Choose the correct option:
In the arrangement shown in Fig. gas is thermally insulated. An ideal gas is filled in the cylinder having pressure `P_(0)` (gt atmospheric pressure `P_(a)`). The spring of force constant `K` is initially un-stretched. The piston of mass `m` and area `S` is frictionless. In equilibrium, the piston rises up by distance `x_(0)`, then

Options

A. Final pressure of the gas is `P_a = (kx0)/S + (mg)/S`
B. Work done by gas is `1/2 kx_0^2 + m dg_0`
C. Decrease in internal energy of gas is `2 (1/2 kx_0^2 +" m gx"_0 + P_a Sx_0)`
D. Decrease in internal energy of gas is `1/2 kx_0^2 -"m gx"_0^2 + P_a Sx_0`

Answer

A

Explanation

Free body diagram of piston has been shown Piston is in equilibrium if `PS = P_a S + m g + kx_0` (P=final pressure of gas ) or
`P = P_a +(mg)/S + (kx_0)/S`
Work done by gas = work done against atmospheric pressure + elastic potential energy stored in the spring + increase in gravitational potential energy of the piston.
`P_a Delta V + 1/2 kx_0^2 + m gx_0`
` W = P_a sx_0 + 1/2 kx_0^2 + m gx_0`

Since the gas in thermally insulated, work is done at the expense of internal energy of the gas. This work done is also decrease in internal energy of gas.
Hence choice C is correct and choice D is wrong.