Question

If two miscible liquids of same volume but different densities ρ₁ and ρ₂ are mixed, then the density of the mixture is given by

(a) \(\frac{\rho _1 + \rho _2}2\)

(b) \(\frac{2\rho _1\rho _2}{\rho_1 + \rho _2}\)

(c) \(\frac{2\rho _1\rho _2}{\rho_1 - \rho _2}\)

(d) \(\frac{\rho _1\rho _2}{\rho_1 + \rho _2}\)

Answer 1

Correct option is (b) \(\frac{2\rho _1\rho _2}{\rho_1 + \rho _2}\)

Density of liquid, A = ρ₁ 

Density of liquid, B = ρ₂

We assume the mass of liquid, A or B = m

Volume of liquid, A = \(\frac m{\rho_1}\)        [\(\because\) Volume = Mass/Density]

Volume of liquid, B = \(\frac m{\rho_2}\)

Total volume of the mixture = \(\frac m{\rho_1}+\frac m{\rho_2}\)

Total mass = m + m = 2m 

Hence, density of mixture = total mass/total volume

\(= \frac{2m}{\left(\frac m{\rho_1}+ \frac m{\rho_2}\right)}\)

\(= \frac{2m}{m\left(\frac 1{\rho_1}+ \frac 1{\rho_2}\right)}\)

\(= \frac{2}{\left(\frac 1{\rho_1}+ \frac 1{\rho_2}\right)}\)

\(= \frac{2}{\left(\frac {\rho_2+\rho_1}{\rho_1\rho_2}\right)}\)

Density of mixture = \(\frac{2\rho_1\rho_2}{\rho_1 + \rho_2}\)