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Question

One mole of an ideal gas is expanded from the state A(P0,V0) to final state B having volume V. The process follows a path represented by a straight line on the P-V diagram (see figure).
If final temperature at B is half the maximum temperature during the process, find volume (V) ?

A
(2+12)V0
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B
V02
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C
2V0
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D
(21)V0
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Solution

The correct option is A (2+12)V0
Line AB is seen to be passing from (2P0,0) & (P0,V0)
so, equation of AB can be written as,
PP0=2P0P0V0(VV0)
PP0=P0VV0+P0
P=P0VV0+2P0
So, equation of the given straight line is the process is
P=(P0V0)V+2P0
For an ideal gas PV=nRT
[P0V0V+2P0]V=nRT
T=P0V0RV2+2P0RV [n=1]...(ii)
This is equation of a parabola.
The vertex of parabola corresponds to dTdV=0
2P0V0RV+2P0R=0V=V0
Temperature in this state is
T0=P0V0nR
n=1T0=P0V0R
This is the maximum temperature during the process.

If final temperature at B is T02=P0V02R then volume can be obtained using (ii)
P0V02R=P0RV0V2+2P0RV
V02=V2V0+2V
2V24V0V+V20=0
V=4V0±16V208V204=4V0±8V04=V0±V02=(2+12)V0
V>V0

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