Question

Two spheres $A$ and $B$ have diameters in the ratio $1:2$, densities in the ratio $2:1$ and thermal capacities in the ratio $1:12$. Find the ratio of their specific heat capacities.

• A. $1:6$
• B. $1:12$
• C. $1:3$
• D. $1:4$

Answer 1

The correct option is C $1:3$

We know that
Mass = Volume $\times$ Density $\left(\rho \right)$
and
Volume of sphere $V=\frac{4}{3}\pi {\left(\frac{D}{2}\right)}^3$
Where $D$ is the diameter of sphere.
$\therefore$ Mass of sphere $A$ $m_A=\frac{4}{3}\pi {\left(\frac{D_A}{2}\right)}^3\times {\rho }_A$
and
Mass of sphere $B$ $m_B=\frac{4}{3}\pi {\left(\frac{D_B}{2}\right)}^3\times {\rho }_B$
$\therefore \frac{m_A}{m_B}=\frac{\frac{4}{3}\pi {\left(\frac{D_A}{2}\right)}^3{\rho }_A}{\frac{4}{3}\pi {\left(\frac{D_B}{2}\right)}^3{\rho }_B}=\frac{D_A^3{\rho }_A}{D_B^3{\rho }_B}$
Also,
We know
Thermal capacity $\left(C\right)=mc$
$\therefore$ where $\left(c\right)$ is specific heat capacity
$\therefore \frac{C_A}{C_B}=\frac{m_A{c}_A}{m_B{c}_B}=\frac{1}{12}$
$\therefore \frac{c_A}{c_B}=\frac{1}{12}\times \frac{m_B}{m_A}$
$\Rightarrow \frac{c_A}{c_B}=\frac{1}{12}\times \frac{D_B^3{\rho }_B}{D_A^3{\rho }_A}$
$\Rightarrow \frac{c_A}{c_B}=\frac{1}{12}\times {\left(\frac{2}{1}\right)}^3\times \frac{1}{2}$
$\Rightarrow \frac{c_A}{c_B}=\frac{1}{3}$