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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 277
(b) For 2s the radial function is R2,0(r) = 8−1/2(Z/a0)3/2[2−Zr/a0]e−Zr/2a0
and so
⟨r−1⟩ = 8−1(Z/a0)3 ∫
∞
0
r(2 − Zr/a0)2e−Zr/a0 dr
= 8−1(Z/a0)3 ∫
∞
0
(4r − 4Zr2/a0 + Z2r3/a20)e−Zr/a0dr
= 8−1(Z/a0)3(a0/Z)2(4 − 8 + 6) = Z/4a0
(c) For a 2p orbital, R2,1(r) = 24−1/2(Z/a0)3/2(Zr/a0)e−Zr/2a0 and so
⟨r−1⟩ = ∫
∞
0
r [24−1/2(Z/a0)3/2(Zr/a0)e−Zr/2a0]
2
dr
= (Z5/24a50)∫
∞
0
r3e−Zr/a0dr = (Z5/24a50) × 3!/(Z/a0)4 = Z/4a0
�e expectation value of 1/r is the same for 2s and 2p.
(d) �ere is no reason to assume that ⟨1/r⟩ = 1/⟨r⟩. To take a speci�c exam-
ple, in the case of a 1s orbital ⟨r⟩ = 3a0/2Z and ⟨1/r⟩ = Z/a0.
P8A.10 (a) �e main di�erence between the Bohr and quantummodels is that in the
Bohrmodel the trajectory (that is the position) of the electron is precisely
de�ned, whereas in quantummechanics only the probability distribution
of the position can be predicted.
(b) In quantum mechanics the magnitude of the orbital angular momentum
of an electron with quantum number l is given by ħ
√
l(l + 1). In con-
trast, in the Bohr model the magnitude depends on n and is nħ. In the
Bohr model the ground state, with n = 1, has angular momentum ħ,
whereas in quantum mechanics the ground state (1s) has zero angular
momentum. As has already been commented on, the predicted distribu-
tion of the electron is quite di�erent in the two cases.
(c) �e angular momentum of the ground state and the spatial distribution
of the electron can be probed experimentally by various kinds of spec-
troscopy.
8B Many-electron atoms
Answers to discussion questions
D8B.2 �is is covered in any introductory or general chemistry text.
D8B.4 See Section 8B.4 on page 325.