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Density of a Mixture

Density of a Mixture of substance in the proportion of mass

Let a number of substances of masses M1, M2 , M3 etc., and densities ρ₁, ρ₂, ρ₃ etc. respectively are mixed together. The total mass of the mixture = M1 + M2 + M3 +….

The total volume = $\displaystyle \frac{{{M}_{1}}}{{{\rho }_{1}}}+\frac{{{M}_{2}}}{{{\rho }_{2}}}+\frac{{{M}_{3}}}{{{\rho }_{3}}}+.....$

So the density of the mixture is $\displaystyle \rho =\frac{{{M}_{1}}+{{M}_{2}}+{{M}_{3}}+....}{\frac{{{M}_{1}}}{{{\rho }_{1}}}+\frac{{{M}_{2}}}{{{\rho }_{2}}}+\frac{{{M}_{3}}}{{{\rho }_{3}}}+.....}$

For two substances the density of the mixture $\displaystyle \rho =\frac{{{\rho }_{1}}{{\rho }_{2}}\left( {{M}_{1}}+{{M}_{2}} \right)}{{{\rho }_{1}}{{M}_{2}}+{{\rho }_{2}}{{M}_{1}}}$

Example
Two liquids of densities 2.5 g/cm³ and 0.8 g/cm³ are taken in the ratio of their masses as 2:3 respectively. Find the average density of the liquid combination.

Solution
Let masses be 2m g & 3m g, then V = V1 + V2 = $\displaystyle \left( \frac{2m}{2.5}+\frac{3m}{0.8} \right)$ cm³

Total mass = 2m + 3m = 5m g

Therefore, the average density $\displaystyle {{\rho }_{avg}}=\frac{5m}{V}=\frac{5m}{\frac{2m}{2.5}+\frac{3m}{0.8}}=\frac{5}{\frac{2}{2.5}+\frac{3}{0.8}}=\frac{10}{9.1}=1.09\text{ gm/c}{{\text{m}}^{\text{3}}}$

Density of a mixture of substance in the proportion of volume

Suppose that a number of substances of volume V1, V2, V3 etc. and densities ρ₁, ρ₂, ρ₃ etc. respectively are mixed.

The total mass of the mixture is = $\displaystyle {{\rho }_{1}}{{V}_{1}}+{{\rho }_{2}}{{V}_{2}}+{{\rho }_{3}}{{V}_{3}}+....$

The total volume of the mixture = V1 + V2 + V3+………

Therefore, the density of the mixture $\displaystyle \rho =\frac{{{\rho }_{1}}{{V}_{1}}+{{\rho }_{2}}{{V}_{2}}+{{\rho }_{3}}{{V}_{3}}}{{{V}_{1}}+{{V}_{2}}+{{V}_{3}}+....}$

For two substances the density of the mixture $\displaystyle \rho =\frac{{{\rho }_{1}}{{V}_{1}}+{{\rho }_{2}}{{V}_{2}}}{{{V}_{1}}+{{V}_{2}}}$

Example
Two miscible liquids of densities 1.2 gm/cc and 1.4 gm/cc are mixed with a proportion ratio of their volumes equal to 3:5. What is the density of resulting liquid?

Solution

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

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

$\displaystyle \rho =\frac{(1.2)(3/5)+1.4}{(3/5)+1}=\frac{3.6+1.4}{8}$

∴ ρ = 1.325 gm/cc

Density of a Mixture

Density of a Mixture

Density of a Mixture of substance in the proportion of mass

Density of a mixture of substance in the proportion of volume

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