Page 49 - Physical Chemistry

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Chapter 1 logarithms and are written as ln x. For practical calculations, one often uses logs to
Thermodynamics the base 10, called common logarithms and written as log x, log x, or lg x. We have
10
ln x log x, log x log x (1.66)*
10
e
s
t
If 10 x, then log x t. If e x, then ln x s. (1.67)
From (1.67), we have
e ln x x and 10 log x x (1.68)
s
x
From (1.67), it follows that ln e s. Since e ln x x ln e , the exponential and nat-
x
ural logarithmic functions are inverses of each other. The function e is often written
x
0
1
as exp x. Thus, exp x e . Since e e, e 1, and e q 0, we have ln e 1,
ln 1 0, and ln 0 q. One can take the logarithm or the exponential of a dimen-
sionless quantity only.
Some identities that follow from the definition (1.67) are
ln xy ln x ln y, ln 1x>y2 ln x ln y (1.69)*
k
ln x k ln x (1.70)*
ln x 1log x2>1log e2 log x ln 10 2.3026 log x (1.71)
10
10
10
10
To find the log of a number greater than 10 100 or less than 10 100 , which cannot
b
be entered on most calculators, we use log(ab) log a log b and log 10 b. For
example,
log 12.75 10 150 2 log 2.75 log 10 150 0.439 150 149.561
10
10
10
To find the antilog of a number greater than 100 or less than 100, we proceed as
follows. If we know that log x 184.585, then
10
x 10 184.585 10 0.585 10 184 0.260 10 184 2.60 10 185
1.9 STUDY SUGGESTIONS
A common reaction to a physical chemistry course is for a student to think, “This
looks like a tough course, so I’d better memorize all the equations, or I won’t do well.”
Such a reaction is understandable, especially since many of us have had teachers who
emphasized rote memory, rather than understanding, as the method of instruction.
Actually, comparatively few equations need to be remembered (they have been
marked with an asterisk), and most of these are simple enough to require little effort
at conscious memorization. Being able to reproduce an equation is no guarantee of
being able to apply that equation to solving problems. To use an equation properly, one
must understand it. Understanding involves not only knowing what the symbols stand
for but also knowing when the equation applies and when it does not apply. Everyone
knows the ideal-gas equation PV nRT, but it’s amazing how often students will use
this equation in problems involving liquids or solids. Another part of understanding an
equation is knowing where the equation comes from. Is it simply a definition? Or is it
a law that represents a generalization of experimental observations? Or is it a rough
empirical rule with only approximate validity? Or is it a deduction from the laws of
thermodynamics made without approximations? Or is it a deduction from the laws of
thermodynamics made using approximations and therefore of limited validity?
As well as understanding the important equations, you should also know the
meanings of the various defined terms (closed system, ideal gas, etc.). Boldface type
(for example, isotherm) is used to mark very important terms when they are first
defined. Terms of lesser importance are printed in italic type (for example, isobar). If
you come across a term whose meaning you have forgotten, consult the index; the
page number where a term is defined is printed in boldface type.