Editing Mole fraction

{{Short description|Proportion of a constituent in a mixture}}
{{Infobox physical quantity
| name = mole fraction
| othernames = molar fraction, amount fraction, amount-of-substance fraction
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| unit = 1
| otherunits = mol/mol
| symbols = ''x''
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}}

In [[chemistry]], the '''[[Mole (unit)|mole]] fraction''' or '''molar fraction''', also called '''mole proportion''' or '''molar proportion''', is a [[quantity (science)|quantity]] defined as the [[ratio]] between the [[amount of substance|amount]] of a constituent substance, ''n<sub>i</sub>'' (expressed in [[unit of measurement|unit]] of [[mole (unit)|moles]], symbol mol), and the total amount of all constituents in a mixture, ''n''<sub>tot</sub> (also expressed in moles):<ref name="goldbook">{{GoldBookRef | file = A00296 | title = amount fraction}}</ref>
:<math>x_i = \frac{n_i}{n_\mathrm{tot}}</math>
It is [[mathematical symbol|denoted]] '''''x<sub>i</sub>''''' (lowercase [[Latin alphabet|Roman]] letter ''[[x]]''), sometimes '''{{mvar|χ<sub>i</sub>}}''' (lowercase [[Greek alphabet|Greek]] letter [[Chi (letter)|chi]]).<ref>{{cite book|last=Zumdahl|first=Steven S.|title=Chemistry|year=2008|publisher=Cengage Learning|isbn=978-0-547-12532-9|edition=8th|page=201}}</ref><ref>{{cite book|last1=Rickard|first1=James N.|last2=Spencer|first2=George M.|last3=Bodner|first3=Lyman H.|title=Chemistry: Structure and Dynamics|year=2010|publisher=Wiley|location=Hoboken, N.J.|isbn=978-0-470-58711-9|edition=5th|page=357}}</ref> (For mixtures of gases, the letter ''y'' is recommended.<ref name="goldbook"/><ref name=ISO/>)

It is a [[dimensionless quantity]] with [[dimension (physics)|dimension]] of <math>\mathsf{N}/\mathsf{N}</math> and [[dimensionless unit]] of '''moles per mole''' ('''mol/mol''' or '''mol{{sdot}}mol<sup>−1</sup>''') or simply 1; [[metric prefixes]] may also be used (e.g., nmol/mol for [[nano-|10<sup>−9</sup>]]).<ref name="BIPM c402">{{cite web | title=SI Brochure | website=BIPM | url=https://www.bipm.org/en/publications/si-brochure | access-date=2023-08-29}}</ref>
When expressed in [[percent]], it is known as the '''mole percent''' or '''molar percentage''' (unit symbol %, sometimes "mol%", equivalent to cmol/mol for [[centi-|10<sup>−2</sup>]]).
The mole fraction is called '''amount fraction''' by the [[International Union of Pure and Applied Chemistry]] (IUPAC)<ref name="goldbook"/> and '''amount-of-substance fraction''' by the U.S. [[National Institute of Standards and Technology]] (NIST).<ref name=NIST>{{cite web|last1=Thompson|first1=A.|last2=Taylor|first2=B. N.|title=The NIST Guide for the use of the International System of Units|date=2 July 2009|url=https://www.nist.gov/pml/pubs/sp811/index.cfm|publisher=National Institute of Standards and Technology|access-date=5 July 2014}}</ref> This nomenclature is part of the [[International System of Quantities]] (ISQ), as standardized in [[ISO 80000-9]],<ref name=ISO>{{cite web | title=ISO 80000-9:2019 Quantities and units — Part 9: Physical chemistry and molecular physics | website=ISO | date=2013-08-20 | url=https://www.iso.org/standard/64979.html | access-date=2023-08-29}}</ref> which deprecates "mole fraction" based on the unacceptability of mixing information with units when expressing the values of quantities.<ref name=NIST/>

The sum of all the mole fractions in a mixture is equal to 1:
:<math>\sum_{i=1}^{N} n_i = n_\mathrm{tot} ; \ \sum_{i=1}^{N} x_i = 1</math>
Mole fraction is numerically identical to the '''number fraction''', which is defined as the [[number of particles]] ([[molecule]]s) of a constituent ''N<sub>i</sub>'' divided by the total number of all molecules ''N''<sub>tot</sub>. 
Whereas mole fraction is a ratio of amounts to amounts (in units of moles per moles), [[molar concentration]] is a quotient of amount to volume (in units of moles per litre).
Other ways of expressing the composition of a mixture as a [[dimensionless quantity]] are [[mass fraction (chemistry)|mass fraction]] and [[volume fraction]].

==Properties==
Mole fraction is used very frequently in the construction of [[phase diagram]]s. 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''&nbsp;=&nbsp;0.1 and ''x''&nbsp;=&nbsp;0.9, the roles of 'solvent' and 'solute' are reversed.
* In a mixture of [[ideal gas]]es, 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:
*: <math>\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}}
\end{align}</math>

Differential quotients can be formed at constant ratios like those above:
: <math>\left(\frac{\partial x_1}{\partial x_2}\right)_{\frac{x_1}{x_3}} = -\frac{x_1}{1 - x_2}</math>

or 
: <math>\left(\frac{\partial x_3}{\partial x_2}\right)_{\frac{x_1}{x_3}} = -\frac{x_3}{1 - x_2}</math>

The ratios ''X'', ''Y'', and ''Z'' of mole fractions can be written for ternary and multicomponent systems:
: <math>\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}
\end{align}</math>

These can be used for solving PDEs like:
: <math>
  \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}
</math>

or 
: <math>
  \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}
</math>

This equality can be rearranged to have differential quotient of mole amounts or fractions on one side.
: <math>
   \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}
</math>

or 
: <math>
   \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}
</math>

Mole amounts can be eliminated by forming ratios:
: <math>
  \left(\frac{\partial n_1}{{\partial n_2}}\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}
</math>

Thus the ratio of chemical potentials becomes:
: <math>
   \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}
</math>

Similarly the ratio for the multicomponents system becomes
: <math>
   \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}}
</math>

==Related quantities==
===Mass fraction===
The [[mass fraction (chemistry)|mass fraction]] ''w<sub>i</sub>'' can be calculated using the formula
:<math>w_i = x_i \frac{M_i}{\bar{M}} = x_i \frac {M_i}{\sum_j x_j M_j}</math>

where ''M<sub>i</sub>'' 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 <math>r_n = \frac{n_2}{n_1}</math>. Then the mole fractions of the components will be:
:<math>\begin{align}
  x_1 &= \frac{1}{1 + r_n} \\[2pt]
  x_2 &= \frac{r_n}{1 + r_n}
\end{align}</math>

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

:<math>\frac{n_2}{n_1} = \frac{x_2}{x_1}</math>

due to division of both numerator and denominator by the sum of molar amounts of components. This property has consequences for representations of [[phase diagram]]s using, for instance, [[ternary plot]]s.

====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<sub>1(123)</sub>, x<sub>2(123)</sub>, x<sub>3(123)</sub> 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.
:<math>x_{1(123)} = \frac{n_{(12)} x_{1(12)} + n_{13} x_{1(13)}}{n_{(12)} + n_{(13)}}</math>

===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 (chemistry)|mass concentration]] ''ρ<sub>i</sub>'' is given by:
:<math>\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}}
\end{align}</math>

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

===Molar concentration===
The conversion to [[molar concentration]] ''c<sub>i</sub>'' is given by:
:<math>\begin{align}
  c_i &= x_i c \\[3pt]
      &= \frac{x_i\rho}{\bar{M}} = \frac{x_i\rho}{\sum_j x_j M_j}
\end{align}</math>

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 [[mass]]es ''m<sub>i</sub>'' and [[molar mass]]es ''M<sub>i</sub>'' of the components:
:<math>x_i = \frac{\frac{m_i}{M_i}}{\sum_j \frac{m_j}{M_j}}</math>

==Spatial variation and gradient==
In a [[inhomogeneous|spatially non-uniform]] mixture, the mole fraction [[gradient]] triggers the phenomenon of [[diffusion]].

==References==
{{Reflist}}

{{Mole concepts}}
{{Chemical solutions}}
{{Authority control}}

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[[Category:Dimensionless quantities of chemistry]]