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109
One more thing:
radial probability distributions
There are two different ways to view the radial probability of the electron in the hydrogen atom.
In the first, we view the distribution in one dimension
.
Let’s take the 1s function as an example:
ψ
1s
(r)
2
=
0
a
/
r
2
3
0
e
a
1
−
π
This function has a maximum at the origin (r = 0), and decreases exponentially with increasing
distance.
In one dimension
, the probability of finding the electron in an element of length dr at distance r
from the origin is given as
ψ
1s
(r)
2
dr
In three dimensions
, the element of length dr sweeps out a spherical shell as
θ
sweeps out its full
range from 0 to
π
, and
φ
varies from 0 to 2
π
.
The probability of finding the electron in that shell
,
with volume dV = 4
π
r
2
dr is just
ψ
1s
(r)
2
dV =
ψ
1s
(r)
2
4
π
r
2
dr
For the onedimensional case, dr does not vary as a function of r.
However, dV = 4
π
r
2
dr does
vary with r.
The shell volume goes to zero as r
→
0.
Angular functions:
When the
l
quantum number is nonzero, the orbitals have an angular dependence.
We will
learn that this angular dependence ultimately determines the shapes
of molecules
1D
3D
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l
= 1:
these are porbitals.
Possible values of m
l
are –1, 0, +1
The porbitals look like the following:
Notice that the three porbitals only differ in spatial orientation.
These orbitals point along
the
coordinate axes.
Alternate lobes have opposite signs.
Each orbital has
one
nodal plane.
l
= 2:
these are dorbitals
.
Possible values of m
l
are –2, –1, 0, +1, +2
Notice that the five dorbitals (almost) only differ in spatial orientation. Alternate lobes have
opposite signs.
Each orbital has
two
nodal planes.
Finally, it is possible to have orbitals with
l
= 3:
these are forbitals
.
Possible values of m
l
are –3, –2, –1, 0, +1, +2, +3
111
Notice that the seven forbitals (almost) only differ in spatial orientation.
Alternate lobes have
opposite signs.
Each orbital has
three
nodal planes.
Remember that in the oneelectron atom, there is no dependence
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This note was uploaded on 04/16/2008 for the course CHM 132 taught by Professor Farrar during the Spring '08 term at Rochester.
 Spring '08
 farrar
 Electron

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