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A thermally insulated chamber of volume $2 V_{0}$ is divided by a frictionless piston of area S into two equal parts A and B. Part A has an ideal gas at pressure $P_{0}$ and temperature $T_{0}$ and in part B is vacuum. A massless spring of force constant k is connected with piston and the wall of the container as shown. Initially spring is unstretched. Gas in chamber A is allowed to expand. Let in equilibrium spring is compressed by $x_{0}$ . Then

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Work done by the gas is $\frac{1}{2} k x_{0}^{2}$

Final pressure of the gas is $\frac{k x_{0}}{S}$

Change in internal energy of the gas is $\frac{1}{2} k x_{0}^{2}$

Temperature of the gas is decreased

answer is A, B, C, D.

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Detailed Solution

Equilibrium of piston gives

$\begin{array}{l} P S = k x_{0} \\ \Rightarrow P = \frac{k x_{0}}{S} \end{array}$

Since the chamber is thermally insulated

$Δ Q = 0$

Elastic P.E of spring=work done by gas

$\Rightarrow$ Work done by gas $= \frac{1}{2} k x_{0}^{2}$

This work is done in the expense of internal energy of the gas.

Therefore, internal energy of the gas is deceased by $= \frac{1}{2} k x_{0}^{2}$

Internal energy of an ideal gas depends on its temperature only. Internal energy of the gas is decreasing. Therefore, temperature of the gas will decrease.

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