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Question

One mole of an ideal monatomic gas has initial temperature T0 is made to go through the cycle abca as shown in the given figure. If U denotes the internal energy, then choose the correct alternative.

A
Uc>Ub>Ua
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B
UcUb=3RT0
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C
UcUa=9RT02
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D
UbUa=3RT02
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Solution

The correct options are
A Uc>Ub>Ua
B UcUb=3RT0
C UcUa=9RT02
D UbUa=3RT02
Internal energy is a state function.(Doesnot depend on path)
We know, ΔU=nCvΔT
From a to b, pressure is contant
VT
Volume changes from Vo to 2Vo
temperature changes from To to 2To .
For a monoatomic gas, Cv=32R
UbUa=nCvΔT
=1×32R(2ToTo)=32RTo....(i)
Option (d) is correct

From b to c, volume is constant.
PT
Pressure changes from Po to 2Po
temperature changes from 2To to 4To.
UcUb=nCvΔT
=1×32R(4To2To)=3RTo....(ii)
Option (b) is correct
Also option (a) is correct

For c to a, we add equations (i) and (ii)
UcUa=(UcUb)+(UbUa)
=3RTo+32RTo=9RT02
Option (c) is also correct.

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