Phys 2101
Homework 16 Solution Spring `11
These solutions use the parameter values from the problems printed in the book, not those that appear in
your personal homework assignment. The logic used to get to the answer is the same, however.
1. (a) Since the gas is ideal, its pressure
p
is given in terms of the number of moles
n
, the volume
V
, and the
temperature
T
by
p = nRT
/
V
. The work done by the gas during the isothermal expansion is
22
11
2
1
ln
.
VV
dV
V
W
pdV
nRT
We substitute
V
2
= 2.00
V
1
to obtain
3
=
ln2.00 = 4.00 mol 8.31 J/mol K 400 K ln2.00 = 9.22 10 J.
W
(b) Since the expansion is isothermal, the change in entropy is given by
1
S
T dQ Q T
,
where
Q
is the heat absorbed. According to the first law of thermodynamics,
E
int
=
Q
W
. Now the internal
energy of an ideal gas depends only on the temperature and not on the pressure and volume. Since the expansion
is isothermal,
E
int
= 0 and
Q = W
. Thus,
3
9.22 10 J
=
=
= 23.1 J/K.
400 K
W
S
T
(c)
Δ
S
= 0 for all reversible adiabatic processes.
2. (a) This may be considered a reversible process (as well as isothermal), so we use
Δ
S = Q
/
T
where
Q = Lm
with
L
= 333 J/g from Table 19-4. Consequently,
.
/
6
.
14
273
)
0
.
12
)(
/
333
(
K
J
K
g
g
J
S
(b) The situation is similar to that described in part (a), except with
L
= 2256 J/g,
m
= 5.00 g, and
T
= 373 K. We
therefore find
Δ
S
= 30.2 J/K.
3. (a) The energy that leaves the aluminum as heat has magnitude
Q = m
a
c
a
(
T
ai
T
f
), where
m
a
is the mass of the
aluminum,
c
a
is the specific heat of aluminum,
T
ai
is the initial temperature of the aluminum, and
T
f
is the final
temperature of the aluminum–water system. The energy that enters the water as heat has magnitude
Q = m
w
c
w
(
T
f
T
wi
), where
m
w
is the mass of the water,
c
w
is the specific heat of water, and
T
wi
is the initial temperature of the
water. The two energies are the same in magnitude, since no energy is lost. Thus,
+
=
=
.
+
a a ai
w w wi
a a
ai
f
w w
f
wi
f
a a
w w
m c T
m c T
m c T
T
m c T
T
T
m c
m c
The specific heat of aluminum is 900 J/kg
K and the specific heat of water is 4190 J/kg
K. Thus,