The Chemistry Maths Book, Second Edition

278 Chapter 9Functions of several variables

Therefore

(ii) PathC

3

1 + 1 C

4

The work done along path C

3

is at constant pressurep 1 = 1 p

2

, and along path C

4

it is

at constant temperatureT 1 = 1 T

1

. ThereforeW

3

1 = 1 nR(T

1

1 − 1 T

2

), W

4

1 = 1 −nRT

1

1 ln(p

1

2 p

2

),

and

The total work done by the gas around the closed pathC 1 = 1 C

1

1 + 1 C

2

1 + 1 C

3

1 + 1 C

4

is then

and this is not zero unless eitherT

2

1 = 1 T

1

orp

2

1 = 1 p

1

.

0 Exercise 61

When forces are conservative and can be derived from a potential-energy function, as

in equation (5.57) of Section 5.7, the work is independent of the path, and no net work

is done around a closed path. In general, the value of a line integral is independent of the

pathif the quantity in the square brackets in equation (9.48) is an exact differential.

8

By the discussion of Section 9.7, there then exists a functionz(x, y)such that

(9.50)

and the line integral (9.48) can be written as

(9.51)

Then, for a path C from A at(x

1

,y

1

)to B at(x

2

, y

2

),

(9.52)

and this depends only on the values of zat the end points (we note that the value of the

integral changes sign if the direction of integration is changed to B to A). In terms of

Z

C

A

B

dz z= z x y z x y













=,−,()()

22 11

ZZ

CC

dz

z

x

dx

z

y

dy

yx

=

∂

∂













• 
  

∂

∂

































F x y dx G x y dy dz

z

x

dx

z

y

yx

() (),+,==

∂

∂











 +

∂

∂











dy

I

C

pdV nR T T=− ()ln()− p p

21 21

Z

CC

3

+

=− − −













4

21 1 21

pdV nR T T T p p()ln()

Z

CC

1

+

=+ = −−













2

12 212 21

pdV W W nR T T T p p()ln()

8

This independence of the path for the line integral of an exact differential was observed by Riemann in 1857.