# Chemistry:Mass fraction

__: Fraction of one substance's mass to the mass of the total mixture__

**Short description**In chemistry, the **mass fraction** of a substance within a mixture is the ratio [math]\displaystyle{ w_i }[/math] (alternatively denoted [math]\displaystyle{ Y_i }[/math]) of the mass [math]\displaystyle{ m_i }[/math] of that substance to the total mass [math]\displaystyle{ m_\text{tot} }[/math] of the mixture.^{[1]} Expressed as a formula, the mass fraction is:

- [math]\displaystyle{ w_i = \frac {m_i}{m_\text{tot}}. }[/math]

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

- [math]\displaystyle{ \sum_{i=1}^{n} w_i = 1. }[/math]

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%*; 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

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*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 until phase change occurs.

## Related quantities

### Mixing ratio

The mixing of two pure components can be expressed introducing the (mass) mixing ratio of them [math]\displaystyle{ r_m = \frac{m_2}{m_1} }[/math]. Then the mass fractions of the components will be

- [math]\displaystyle{ \begin{align} w_1 &= \frac{1}{1 + r_m}, \\ w_2 &= \frac{r_m}{1 + r_m}. \end{align} }[/math]

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

- [math]\displaystyle{ \frac{m_2}{m_1} = \frac{w_2}{w_1} }[/math]

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 [math]\displaystyle{ \rho }[/math].

- [math]\displaystyle{ w_i = \frac{\rho_i}{\rho}. }[/math]

### Molar concentration

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

- [math]\displaystyle{ w_i = \frac{\rho_i}{\rho} = \frac{c_i M_i}{\rho}, }[/math]

where [math]\displaystyle{ c_i }[/math] is the molar concentration, and [math]\displaystyle{ M_i }[/math] is the molar mass of the component [math]\displaystyle{ i }[/math].

### Mass percentage

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**Percentage by mass** can also be expressed as *percentage by weight*, abbreviated *wt%*, or weight–weight percentage.

### Mole fraction

The mole fraction [math]\displaystyle{ x_i }[/math] can be calculated using the formula

- [math]\displaystyle{ x_i = \frac{w_i}{M_i} \bar{M}, }[/math]

where [math]\displaystyle{ M_i }[/math] is the molar mass of the component [math]\displaystyle{ i }[/math], and [math]\displaystyle{ \bar{M} }[/math] is the average molar mass of the mixture.

Replacing the expression of the molar-mass products,

- [math]\displaystyle{ x_i = \frac{\frac{w_i}{M_i}}{\sum_j \frac{w_j}{M_j}}. }[/math]

## Spatial variation and gradient

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

## See also

## References

- ↑ IUPAC,
*Compendium of Chemical Terminology*, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "mass fraction". doi:10.1351/goldbook.M03722 - ↑ Formula from Mass Composition.
- ↑ "How to Calculate Mass Percent Composition".
*ThoughtCo*. http://chemistry.about.com/od/workedchemistryproblems/a/mass-percent-worked-problem.htm.