74. The work (the “area under the curve”) for process 1 is 4p

i

V

i

, so that U

b

− U

a

= Q

1

− W

1

= 6p

i

V

i

by the

First Law of Thermodynamics.

(a) Path 2 involves more work than path 1 (note the triangle in the figure of area

1
2

(4V

i

)(p

i

/2) = p

i

V

i

).

With W

2

= 4p

i

V

i

+ p

i

V

i

= 5p

i

V

i

, we obtain

Q

2

= W

2

+ U

b

− U

a

= 5p

i

V

i

+ 6p

i

V

i

= 11p

i

V

i

.

(b) Path 3 starts at a and ends at b so that ∆U = U

b

− U

a

= 6p

i

V

i

.