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 the heat required in the process in energy units is

• A. $5\times {10}^{4}Dm$
• 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

Answer: (C)

$\left(65/4\right)\times {10}^{4}Dm$

Given : $c=D{T}^3$ $T_1=20$ K $T_2=30$ K

Heat required $Q=\int mcdT$

$\therefore$ $Q=\int mD{T}^{3}dT=Dm\int T^{3}dT$

OR $Q=Dm\times \frac{T^4}{4}{|}_{T_1}^{T_2}$

OR $Q=Dm\times \frac{T_2^4-T_1^4}{4}$

OR $Q=Dm\times \frac{(30{)}^4-(20{)}^4}{4}=\frac{65}{4}\times {10}^{4}Dm$