Page 34 - Essentials of physical chemistry

xxxii Introduction: Mathematics and Physics Review

Then we can define a second derivative as a dot product of the gradient with itself as

q q q q q q 2 2 2
~ ~ ^ i þ j ^ þ k ^ i ^ þ j ^ þ k q v q v q v :
^
r rv ^ v ¼ 2 þ 2 þ 2
qx qy qz qx qy qz qx qy qz
Note the result is a scalar second derivative in three variables. There is also another type of vector
product one may encounter when dealing with magnetic fields called a cross product.
^ ^

i j k ^
~ ^ ^ ^
~ a b a x a y ¼ i(a y b z b y a z ) j(a x b z b x a z ) þ k(a x b y b x a y ):

a z

b x b y b z
~
A similar directional cross product called the ‘‘curl’’ exists as r ~ v but we will not need it for
this course.
REVIEW OF CLASSICAL PHYSICS
Although our review of calculus should be adequate for the remainder of the text, we still need to
review physics for that part of physical chemistry. Newton’s second law is used heavily in the form:
F ¼ ma, but we need to remind ourselves that this is a vector equation.
* 2 *
* * * dv d r
F ¼ ma, where F x ¼ ma x , F y ¼ ma y , and a ¼ ¼
dt dt 2
In units: Force ¼ Newtons in mks; dynes in cgs.
2
1 Newton 1kg m=s , 1 dyne 1g cm=s 2
In units: Energy 1 joule ¼ 1 Newton meter, 1 erg 1 dyne cm (Figure I.2).
5
7
Simple conversions: 1 Newton ¼ 10 dynes, 1 joule ¼ 10 ergs
Recall that mass (kg or g) is an intrinsic property of matter but weight is a force on planet Earth
(Anyone here planning a vacation on another planet?) using the acceleration of gravity measured on
Earth (which is an average value since Earth is not exactly spherical) so instead of F ¼ ma we use
w ¼ mg (small w for weight and capital W for work) with the average value of g usually given as

2
g ¼ 980 cm=s ¼ 9:80 m=s 2
An important concept is Work (Force)(distance) or W ¼ Fd. In mks units work is given in joules
and in cgs units in ergs.
7
1 joule 10 ergs
[‘‘Erg!’’ is the last word of a ‘‘dying centipede,’’ that is, a dyne-cm]
Chemists often define heat energy in calories.
3
1 calorie the heat required to raise the temperature of 1 cm of water (1 c.c.) 18C (from 48C
to 58C). Later, we will discuss an experiment by Joule to convert heat energy to mechanical
energy; he found
1 calorie ¼ 4.184 joules

Erg !

Dyne – cm
*

FIGURE I.2 Humorous cartoon to remember the units of energy in the cgs system.