1 (281)

Prévia do material em texto

SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 273
Setting the term in the parentheses to zero and solving the resulting quadratic
gives ρ = 5 ±
√
7, so extrema occur at r = (3a0/2Z)(5 ±
√
7) .
�e simplest way to identify the nature of the extrema is to make a plot of
R3,0(ρ), from which it is immediately evident that ρ = 5 −
√
7 is a minimum
and ρ = 5 +
√
7 is a maximum.
E8A.5(b) Assuming that the electron is in the ground state, the wavefunction is ψ =
Ne−r/a0 , and so the radial distribution function, given by [8A.17a–312], isR(r) =
4πr2ψ2 = 4πN2r2e−2r/a0 .�e �rst step is to �nd the value of r at which this is
a maximum, and this is done by solving dR(r)/dr = 0; for such a calculation
the constants 4πN2 can be discarded.
dR(r)
dr
= 2r e−2r/a0 − (2r2/a0) e−2r/a0
�e derivative is zero at r = 0 and r = a0, with the latter being the maximum.
�e radial distribution function falls to a fraction f of its maximum at radius
r′ given by R(r′)/R(a0) = f , hence
f = R(r
′)
R(a0)
= 4πN
2r′2e−2r
′/a0
4πN2a20e−2a0/a0
= r
′2e−2r
′/a0
a20e−2
�e solutions to this equation need to be found numerically using mathemat-
ical so�ware. For f = 0.5 the solutions are r′ = 0.381a0 and 2.08a0 . For
f = 0.75 the solutions are r′ = 0.555a0 and 1.64a0 .
E8A.6(b) �e radial wavefunction is R4,1 = N(20 − 10ρ + ρ2)e−ρ/2 where ρ = Zr/2a0.
Radial nodes occur when the wavefunction passes through 0, which is when
20 − 10ρ + ρ2 = 0. �e roots of this quadratic equation are at ρ = 5 ±
√
5 and
hence the nodes are at r = (2a0/Z)(5 ±
√
5) . �e wavefunction goes to zero
as ρ →∞, but this does not count as a node as the wavefunction does not pass
through zero.
E8A.7(b) Angular nodes occurwhen sin2 θ sin 2ϕ = 0, which occurswhen either of sin2 θ
or sin 2ϕ is equal to zero; recall that the range of θ is 0→ π and of ϕ is 0→ 2π.
Although the function is zero for θ = 0 this does not describe a plane, and so
is discounted.�e function is zero for ϕ = 0 with any value of θ: this is the xz
plane (the solution ϕ = π corresponds to the same plane).�e function is also
zero for ϕ = π/2 with any value of θ: this is the yz plane.�ere are two nodal
planes, as expected for a d orbital.
E8A.8(b) �e radial distribution function is de�ned in [8A.17b–312], P(r) = r2R(r)2.
For the 3s orbital R(r) is given in Table 8A.1 on page 306 as R2,0 = N(6 −
6ρ + ρ2)e−ρ/2 where ρ = 2Zr/na0, which for n = 3 is ρ = 2Zr/3a0. With
the substitution r2 = ρ2(3a0/2Z)2, the radial distribution function is therefore
P(ρ) = N2(3a0/2Z)2ρ2(6 − 6ρ + ρ2)2e−ρ .