Question

The densities of two substances are in the ratio $5:6$ and their specific heats are in the ratio $3:5$ respectively. Then their thermal capacities per unit volume will be in the ratio
(A) $2:1$
(B) $1:2$
(C) $25:18$
(D) $18:25$

Answer 1

Hint: Use the formula of the thermal capacity to find the thermal capacity of the two substances in it. Substitute the formula of the mass in both the formulas. Divide the both the obtained equation to find the ratio of the thermal capacity of the two substances.
Formula used:
(1) The formula of the thermal capacity of the substance is given by
$T = sh \times \rho $
Where $T$ is the thermal capacity of the substance, $sh$ is the specific heat of the substance and $\rho $ is the density of the substance.
(2) The formula of the mass is given by
$m = \rho V$
Where $m$ is the mass of the substance, $V$ is the volume of the given substance.

Complete answer:
The ratio of the density of the two substances, $\dfrac{{{\rho _1}}}{{{\rho _2}}} = \dfrac{5}{6}$
The ratio of the specific heat of the two substance, $\dfrac{{s{h_1}}}{{s{h_2}}} = \dfrac{3}{5}$
Use the formula of the thermal capacity of the first substance,
${T_1} = s{h_1} \times {\rho _1}$ ……………………………….(1)
Similarly, the thermal capacity of the second substance is calculated.
${T_2} = s{h_2} \times {\rho _2}$ ……………………………….(2)
Dividing the equation (1) and (2), to find the ratio of the thermal capacity of the two substance, we get
$
  \dfrac{{{T_1}}}{{{T_2}}} = \dfrac{{s{h_1} \times {\rho _1}}}{{s{h_2} \times {\rho _2}}} \\
  \dfrac{{{T_1}}}{{{T_2}}} = \dfrac{{s{h_1}}}{{s{h_2}}} \times \dfrac{{{\rho _1}}}{{{\rho _2}}} \\
 $
Substituting the ratio of the specific heat and the density of the two substance in the above equation, we get
$\dfrac{{{T_1}}}{{{T_2}}} = \dfrac{3}{5} \times \dfrac{5}{6}$
By simplification of the above step, we get
$\dfrac{{{T_1}}}{{{T_2}}} = \dfrac{1}{2}$
Hence the ratio of the thermal capacity of the two substances is obtained as $\dfrac{1}{2}$ .

Thus the option (B) is correct.

Note:
Remember the formula of the thermal capacity and the mass of the body. The thermal capacity is the amount of the heat needed for the change in the unit temperature of the body. It depends on the density of the body and the specific heat of it.