Question

The specific heat of an ideal gas varies according to $S=a{T}^3$, where, $a$ is a constant and $T$ is absolute temperature. The heat energy needed to raise the temperature of a unit mass of the gas from $1\text{K}$ to $2\text{K}$ will be

• A. $\frac{15a}{2}$
• B. $\frac{15a}{4}$
• C. $\frac{15a}{7}$
• D. $\frac{15a}{8}$

Answer 1

The correct option is B $\frac{15a}{4}$

Here, specific heat of gas is not constant, thus we need to use integration method to find heat supplied.
Amount of heat required to raise temperature of a unit mass by a very small amount,
$Q=\int SdT$ $[\because dm=1]$
$\Rightarrow Q=\int a{T}^{3}dT$
$\Rightarrow Q=a{\left[\frac{T^4}{4}\right]}_1^2$
$\Rightarrow Q=\frac{a}{4}[16-1]$
$\Rightarrow Q=\frac{15a}{4}$

Why this question? Tips: Use technique of integration since we can't apply $Q=ms\Delta T$ directly because here $S$ is variable.