Physical Chemistry Third Edition

752 17 The Electronic States of Atoms. I. The Hydrogen Atom

The locations that lie at distances from the nucleus betweenr′andr′+drconstitute

a spherical shell of radiusr′and thicknessdr, as shown crudely in Figure 17.11a. The

probability of finding the electron in this shell is obtained by integrating overθandφ:

fr(r)dr

(∫

π

0

∫ 2 π

0

|ψ(r,θ,φ)|^2 r^2 sin(θ)φdθ

)

dr (17.5-11)

The integral can be factored, and theθandφintegrals give factors of unity due to the

separate normalizations ofΘandΦ:

fr(r)drR(r)∗R(r)

(∫

π

0

|Θ|^2 sin(θ)dθ

∫ 2 π

0

|Φ|^2 dφ

)

r^2 drR(r)∗R(r)r^2 dr

(17.5-12)

so that

fr(r)r^2 R(r)∗R(r)r^2 R(r)^2 (17.5-13)

where we used the fact that theRfunction is real so that it equals its complex conjugate.

The expectation value of a quantity depending only onrcan be computed using

the radial distribution function. For example, the result of Example 17.7 can also be

obtained from

〈 1

r

〉



∫∞

0

1

r

R(r)^2 r^2 dr

∫∞

0

1

r

fr(r)dr (17.5-14)

Figure 17.11b shows graphs of the radial distribution function for several states. All

of the states of a given subshell have the same radial distribution function because

they have the same radial factor in their wave functions. A graph of a given radial

distribution function can be sketched from the properties of theRfunction. All radial

distribution functions vanish atr0 because of ther^2 factor and vanish atr→∞

because the exponential factor inRoverwhelms any finite power ofrasr→∞.

r

z

Thicknessdr

x

y

0.6

0.5

0.4

0.3

(Radial distribution function)/

a

0.2

0.1

0.0

(a) (b)

0

1 s

2 s 3 s

2 p

4

r/a

812

Figure 17.11 The Probability Distribution for Electron-Nucleus Distances.(a) A

spherical shell of radiusr, and thicknessdr. (b) Radial distribution functions for hydrogen

orbitals.