# Mass fraction

The mass fraction ( symbol : w , sometimes also ω , y or ξ ), formerly also referred to as mass fraction , is a content quantity according to DIN 1310 , i.e. a physical-chemical quantity for the quantitative description of the composition of substance mixtures / mixed phases . Here, the mass of a mixture component under consideration is related to the sum of the masses of all mixture components , the mass fraction thus indicates the relative proportion of the mass of a mixture component under consideration in the total mass of the mixture.

## Definition and characteristics

The following table is in the size equations distinguish between

• the simple case of a binary mixture ( Z = 2, two-substance mixture of components i and j , for example the solution of a single substance i in a solvent j ) and
• the generally applicable formulation for a mixture of substances made up of a total of Z components (index z as a general index for the sums , includes i and possibly j ).
binary mixture ( Z = 2) general mixture ( Z components)
definition ${\ displaystyle w_ {i} = {\ frac {m_ {i}} {m_ {i} + m_ {j}}}}$ ${\ displaystyle w_ {i} = {\ frac {m_ {i}} {m}} \ \ {\ text {with}} \ \ m = \ sum _ {z = 1} ^ {Z} m_ {z} }$
Range of values ${\ displaystyle 0 \ leq w_ {i} \ leq 1}$
Sum criterion ${\ displaystyle w_ {i} + w_ {j} = 1 \ \ Rightarrow \ w_ {j} = 1-w_ {i}}$ ${\ displaystyle \ sum _ {z = 1} ^ {Z} w_ {z} = 1 \ \ Rightarrow \ w_ {Z} = 1- \ sum _ {z = 1} ^ {Z-1} w_ {z} }$

The mass fraction w i is defined as the value of the quotient of the mass m i of the considered mixture component i and the total mass m of the mixture. The latter is the sum of the masses of all components ( including i ) of the mixture.

As the quotient of two dimensions of the same size, the mass fraction is a quantity of the dimension number and can be specified, as in the table above, by a pure decimal number without a unit of measure , alternatively also with the addition of a fraction of the same units ( kg / kg or g / g), possibly with different Decimal prefixes (e.g. g / kg), or with auxiliary units such as percent (% = 1/100), per thousand (‰ = 1/1000) or parts per million (1 ppm = 1/1 000 000). However, outdated information that is no longer in accordance with the standard should be avoided, such as B. mass percent (abbreviations for example Ma% or m% ) or weight percent (abbreviations for example wt .-% , wt.% Or wt% ) or weight percent ; also the designation of weight percent ( English for weight percent ) and their abbreviations ( wt.% or wt% ) are sometimes encountered, further realizations also as % (m / m) or % (w / w) . Instead, the intended content must be clearly described, for example, instead of “13.5 percent by weight”, the following should be formulated today: “The mass fraction of the mixture component i is 13.5%” or in the form of an equation: “ w i  = 13.5%”.

The mass fraction w i of a considered mixture component i can assume numerical values ​​between 0 = 0% (component i is not contained in the mixture) and 1 = 100% (component i is present as a pure substance ).

The mass fractions of all components of a mixture add up to 1 = 100%. From this it follows that the knowledge or determination of the mass fractions of Z  - 1 components is sufficient (in the case of a two-substance mixture, the mass fraction of one component), since the mass fraction of the remaining component can be calculated simply by calculating the difference to 1 = 100%.

The values ​​of the mass fractions for a substance mixture of a given composition are - in contrast to the volume-related content quantities ( concentrations , volume fraction , volume ratio ) - independent of temperature and pressure , since the masses of the mixture components, in contrast to the volumes, do not vary with temperature or pressure change, provided no material conversions occur.

The mass fraction is used in numerous application areas in various fields, especially chemistry , but also mineralogy , petrology , materials science and materials science , for example to describe the composition of rocks , minerals ( mixed crystals ) and alloys or to set up Tw phase diagrams .

## Relationships with other salary levels

The following table shows the relationships between the mass fraction w i and the other content values ​​defined in DIN 1310 in the form of size equations . M stands for the molar mass , ρ for the density of the respective pure substance determined by the index (at the same pressure and temperature as in the substance mixture). The index z in turn serves as a general index for the sums and includes i .

Relationships of the mass fraction w i with other salary quantities
Masses - ... Amount of substance - ... Particle number - ... Volume - ...
... - share Mass fraction w Amount of substance fraction x Particle number fraction X Volume fraction φ
${\ displaystyle w_ {i}}$ ${\ displaystyle w_ {i} = {\ frac {x_ {i} \ cdot M_ {i}} {\ sum _ {z = 1} ^ {Z} (x_ {z} \ cdot M_ {z})}} }$ ${\ displaystyle w_ {i} = {\ frac {X_ {i} \ cdot M_ {i}} {\ sum _ {z = 1} ^ {Z} (X_ {z} \ cdot M_ {z})}} }$ ${\ displaystyle w_ {i} = {\ frac {\ varphi _ {i} \ cdot \ rho _ {i}} {\ sum _ {z = 1} ^ {Z} (\ varphi _ {z} \ cdot \ rho _ {z})}}}$
… - concentration Mass concentration β Molar concentration c Particle number concentration C Volume concentration σ
${\ displaystyle w_ {i} = {\ frac {\ beta _ {i}} {\ sum _ {z = 1} ^ {Z} \ beta _ {z}}}}$ ${\ displaystyle w_ {i} = {\ frac {c_ {i} \ cdot M_ {i}} {\ sum _ {z = 1} ^ {Z} (c_ {z} \ cdot M_ {z})}} }$ ${\ displaystyle w_ {i} = {\ frac {C_ {i} \ cdot M_ {i}} {\ sum _ {z = 1} ^ {Z} (C_ {z} \ cdot M_ {z})}} }$ ${\ displaystyle w_ {i} = {\ frac {\ sigma _ {i} \ cdot \ rho _ {i}} {\ sum _ {z = 1} ^ {Z} (\ sigma _ {z} \ cdot \ rho _ {z})}}}$
... - ratio Mass ratio ζ Molar ratio r Particle number ratio R Volume ratio ψ
${\ displaystyle w_ {i} = {\ frac {1} {\ sum _ {z = 1} ^ {Z} \ zeta _ {zi}}}}$ ${\ displaystyle w_ {i} = {\ frac {M_ {i}} {\ sum _ {z = 1} ^ {Z} (r_ {zi} \ cdot M_ {z})}}}$ ${\ displaystyle w_ {i} = {\ frac {M_ {i}} {\ sum _ {z = 1} ^ {Z} (R_ {zi} \ cdot M_ {z})}}}$ ${\ displaystyle w_ {i} = {\ frac {\ rho _ {i}} {\ sum _ {z = 1} ^ {Z} (\ psi _ {zi} \ cdot \ rho _ {z})}} }$
Quotient
amount of substance / mass
Molality b
${\ displaystyle w_ {i} = b_ {i} \ cdot M_ {i} \ cdot w_ {j}}$ ( i = solute, j = solvent)
specific amount of partial substances q
${\ displaystyle w_ {i} = q_ {i} \ cdot M_ {i}}$

In the table above in the equations in the mole fraction x and Teilchenzahlanteil X occurring denominator - Terme are equal to the average molar mass of the material mixture and can be replaced in accordance with: ${\ displaystyle {\ overline {M}}}$

${\ displaystyle \ sum _ {z = 1} ^ {Z} (x_ {z} \ cdot M_ {z}) = \ sum _ {z = 1} ^ {Z} (X_ {z} \ cdot M_ {z }) = {\ overline {M}}}$

Furthermore, the denominator terms in the equations for the mass concentration β , the molar concentration c and the volume concentration σ correspond to the density ρ of the mixed phase, and for the particle number concentration C the product of Avogadro's constant N A and mixed phase density ρ :

${\ displaystyle \ sum _ {z = 1} ^ {Z} \ beta _ {z} = \ sum _ {z = 1} ^ {Z} (c_ {z} \ cdot M_ {z}) = \ sum _ {z = 1} ^ {Z} (\ sigma _ {z} \ cdot \ rho _ {z}) = \ rho \ quad {\ text {or}} \ quad \ sum _ {z = 1} ^ { Z} (C_ {z} \ cdot M_ {z}) = N _ {\ mathrm {A}} \ cdot \ rho}$

## Examples

### Solution of sodium chloride in water

An aqueous solution of common salt ( sodium chloride NaCl) is prepared from 6 grams of NaCl and 194 grams of water (H 2 O); the total mass of the solution is thus 200 grams. The mass fractions of NaCl or H 2 O in this solution are then:

{\ displaystyle {\ begin {aligned} w _ {\ mathrm {NaCl}} & = {\ frac {m _ {\ mathrm {NaCl}}} {m _ {\ mathrm {NaCl}} + m _ {\ mathrm {H_ {2 } O}}}} = \ mathrm {\ frac {6 \ g} {6 \ g + 194 \ g}} = \ mathrm {\ frac {6 \ g} {200 \ g}} = 0 {,} 03 = 3 \ \% \\ w _ {\ mathrm {H_ {2} O}} & = 1-w _ {\ mathrm {NaCl}} = 0 {,} 97 = 97 \ \% \ end {aligned}}}

### Nitrogen and oxygen in air

Air as the gas mixture of the earth's atmosphere contains the two main components nitrogen (particles: N 2 molecules) and oxygen (particles: O 2 molecules). When viewed approximately as a mixture of ideal gases , the usually tabulated mean volume fractions of the individual gases in dry air at sea level (N 2 : approx. 78.1%; O 2 : approx. 20.9%) are to be equated with the molar proportions x , so:

${\ displaystyle x _ {\ mathrm {N_ {2}}} \ approx 0 {,} 781 = 78 {,} 1 \ \% \ qquad x _ {\ mathrm {O_ {2}}} \ approx 0 {,} 209 = 20 {,} 9 \ \%}$

With the help of the molar masses of nitrogen and oxygen and the mean molar mass of dry air of about 28.95 g mol −1 , the mass fractions w of nitrogen and oxygen can be determined:

${\ displaystyle w _ {\ mathrm {N_ {2}}} = {\ frac {x _ {\ mathrm {N_ {2}}} \ cdot M _ {\ mathrm {N_ {2}}}} {{\ overline {M }} _ {\ mathrm {air}}}} \ approx \ mathrm {\ frac {0 {,} 781 \ cdot 28 {,} 01 \ g \ cdot mol ^ {- 1}} {28 {,} 95 \ g \ cdot mol ^ {- 1}}} \ approx 0 {,} 756 = 75 {,} 6 \ \%}$
${\ displaystyle w _ {\ mathrm {O_ {2}}} = {\ frac {x _ {\ mathrm {O_ {2}}} \ cdot M _ {\ mathrm {O_ {2}}}} {{\ overline {M }} _ {\ mathrm {air}}}} \ approx \ mathrm {\ frac {0 {,} 209 \ times 32 {,} 00 \ g \ times mol ^ {- 1}} {28 {,} 95 \ g \ cdot mol ^ {- 1}}} \ approx 0 {,} 231 = 23 {,} 1 \ \%}$

In reality the air is not completely dry; Due to the water vapor as an additional component in the mixture, the proportions and mass of nitrogen and oxygen are somewhat smaller.

## Individual evidence

1. a b c Standard DIN 1310 : Composition of mixed phases (gas mixtures, solutions, mixed crystals); Terms, symbols. February 1984.
2. a b Standard DIN EN ISO 80000-9 : Quantities and units - Part 9: Physical chemistry and molecular physics. August 2013. Section 3: Terms, symbols and definitions. Table entry no. 9–12.
3. a b P. Kurzweil: The Vieweg unit lexicon: terms, formulas and constants from natural sciences, technology and medicine . 2nd Edition. Springer Vieweg, 2000, ISBN 978-3-322-83212-2 , p. 34, 164, 225, 427 , doi : 10.1007 / 978-3-322-83211-5 ( lexical part as PDF file, 71.3 MB ; limited preview in Google Book Search - softcover reprint 2013).
4. a b Entry on mass fraction . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.M03722 Version: 2.3.3.