JEE Advanced · Physics · 13. Thermodynamics

An ideal monoatomic gas is confined in a horizontal cylinder by a spring loaded piston (as shown in the figure).Initially the gas is at temperature $T_{1}$ , pressure $P_{1}$ and volume $V_{1}$ and the spring is in its relaxed state. The gas is then heated very slowly to temperature $T_{2}$ , pressure $P_{2}$ and volume $V_{2}$ . During this process the piston moves out by a distance $x$ . Ignoring the friction between the piston the cylinder, the correct statement(s) is (are)

A If $V_{2} = 2 V_{1}$ and $T_{2} = 3 T_{1}$ , then the energy stored in the spring is $\frac{1}{4} P_{1} V_{1}$
B If $V_{2} = 2 V_{1}$ and $T_{2} = 3 T_{1}$ , then the change in internal energy is $3 P_{1} V_{1}$
C If $V_{2} = 3 V_{1}$ and $T_{2} = 4 T_{1}$ , then the work done by the gas is $\frac{7}{3} P_{1} V_{1}$
D If $V_{2} = 3 V_{1}$ and $T_{2} = 4 T_{1}$ , then the heat supplied to the gas is $\frac{17}{6} P_{1} V_{1}$

Verified Solution

Answer & Solution

Correct Answer

(A) If $V_{2} = 2 V_{1}$ and $T_{2} = 3 T_{1}$ , then the energy stored in the spring is $\frac{1}{4} P_{1} V_{1}$

Step-by-step Solution

Detailed explanation

(i)

P = P_{1} + \frac{K x}{A}

P_{2} = \frac{3}{2} P_{1} \Rightarrow x = \frac{V_{1}}{A}

\frac{3 P_{1}}{2} = P_{1} + \frac{K x}{A}

K x = \frac{P_{1} A}{2}

Energy of spring

\frac{1}{2} K x^{2} = \frac{P_{1} A}{4} x = \frac{P_{1} V_{1}}{4}

(ii)

∆ U = \frac{f}{2} (P_{2} V_{2} - P_{2} V_{1})

= 3 P_{1} V_{1}

(iii)

P_{f} = \frac{4 P_{1}}{3}

K X = \frac{P_{1}}{3} A

X = \frac{2 V_{1}}{A}

W_{g a s} = - (W_{P_{a t m}} + W_{s p r i n g})

= (P_{1} A x + \frac{1}{2} K x . x)

= + (P_{1} A . \frac{2 V_{1}}{A} + \frac{1}{2} . \frac{P_{1} A}{3} . \frac{2 V_{1}}{A})

= 2 P_{1} V_{1} + \frac{P_{1} V_{1}}{3} = \frac{7 P_{1} V_{1}}{3}

(iv)

∆ Q = W + ∆ U

= \frac{7 P_{1} V_{1}}{3} + \frac{3}{2} (P_{2} V_{2} - P_{1} V_{1})

= \frac{7 P_{1} V_{1}}{3} + \frac{3}{2} (\frac{4}{3} P_{1} . 3 V_{1} - P_{1} V_{1})

= \frac{7 P_{1} V_{1}}{3} + \frac{9}{2} P_{1} V_{1} = \frac{41 P_{1} V_{1}}{6}

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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