# Particle number ratio

The Teilchenzahlverhältnis ( symbols : R ) according to DIN 1310 , a physico-chemical quantity for quantitative description of the composition of mixtures / mixed phase , a so-called content size . It gives the ratio of the particle numbers of two considered mixture components to each other.

## Definition and characteristics

The particle number ratio is defined as the value of the quotient of the particle number of the one considered mixture component and the particle number of the other considered mixture component : ${\ displaystyle R_ {ij}}$ ${\ displaystyle N_ {i}}$${\ displaystyle i}$${\ displaystyle N_ {j}}$${\ displaystyle j}$

${\ displaystyle R_ {ij} = {\ frac {N_ {i}} {N_ {j}}}}$

To avoid ambiguity when specifying particle number ratios, the numerator component and denominator component must always be specified, e.g. B. by the specified index notation. Interchanging the numerator and denominator components leads to the reciprocal value . In multicomponent mixtures, a corresponding number of particle number ratios can be formulated: with a total of components piece, if the respective reciprocal values ​​and trivial particle number ratios count as well ( variation with repetition ), otherwise piece ( combination without repetition ). ${\ displaystyle R_ {ji} = {\ tfrac {1} {R_ {ij}}} = {\ tfrac {N_ {j}} {N_ {i}}}}$${\ displaystyle Z}$${\ displaystyle Z ^ {2}}$${\ displaystyle R_ {ii} = {\ tfrac {N_ {i}} {N_ {i}}} = 1}$${\ displaystyle {\ tbinom {Z} {2}}}$

In the case of solutions, which is a common case of chemical mixtures, the component can be, for example, a dissolved substance and the solvent or another dissolved substance. “ Particles ” can be material elementary objects such as atoms , molecules , ions or formula units . ${\ displaystyle i}$${\ displaystyle j}$

As the quotient of two dimensionless quantities , the particle number ratio is itself also a dimensionless quantity and can assume numerical values ​​≥ 0. If the mixture component is not present (i.e. if ), the minimum value is obtained . If the mixture component is not present ( if, for example, there is no mixture but a pure substance ) the particle number ratio is not defined . ${\ displaystyle i}$${\ displaystyle N_ {i} = 0}$${\ displaystyle R_ {ij} = 0}$${\ displaystyle j}$${\ displaystyle N_ {j} = 0}$ ${\ displaystyle i}$${\ displaystyle R_ {ij}}$

## Relationships with other salary levels

Because of the proportionality between the number of particles and the amount of substance (assuming reference to the same type of particle; the conversion factor is Avogadro's constant ) the value of the particle number ratio is equal to the value of the molar ratio : ${\ displaystyle N}$ ${\ displaystyle n}$ ${\ displaystyle N _ {\ mathrm {A}} \ approx 6 {,} 022 \ cdot 10 ^ {23} \ \ mathrm {mol} ^ {- 1}}$${\ displaystyle R_ {ij}}$ ${\ displaystyle r_ {ij}}$

${\ displaystyle R_ {ij} = {\ frac {N_ {i}} {N_ {j}}} = {\ frac {n_ {i} \ cdot N _ {\ mathrm {A}}} {n_ {j} \ cdot N _ {\ mathrm {A}}}} = {\ frac {n_ {i}} {n_ {j}}} = r_ {ij}}$

The following table shows the relationships between the ratio of the number of particles and the other content values ​​defined in DIN 1310 in the form of size equations . Are thereby or for the respective molar masses , or for the respective densities of the pure substances or (at the same pressure and the same temperature as in the mixture). ${\ displaystyle R_ {ij}}$${\ displaystyle M_ {i}}$${\ displaystyle M_ {j}}$${\ displaystyle \ rho _ {i}}$${\ displaystyle \ rho _ {j}}$ ${\ displaystyle i}$${\ displaystyle j}$

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

Summing for all mixture components Teilchenzahlverhältnisse to a fixed component of the mixture , one obtains the reciprocal of the Teilchenzahlanteils the fixed component of the mixture (mixture of a total of components Index as a general running index for the summation , involving the trivial Teilchenzahlverhältnisses in the sum): ${\ displaystyle R_ {zi}}$${\ displaystyle i}$${\ displaystyle i}$${\ displaystyle Z}$${\ displaystyle z}$${\ displaystyle R_ {ii} = {\ tfrac {N_ {i}} {N_ {i}}} = 1}$

${\ displaystyle \ sum _ {z = 1} ^ {Z} R_ {zi} = \ sum _ {z = 1} ^ {Z} {\ frac {N_ {z}} {N_ {i}}} = { \ frac {1} {X_ {i}}}}$

## 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. This results in the ratio of particles between nitrogen and oxygen:

${\ displaystyle R _ {\ mathrm {{N_ {2}} / {O_ {2}}}} = {\ frac {N _ {\ mathrm {N_ {2}}}} {N _ {\ mathrm {O_ {2} }}}} = {\ frac {x _ {\ mathrm {N_ {2}}}} {x _ {\ mathrm {O_ {2}}}}}} \ approx {\ frac {78 {,} 1 \ \%} {20 {,} 9 \ \%}} \ approx 3 {,} 74}$

So air contains around four times as many N 2 molecules as O 2 molecules.

### Relationship formulas of chemical compounds

Content parameters such as the particle number ratio can also be transferred accordingly when it comes to tracing a chemical compound back to the chemical elements involved . From the ratio formula , the Teilchenzahlverhältnisse of atoms of chemical elements involved in a chemical compound can be read directly, for example, acetic acid : molecular formula C 2 H 4 O 2 , empirical formula CH 2 O . ${\ displaystyle \ Rightarrow R _ {\ mathrm {H / C}} = 2, \ R _ {\ mathrm {O / C}} = 1, \ R _ {\ mathrm {H / O}} = 2}$

## 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, 2013, ISBN 978-3-322-83212-2 , p. 224, 225, 419 , 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).