Page 29 - Essentials of physical chemistry

Introduction: Mathematics and Physics Review xxvii

under the curve between the limits and comparing to the weight of a known rectangle area of graph
paper. Since chemists have access to very accurate balances, it is relatively easy to achieve 1%
accuracy by this graphical-weighing method.
ð
dg(x)
Indefinite integral: f (x)dx ¼ g(x) þ C, where ¼ f (x) and C is an unknown constant. For
dx
homework problems, an indefinite integral is incorrect unless the þC term is given.
ð (nþ1) ð ax
x e
n ax
Examples: x dx ¼ þ C and e dx ¼ þ C.
(n þ 1) a
We can treat an important but more complicated case that we will use to justify formulas we merely
memorize later but we need to be aware of a process called integration by parts, which makes use of
definite integration. Consider the integral that is the reverse process of the product derivative rule.
Simply put, we wrap a definite integration process around the product rule formula, that is, we
perform the definite integration term by term on both sides of the product equation and use the same
limits on all the terms. Note that d(UV) ¼ UdV þ VdU does not have the dx denominator. This form
of a derivative of just the numerator is called the ‘‘differential’’ and is valid for whatever variable is
in the denominator. This concept is often used in thermodynamics.
ð b ð b ð b ð b ð b ð b
d(UV) ¼ UdV þ VdU, then UdV ¼ d(UV) VdU
a a a a a a
This simple formal trick can be used in a tedious way to evaluate definite integrals, which cannot be
integrated in a single step. This process can seem confusing at first but can be learned. However,
there is a problem in that if you arbitrarily choose U and V by separating the UV product in a correct
but not efficient way, the process will actually make the integration step more complicated. The
strategy is to choose U and V in such a way that the next step is simpler so that multiple applications
will lead to a form that is easy to integrate; usually that direction is one that reduces a power in the
integrand of the original integral to a lower form.

Example:

1
ð 1 ð 1
xe ax 1 e 1 ax
ð ax
ax
xe dx ¼ dx ¼ (0 0) þ e dx
a 0 a a
0 0 0

1 1
[e ax 1
]
a a
¼ 2 0 ¼ 2
Now that is a very tedious process (however, it is used in research if necessary) so we only show
this once and then generalize to a key formula we can memorize. This is a well-established
procedure where tables of ‘‘integral formulas’’ fill several volumes. Fortunately for us we can get
around this in this first chapter and simply memorize the final formula.
ð
1 n!
n ax
Key formula: x e dx ¼
0 a nþ1
Although we are packing a lot of information into this first chapter, students have absorbed this in my
class for many years and I usually spend two or three of the first classes on this material. In my
experience, this review is essential for mental comfort in all the later chapter topics. So far all the
material mentioned earlier may have been treated in one or two semesters of calculus but to go ahead in
thermodynamics we have to understand how to treat more than one variable in the presence of the
others. A simple example is the set of basic physical variables (P, V, T, n). While this topic would
normally be treated in a course in multivariant calculus, it is unlikely that biology majors will have