Question

The specific heat of a metal at low temperature varies according to $s=\alpha T^3$, where $\alpha$ is a constant and $T$ is absolute temperature. The heat energy needed to raise the temperature of unit mass of the metal from $T=1\text{K}$ to $T=2\text{K}$ is

• A. $3\alpha$
• B. $\frac{15\alpha }{4}$
• C. $\frac{2\alpha }{3}$
• D. $\frac{13\alpha }{4}$

Answer 1

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

We know that heat required to raise the temperature of a body by a small amount $dT$ is given by,
$dQ=msdT$
The amount of heat required to raise temperature by finite amount is obtained by integration with proper limits.
$Q=\int dQ=\int msdT$
$Q={\int }_1^2\left(1\right)\times \alpha T^3\times dT$
$[m=1(\text{unit mass}\left)\right]$
$\Rightarrow Q={\left[\frac{\alpha T^4}{4}\right]}_1^2$
$\Rightarrow Q=\frac{\alpha (2{)}^4}{4}-\frac{\alpha (1{)}^4}{4}$
$\therefore Q=\frac{15\alpha }{4}$