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  2. Question
  3. Physics
  4. The specific heat c of a solid at low temperature shows temperature dependence according to the relation c = DT3 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 20 K to 30 K. The amount of the heat required in the process in energy units is

Q. The specific heat $c$ of a solid at low temperature shows temperature dependence according to the relation $c = DT^3$ 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 $20\, K$ to $30\, K$. The amount of the heat required in the process in energy units is

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Solution:

Amount of heat required, $Q=\int d Q$
$=\int_\limits{T_{1}=20}^{T_{2}=30} m c d T $
Given, $ C =D T^{3}$
$\therefore Q =\int_\limits{20}^{30} m D T^{3} d T=m D \int_\limits{20}^{30} T^{3} d T $
$=m D \frac{1}{4}\left[(30)^{2}-(20)^{2}\right]=\frac{65}{4} \times 10^{4} mD$

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