Page 277 - Physical Chemistry

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Chapter 8 1/(1 x) expanded about a 0 gives the Taylor series (8.8). The nearest real singu-
Real Gases larity to x 0 is at x 1, since 1/(1 x) becomes infinite at x 1. For this function,
c 1, and the Taylor series (8.8) converges to 1/(1 x) for all x in the range 1
x 1. In some cases, c is less than the distance to the nearest real singularity. The gen-
eral method of finding c is given in Prob. 8.33.

EXAMPLE 8.2 Taylor series

Find the Taylor series for sin x with a 0.
(n)
To find f (a) in (8.32), we differentiate f(x) n times and then set x a. For
f(x) sin x and a 0, we get
f1x2 sin x f1a2 sin 0 0
f¿1x2 cos x f¿1a2 cos 0 1
f–1x2 sin x f–1a2 sin 0 0
f‡1x2 cos x f‡1a2 cos 0 1
f 1iv2 1x2 sin x f 1iv2 1a2 sin 0 0
.........................................
(n)
The values of f (a) are the set of numbers 0, 1, 0, 1 repeated again and again.
The Taylor series (8.32) is

11x 02 01x 02 2 1 121x 02 3 01x 02 4
sin x 0 p
1! 2! 3! 4!
3
5
7
sin x x x >3! x >5! x >7! p for all x (8.35)
The function sin x has no singularities for real values of x. A full mathematical
investigation shows that (8.35) is valid for all values of x.
Exercise
Use (8.32) to find the first four nonzero terms of the Taylor series for cos x with
4
2
6
a 0. (Answer: 1 x /2! x /4! x /6!

.)
Another example is ln x. Since ln 0 doesn’t exist, we cannot take a 0 in (8.32).
A convenient choice is a 1. We find (Prob. 8.29)

2
3
ln x 1x 12 1x 12 >2 1x 12 >3 p for 0 6 x 6 2 (8.36)
The nearest singularity to a 1 is at x 0 (where f doesn’t exist), and the series (8.36)
converges to ln x for 0 x 2. Two other important Taylor series are
x 2 x 3 q x n
x
e 1 x p a for all x (8.37)
2! 3! n 0 n!
2
6
4
cos x 1 x >2! x >4! x >6! p for all x (8.38)
Taylor series are useful in physical chemistry when x in (8.32) is close to a, so that
only the first few terms in the series need be included. For example, at low pressures,
V of a gas is large and b/V ( x) in (8.9) is close to zero. In general, Taylor series
m
m
are useful under limiting conditions such as low P in a gas or low concentration in a
solution.