SME
Physical Chemistry II: Quantum & Spectroscopy
Hydrogen atom
Use this for “For the 2p orbital of He⁺, find the energy, the radial and angular nodes, and how many orbitals share that energy.”
Diagram
He⁺,
n
=
2
(2p):
E
=
−13.6
×
2
2
2
2
=
−13.6
eV.
Radial nodes n − l − 1 = 2 − 1 − 1 = 0; angular nodes l = 1.
Every subshell of a shell has the same energy here: g = n² = 4 orbitals.
The slice cuts one of the 3 2p orbitals; + and − are the signs of ψ.
This is an example. Type your own values to start a new problem.
Eₙ
=
−13.6
eV
×
Z
2
n
2
=
E (eV)
l
Angular momentum quantum number
example
radial
Radial nodes
calculated
angular
Angular nodes
calculated
g
Orbitals at that energy
calculated
Formulas
E
=
−13.6
×
Z
2
n
2
−13.6
=
−13.6
×
2
2
2
2
radial
=
n
−
l
−
1
0 = 2 − 1 − 1
angular
=
l
1 = 1
g
=
n
2
4
=
2
2
Assumptions
•
One electron only (H, He⁺, Li²⁺), so every subshell of a shell has the same energy: E = −13.6Z² ÷ n² eV.
•
l runs from 0 to n − 1 (s, p, d, f); there are n − 1 nodes in all, l of them angular.
•
The nucleus is held still; the Rydberg energy is 13.6 eV.
Step-by-step
WE KNOW
Z = 2, n = 2, l = 1
FIND
energy of the level (E), radial nodes (radial), angular nodes (angular), orbitals at that energy (g)
Step 1 · Find energy of the level (E)
Use E = −13.6 × Z² ÷ n²
One
electron:
the
levels
are
hydrogen’s
−13.6
eV
n
2
,
times
Z
2
for
the
larger
charge.
E
=
−13.6
×
2
2
2
2
E
=
−13.6
×
4
4
E
=
−
54.4
4
E
=
−13.6
eV
Step 2 · Find radial nodes (radial)
Use radial = n − l − 1
There
are
n
−
1
nodes
in
all;
l
of
them
are
angular,
the
rest
are
spheres.
radial = 2 − 1 − 1
radial = 0
Step 3 · Find angular nodes (angular)
Use angular = l
Each
unit
of
l
is
one
nodal
plane
or
cone
through
the
nucleus.
angular = 1
Step 4 · Find orbitals at that energy (g)
Use g = n²
Each
l
from
0
to
n
−
1
has
2
l
+
1
orbitals,
and
they
add
up
to
n
2
.
g
=
2
2
g
=
4
Check
Put the numbers back in. Both sides should match.
−13.6
=
−13.6
×
2
2
2
2
✓
0 = 2 − 1 − 1 ✓
1 = 1 ✓
4
=
2
2
✓
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