Question

According to Debye's $ {{T}^{3}} $ law, the specific heat of many solids at low temperature T vanes acceding to the relation $ C=2{{T}^{3}}, $ where a is constant. The heat energy required to raise the temperature of 4kg mass from $ T=1 $ to $ T=3K $

• A. $ 208\text{ }\alpha $
• B. $ 20\text{ }\alpha $
• C. $ 80\text{ }\alpha $
• D. $ \text{8 }\alpha $

Answer 1

Answer: (C)

Since, $ dQ=mc\,dT $
$ \Rightarrow $ $ Q\int{dQ}=m\alpha \int_{1}^{3}{{{T}^{3}}dT} $
$ \Rightarrow $ $ Q=4\alpha \left[ \frac{{{T}^{4}}}{4} \right]_{1}^{3} $
$ \Rightarrow $ $ Q=\alpha [{{(3)}^{4}}-{{(1)}^{4}}]=\alpha (81-1) $
$ \Rightarrow $ $ Q=80\,\alpha $