The Chemistry Maths Book, Second Edition

9.9 Multiple integrals 281

Also, the Maxwell relation derived from the differential Gibbs energy is

Therefore,

Experimentally, the heat capacity is measured at several temperatures betweenT

1

and

T

2

at pressurep

1

, and the expansivity is measured at several pressures betweenp

1

and

p

2

at temperature T

2

. The integrals are evaluated either by plottingC

p

2 TagainstT

andαVagainst pand measuring the area under the curve in each case, or by using a

numerical integration method (see Chapter 20).

9.9 Multiple integrals

A function of two independent variables,f(x, y), can be integrated with respect to

either, whilst keeping the other constant:

EXAMPLE 9.28Integrate f(x, 1 y) 1 = 12 xy 1 + 13 y

2

with respect to x in the interval

a 1 ≤ 1 x 1 ≤ 1 b, and with respect to yin the intervalc 1 ≤ 1 y 1 ≤ 1 d.

(i)

(ii)

In addition to these ‘partial integrals’, the integral with respect to both xand yis also

defined:

(9.53)

ZZ Z Z

c

d

a

b

c

d

a

b

f x y dx dy(),=f x y dx dy(),





















==,





















ZZ

a

b

c

d

fxydydx()

Z

c

d

c

d

()xy y dy xy y x d c()(23d

223223

+=+













=−+−cc

3

)

Z

a

b

a

b

() 23 xy y dx x y y x 3 y b a() 3 y

222 22

+=+













=−+

22

()ba−

ZZ

a

b

c

d

fxydx(),,or fxydy()

∆∆S

C

T

dT p p S V dp

T

T

p

p

p

13 1 32

1

2

1

2

→→

==ZZ()at ,=−α (attTT=

2

)

Tp

S

p

V

T

V

∂

∂













=−

∂

∂













=−α.

pp

H

T

T

S

T

C

p

∂

∂











 =

∂

∂











 =