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

In order to answer the given question we need to know the relation between thermal capacity, specific heat and the density of a substance. Also we need to know the relation between mass, density and volume of a substance. After that we need to solve the equation to conclude with the correct solution of the given question.

Complete step by step solution:
First of all let us write the formula for the thermal capacity.
Thermal capacity is the product of specific heat and mass of a given substance. Mathematically, we can write it as, $T = Q \times m$ ……………(i)
Now, we need to find the thermal capacity per unit volume. So, we need to write the mass of the substance in terms of volume. We know that, density of a substance is mass per unit volume of that substance, i.e.
$\Rightarrow \rho = \dfrac{m}{V}$
$\therefore m = \rho V$
Substituting the value of mass in equation (i), we get,
$\Rightarrow T = Q \times \rho V$
Now, let us find the ratios of the thermal capacities. So, we can write,
$\Rightarrow \dfrac{{{T_1}}}{{{T_2}}} = \dfrac{{{Q_1}}}{{{Q_2}}} \times \dfrac{{{\rho _1}}}{{{\rho _2}}} \times \dfrac{V}{V}$
$\Rightarrow \dfrac{{{T_1}}}{{{T_2}}} = \dfrac{{5 \times 3}}{{6 \times 5}} = \dfrac{1}{2}$
Therefore, the required ratio of thermal capacities per unit volume is $1:2$.

Hence, option (B), i.e. $1:2$ is the correct choice of the given question.

Note: Thermal capacity of a substance is the property of a material to absorb and release heat. We can define thermal capacity of a body as the heat required to raise the temperature of a body by one degree. The specific heat capacity of a body is the measure of the heat energy that a body in a unit quality absorbs or releases when there is an increase or decrease of one Kelvin.