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 .