The Chemistry Maths Book, Second Edition

9.8 Line integrals 277

(see Example 6.10 for ). The result is different from that obtained in

Example 9.23.

0 Exercises 59, 60

The dependence of the line integral on the path can be understood, for example, in

terms of the work done on a body in moving it from A to B along the path. From the

discussion of work and potential energy in Section 5.7, we expect the work done to

depend on the path when the forces acting on the body are not conservative forces;

for example, when work is done against friction. In particular, net work is done

in moving the body around a closed path. A dependence on path is also found in

thermodynamics.

EXAMPLE 9.25Work in thermodynamics

When changes in a thermodynamic system are reversible (Section 5.8), the quantity

TdSin equation (9.40),

dU 1 = 1 TdS 1 − 1 pdV

is identified with the heat absorbed by the system, andpdVwith the (mechanical)

work done by the system. We consider the work done by the ideal gas on (i) expansion

fromV

1

1 = 1 V(p

1

, T

1

), at point A in Figure 9.10, along pathC

1

1 + 1 C

2

toV

2

1 = 1 V(p

2

, T

2

)

at point B, and (ii) the return to A alongC

3

1 + 1 C

4

.

(i) PathC

1

1 + 1 C

2

The work done along path C

1

is at constant pressurep 1 = 1 p

1

and, by equation (5.71),

W

1

1 = 1 nR(T

2

1 − 1 T

1

).

The work done along path C

2

is at constant temperatureT 1 = 1 T

2

and, by equation

(5.72),

W

2

1 = 1 −nRT

2

1 ln(p

2

2 p

1

).

Z

0

1

2

1 −xdx

0 T

1

T

2

p

1

p

2

• 
  
  
  
  • 
    
  
  
  • 
    
  
  
  
  

a

b

c

1

c

2

c

3

c

4

T

p

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Figure 9.10