# Mole fraction

In chemistry , the unit of mole fraction or molar fraction ( Xi ) is defined as the sum of a component (expressed in moles ) , Ni  , the total sum of all constituents of a mixture (also expressed in moles). Divided by, n small child . [1] This expression is given below:

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


The sum of all the mole fractions equals 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 percentage , the molar percentage , or the molar ratio ( mol% ).

The mole fraction is also called the amount fraction . [1] This is similar to the number fraction , which is defined as the number of molecules of a component n i divided by the total number of all molecules n tot . The mole fraction is sometimes denoted by the lowercase Greek letter ( chi ) instead of the Roman x . [2] [3] For mixtures of gases, IUPAC recommends the letter y .

The United States National Institute of Standards a

nd Technology prefers the quantity fraction of a substance to the mole fraction because it does not have a unit mole name.

While mole fraction is the ratio of moles to moles, molar concentration is the quotient of moles to volume.

The mole fraction is a way of expressing the composition of a mixture of a dimensionless quantity ; Mass fraction (percentage by weight, wt%) and volume fraction ( percentage by volume, vol%) are others.

## Property

• It is not temperature dependent (such as the molar concentration ) and does not require knowledge of the density of the phase(s) involved.
• A mixture of known mole fraction can be prepared by weighing the appropriate mass of the components
• The measurement 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 the partial pressure and the total pressure of the mixture.
• The mole fractions of one component in a ternary mixture can be expressed as functions of the other components’ mole fractions and the binary mole ratio:
{\displaystyle {\begin{aligned}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{aligned}}}

Differential quotients can be made in a constant ratio like the ones given above:

{\displaystyle \left({\frac {\partial x_{1}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{1}}{1-x_{2}}}}


Come on

{\displaystyle \left({\frac {\partial x_{3}}{\partial x_{2}}}\right)_{\frac {x_{1}}{x_{3}}}=-{\frac {x_{3}}{1-x_{2}}}}


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

{\displaystyle {\begin{aligned}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{aligned}}}

These can be used to solve PDEs like:

{\displaystyle \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}}}


Come On

{\displaystyle \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 on the one hand for mole quantities or differential quotients of fractions.

{\displaystyle \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}}}


come on

{\displaystyle \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}}}

The amount of mole can be eliminated by making the ratio:

{\displaystyle \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}}}


Thus the ratio of chemical potentials becomes:

{\displaystyle \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 for a multicomponent system, the ratio becomes

{\displaystyle \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}}}}}

## related quantity

### mass fraction

The mass fraction Wi can be calculated using the formula

{\displaystyle w_{i}=x_{i}{\frac {M_{i}}{\bar {M}}}=x_{i}{\frac {M_{i}}{\sum _{j}x_{j}M_{j}}}}


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

### molar mixing ratio

A mixture of two pure components can be expressed by introducing their amounts or molar mixing ratios . Then the moles of the components will differ:

{\displaystyle r_{n}={\frac {n_{2}}{n_{1}}}}

{\displaystyle {\begin{aligned}x_{1}&={\frac {1}{1+r_{n}}}\\[2pt]x_{2}&={\frac {r_{n}}{1+r_{n}}}\end{aligned}}}


The sum ratio is equal to the ratio of the mole fractions of the components:

{\displaystyle {\frac {n_{2}}{n_{1}}}={\frac {x_{2}}{x_{1}}}}

Because of the division of both the numerator and the denominator by the sum of the molar amounts of the components. For example, this property has consequences for the representation of phase diagrams using ternary plots .

#### A ternary mixture is formed by mixing a binary mixture with a common component.

Mixing a binary mixture with a common component gives a ternary mixture with some mixing ratio between the three components. These mixture ratios from the ternary and related mole fractions of a ternary mixture x 1(123) , x 2(123) , x 3(123) can be expressed as a function of several mixture ratios, the mixing ratio between the components. To make the mixture ratio of a binary mixture and a binary mixture ternary.

{\displaystyle x_{1(123)}={\frac {n_{(12)}x_{1(12)}+n_{13}x_{1(13)}}{n_{(12)}+n_{(13)}}}}


### mole percentage

Multiplying the mole fraction by 100 gives the mole percentage, also called the amount/amount percentage [abbreviated as (n/n)%].

### mass concentration

The conversion mass concentration to and from is given by:

{\displaystyle {\begin{aligned}x_{i}&={\frac {\rho _{i}}{\rho }}{\frac {\bar {M}}{M_{i}}}\\[3pt]\Leftrightarrow \rho _{i}&=x_{i}\rho {\frac {M_{i}}{\bar {M}}}\end{aligned}}}

where  is the mean molar mass of the mixture.

### molar concentration

The conversion to molar concentration i is given by:

{\displaystyle {\begin{aligned}c_{i}&=x_{i}c\\[3pt]&={\frac {x_{i}\rho }{\bar {M}}}={\frac {x_{i}\rho }{\sum _{j}x_{j}M_{j}}}\end{aligned}}}


where m is the mean molar mass of the solution, c is the total molar concentration and is the density of the solution.

### mass and molar mass

The mole fraction can be calculated from the mass m i and the molar mass M i of the components:

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