Page 33 - Essentials of physical chemistry

Introduction: Mathematics and Physics Review xxxi

p
ð
p
Now consider the somewhat tricky integral over u. We see that
0
½
sin(u)du ¼ cos(u)Š ¼
0

3
a
( 1) ( 1)Š ¼ 2 so we obtain V ¼ 4p , which is the familiar formula for the volume of
½
3
a sphere of radius a. The key point here is the product of integrals over u and f. Please note for
future reference the integral over the angles is 4p. For the remainder of this text, we do not want to
repeat this derivation and when we say ‘‘sphere’’ you should automatically think ‘‘4p’’ over angles.
For students who have not encountered three-dimensional integrals, we appeal to a picture for the
volume element and a shortcut for future treatment of spherical systems.
[Three friends, a biologist, a statistician and a physical chemist, went to a horse race and wanted
to bet on the winner. The biologist said he wanted to know what the horses ate for breakfast and the
statistician wanted to know how the various horses finished in previous races but the physical
chemist said nothing and was in deep thought for a long while. Finally the biologist asked what he
was thinking and the physical chemist said, ‘‘First you assume a spherical horse . . . ’’!]
By now you may need more humor beyond the ‘‘spherical horse joke’’ so there is one more
integral that we will need which is related to the Napier–Bernoulli–Euler constant e. Assuming you
have found your calculus book by now you should look up the natural logarithm formulas and find a
ð
dx
key integral as: ¼ ln (x) þ C.
x
The mnemonic to remember this integral is to set up the variable as x ¼ cabin so that we can
obtain an easily remembered formula:
ð
d(cabin)
¼ ln(cabin) þ ‘‘sea’’ ¼ A boat? ¼ Noah’s Ark?
cabin
Maybe that sort of humorous memory device will help you later on when we encounter this integral
many times?

SHORT REVIEW OF VECTOR ALGEBRA=CALCULUS
It may be important to review some basic facts about vectors since this topic comes up in the ‘‘dot
^ ^ ^
product’’ of a velocity vector. The unit vectors (i, j, k) point in the (x, y, z) directions and have a
~
^ ^ ^ ^ ^ ^
length of 1. Then according to the definition ~ a b ¼jajjbj cos (u ab ), we have i i ¼ j j ¼ k k ¼ 1
^ ^ ^
^ ^ ^ ^ ^ ^
(normalized) as well as i j ¼ i k ¼ j k ¼ 0 (mutually orthogonal). The (i, j, k) are the building
blocks for vectors in three-dimensional space and form an orthonomal basis set for (x, y, z) space.
We will need the idea of a ‘‘basis set’’ later on as a way to use components to build a linear
combination and ‘‘orthonormality’’ can lead to several time-saving steps later. We especially need
^
^
^
the idea of projecting a component out of a linear combination. Suppose we have ~ v ¼ ai þ bj þ ck
^
and we want to know what part or how much of that vector is the j part. We can take the dot product
of the component we want with the linear combination to find
^
^
^ j ~ v ¼ j (ai þ bj þ ck) ¼ 0 þ b þ 0
^
^
^ ^ ^
using the orthonormality of (i, j, k).
That concept is extremely important in several applications later in this text to greatly simplify
tedious spectroscopic concepts involving the Fourier transform. That step allows us to use the idea
of projection of an orthonormal component without doing a complicated integral! There are also
derivatives of vectors. A vector is a length with an associated direction. The speedometer reading of
an automobile gives a scalar speed (40 mph), but you need a dashboard compass to use that speed as
a directional vector as (40 mph, west). We can also define a directional ‘‘gradient vector’’ as

~ ^ qv ^ qv ^ qv :
rv i þ j þ k
qx qy qz