Proportion of a constituent in a mixture
mole fraction Other names
molar fraction, amount fraction, amount-of-substance fraction Common symbols
x SI unit 1 Other units
mol/mol
In chemistry , the mole fraction or molar fraction , also called mole proportion or molar proportion , is a quantity defined as the ratio between the amount of a constituent substance, ni (expressed in unit of moles , symbol mol), and the total amount of all constituents in a mixture, n tot (also expressed in moles):[ 1]
x
i
=
n
i
n
t
o
t
{\displaystyle x_{i}={\frac {n_{i)){n_{\mathrm {tot} ))))
It is denoted xi (lowercase Roman letter x ), sometimes χi (lowercase Greek letter chi ).[ 2] [ 3] (For mixtures of gases, the letter y is recommended.[ 1] [ 4] )
It is a dimensionless quantity with dimension of
N
/
N
{\displaystyle {\mathsf {N))/{\mathsf {N))}
and dimensionless unit of moles per mole (mol/mol or mol ⋅ mol-1 ) or simply 1; metric prefixes may also be used (e.g., nmol/mol for 10-9 ).[ 5]
When expressed in percent , it is known as the mole percent or molar percentage (unit symbol %, sometimes "mol%", equivalent to cmol/mol for 10-2 ).
The mole fraction is called amount fraction by the International Union of Pure and Applied Chemistry (IUPAC)[ 1] and amount-of-substance fraction by the U.S. National Institute of Standards and Technology (NIST).[ 6] This nomenclature is part of the International System of Quantities (ISQ), as standardized in ISO 80000-9 ,[ 4] which deprecates "mole fraction" based on the unacceptability of mixing information with units when expressing the values of quantities.[ 6]
The sum of all the mole fractions in a mixture is equal to 1:
∑
i
=
1
N
n
i
=
n
t
o
t
;
∑
i
=
1
N
x
i
=
1
{\displaystyle \sum _{i=1}^{N}n_{i}=n_{\mathrm {tot} };\ \sum _{i=1}^{N}x_{i}=1}
Mole fraction is numerically identical to the number fraction , which is defined as the number of particles (molecules ) of a constituent Ni divided by the total number of all molecules N tot .
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 and volume fraction are others.
Mole fraction is used very frequently in the construction of phase diagrams . 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 = 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
In a ternary mixture one can express mole fractions of a component as functions of other components mole fraction and binary mole ratios:
x
1
=
1
−
x
2
1
+
x
3
x
1
x
3
=
1
−
x
2
1
+
x
1
x
3
{\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 formed at constant ratios like those above:
(
∂
x
1
∂
x
2
)
x
1
x
3
=
−
x
1
1
−
x
2
{\displaystyle \left({\frac {\partial x_{1)){\partial x_{2))}\right)_{\frac {x_{1)){x_{3))}=-{\frac {x_{1)){1-x_{2))))
or
(
∂
x
3
∂
x
2
)
x
1
x
3
=
−
x
3
1
−
x
2
{\displaystyle \left({\frac {\partial x_{3)){\partial x_{2))}\right)_{\frac {x_{1)){x_{3))}=-{\frac {x_{3)){1-x_{2))))
The ratios X , Y , and Z of mole fractions can be written for ternary and multicomponent systems:
X
=
x
3
x
1
+
x
3
Y
=
x
3
x
2
+
x
3
Z
=
x
2
x
1
+
x
2
{\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 for solving PDEs like:
(
∂
μ
2
∂
n
1
)
n
2
,
n
3
=
(
∂
μ
1
∂
n
2
)
n
1
,
n
3
{\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))}
or
(
∂
μ
2
∂
n
1
)
n
2
,
n
3
,
n
4
,
…
,
n
i
=
(
∂
μ
1
∂
n
2
)
n
1
,
n
3
,
n
4
,
…
,
n
i
{\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 to have differential quotient of mole amounts or fractions on one side.
(
∂
μ
2
∂
μ
1
)
n
2
,
n
3
=
−
(
∂
n
1
∂
n
2
)
μ
1
,
n
3
=
−
(
∂
x
1
∂
x
2
)
μ
1
,
n
3
{\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))}
or
(
∂
μ
2
∂
μ
1
)
n
2
,
n
3
,
n
4
,
…
,
n
i
=
−
(
∂
n
1
∂
n
2
)
μ
1
,
n
2
,
n
4
,
…
,
n
i
{\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))}
Mole amounts can be eliminated by forming ratios:
(
∂
n
1
∂
n
2
)
n
3
=
(
∂
n
1
n
3
∂
n
2
n
3
)
n
3
=
(
∂
x
1
x
3
∂
x
2
x
3
)
n
3
{\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:
(
∂
μ
2
∂
μ
1
)
n
2
n
3
=
−
(
∂
x
1
x
3
∂
x
2
x
3
)
μ
1
{\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 the ratio for the multicomponents system becomes
(
∂
μ
2
∂
μ
1
)
n
2
n
3
,
n
3
n
4
,
…
,
n
i
−
1
n
i
=
−
(
∂
x
1
x
3
∂
x
2
x
3
)
μ
1
,
n
3
n
4
,
…
,
n
i
−
1
n
i
{\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))))}
The mass fraction wi can be calculated using the formula
w
i
=
x
i
M
i
M
¯
=
x
i
M
i
∑
j
x
j
M
j
{\displaystyle w_{i}=x_{i}{\frac {M_{i)){\bar {M))}=x_{i}{\frac {M_{i)){\sum _{j}x_{j}M_{j))))
where Mi is the molar mass of the component i and M̄ is the average molar mass of the mixture.
The mixing of two pure components can be expressed introducing the amount or molar mixing ratio of them
r
n
=
n
2
n
1
{\displaystyle r_{n}={\frac {n_{2)){n_{1))))
. Then the mole fractions of the components will be:
x
1
=
1
1
+
r
n
x
2
=
r
n
1
+
r
n
{\displaystyle {\begin{aligned}x_{1}&={\frac {1}{1+r_{n))}\\[2pt]x_{2}&={\frac {r_{n)){1+r_{n))}\end{aligned))}
The amount ratio equals the ratio of mole fractions of components:
n
2
n
1
=
x
2
x
1
{\displaystyle {\frac {n_{2)){n_{1))}={\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 phase diagrams using, for instance, ternary plots .
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 x1(123) , x2(123) , x3(123) 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.
x
1
(
123
)
=
n
(
12
)
x
1
(
12
)
+
n
13
x
1
(
13
)
n
(
12
)
+
n
(
13
)
{\displaystyle x_{1(123)}={\frac {n_{(12)}x_{1(12)}+n_{13}x_{1(13))){n_{(12)}+n_{(13)))))
Multiplying mole fraction by 100 gives the mole percentage, also referred as amount/amount percent [abbreviated as (n/n)% or mol %].
The conversion to and from mass concentration ρi is given by:
x
i
=
ρ
i
ρ
M
¯
M
i
⇔
ρ
i
=
x
i
ρ
M
i
M
¯
{\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 M̄ is the average molar mass of the mixture.
Molar concentration [ edit ] The conversion to molar concentration ci is given by:
c
i
=
x
i
c
=
x
i
ρ
M
¯
=
x
i
ρ
∑
j
x
j
M
j
{\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 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 [ edit ] The mole fraction can be calculated from the masses mi and molar masses Mi of the components:
x
i
=
m
i
M
i
∑
j
m
j
M
j
{\displaystyle x_{i}={\frac {\frac {m_{i)){M_{i))}{\sum _{j}{\frac {m_{j)){M_{j))))))
Spatial variation and gradient [ edit ] In a spatially non-uniform mixture, the mole fraction gradient triggers the phenomenon of diffusion .
Constants Physical quantities Laws