Question

When two metals with equal volumes are mixed together, the density of the mixture is $4$ $kg/m^3$. When equal masses of the same two metals are mixed together, the mixture density is $3$ $kg/m^3$. Calculate the densities of each metal.

Answer 1

Mass/Volume = density.

Let the mass, volume and density of the first metal and second metal be $m_1,V_1,{\rho }_1,m_2,V_2,{\rho }_2$

Case 1: Volume of both metals are same and let it be equal to $V$

Hence, density = $\frac{m_1+m_2}{2V}=\frac{V{\rho }_1+V{\rho }_2}{2V}=\frac{{\rho }_1+{\rho }_2}{2}=4kg{m}^{-3}$ -----------Eqn (1)

Case 2: Mass of both metals are same and let it be equal to $M$

Hence, density =$\frac{2m}{V_1+V_2}$ = $\frac{2m}{\frac{m}{{\rho }_1}+\frac{m}{{\rho }_2}}$ = $\frac{2{\rho }_1{\rho }_2}{{\rho }_1+{\rho }_2}$= $3kg{m}^{-3}$ -----------Eqn (2)

Substitute (1) in (2),

$3=\frac{2{\rho }_1{\rho }_2}{{\rho }_1+{\rho }_2}$ gives ${\rho }_1{\rho }_2=12$. Hence, ${\rho }_1$ = $\frac{12}{{\rho }_2}$

Solving, we get; ${\rho }_1=6or2$.

Hence, ${\rho }_2=2or6$