2016
Chapter 20
20.49.
IDENTIFY:
Use
UQW
Δ=−
and the appropriate expressions for
Q
,
W
and
U
Δ
for each type of process.
p
Vn
R
T
=
relates
T
Δ
to
p
and
V
values.
H
,
W
e
Q
=
where
H
Q
is the heat that enters the gas during the cycle.
SET UP:
For a monatomic ideal gas,
53
22
and C
.
PV
CR
R
==
(a)
ab
: The temperature changes by the same factor as the volume, and so
5
(
)
(2.5)(3.00 10 Pa)(0.300 m )
2.25 10 J.
P
Pa
a
b
C
Qn
CT
pV V
R
=Δ
=
−=
×
=
×
The work
p V
Δ
is the same except for the factor of
5
5
2
, so
0.90 10 J.
W
=×
5
1.35 10 J.
Δ=− =
×
bc
: The temperature now changes in proportion to the pressure change, and
3
53 5
2
(
)
(1.5)( 2.00 10 Pa)(0.800 m )
2.40 10 J,
cb
b
Qp
p
V
=−
×
=
−
×
and the work is zero
5
(
0).
2.40 10 J.
VU
Q
W
Δ= Δ= − =
−
×
ca
: The easiest way to do this is to find the work done first;
W
will be the negative of area in the
p

V
plane
bounded by the line representing the process
ca
and the verticals from points
a
and
c
. The area of this trapezoid is
55
3
3
4
1
2
(3.00 10 Pa 1.00 10 Pa)(0.800 m
0.500 m )
6.00 10 J
×+
×
−
=
×
and so the work is
5
0.60 10 J.
−×
U
Δ
must
be
5
1.05 10 J (since
0
U
×Δ
=
for the cycle, anticipating part (b)), and so
Q
must be
5
0.45 10 J.
UW
Δ+ =
×
(b)
See above;
5
0.30 10 J,
0.
QW
U
== ×
Δ=
(c)
The heat added, during process
ab
and
ca
, is 2.25
10 J
0.45 10 J
×
5
2.70 10 J
and the efficiency is
5
5
H
0.30 10
0.111 11.1%.
2.70 10
W
e
Q
×
=
=
×
EVALUATE:
For any cycle,
0
U
and
.
=
20.50.
IDENTIFY:
Use the appropriate expressions for
Q
,
W
and
U
Δ
for each process.
H
/
eWQ
=
and
Carnot
C
H
1/
.
eT
T
SET UP:
For this cycle,
H2
TT
=
and
C1
=
EXECUTE:
(a)
ab
: For the isothermal process,
0
T
Δ
=
and
0.
U
Δ
=
11
1
ln(
)
ln(1/ )
ln( )
ba
Wn
R
T VV n
R
T r n
R
Tr
=
−
and
1
ln( ).
QW n
R
T r
−
bc
: For the isochoric process,
0
V
and
0.
W
=
21
()
.
VV
QU
n
C
T
n
C
T
T
=
Δ =
−
cd
: As in the process
ab
,
2
0 and
ln( ).
Q
n
R
T
r
da
: As in process
bc
,
0 and
0;
VW
=
12
.
V
UQn
CTT
Δ==
−
(b)
The values of
Q
for the processes are the negatives of each other.
(c)
The net work for one cycle is
net
2
1
l
n
(
)
,
R
T
and the heat added (neglecting the heat exchanged during
the isochoric expansion and compression, as mentioned in part (b)) is
cd
2
ln( ),
R
=
and the efficiency is
net
1(
)
.
cd
W
T
Q
−
This is the same as the efficiency of a Carnotcycle engine operating between the two
temperatures.
EVALUATE:
For a Carnot cycle two steps in the cycle are isothermal and two are adiabatic and all the heat flow
occurs in the isothermal processes. For the Stirling cycle all the heat flow is also in the isothermal steps, since the
net heat flow in the two constant volume steps is zero.
20.51.
IDENTIFY:
The efficiency of the composite engine is
12
H1
,
WW
e
Q
+
=
where
H1
Q
is the heat input to the first engine
and
1
W
and
2
W
are the work outputs of the two engines. For any heat engine,
CH
,
WQ Q
=+
and for a Carnot engine,
low
low
high
high
,
QT
where
low
Q
and
high
Q
are the heat flows at the two reservoirs that have temperatures
low
T
and
high
.
T
SET UP:
high,2
low,1
.