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Question

An ideal monoatomic gas is confined in a cylinder by a spring-loaded piston of cross-sectional area 8×103 m2. Initially the gas is at 300 K and occupies a volume of 2.4×103 m3. The spring is in a relaxed state. The gas is heated by a small heater coil H. The force constant of the spring is 8000 N/m and the atmospheric pressure is 1.0×105 Pa. The cylinder and piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and cylinder. There is no heat loss through the heater coil wire and thermal capacity of the heater coil is negligible. With all the above assumptions, if the gas is heated by the heater until the piston moves out slowly by 0.1 m, then the final temperature is -


A
400 K
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B
800 K
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C
1200 K
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D
300 K
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Solution

The correct option is B 800 K
Given:
Cross sectional area of piston, A=8×103 m2
Initial temperature, T1=300 K
Initial volume, V1=2.4×103 m3
Force constant of spring, k=8000 N/m
Atmospheric pressure, Po=1.0×105 Pa
Distance moved out by piston, x=0.1 m
To find:
Final temperature, T2=?

Initial pressure will be equal to the atmospheric pressure as spring is in natural state.
P1=P0=1.0×105 Pa
And, final pressure will balance the pressure due to atmosphere and spring.
P2=kxA+P0=8000×0.18×103+1.0×105
=2×105 Pa
Also, final volume will be equal to initial volume plus increase in volume.
V2=V1+Ax=2.4×103+8×103×0.1
=3.2×103 m3

We know that, ideal gas equation is given by
PV=nRT
As number of moles will remain constant, so,
P1V1T1=P2V2T2
T2=P2V2T1P1V1
T2=2×105×3.2×103×3001.0×105×2.4×103
T2=800 K

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