# Mass ratio

The mass ratio ( Symbol : ζ ) 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 masses of two considered mixture components to each other.

## Definition and characteristics

The mass ratio ζ ij is defined as the value of the quotient of the mass m i of the one considered mixture component i and the mass m j of the other considered mixture component j :

${\ displaystyle \ zeta _ {ij} = {\ frac {m_ {i}} {m_ {j}}}}$

To avoid ambiguity when specifying mass 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 mass ratios can be formulated: with a total of Z components, Z 2 pieces, if the respective reciprocal values ​​and trivial mass ratios count as well ( variation with repetition ), otherwise pieces ( combination without repetition ). ${\ displaystyle \ zeta _ {ji} = {\ tfrac {1} {\ zeta _ {ij}}} = {\ tfrac {m_ {j}} {m_ {i}}}}$${\ displaystyle \ zeta _ {ii} = {\ tfrac {m_ {i}} {m_ {i}}} = 1}$${\ displaystyle {\ tbinom {Z} {2}}}$

In the case of solutions, which are a common case of chemical mixtures, component i can be, for example, a dissolved substance and j the solvent or another dissolved substance.

As the quotient of two dimensions of equal dimensions, the mass ratio is a quantity of the dimension number and can assume numerical values ​​≥ 0. It can be specified as a pure decimal number without a unit of measurement , 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 mille (‰ = 1 / 1,000) or parts per million (1 ppm = 1 / 1,000,000). In this case, however, the outdated, no longer standard specification “mass percent” (or “weight percent”) should be avoided. If the mixture  component i is not present (i.e. if m i = 0), the minimum value is ζ ij  = 0. If the mixture component j is not present ( m j  = 0, for example if there is no mixture but a pure substance i ), the mass ratio ζ ij is not defined .

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

## Relationships with other salary levels

The following table shows the relationships between the mass ratio ζ ij and the other content quantities defined in DIN 1310 in the form of size equations . M i and M j stand for the respective molar masses , ρ i and ρ j for the respective densities of the pure substances i and j (at the same pressure and the same temperature as in the substance mixture).

Relationship between the mass ratio ζ ij and 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
${\ displaystyle \ zeta _ {ij} = {\ frac {w_ {i}} {w_ {j}}}}$ ${\ displaystyle \ zeta _ {ij} = {\ frac {x_ {i}} {x_ {j}}} \ cdot {\ frac {M_ {i}} {M_ {j}}}}$ ${\ displaystyle \ zeta _ {ij} = {\ frac {X_ {i}} {X_ {j}}} \ cdot {\ frac {M_ {i}} {M_ {j}}}}$ ${\ displaystyle \ zeta _ {ij} = {\ frac {\ varphi _ {i}} {\ varphi _ {j}}} \ cdot {\ frac {\ rho _ {i}} {\ rho _ {j} }}}$
… - concentration Mass concentration β Molar concentration c Particle number concentration C Volume concentration σ
${\ displaystyle \ zeta _ {ij} = {\ frac {\ beta _ {i}} {\ beta _ {j}}}}$ ${\ displaystyle \ zeta _ {ij} = {\ frac {c_ {i}} {c_ {j}}} \ cdot {\ frac {M_ {i}} {M_ {j}}}}$ ${\ displaystyle \ zeta _ {ij} = {\ frac {C_ {i}} {C_ {j}}} \ cdot {\ frac {M_ {i}} {M_ {j}}}}$ ${\ displaystyle \ zeta _ {ij} = {\ frac {\ sigma _ {i}} {\ sigma _ {j}}} \ cdot {\ frac {\ rho _ {i}} {\ rho _ {j} }}}$
... - ratio Mass ratio ζ Molar ratio r Particle number ratio R Volume ratio ψ
${\ displaystyle \ zeta _ {ij}}$ ${\ displaystyle \ zeta _ {ij} = r_ {ij} \ cdot {\ frac {M_ {i}} {M_ {j}}}}$ ${\ displaystyle \ zeta _ {ij} = R_ {ij} \ cdot {\ frac {M_ {i}} {M_ {j}}}}$ ${\ displaystyle \ zeta _ {ij} = \ psi _ {ij} \ cdot {\ frac {\ rho _ {i}} {\ rho _ {j}}}}$
Quotient
amount of substance / mass
Molality b
${\ displaystyle \ zeta _ {ij} = b_ {i} \ cdot M_ {i}}$ (i = solute, j = solvent)
specific amount of partial substances q
${\ displaystyle \ zeta _ {ij} = {\ frac {q_ {i}} {q_ {j}}} \ cdot {\ frac {M_ {i}} {M_ {j}}}}$

Summing for all mixture components, the mass ratios ζ zi to a fixed component of the mixture i , one obtains the reciprocal of the mass fraction of the fixed component of the mixture i (mixture of a total of Z components index such as a general running index for the summation , integration of the trivial mass ratio in the sum) : ${\ displaystyle \ zeta _ {ii} = {\ tfrac {m_ {i}} {m_ {i}}} = 1}$

${\ displaystyle \ sum _ {z = 1} ^ {Z} \ zeta _ {zi} = \ sum _ {z = 1} ^ {Z} {\ frac {m_ {z}} {m_ {i}}} = {\ frac {1} {w_ {i}}}}$

## 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 amounts of substance or number of particles . Taking into account the molar masses, the mass ratio of nitrogen to oxygen is:

${\ displaystyle \ zeta _ {\ mathrm {{N_ {2}} / {O_ {2}}}} = {\ frac {x _ {\ mathrm {N_ {2}}}} {x _ {\ mathrm {O_ { 2}}}}} \ cdot {\ frac {M _ {\ mathrm {N_ {2}}}} {M _ {\ mathrm {O_ {2}}}}}} = {\ frac {0 {,} 781} { 0 {,} 209}} \ cdot {\ frac {28 {,} 014 \ \ mathrm {g \ cdot mol ^ {- 1}}} {31 {,} 998 \ \ mathrm {g \ cdot mol ^ {- 1}}}} = 3 {,} 27}$

### Element-mass ratios in chemical compounds

Content values ​​such as the mass ratio can also be transferred accordingly when it comes to considering chemical elements as components of chemical compounds . The molecular formula of a chemical compound can be used to derive the particle number ratios of the atoms of the chemical elements involved; the mass ratios are obtained by linking them with the molar masses. The mass ratio of oxygen to hydrogen in water H 2 O or in hydrogen peroxide H 2 O 2 serve as an example :

${\ displaystyle \ zeta _ {\ mathrm {O / H}} (\ mathrm {H_ {2} O}) = R _ {\ mathrm {O / H}} (\ mathrm {H_ {2} O}) \ cdot {\ frac {M _ {\ mathrm {O}}} {M _ {\ mathrm {H}}}} = {\ frac {1} {2}} \ cdot {\ frac {15 {,} 999 \ \ mathrm { g \ cdot mol ^ {- 1}}} {1 {,} 008 \ \ mathrm {g \ cdot mol ^ {- 1}}}} = 7 {,} 936 \ approx 8}$
${\ displaystyle \ zeta _ {\ mathrm {O / H}} (\ mathrm {H_ {2} O_ {2}}) = R _ {\ mathrm {O / H}} (\ mathrm {H_ {2} O_ { 2}}) \ cdot {\ frac {M _ {\ mathrm {O}}} {M _ {\ mathrm {H}}}} = {\ frac {2} {2}} \ cdot {\ frac {15 {, } 999 \ \ mathrm {g \ cdot mol ^ {- 1}}} {1 {,} 008 \ \ mathrm {g \ cdot mol ^ {- 1}}}} = 15 {,} 872 \ approx 16}$

The law of constant proportions (in a certain chemical compound the chemical elements that constitute it always appear in the same mass ratio) and the law of multiple proportions (if two chemical elements form different compounds with one another, this is how the individual mass ratios of the two chemical elements stand) in these connections to each other in the ratio of small whole numbers).

## 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 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. 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).