# Molar ratio

The molar ratio ( Symbols : r ) is in accordance with DIN 1310 a so-called content size , which is a physico-chemical quantity for quantitative description of the composition of mixtures / mixed phases . It gives the ratio of the amounts of substance of two considered mixture components to each other.

## Definition and characteristics

The molar ratio r ij is defined as the value of the quotient of the substance amount n i of the one considered mixture component i and the substance amount n j of the other considered mixture component j :

${\ displaystyle r_ {ij} = {\ frac {n_ {i}} {n_ {j}}}}$

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

As the quotient of two dimensionally identical quantities, the molar ratio is a dimensionless quantity 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 ( mol / mol), possibly combined with decimal prefixes (e.g. mmol / mol), or with auxiliary units such as percent (% = 1 / 100), per mille (‰ = 1 / 1,000) or parts per million (1 ppm = 1 / 1,000,000). However, outdated terms such as molar percentages , molar percentages or atomic percentages should be avoided here; instead, the intended content must be clearly identified.

If the mixture  component i is not present (i.e. if n i = 0), the minimum value r ij  = 0. If the mixture component j is not present ( n j  = 0, for example if there is no mixture but a pure substance i ), the molar ratio r ij is not defined .

The values ​​of the molar ratios for a substance mixture of a given composition are - in contrast to the volume-related contents ( concentrations , volume fraction , volume ratio ) - independent of temperature and pressure , since the substance amounts of the mixture components, in contrast to the volumes, do not change with the 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 molar ratio r ij is equal to the value of the particle number ratio 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 molar ratio r ij and the other content values ​​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 of the molar ratio r ij with 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
${\ displaystyle r_ {ij} = {\ frac {w_ {i}} {w_ {j}}} \ cdot {\ frac {M_ {j}} {M_ {i}}}}$ ${\ displaystyle r_ {ij} = {\ frac {x_ {i}} {x_ {j}}}}$ ${\ displaystyle r_ {ij} = {\ frac {X_ {i}} {X_ {j}}}}$ ${\ displaystyle r_ {ij} = {\ frac {\ varphi _ {i}} {\ varphi _ {j}}} \ cdot {\ frac {M_ {j}} {M_ {i}}} \ cdot {\ frac {\ rho _ {i}} {\ rho _ {j}}}}$
… - concentration Mass concentration β Molar concentration c Particle number concentration C Volume concentration σ
${\ displaystyle r_ {ij} = {\ frac {\ beta _ {i}} {\ beta _ {j}}} \ cdot {\ frac {M_ {j}} {M_ {i}}}}$ ${\ displaystyle r_ {ij} = {\ frac {c_ {i}} {c_ {j}}}}$ ${\ displaystyle r_ {ij} = {\ frac {C_ {i}} {C_ {j}}}}$ ${\ displaystyle r_ {ij} = {\ frac {\ sigma _ {i}} {\ sigma _ {j}}} \ cdot {\ frac {M_ {j}} {M_ {i}}} \ cdot {\ frac {\ rho _ {i}} {\ rho _ {j}}}}$
... - ratio Mass ratio ζ Molar ratio r Particle number ratio R Volume ratio ψ
${\ displaystyle r_ {ij} = \ zeta _ {ij} \ cdot {\ frac {M_ {j}} {M_ {i}}}}$ ${\ displaystyle r_ {ij}}$ ${\ displaystyle r_ {ij} = R_ {ij}}$ ${\ displaystyle r_ {ij} = \ psi _ {ij} \ cdot {\ frac {M_ {j}} {M_ {i}}} \ cdot {\ frac {\ rho _ {i}} {\ rho _ { j}}}}$
Quotient
amount of substance / mass
Molality b
${\ displaystyle r_ {ij} = b_ {i} \ cdot M_ {j}}$ ( i = solute, j = solvent)
specific amount of partial substances q
${\ displaystyle r_ {ij} = {\ frac {q_ {i}} {q_ {j}}}}$

Summing for all mixture components, the molar ratios r zi to a fixed component of the mixture i , one obtains the reciprocal of the mole fraction x i of the fixed mixing component i (mixture of a total of Z components index such as a general running index for the summation , integration of the trivial molar ratio in the Total): ${\ displaystyle r_ {ii} = {\ tfrac {n_ {i}} {n_ {i}}} = 1}$

${\ displaystyle \ sum _ {z = 1} ^ {Z} r_ {zi} = \ sum _ {z = 1} ^ {Z} {\ frac {n_ {z}} {n_ {i}}} = { \ frac {1} {x_ {i}}}}$

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 pressure), the terms appearing in some of the equations in the table above (ratio of molar masses multiplied by the inverse ratio of Densities) can also be replaced by the ratio of the molar volumes:

${\ displaystyle {\ frac {M_ {j}} {M_ {i}}} \ cdot {\ frac {\ rho _ {i}} {\ rho _ {j}}} = {\ frac {V _ {\ mathrm {m}, j}} {V _ {\ mathrm {m}, i}}}}$

If the mixture components i and j are ideal gases , the molar volumes are the same and their ratio is therefore equal to one. From this it follows from the table above that with mixtures of ideal gases not only do the values ​​of the molar ratio r ij and the particle number ratio R ij match, but there is also equality with the volume ratio ψ ij :

${\ displaystyle r_ {ij} = R_ {ij} = \ psi _ {ij} \ \ {\ text {for ideal gases}} i, j}$

## Examples

### 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. The mass ratio ζ NaCl / H 2 O is:

${\ displaystyle \ zeta _ {\ mathrm {NaCl / H_ {2} O}} = {\ frac {w _ {\ mathrm {NaCl}}} {w _ {\ mathrm {H_ {2} O}}}} = { \ frac {0 {,} 03} {0 {,} 97}} = 0 {,} 0309}$

Taking into account the molar masses , the molar ratio of NaCl formula units to H 2 O molecules results :

${\ displaystyle r _ {\ mathrm {NaCl / H_ {2} O}} = \ zeta _ {\ mathrm {NaCl / H_ {2} O}} \ cdot {\ frac {M _ {\ mathrm {H_ {2} O }}} {M _ {\ mathrm {NaCl}}}} = 0 {,} 0309 \ cdot \ mathrm {\ frac {18 {,} 02 \ g \ cdot mol ^ {- 1}} {58 {,} 44 \ g \ cdot mol ^ {- 1}}} = 0 {,} 009537}$

### Stoichiometry

Mole ratios also have an extended meaning beyond their use as a content specification for mixtures of substances. They often occur in connection with stoichiometric considerations. In the case of chemical reactions described by reaction equations , the stoichiometric coefficients can be put in relation to one another in pairs and then represent molar ratios. In chemical experiment instructions or synthesis instructions, it happens that molar ratios (based on a selected starting material) are specified with regard to the amounts of the starting materials to be used. So z. B. in the oxyhydrogen reaction, hydrogen is reacted with oxygen in a molar ratio of 2: 1; When referring to oxygen, the term “2 equivalents of hydrogen” is often used, which is derived from the term equivalent .

## 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 3-322-83212-0 , p. 224, 225, 419 , doi : 10.1007 / 978-3-322-83211-5 ( lexical part as PDF file, 71.3 MB ; limited preview in Google Book search).