Mid Term 105A
10/31/06
1/5
Mid Term #1
105A Fall 2006
Question 1
One mole of gas in a chamber is compressed and decompressed using a piston which
slides without friction. The initial state is at p
1
= 2x10
5
Pa and V
1
=14x10
3
m
3
. This gas
goes successively through:
1.
An isobaric decompression, which double its volume;
2.
An isothermal compression, which bring the volume back to V
1
;
3.
An isochoric (constant volume) cooling, which brings it back to p
1
.
We suppose this gas ideal.
a
What is the temperature at which the compression happens?
b
Draw the different processes on a (p,V) diagram. How is this type of
transformation called?
c
What is the work and heat exchanged by this system during one of these
transformations?
a (5 pts) We start with one mole of gas at 2x10
5
Pa and 14x10
3
m
3
. Using
the ideal gas law the temperature is found to
=
=
=
2 2
1
1
2
1
2
2
pV
p V
T
T
R
R
or
T
2
= 673.56 K.
b (10 pts)
Using the ideal gas law at constant
temperature for closed system we
also find that
=
3 3
2 2
pV
pV
. Since the
isothermal process brings us back
to
V
1
we
have
that
V
3
= V
1
. So
=
3 1
2
1
2
pV
p V
and because
p
2
= p
1
we finally find
=
3
1
2
p
p
Since we come back to the original
state this transformation is called
a cycle.
c (15 pts) Using the first law of thermodynamics we have for the whole
cycle
1
2
3
p
V
V
1
2V
1
p
1
2p
1
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Mid Term 105A
10/31/06
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∆
=
−
cycle
U
Q
W
.Since
∆
=
0
cycle
U
,
Q
=
W
. The elementary work for a simple
compressional system such as this one is pdV and the total work is
→
→
→
=
+
+
1
2
2
3
3
1
W
W
W
W
. So
=
+
+
∫
∫
∫
2
3
1
1
2
3
W
pdV
pdV
pdV
. Since the process
3
→
1
is
isochoric
→
=
3
1
0
W
,
since
the
process
1
→
2
is
isobaric
→
=
−
=
1
2
1
2
1
1 1
(
)
W
p V
V
pV
and
since
2
→
3
is
isothermal
→
=
=
∫
∫
3
3
2 2
2
3
1
1
2
2
2
pV
dV
W
dV
p V
V
V
or
→
=
= −
3
2
3
1
1
1 1
2
2 ln
2ln2
V
W
p V
pV
V
. The total work
is
(
)
=
−
<
1 1
1
2ln2
0
W
pV
or
W
= 1081.6 J = 
Q
Question 2
A closed chamber has adiabatic wall (no
exchange of heat with the surroundings).
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 Fall '06
 PierreGourdain
 Thermodynamics

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