Question

The specific heat $C$ of a solid at low temperature shows temperature dependence according to the relation $C=D{T}^3$, where $D$ is a constant and $T$ is the temperature in kelvin. A piece of this solid of mass $m$ $kg$ is taken and its temperature is raised from $20K$ to $30K$. The amount of heat required in the process in energy units is:

• A. $5\times {10}^{2}Dm$
• B. $\left(33/4\right)\times {10}^{4}Dm$
• C. $\left(65/4\right)\times {10}^{4}Dm$
• D. $\left(5/4\right)\times {10}^{4}Dm$

Answer 1

The correct option is C $\left(65/4\right)\times {10}^{4}Dm$

Amount of heat required, $Q=\int dQ$
$={\int }_{T_1=20}^{T_2=30}mcdT$
given $C=D{T}^3$
$\therefore Q={\int }_{20}^{30}mD{T}^{3}dT=mD{\int }_{20}^{30}{T}^{3}dT$
$=mD\frac{1}{4}[{\left(30\right)}^4-{\left(20\right)}^4]=\frac{65}{4}\times {10}^{4}mD$