Question

A thermally insulated chamber of volume $2v_0$ 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 there 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. Consider the compression in spring to be $x_0$ in equilibrium.

• A. Final pressure of the gas is $\frac{k{x}_0}{2s}$
• B. Work done by the gas is $k{x}_0^2$
• C. Change in internal energy of the gas is $\frac{1}{2}k{x}_0^2$
• D. Temperature of the gas is increased

Answer 1

The correct option is C

Change in internal energy of the gas is $\frac{1}{2}k{x}_0^2$

Analysing equilibrium of piston,

$PS=k{x}_0$

$\Rightarrow P=\frac{k{x}_0}{S}$

Since, the chamber is thermally insulated,

$\Delta Q=0$

$\therefore$ Elastic potential energy of the spring

$=\frac{1}{2}k{x}_0^2$

= work done by the gas

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

Therefore, internal energy of the gas is decreased by $\frac{1}{2}k{x}_0^2$