Question

The specific heat of metals at low temperature varies according to $ S = a{T^3} $ where $ a $ is a constant and $ T $ is the absolute temperature. The heat energy needed to raise unit mass of the metal from $ T = 1K $ to $ T = 2K $ is-
(A) $ 3a $
(B) $ \dfrac{{15a}}{4} $
(C) $ \dfrac{{2a}}{3} $
(D) $ \dfrac{{12a}}{5} $

Answer 1

Hint
The heat required to raise the temperature of a substance is the product of the mass, specific heat, and the temperature change. So we can find the small amount of heat required to raise the temperature by $ dT $ the amount and then integrate it over the limit from $ T = 1K $ to $ T = 2K $ .
In the solution, we will be using the following formula,
$\Rightarrow Q = MS\Delta T $
where $ Q $ is the amount of heat required to raise the temperature, $ M $ is the mass of the substance, $ S $ is the specific heat of metals and $ \Delta T $ is the temperature change.

Complete step by step answer
The amount of heat that is required to raise the temperature of a metal is given by
$\Rightarrow Q = MS\Delta T $
In the question, we are provided that the specific heat of metals at low temperature is given by, $ S = a{T^3} $
So the heat required to raise the temperature of a unit mass of the substance by an amount $ dT $ is given by,
$\Rightarrow dQ = 1 \times SdT $
$\Rightarrow dQ = 1 \times a{T^3}dT $
Since it is for unit mass.
So the total heat required to raise the temperature of this unit mass from $ 1K $ to $ 2K $ can be calculated by integrating $ dQ $ over the limits $ 1K $ to $ 2K $ .
$ \therefore \int {dQ} = \int_{1K}^{2K} {a{T^3}dT} $
So we get on calculating,
$\Rightarrow Q = \int_{1K}^{2K} {a{T^3}dT} $
On integrating the R.H.S of the above equation we get,
$\Rightarrow Q = a\left. {\dfrac{{{T^4}}}{4}} \right|_{1K}^{2K} $
So substituting the limits we get,
$\Rightarrow Q = \dfrac{a}{4}\left[ {{{\left( 2 \right)}^4} - {{\left( 1 \right)}^4}} \right] $
By calculating we get the value as,
$\Rightarrow Q = \dfrac{{15a}}{4} $
So the heat required to raise the unit mass of the metal from temperature $ 1K $ to $ 2K $ is $ \dfrac{{15a}}{4} $ .
Therefore, the correct answer is option (B). $ \dfrac{{15a}}{4} $ .

Note
The specific heat capacity of metals is the amount of heat that is required to raise the temperature of a unit mass of that metal by a unit amount. The relationship between the heat and temperature of substances is expressed in the form of specific heat.