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.