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Question

An ideal monatomic gas is confined in a horizontal cylinder by a spring - loaded piston (as shown in the figure). Initially the gas is at temperature T1, pressure p1 and volume V1 and the spring is in its relaxed state. The gas is then heated very slowly to temperature T2, pressure p2 and volume V2.
During this process the piston moves out by a distance x. Which of the following statements is /are correct?
(Ignore the friction between the piston and the cylinder).


A
If V2=2V1 and T2=3T1, then the energy stored in the spring is 14p1V1.
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B
If V2=2V1 and T2=3T1, then the change in internal energy is 3p1V1
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C
If V2=3V1 and T2=4T1, then the work done by the gas is 73p1V1.
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D
If V2=3V1 and T2=4T1, then the heat supplied to the gas is 176p1V1.
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Solution

The correct options are
A If V2=2V1 and T2=3T1, then the energy stored in the spring is 14p1V1.
B If V2=2V1 and T2=3T1, then the change in internal energy is 3p1V1
C If V2=3V1 and T2=4T1, then the work done by the gas is 73p1V1.
For (a),(b)
V2=2V1 and T2=3T1 p2=32p1
p=p1+KxA
p2=32p1x=V1A
32p1=p1+KxA
Kx=p1A2
Energy stored in the spring =12Kx2=12(Kx)x=14p1V1.
Now, ΔU=f2(p2V2p1V1)=3p1V1.

In option (c),(d):
If V2=3V1 and T2=4T1
p2=43p1Kx=p13A,
so x=2V1A
Now, Wgas=(Watm+Wspring)=p1Ax+12Kx2
=p1A.2V1A+12.p1A3.2V1A=73p1V1.
Heat supplied to the system,
ΔQ=W+ΔU=73p1V1+32(p2V2p1V1)=416p1V1.

Hence, the correct options are (a), (b) and (c)

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