# Substance proportion

The mole fraction ( Symbols : x , in gas mixtures optional y , besides also ), formerly known as mole fraction or falsely molar fraction referred to, according to DIN 1310 a so-called content size , which is a physico-chemical quantity for quantitative description of the composition of mixtures / mixed phases . Here, the amount of substance of a mixture component under consideration is related to the sum of the amounts of substance of all mixture components , so the amount of substance indicates the relative proportion of the amount of substance of a mixture component under consideration in the total amount of substance of the mixture. ${\ displaystyle \ chi}$

## 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 ${\ displaystyle x_ {i} = {\ frac {n_ {i}} {n_ {i} + n_ {j}}}}$ ${\ displaystyle x_ {i} = {\ frac {n_ {i}} {n}} \ \ {\ text {with}} \ \ n = \ sum _ {z = 1} ^ {Z} n_ {z} }$
Range of values ${\ displaystyle 0 \ leq x_ {i} \ leq 1}$
Sum criterion ${\ displaystyle x_ {i} + x_ {j} = 1 \ \ Rightarrow \ x_ {j} = 1-x_ {i}}$ ${\ displaystyle \ sum _ {z = 1} ^ {Z} x_ {z} = 1 \ \ Rightarrow \ x_ {Z} = 1- \ sum _ {z = 1} ^ {Z-1} x_ {z} }$

The mole fraction x i is defined as the value of the quotient of the amount of substance n i of the considered mixture component i and the total amount of substance n of the mixture. The latter is the sum of the mole quantities of all components ( including i ) of the mixture. The " particles " on which the concept of the amount of substance is based can be material elementary objects such as atoms , molecules , ions or formula units and must be specified for all mixture components.

As the quotient of two dimensions of the same size, the mole 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 ( mol / mol), possibly combined with decimal prefixes (e.g. B. mmol / mol), or with auxiliary units such as percent (% = 1/100), per thousand (‰ = 1 / 1,000) or parts per million (1 ppm = 1 / 1,000,000). However, this is outdated designations as mole percent , mol percent (abbreviation for example mol%) or atomic percent (abbreviation for example at .-%) to avoid instead is meant to refer to the content clearly size. For example, instead of “73.8 mol%”, the following should be formulated today: “The mole fraction of the mixture component i is 73.8%” or in the form of an equation: “ x i  = 73.8%”.

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

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

The amount of substance is used in numerous areas of application 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 Tx phase diagrams .

## Relationships with other salary levels

Because of the proportionality between the number of particles N and the amount of substance n (based on the same type of particle; the conversion factor is Avogadro's constant N A  ≈ 6.022 · 10 23  mol −1 ) the value of the amount of substance x i is equal to the value of the amount of particle X i :

${\ displaystyle x_ {i} = {\ frac {n_ {i}} {n}} = {\ frac {n_ {i} \ cdot N _ {\ mathrm {A}}} {n \ cdot N _ {\ mathrm { A}}}} = {\ frac {N_ {i}} {N}} = X_ {i}}$

The following table shows the relationships between the amount of substance x 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 (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 .

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

Since the molar volume V m of a pure substance is equal to the quotient of its molar mass M and density ρ (at a given temperature and a given pressure), the reciprocal terms in some equations in the table above can be replaced accordingly:

${\ displaystyle {\ frac {M_ {i}} {\ rho _ {i}}} = V _ {\ mathrm {m}, i} \ \ Leftrightarrow \ {\ frac {\ rho _ {i}} {M_ { i}}} = {\ frac {1} {V _ {\ mathrm {m}, i}}}}$

In the case of mixtures of ideal gases, not only do the values ​​of the molar fraction x i and the particle number fraction X i match, but because of the uniform molar volumes and the ideal mixture character, there is also equality with the volume fraction φ i and the volume concentration σ i :

${\ displaystyle x_ {i} = X_ {i} = \ varphi _ {i} = \ sigma _ {i} \ \ {\ text {for mixtures of ideal gases}}}$

## Examples

### 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, so:

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

### Solution of sodium chloride in water

A solution of sodium chloride (common salt) NaCl in water H 2 O with the mass fractions w NaCl = 0.03 = 3% and correspondingly w H 2 O = 1 - w NaCl = 0.97 = 97% is considered. Taking into account the molar masses , the molar proportions of NaCl formula units or H 2 O molecules are:

${\ displaystyle x _ {\ mathrm {NaCl}} = {\ frac {w _ {\ mathrm {NaCl}} / M _ {\ mathrm {NaCl}}} {w _ {\ mathrm {NaCl}} / M _ {\ mathrm {NaCl }} + w _ {\ mathrm {H_ {2} O}} / M _ {\ mathrm {H_ {2} O}}}} = \ mathrm {\ frac {0 {,} 03 / (58 {,} 44 \ g \ cdot mol ^ {- 1})} {0 {,} 03 / (58 {,} 44 \ g \ cdot mol ^ {- 1}) + 0 {,} 97 / (18 {,} 02 \ g \ cdot mol ^ {- 1})}} \ approx 0 {,} 009 = 0 {,} 9 \ \%}$
${\ displaystyle x _ {\ mathrm {H_ {2} O}} = 1-x _ {\ mathrm {NaCl}} \ approx 0 {,} 991 = 99 {,} 1 \ \%}$

## Individual evidence

1. a b c d Standard DIN 1310 : Composition of mixed phases (gas mixtures, solutions, mixed crystals); Terms, symbols. February 1984.
2. a b c 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, 224, 225, 281, 444 , 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 ).
3. a b c 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. 9–14.
4. a b c Entry on amount fraction . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.A00296 Version: 2.3.3.
5. a b Entry on substance fraction . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.S06075 Version: 2.3.3.