SME
Physical Chemistry II: Quantum & Spectroscopy · Problem type
The mean and most probable radius
Use this for “Find the mean radius and the most probable radius of the electron in hydrogen’s 2p orbital.”
Diagram
H 2p: P(r) = r²R²(r), the chance per a₀ of finding the electron at r; the shaded area is 1.
Radial nodes n − l − 1 = 0.
⟨r⟩
=
3n² − l(l + 1)
2Z
⟨r⟩
=
3 ×
2
2
− 1 × 2
2
=
5
a₀
=
0.265
nm
rₘₚ
=
n
2
a₀
Z
=
2
2
1
=
4
a₀
=
0.212
nm
(one
hump).
This is an example. Type your own values to start a new problem.
⟨r⟩
=
3
×
n
2
−
(
+
1)
2
×
a₀
=
a₀
⟨r⟩ₙₘ
Mean radius in nanometers
calculated · nm
radial
Radial nodes
calculated
rₘₚ
Most probable radius in Bohr radii
worked out (not typed)
rmp,nm
Most probable radius in nanometers
worked out (not typed) · nm
Formulas
⟨r⟩
=
3 ×
n
2
−
l
× (
l
+ 1)
2 ×
Z
5
=
3 ×
2
2
− 1 × (1 + 1)
2 × 1
⟨r⟩ₙₘ = ⟨r⟩ × 0.0529177
0.2646 = 5 × 0.0529177
radial
=
n
−
l
−
1
0 = 2 − 1 − 1
rₘₚ
=
n
2
Z
(for
l
=
n
−
1)
4
=
2
2
1
(for
1
=
2
−
1)
rmp,nm = rₘₚ × 0.0529177
0.2117 = 4 × 0.0529177
Assumptions
•
One electron; radii are in Bohr radii, a₀ = 0.0529177 nm, and shrink as 1 ÷ Z.
•
P(r) = r²R²(r) is the chance per unit r of finding the electron that far out; ⟨r⟩ is its average.
•
rₘₚ = n²a₀ ÷ Z holds only when l = n − 1 (1s, 2p, 3d), where P(r) has one hump.
Step-by-step
WE KNOW
Z = 1, n = 2, l = 1
FIND
mean radius in Bohr radii (⟨r⟩), mean radius in nanometers (⟨r⟩ₙₘ), radial nodes (radial), most probable radius in Bohr radii (rₘₚ), most probable radius in nanometers (rmp,nm)
Step 1 · Find mean radius in Bohr radii (⟨r⟩)
Use ⟨r⟩ = (3 × n² − l × (l + 1)) ÷ (2 × Z)
Average
r
over
P(r):
it
grows
as
n
2
and
shrinks
as
Z
,
a
little
less
for
higher
l
.
⟨r⟩
=
3 ×
2
2
− 1 × (1 + 1)
2 × 1
⟨r⟩
=
3 ×
2
2
− 2
2
⟨r⟩
=
3 × 4 − 2
2
⟨r⟩
=
12 − 2
2
⟨r⟩
=
10
2
⟨r⟩ = 5
Step 2 · Find mean radius in nanometers (⟨r⟩ₙₘ)
Use ⟨r⟩ₙₘ = ⟨r⟩ × 0.0529177
One Bohr radius a₀ is 0.0529177 nm.
⟨r⟩ₙₘ = 5 × 0.0529177
⟨r⟩ₙₘ = 0.2646 nm
Step 3 · Find radial nodes (radial)
Use radial = n − l − 1
P(r)
drops
to
zero
between
its
humps
n
−
l
−
1
times.
radial = 2 − 1 − 1
radial = 0
Step 4 · Find most probable radius in Bohr radii (rₘₚ)
Use rₘₚ = n² ÷ Z (for l = n − 1)
With
l
=
n
−
1,
P(r)
has
one
hump;
setting
its
slope
to
zero
puts
it
at
n
2
a₀
Z
.
rₘₚ
=
n
2
Z
rₘₚ
=
2
2
1
rₘₚ = 4
Step 5 · Find most probable radius in nanometers (rmp,nm)
Use rmp,nm = rₘₚ × 0.0529177
One Bohr radius a₀ is 0.0529177 nm.
rmp,nm = 4 × 0.0529177
rmp,nm = 0.2117 nm
Check
Put the numbers back in. Both sides should match.
5
=
3 ×
2
2
− 1 × (1 + 1)
2 × 1
✓
0.2646 = 5 × 0.0529177 ✓
0 = 2 − 1 − 1 ✓
4
=
2
2
1
(for
1
=
2
−
1)
✓
0.2117 = 4 × 0.0529177 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.