# Mass fraction (chemistry)

In chemistry, the mass fraction ${\displaystyle w_{i}}$ is the ratio of one substance with mass ${\displaystyle m_{i}}$ to the mass of the total mixture ${\displaystyle m_{\text{tot}}}$, defined as[1]

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

The symbol ${\displaystyle Y_{i}}$ is also used to denote mass fraction. The sum of all the mass fractions is equal to 1:

${\displaystyle \sum _{i=1}^{N}m_{i}=m_{\text{tot}};\sum _{i=1}^{N}w_{i}=1}$

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%; 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.

For elemental analysis, mass fraction (or mass percent composition) can also refer to the ratio of the mass of one element to the total mass of a compound. It can be calculated for any compound using its 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% = mass solute/mass total solution after mixing), or "volume percent solutions" (v/v = v% = 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}}}\\[3pt]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 ρi (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

The mass percentage is sometimes called weight percent (wt%) or weight-weight percentage.

### Mole fraction

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

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

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

Replacing the expression of the molar mass-products:

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

## Spatial variation and gradient

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

## References

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

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