Mixing ratio

In chemistry and physics, the dimensionless mixing ratio is the abundance of one component of a mixture relative to that of all other components. The term can refer either to mole ratio (see concentration) or mass ratio (see stoichiometry).^[1]

In atmospheric chemistry and meteorology

Mole ratio

In atmospheric chemistry, mixing ratio usually refers to the mole ratio r_i, which is defined as the amount of a constituent n_i divided by the total amount of all other constituents in a mixture:

${\displaystyle r_{i}={\frac {n_{i}}{n_{\mathrm {tot} }-n_{i}}}}$ 

The mole ratio is also called amount ratio.^[2] If n_i is much smaller than n_tot (which is the case for atmospheric trace constituents), the mole ratio is almost identical to the mole fraction.

Mass ratio

In meteorology, mixing ratio usually refers to the mass ratio of water ${\displaystyle \zeta }$ , which is defined as the mass of water ${\displaystyle m_{\mathrm {H2O} }}$  divided by the mass of dry air (${\displaystyle m_{\mathrm {air} }-m_{\mathrm {H2O} }}$ ) in a given air parcel:^[3]

${\displaystyle \zeta ={\frac {m_{\mathrm {H2O} }}{m_{\mathrm {air} }-m_{\mathrm {H2O} }}}}$ 

The unit is typically given in ${\displaystyle \mathrm {g} \,\mathrm {kg} ^{-1}}$ . The definition is similar to that of specific humidity.

Mixing ratio of mixtures or solutions

Two binary solutions of different compositions or even two pure components can be mixed with various mixing ratios by masses, moles, or volumes.

The mass fraction of the resulting solution from mixing solutions with masses m₁ and m₂ and mass fractions w₁ and w₂ is given by:

${\displaystyle w={\frac {w_{1}m_{1}+w_{2}m_{1}r_{m}}{m_{1}+m_{1}r_{m}}}}$ 

where m₁ can be simplified from numerator and denominator

${\displaystyle w={\frac {w_{1}+w_{2}r_{m}}{1+r_{m}}}}$ 

and

${\displaystyle r_{m}={\frac {m_{2}}{m_{1}}}}$ 

is the mass mixing ratio of the two solutions.

By substituting the densities ρ_i(w_i) and considering equal volumes of different concentrations one gets:

${\displaystyle w={\frac {w_{1}\rho _{1}(w_{1})+w_{2}\rho _{2}(w_{2})}{\rho _{1}(w_{1})+\rho _{2}(w_{2})}}}$ 

Considering a volume mixing ratio r_V(21)

${\displaystyle w={\frac {w_{1}\rho _{1}(w_{1})+w_{2}\rho _{2}(w_{2})r_{V}}{\rho _{1}(w_{1})+\rho _{2}(w_{2})r_{V}}}}$ 

The formula can be extended to more than two solutions with mass mixing ratios

${\displaystyle r_{m1}={\frac {m_{2}}{m_{1}}}\quad r_{m2}={\frac {m_{3}}{m_{1}}}}$ 

to be mixed giving:

${\displaystyle w={\frac {w_{1}m_{1}+w_{2}m_{1}r_{m1}+w_{3}m_{1}r_{m2}}{m_{1}+m_{1}r_{m1}+m_{1}r_{m2}}}={\frac {w_{1}+w_{2}r_{m1}+w_{3}r_{m2}}{1+r_{m1}+r_{m2}}}}$ 

Volume additivity

The condition to get a partially ideal solution on mixing is that the volume of the resulting mixture V to equal double the volume V_s of each solution mixed in equal volumes due to the additivity of volumes. The resulting volume can be found from the mass balance equation involving densities of the mixed and resulting solutions and equalising it to 2:

${\displaystyle V={\frac {(\rho _{1}+\rho _{2})V_{\mathrm {s} }}{\rho }},V=2V_{\mathrm {s} }}$ 

implies

${\displaystyle {\frac {\rho _{1}+\rho _{2}}{\rho }}=2}$ 

Of course for real solutions inequalities appear instead of the last equality.

Solvent mixtures mixing ratios

Mixtures of different solvents can have interesting features like anomalous conductivity (electrolytic) of particular lyonium ions and lyate ions generated by molecular autoionization of protic and aprotic solvents due to Grotthuss mechanism of ion hopping depending on the mixing ratios. Examples may include hydronium and hydroxide ions in water and water alcohol mixtures, alkoxonium and alkoxide ions in the same mixtures, ammonium and amide ions in liquid and supercritical ammonia, alkylammonium and alkylamide ions in ammines mixtures, etc....

References

1. IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "mixing ratio". doi:10.1351/goldbook.M03948
2. "Pure and Applied Chemistry, 2008, Volume 80, No. 2, pp. 233-276". Iupac.org. 2016-06-14. Retrieved 2016-06-30.
3. Whiteman, D.N. (2015). Encyclopedia of Atmospheric Sciences (Second Edition, Volume 3 ed.). Elsevier Ltd. p. 298. ISBN 978-0-12-382225-3.