \\[3pt]
 \Leftrightarrow \rho_i &= x_i \rho \frac{M_i}{\bar{M}}
     

where M̄ is the average molar mass of the mixture.

$\backslash begin\{align\}$

 c_i &= x_i c \\[3pt]
     &= \frac{x_i\rho}{\bar{M}} = \frac{x_i\rho}{\sum_j x_j M_j}
     

where M̄ is the average molar mass of the solution, c is the total molar concentration and ρ is the density of the solution.

$x\_i\; =\; \backslash frac\{\backslash frac\{m\_i\}\{M\_i\}\}\{\backslash sum\_j\; \backslash frac\{m\_j\}\{M\_j\}\}$