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274 8 ATOMIC STRUCTURE AND SPECTRA
Mathematical so�ware is used to �nd the values of ρ for which dP(ρ)/dρ = 0,
giving the results ρ = 0, 0.493, 1.27, 2.79, 4.73, 8.72. �e simplest way to
identify which of these is a maximum is to plot P(ρ) against ρ, fromwhich it is
evident that ρ = 0.493, 2.79, 8.72 are all maxima, with the principal maximum
being at ρ = 8.72.�emaximum in the radial distribution function is therefore
at r = 8.72 × (3a0/2Z) .
E8A.9(b) �e radius at which the electron is most likely to be found is that at which the
radial distribution function is a maximum.�e radial distribution function is
de�ned in [8A.17b–312], P(r) = r2R(r)2. For the 3p orbital R(r) is given in
Table 8A.1 on page 306 as R3,0 = N(4 − ρ)ρe−ρ/2 where ρ = 2Zr/na0, which
for n = 3 is ρ = 2Zr/3a0. With the substitution r2 = ρ2(3a0/2Z)2,�e radial
distribution function is therefore P(ρ) = N2(3a0/2Z)2(4 − ρ)2ρ4e−ρ .
To �nd the maximum in this function the derivative is set to zero and the re-
sulting equation solved for ρ. Mathematical so�ware gives the following values
of ρ: 0, 2, 4, 8. It is evident that P(ρ) is zero at ρ = 0 and ρ = 4, and that
P(ρ) tends to zero as ρ → ∞. �erefore ρ = 2 and ρ = 8 must correspond
to maxima; a plot of P(ρ) shows that the latter is the principal maximum; this
occurs at r = 8(3a0/2Z) .
E8A.10(b) �e N shell has n = 4.�e possible values of l (subshells) are 0, corresponding
to the s orbital, l = 1 corresponding to the p orbitals, l = 2 corresponding to
the d orbitals, and l = 3 corresponding to the f orbitals; there are therefore
4 subshells . As there is one s orbital, 3 p orbitals, 5 d orbitals and 7 p orbitals,
there are 16 orbitals in total.
E8A.11(b) �e magnitude of the orbital angular momentum of an orbital with quantum
number l is
√
l(l + 1)ħ.�e total number of nodes for an orbital with quantum
number n is n − 1, l of these are angular and so the number of radial nodes is
n − l − 1.
orbital n l ang. mom. angular nodes radial nodes
4d 4 2
√
6ħ 2 1
2p 2 1
√
2ħ 1 0
3p 3 1
√
2ħ 1 1
E8A.12(b) All the 3d orbitals have the same value of n and l , and hence have the same
radial function, which is given in Table 8A.1 on page 306 as R3,2 = Nρ2e−ρ/2
where ρ = 2Zr/na0, which for n = 3 is ρ = 2Zr/3a0. Radial nodes occur when
the wavefunction passes through zero.�e function goes to zero at ρ = 0 and as
ρ →∞, but it does not pass through zero at these points so they are not nodes.
�e number of radial nodes is therefore 0.