# Particle number fraction

The Teilchenzahlanteil ( Symbol : X ) according to DIN 1310 , a physico-chemical quantity for quantitative description of the composition of mixtures / mixed phase , a so-called content size . It indicates the relative proportion of the number of particles in a particular mixture component in relation to the total number of particles in 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 ${\ displaystyle X_ {i} = {\ frac {N_ {i}} {N_ {i} + N_ {j}}}}$ ${\ displaystyle X_ {i} = {\ frac {N_ {i}} {N}} \ \ {\ text {with}} \ \ N = \ sum _ {z = 1} ^ {Z} N_ {z} }$
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 Teilchenzahlanteil X i is defined as the value of the quotient from the particle number N i of the considered mixture component i and the total particle number N of the mixture. The latter is the sum of the particle numbers of all components ( including i ) of the mixture. “ Particles ” 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 dimensionless quantities , the number of particles is itself also a dimensionless quantity and can be specified as a pure decimal number as in the table above or with auxiliary units such as percent (% = 1/100), per mil (‰ = 1 / 1,000) or parts per million (1 ppm = 1 / 1,000,000).

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

The values ​​of the particle number 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 particle numbers of the mixture components, in contrast to the volumes, do not vary with temperature or pressure change, provided no material conversions occur.

## 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 number of particles X i is equal to the value of the amount of substance 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 particle number fraction 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 number of particles 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
${\ displaystyle X_ {i} = {\ frac {w_ {i} / M_ {i}} {\ sum _ {z = 1} ^ {Z} {(w_ {z} / M_ {z})}}} }$ ${\ displaystyle X_ {i} = x_ {i}}$ ${\ displaystyle X_ {i}}$ ${\ displaystyle X_ {i} = {\ frac {\ varphi _ {i} \ cdot \ rho _ {i} / M_ {i}} {\ sum _ {z = 1} ^ {Z} {(\ varphi _ {z} \ cdot \ rho _ {z} / M_ {z})}}}}$
… - 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 particle number fraction X i and the molar 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 mole proportions, which in turn equal the Particle number fractions are (see above):

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

### Solution of glucose in water

A solution of glucose (Glc) in water (H 2 O) with the mass fractions w Glc = 0.01 = 1% and correspondingly w H 2 O = 1 - w Glc = 0.99 = 99% is considered. Taking into account the molar masses , the particle number fractions of glucose molecules and water molecules are:

${\ displaystyle X _ {\ mathrm {Glc}} = {\ frac {w _ {\ mathrm {Glc}} / M _ {\ mathrm {Glc}}} {w _ {\ mathrm {Glc}} / M _ {\ mathrm {Glc }} + w _ {\ mathrm {H_ {2} O}} / M _ {\ mathrm {H_ {2} O}}}} = \ mathrm {\ frac {0 {,} 01 / (180 {,} 16 \ g \ cdot mol ^ {- 1})} {0 {,} 01 / (180 {,} 16 \ g \ cdot mol ^ {- 1}) + 0 {,} 99 / (18 {,} 02 \ g \ cdot mol ^ {- 1})}} \ approx 0 {,} 001 = 0 {,} 1 \ \%}$
${\ displaystyle X _ {\ mathrm {H_ {2} O}} = 1-X _ {\ mathrm {Glc}} \ approx 0 {,} 999 = 99 {,} 9 \ \%}$

## 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 c P. Kurzweil: The Vieweg unit lexicon: terms, formulas and constants from natural sciences, technology and medicine . 2nd Edition. Springer Vieweg, 2013, ISBN 978-3-322-83212-2 , p. 34, 224, 225, 444 , doi : 10.1007 / 978-3-322-83211-5 ( limited preview in the Google book search - softcover reprint of the 2nd edition 2000). Lexical part (PDF; 71.3 MB).