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).

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


 * $$r_i = \frac{n_i}{n_\mathrm{tot}-n_i}$$

The mole ratio is also called amount ratio. If ni is much smaller than ntot (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 $$\zeta$$, which is defined as the mass of water $$m_\mathrm{H2O}$$ divided by the mass of dry air ($$m_\mathrm{air}-m_\mathrm{H2O}$$) in a given air parcel:


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

The unit is typically given in $$\mathrm{g}\,\mathrm{kg}^{-1}$$. 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:


 * $$w = \frac{w_1 m_1 + w_2 m_1 r_m}{m_1 + m_1 r_m}$$

where m1 can be simplified from numerator and denominator


 * $$w = \frac{w_1 + w_2 r_m}{1 + r_m}$$

and


 * $$r_m = \frac{m_2}{m_1}$$

is the mass mixing ratio of the two solutions.

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


 * $$w = \frac{w_1\rho_1(w_1) + w_2\rho_2(w_2)}{\rho_1(w_1) + \rho_2(w_2)} $$

Considering a volume mixing ratio rV(21)


 * $$w = \frac{w_1\rho_1(w_1) + w_2\rho_2(w_2) r_V}{\rho_1(w_1) + \rho_2(w_2) r_V} $$

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


 * $$r_{m1} = \frac{m_2}{m_1} \quad r_{m2} = \frac{m_3}{m_1}$$

to be mixed giving:


 * $$w = \frac{w_1 m_1 + w_2 m_1 r_{m1} + w_3 m_1 r_{m2}}{m_1 + m_1 r_{m1} + m_1 r_{m2}} = \frac{w_1 + w_2 r_{m1} + w_3 r_{m2}}{1 + r_{m1} + r_{m2}}$$

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 Vs 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:


 * $$V = \frac{(\rho_1 + \rho_2) V_\mathrm{s}}{\rho}, V =2V_\mathrm{s}$$

implies
 * $$\frac{\rho_1 + \rho_2}{\rho} = 2 $$

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....