Question

The ratio of densities of two substances is 2:3 and that of specific heats is 1:2 the ratio of thermal capacities per unit volume is:
$\begin{align}
  & A.\text{ }1:2 \\
 & B.\text{ }2:1 \\
 & C.\text{ }1:3 \\
 & D.\text{ }3:1 \\
\end{align}$

Answer 1

Hint: The thermal heat capacity per unit volume is directly proportional to the product of the density and the specific heat of the substance by using this relation to find the values of the thermal capacities per unit volume.
Formula used: $H=\rho VS$

Complete answer:
Given that:
The ratio of the densities = 2:3
The ratio of the specific heats = 1:2
Find the ratio of the thermal heat capacities for unit volume using formula
$H=\rho VS$
Where, ‘H’ is the thermal heat capacity, ‘ρ’ is the density of the substance, ‘V’ is the volume of the substance which is given and ‘S’ is the specific heat.
We know that, it can be written as
$\dfrac{H}{V}=\rho S$
We know it is given in question that we have the ratios of two substances.
Now, $\dfrac{{{H}_{1}}}{{{V}_{1}}}={{\rho }_{1}}{{S}_{1}}.....\left( 1 \right)$
And $\dfrac{{{H}_{2}}}{{{V}_{2}}}={{\rho }_{2}}{{S}_{2}}.....\left( 2 \right)$
By dividing (1) and (2) equation we get,
$\dfrac{\dfrac{{{H}_{1}}}{{{V}_{1}}}}{\dfrac{{{H}_{2}}}{{{V}_{2}}}}=\dfrac{{{\rho }_{1}}}{{{\rho }_{2}}}\times \dfrac{{{S}_{1}}}{{{S}_{2}}}$
Now, substitute the values of densities and specific heats, we get,
$\dfrac{\dfrac{{{H}_{1}}}{{{V}_{1}}}}{\dfrac{{{H}_{2}}}{{{V}_{2}}}}=\dfrac{2}{3}\times \dfrac{1}{2}$
We have to simplify this,
$\dfrac{\dfrac{{{H}_{1}}}{{{V}_{1}}}}{\dfrac{{{H}_{2}}}{{{V}_{2}}}}=\dfrac{1}{3}$
The ratio of thermal capacities of two substances for unit volume is 1:3.

Therefore, the correct option is (C).

Additional Information:
The specific heat capacity is constant for a given material. When a solid or liquid is kept in an open atmosphere and heated, the pressure remains constant. The specific heat capacity is also known as the specific heat in short.

Note:
Thermal capacity is property of matter, defined as the amount of the heat to be supplied to a given mass of a material to produce a unit change in its temperature the S.I unit of heat capacity is the joule per kelvin $(J{{K}^{-1}})$.