78. It is straightforward to show, from Eq. 20-11, that for any process that is depicted as a straight line on
the pV diagram, the work is
W
straight
=
p
i
+ p
f
2
∆V
which includes, as special cases, W = p∆V for constant-pressure processes and W = 0 for constant-
volume processes. Also, from the ideal gas law in ratio form (see Sample Problem 1), we find the final
temperature:
T
2
= T
1
p
2
p
1
V
2
V
1
= 4T
1
.
(a) With ∆V = V
2
− V
1
= 2V
1
− V
1
= V
1
and p
1
+ p
2
= p
1
+ 2p
1
= 3p
1
, we obtain
W
straight
=
3
2
(p
1
V
1
) =
3
2
nRT
1
where the ideal gas law is used in that final step.
(b) With ∆T = T
2
− T
1
= 4T
1
− T
1
= 3T
1
and C
V
=
3
2
R, we find
∆E
int
= n
3
2
R
(3T
1
) =
9
2
nRT
1
.
(c) The energy added as heat is Q = ∆E
int
+ W
straight
= 6nRT
1
.
(d) The molar specific heat for this process may be defined by
C
straight
=
Q
n∆T
=
6nRT
1
n (3T
1
)
= 2R .