One mole of a monatomic ideal gas is taken from an initial state of pressure po and volume Vo to a final state of pressure 2po and volume 2V by two different processes. (1) It expands isothermally until its volume is doubled, and then its pressure is increased at constant volume to the final state. (1) It is compressed isothermally until its pressure is doubled, and then its volume is increased at constant pressure to the final state. Show the path of each process on a pV diagram. For each process calculate in terms of po and Vo: (a) the heat absorbed by the gas in each part of the process; (b) the work done on the gas in each part of the process; (c) the change in internal energy of the gas, Eints - Eint, and (d) the change in entropy of the gas, St-S.
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(a) We have PV=nRT
If T
1
and T
2
are initial and final temperatures,
P
o
V
o
=nRT
1
and
2
P
o
2V
o
=nRT
2
⟹T
1
=T
2
(b) T
1
=T
2
⟹ΔU=0
(c) Work done = area under the curve
W=
2
1
×P
o
/2×(2V
o
−V
o
)+P
o
/2×(2V
o
−V
o
)
=
4
1
×P
o
×V
o
+
2
1
×P
o
×V
o
=
4
3
×P
o
×V
o
(d) We have,
ΔQ−ΔW=ΔU
Since ΔU=0 and ΔW is positive (i.e.s work is done by the system),
ΔQ>0, heat is absorbed in the process.
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