The Chemistry Maths Book, Second Edition

82 Chapter 3Transcendental functions

with the 1 + 1 sign for growth and the 1 − 1 sign for decay. The proportionality factor kis

called the rate constant. The solution of the differential equation is

x(t) 1 = 1 x

0

e

±kt

where x

0

is the size at timet 1 = 10. As an example, consider a system whose size xis

doubled after every time interval τ. Starting with sizex 1 = 1 x

0

, the size after time τis 2 x

0

,

after time 2 τit is 4 x

0

, after time 3 τit is 8 x

0

, and so on. After time t,

x 1 = 12

t 2 τ

x

0

Equating this with the solution of the differential equation shows that the rate

constant kis inversely proportional to the time interval τ:k 1 = 1 (ln 1 2) 2 τ, whereln 12 is

the natural logarithm of the number 2 (see Section 3.7).

EXAMPLE 3.21Atomic orbitals

The 1sorbital for an electron in the ground state of the hydrogen atom is

ψ 1 = 1 e

−r

where ris the distance of the electron from the nucleus. All the orbitals for the

hydrogen atom have the form

ψ 1 = 1 f (x, y, z)e

−ar

where(x, y, z)are the cartesian coordinates of the electron relative to the nucleus at

the origin, and ais a constant. The functionf(x, y, z)is a polynomial in x,y, and z,

and determines the shape of the orbital; for example,f 1 = 1 zgives a p

z

orbital.

EXAMPLE 3.22The normal distribution

The normal or Gaussian distribution in statistics is described by the probability

density function

where μis the mean and σis the standard deviation of the distribution (see Section

21.8). The probability function forms the basis for the statistical analysis of a wide

range of phenomena; for example, error analysis of the results of experiments in the

physical sciences, sample analysis in population studies, sample analysis for quality

control in the manufacturing industry.

0 Exercise 36

px

x

()=−exp

−

































1

2

1

2

2

σ

μ

π σ