A thermally insulated chamber of volume `2V_(0)` is divided by a frictionless piston of area `S` into two equal part `A` and `B`. Part `A` has an ideal gas at pressrue `P_(0)` and temperature `T_(0)` and part `B` is vacuum. A massless spring of force constant `K` is connected with the piston and the wall of the container as shown. Initially the spring is unstretched. The gas inside chamber `A` is allowed to expand. Let in equilibrium the spring be compressed by `x_(0)`. Then

Options

A. Final pressure of the gas is `(kx_(0))/(S)`
B. Work done by the gas is `(1)/(2)kx_(0)^(2)`
C. Change in internal energy of the gas is `(1)/(2)kx_(0)^(2)`
D. Temeperature of the gas decreased

Explanation

Equilibrium piston gives
`PS=k x_(0)` or `P=(k x_(0))/(S)`
Since, the chamber is thermally insulted
`DeltaQ=0`
`therefore` Elastic potential energy of spring=work done by gas or work done by gas `=(1)/(2)kx_(0)^(2)`
This work is done in the expense of internal energy of the gas. Therefore, internal energy of the gas is decreased by `=(1)/(2)kx_(0)^(2)`
Internal energy of an ideal gas depedns on its temperature only. Internal energy of the gas is decreasing. Therefore, temperature of the gas will decrease.