Mass fraction (chemistry)

Not to be confused with Mass concentration (chemistry).

In chemistry, the mass fraction of a substance within a mixture is the ratio ${\displaystyle w_{i}}$ (alternatively denoted ${\displaystyle Y_{i}}$) of the mass ${\displaystyle m_{i}}$ of that substance to the total mass ${\displaystyle m_{\text{tot}}}$ of the mixture.^[1] Expressed as a formula, the mass fraction is:

${\displaystyle w_{i}={\frac {m_{i}}{m_{\text{tot}}}}.}$

Because the individual masses of the ingredients of a mixture sum to ${\displaystyle m_{\text{tot}}}$, their mass fractions sum to unity:

${\displaystyle \sum _{i=1}^{n}w_{i}=1.}$

Mass fraction can also be expressed, with a denominator of 100, as percentage by mass (in commercial contexts often called percentage by weight, abbreviated wt.% or % w/w; see mass versus weight). It is one way of expressing the composition of a mixture in a dimensionless size; mole fraction (percentage by moles, mol%) and volume fraction (percentage by volume, vol%) are others.

When the prevalences of interest are those of individual chemical elements, rather than of compounds or other substances, the term mass fraction can also refer to the ratio of the mass of an element to the total mass of a sample. In these contexts an alternative term is mass percent composition. The mass fraction of an element in a compound can be calculated from the compound's empirical formula^[2] or its chemical formula.^[3]

Terminology

Percent concentration does not refer to this quantity. This improper name persists, especially in elementary textbooks. In biology, the unit "%" is sometimes (incorrectly) used to denote mass concentration, also called mass/volume percentage. A solution with 1 g of solute dissolved in a final volume of 100 mL of solution would be labeled as "1%" or "1% m/v" (mass/volume). This is incorrect because the unit "%" can only be used for dimensionless quantities. Instead, the concentration should simply be given in units of g/mL. Percent solution or percentage solution are thus terms best reserved for mass percent solutions (m/m, m%, or mass solute/mass total solution after mixing), or volume percent solutions (v/v, v%, or volume solute per volume of total solution after mixing). The very ambiguous terms percent solution and percentage solutions with no other qualifiers continue to occasionally be encountered.

In thermal engineering, vapor quality is used for the mass fraction of vapor in the steam.

In alloys, especially those of noble metals, the term fineness is used for the mass fraction of the noble metal in the alloy.

Properties

The mass fraction is independent of temperature.

Related quantities

Mixing ratio

The mixing of two pure components can be expressed introducing the (mass) mixing ratio of them ${\displaystyle r_{m}={\frac {m_{2}}{m_{1}}}}$. Then the mass fractions of the components will be

${\displaystyle {\begin{aligned}w_{1}&={\frac {1}{1+r_{m}}},\\w_{2}&={\frac {r_{m}}{1+r_{m}}}.\end{aligned}}}$

The mass ratio equals the ratio of mass fractions of components:

${\displaystyle {\frac {m_{2}}{m_{1}}}={\frac {w_{2}}{w_{1}}}}$

due to division of both numerator and denominator by the sum of masses of components.

Mass concentration

The mass fraction of a component in a solution is the ratio of the mass concentration of that component ρ_i (density of that component in the mixture) to the density of solution ${\displaystyle \rho }$.

${\displaystyle w_{i}={\frac {\rho _{i}}{\rho }}.}$

Molar concentration

The relation to molar concentration is like that from above substituting the relation between mass and molar concentration:

${\displaystyle w_{i}={\frac {\rho _{i}}{\rho }}={\frac {c_{i}M_{i}}{\rho }},}$

where ${\displaystyle c_{i}}$ is the molar concentration, and ${\displaystyle M_{i}}$ is the molar mass of the component ${\displaystyle i}$.

Mass percentage

Mass percentage is defined as the mass fraction multiplied by 100.

Mole fraction

The mole fraction ${\displaystyle x_{i}}$ can be calculated using the formula

${\displaystyle x_{i}={\frac {w_{i}}{M_{i}}}{\bar {M}},}$

where ${\displaystyle M_{i}}$ is the molar mass of the component ${\displaystyle i}$, and ${\displaystyle {\bar {M}}}$ is the average molar mass of the mixture.

Replacing the expression of the molar-mass products,

${\displaystyle x_{i}={\frac {\frac {w_{i}}{M_{i}}}{\sum _{j}{\frac {w_{j}}{M_{j}}}}}.}$

Spatial variation and gradient

In a spatially non-uniform mixture, the mass fraction gradient gives rise to the phenomenon of diffusion.

See also

• Mass-flux fraction

References

1. IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "mass fraction". doi:10.1351/goldbook.M03722
2. Formula from Mass Composition.
3. "How to Calculate Mass Percent Composition". ThoughtCo. Retrieved 2018-01-05.