Question

The specific heat of a metal at low temperatures 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. $\frac{15a}{4}$
• C. $\frac{2a}{3}$
• D. $\frac{12a}{5}$

Answer 1

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

At temperature $t$ the heat energy required to raise temperature of unit mass by $dt$ is $dq=a{t}^3\times 1\times dt$

So heat required to raise temperature from $1K$ to $2K$ is

${\int }_0^{Q}dq={\int }_1^{2}a{t}^{3}dt\Rightarrow Q=a\frac{t^4}{4}{|}_1^2=a(16-1)/4$

$\Rightarrow Q=15a/4$