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Question

In the arrangement shown in figure, gas is thermally insulated. An ideal gas is filled in the cylinder having pressure P0(> atmospheric pressure Pa). The spring of force constant K is initially unstretched. The piston of mass m and area S is frictionless. In equilibrium, the piston rises up by distance x0, then

A
final pressure of the gas is P0+Kx0S+mgS
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B
work done by the gas is 12Kx20+mgx0
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C
decrease in internal energy of the gas is 12Kx20+mgx0+PaSx0
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D
all of the above
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Solution

The correct option is C decrease in internal energy of the gas is 12Kx20+mgx0+PaSx0
Equilibrium of piston gives


PS=PaS+mg+Kx0
P=Pa+mgS+Kx0S
(P=final pressure of gas)
Work done by the gas
= work done against atmospheric pressure + elastic potential energy stored in the spring + increase in gravitational potential energy of the piston
=PaΔV+12Kx20+mgx0=PaSx0+12Kx20+mgx0
There occurs decrease in internal energy of the gas, because the gas is thermally insulated and hence, work is done at the expense of internal energy of the gas.

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