WBJEE · Physics · Thermal Properties of Matter

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 is kelvin. A piece of this solid of mass \(m \mathrm{kg}\) is taken and its temperature is raised from \(20 \mathrm{K}\) to \(30 \mathrm{K}\). The amount of heat required in the process in energy units is

A \(5 \times 10^{4} \mathrm{Dm}\)
B \((33 / 4) \times 10^{4} D m\)
C \((65 / 4) \times 10^{4} D m\)
D \((5 / 4) \times 10^{4} D m\)

Verified Solution

Answer & Solution

Correct Answer

(C) \((65 / 4) \times 10^{4} D m\)

Step-by-step Solution

Detailed explanation

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

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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