Page 26 - Essentials of physical chemistry

xxiv Introduction: Mathematics and Physics Review

of this text is to use your mental synthesis skills learning massive amounts of chemistry to ‘‘prime the
pump’’ of your mind using a few examples at a level of two semesters of calculus even though you
may have only had one semester! We are going to show the mathematical details (calculus nuggets) of
selected cases to provide depth in a few areas but you have to do your part! The author earned a grade
of D- in the first quarter of organic chemistry taken in summer school by lounging in a hammock and
just moving his eyes over class notes. The second and third quarters of organic chemistry led to grades
of A because the present author wrote each reaction over 20 times! Thus it is very important that you
use a pencil and paper and copy over the examples and proofs given in this text. Would you expect to
learn to swim by reading a book or learn to shoot basketball foul shots by reading about it? This author
firmly believes that hand-eye activity in writing equations is a valid way to study physical chemistry
as well as organic chemistry. If you expect to just read this text or highlight key passages as if it were a
history book, you are already in trouble. However, if you follow along with pencil and paper you will
be amazed that priming your mental pump with detailed equations will lead to increased confidence at
what you can do with a minimum of mathematics! This approach has worked with timid young ladies
as well as overconfident male athletes and the result has always been great morale in the class to the
point of actually having ‘‘fun’’ in physical chemistry!
The overwhelming obstacle to the study of physical chemistry is a lack of skill in mathematics
whether in calculus or just basic competence in careful addition. Overall, the language of all the
physical sciences is mathematics! It is amazing that if you check on journal articles in Chinese,
Russian, Turkish, French, German, or English, the equations and tables of numbers are the same! In
fact, we will start slowly with calculus examples but the real language of this course is ‘‘Calculus’’!
The author has taught this material to hundreds of students from all over the world whose use of
English is a second or third language. This is not a course on the history of science; we seek to form
mathematical relationships in your mind. The common language really is Calculus at a level of two
semesters of that topic. The prerequisite for this text is at least one semester of calculus and presumably
some exposure to trigonometry. That means you know the basic ideas of calculus and have some
experience with derivatives but perhaps have not worked with integrals or certainly not integrals over
more than one variable. That can be a problem for this course so this ‘‘Introduction’’ should be studied
or presented in class before the first ‘‘real’’ chapter. Hopefully your teacher will spend a lecture or two
in the first week on this material, but if not it will be in your best interest to study this chapter on your
own. We just have to get over the initial barrier to review derivatives and then explore analytical
integration (the reverse of taking a derivative) in a brief way that will get students into the game with at
least a fighting chance of winning that A grade in physical chemistry! Along the way we might as well
review some key topics from sophomore physics, which is another basic prerequisite, and introduce
the use of partial derivatives, which we will use a LOT in thermodynamics!
Let us start with a review of derivatives. It is important to understand for future use that both dx
and dy are individually small quantities that may be manipulated using algebra and the derivative is
the limiting case of the ratio of the two quantities in the limit of infinitesimal size!

dy Dy y(x þ h) y(x)
lim ¼ lim :
dx Dx!0 Dx h!0 h
Note the use of the ‘‘super equal’’ sign with three lines, which indicates a definition of a term rather
than equal values of two expressions.
2
Example: Let y ¼ x , then we find (as expected)
2 2 2 2 2 2
dy (x þ h) x x þ 2xh þ h x 2xh þ h
¼ lim ¼ lim ¼ lim ¼ 2x:
dx h!0 h h!0 h h!0 h
n
This can be generalized using the binomial expansion of (x þ h) .