MHT CET · Physics · Thermodynamics

Two cylinders A and B fitted with piston contain equal amount of an ideal diatomic as at temperature ' \(T\) ' K . The piston of cylinder A is free to move while that of B is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise temperature of the gas in A is ' \(\mathrm{dT}_{\mathrm{A}}\) ', then the rise in temperature of the gas in cylinder B is \(\left(\gamma=\frac{\dot{\mathrm{C}}_{\mathrm{P}}}{\mathrm{C}_{\mathrm{V}}}\right)\)

A \(\quad 2 \mathrm{dT}_{\mathrm{A}}\)
B \(\frac{\mathrm{dT}_{\mathrm{A}}}{2}\)
C \(\gamma \mathrm{dT}_{\mathrm{A}}\)
D \(\frac{\mathrm{dT}_{\mathrm{A}}}{\gamma}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\gamma \mathrm{dT}_{\mathrm{A}}\)

Step-by-step Solution

Detailed explanation

In cylinder A , heat is supplied at constant pressure while in B at constant volume
\(\begin{aligned}
& \mathrm{Q}_{\mathrm{A}}=\mathrm{Q}_{\mathrm{B}} \\
& \mathrm{nC}_{\mathrm{p}} \mathrm{dT}_{\mathrm{A}}=\mathrm{nC}_{\mathrm{v}} \mathrm{dT}_{\mathrm{B}} \\
& \mathrm{dT}_{\mathrm{B}}=\frac{\mathrm{C}_{\mathrm{p}}}{\mathrm{C}_{\mathrm{v}}} \mathrm{dT}_{\mathrm{A}}=\gamma \mathrm{dT}_{\mathrm{A}}
\end{aligned}\)

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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