Page 32 - Essentials of physical chemistry

xxx Introduction: Mathematics and Physics Review

this method of initial rapid introduction to necessary mathematics has worked in classes for over
30 years. Actually, we have no choice if you are limited to only one semester of calculus and we
need to do a thorough treatment of physical chemistry in one basic semester course. By showing the
applications of the mathematics in the laboratory topics we want to treat, it has been found that
students accept the mathematics, and this is an adequate method of learning, although taking the
mathematics in separate courses would be desirable.
Now there is another mathematics topic we need to demonstrate, that of spherical polar
coordinates, which occur frequently in physical chemistry. We will often need to reason in three
dimensions and use spherical polar coordinates.
Suppose we want to calculate the volume of a rectangular box with dimensions (L W H), we
could set up an integral for the volume as

L
ð ð W ð H
W
H
L
V ¼ dx dy dz ¼ x½Š y½Š z½Š ¼ (L 0)(W 0)(H 0) ¼ LWH, as expected:
0
0
0
0 0 0
However, when we need to treat a spherical system, we have to convert (x, y, z)to (r,u, f) coordinates. It
is easily seen that z ¼ r cos(u), x ¼ r sin(u) cos(f), and y¼ r sin(u) sin(f) if you draw the projections of
the vector r on the (x, y,z) axis system. That is easy but it is more difficult to convert the volume element
to polar coordinates (Figure I.1). This is difficult to visualize but you should be able to see a quasi-cube
with dimension (rdu) as a short arc caused by a small change in u with fixed r as if the vector acts like a
crane arm moving up or down with the base of the crane at the origin. Then another small arc can be
generated if the crane arm swings in the f angle but the effective length of the crane arm in the (x, y)
plane (the dotted line) is r sin(u) and when there is a small change in f, the arc length is the product
(radius)(arc) or r sin(u)df. Finally, the thickness of this small quasi-cube is dr and so in the limit of
2
infinitesimals we obtain r sin(u)drdudf in place of dxdydz. Note that the range of the f angle is 0 to 2p,
so the crane arm can spin all the way around the z-axis, but the u angle only needs to vary between 0 and
p when coordinated with the f angle to reach any coordinate in the original (x,y,z) coordinate system.
Consider the volumeof a sphere of radius a inthe spherical polar coordinatesystemas a key example.

ð a ð 2p ð p ð p ð p
a 2p 3
2
3

V ¼ r dr df sin(u)du ¼ r =3 [f] sin(u)du ¼ 2pa =3 sin(u)du:
0 0
0 0 0 0 0
z

ρ sin θ dφ

ρ sin θ
ρ dθ
dρ
θ
O y

φ
dφ
x
FIGURE I.1 Volume element in spherical polar coordinates. (Adapted from Thomas, G.B., Calculus and
Analytic Geometry, Addison-Wesley Publishing Co., Reading, MA, 1953, p. 549.)