Physical Chemistry Third Edition

B Some Useful Mathematics 1239

IfT,V, andnare the independent variables in the exact differential thenzis a function

ofT,V, andn. The theorem states that a line integral ofdzis equal to the value ofzat

the end point of the integration minus the value ofzat the starting point:

∫

c

dz

∫

c

[(

∂z

∂T

)

V,n

dT+

(

∂z

∂V

)

T,n

dV+

(

∂z

∂n

)

T,V

dn

]

z(T 2 ,V 2 ,n 2 )−z(T 1 ,V 1 ,n 1 ) (B-23)

whereT 2 ,V 2 , andn 2 are the values of the independent variables at the final point of

the curve, andT 1 ,V 1 , andn 1 are the values at the initial point of the curve. Since many

different curves can have the same initial and final points, Eq. (B-23) means that the

line integral depends only on the initial point and the final point, and is independent

of the curve between these points. It is said to bepath-independent. The line integral

of an inexact differential is generallypath-dependent. That is, one can always find two

or more paths between a given initial point and a given final point for which the line

integrals are not equal.

Multiple Integrals Ifff(x,y,z) is an integrand function, a multiple integral with

constant limits is denoted by

I(a 1 ,a 2 ,b 1 ,b 2 ,c 1 ,c 2 )

∫a 2

a 1

∫b 2

b 1

∫c 2

c 1

f(x,y,z)dzdydx (B-24)

The integrations are carried out sequentially. The leftmost differential and the rightmost

integral sign belong together, and this integration is done first, and so on. Variables

not yet integrated are treated as constants during the integrations. In Eq. (B-24),zis

first integrated fromc 1 toc 2 , treatingxandyas constants during this integration.

The result is a function ofxandy, which is the integrand whenyis then integrated

fromb 1 tob 2 , treatingxas a constant. The result is a function ofx, which is the

integrand whenxis then integrated froma 1 toa 2. In this multiple integral the limits

of thezintegration can be replaced by functions ofxandy, and the limits of they

integration can be replaced by functions ofx. The limit functions are substituted into the

indefinite integral in exactly the same way as are constants when the indefinite integral is

evaluated at the limits.

If the variables are Cartesian coordinates and the limits are constants, the region of

integration is a rectangular parallelepiped (box) as shown in Figure B.2. If the limits

for the first two integrations are not constants, the region of integration can have a more

complicated shape.

Region of integration

ofx from a 1 to a 2

y from b 1 to b 2

z from c 1 to c 2

z

y

x

c 2

c 1

a 1

a 2

b 1 b^2

Figure B.2 An Integration Region
in Cartesian Coordinates with Con-
stant Limits.

The integration process can be depicted geometrically as follows: The product

dxdydzis avolume elementand is depicted in Figure B.3 (although the box pictured

has finite dimensions and the volume elementdxdydzis infinitesimal). This volume

element is also denoted asd^3 r.If(x,y,z) represents a point in the volume element,

then the contribution of the element of volume to the integral is equal to the value of

the function at (x,y,z) times the volume of the volume element:

(contribution of the volume elementdxdydz)f(x,y,z)dxdydz (B-25)

The integral is the sum of the contributions of all the volume elements in the region of

integration.

If an integral over a volume in three-dimensional space is needed and spherical

polar coordinates are used, the volume element is as depicted in Figure B.4. The length