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. temperature of the gas is decreased

Answer

A, B, C, D

Explanation

Equilibrium of piston gives `PS=Kx_(0)`
`P=(Kx_(0))/( S)`
Since the chamber is thermally insulated, `DeltaQ=0`
`:.` Elastic `PE` of spring `=` Work done by gas
or, Work done by gas `=(1)/(2) Kx_(0)^(2)`
This work is done at the expense of internal energy of the gas. Therefore, internal energy of the gas is decreased by `(1//2)Kx_(0)^(2)`.