Chemistry Reference
In-Depth Information
so that the average value of the radial position for the electron in the H 1s orbital is actually
one and a half times the Bohr radius. Looking again at the probability plot in Figure 7.11b
shows that the distribution beyond
a
0
shows a tailing toward large
r
; this explains why the
average value is beyond the most probable.
We can use the same approach to compare the average radial distance of 2s and 2p
functions.
For the 2s function, Tables A9.1 and A9.2 give
Z
eff
a
0
3
/
2
r
)exp
−
2
r
1
4
√
2
χ
200
=
R
20
Y
00
=
A
(2
−
ρ
with
A
=−
(A9.64)
π
So the expectation value for the radial coordinate of the electron in this case is
⎡
∞
π
π
⎣
4
r
200
=
A
2
d
φ
sin(
θ
)d
θ
r
3
exp(
−
ρ
r
)d
r
−
π
0
0
⎤
∞
∞
⎦
r
4
exp(
2
r
5
exp(
−
4
ρ
−
ρ
r
)d
r
+
ρ
−
ρ
r
)d
r
(A9.65)
0
0
The angular integrals are the same as before, but we now require three of the standard
radial integrals from Table A9.4:
A
2
4
6
ρ
24
ρ
A
2
48
ρ
2
120
ρ
r
200
=
4
−
4
ρ
+
ρ
=
4
(A9.66)
π
π
4
5
6
4
Remembering that
depends on the principal quantum number (see Table A9.2), and
substituting back for
A
, we obtain
ρ
a
0
Z
eff
r
200
=
6
(A9.67)
So an electron in a 2s orbital is, on average, much further from the nucleus than a 1s
electron.
For the 2p function we will take
n
=2,
l
= 1 and
m
l
= 1. Hence, reading the corresponding
functions from Tables A9.1 and A9.2:
Ar
exp
−
2
r
sin(
χ
211
=
R
21
Y
11
=
θ
)exp(i
φ
)
with
Z
eff
a
0
3
/
2
3
2
1
/
2
1
4
√
6
A
=−
(A9.68)
ρ
π
