SME
University Physics III: Waves, Optics & Modern
Atomic structure
Use this for “For He⁺ (Z = 2), find the energies of n = 3 and n = 2, the wavelength of the 3 → 2 line and the radius of the n = 2 orbit.”
Diagram
He⁺,
n
=
2:
E
=
−13.6
×
2
2
2
2
=
−13.6
eV.
Every subshell of a shell has the same energy here: g = n² = 4 orbitals.
Upper level nᵤ = 3
Energy of the upper level Eᵤ = −6.044 eV
Photon energy ΔE = 7.556 eV
Wavelength λ = 164.1 nm
This is an example. Type your own values to start a new problem.
Z
Nuclear charge
example
nᵤ
Upper level
example
nₗ
Lower level
example
Eᵤ
Energy of the upper level
calculated · eV
Eₗ
Energy of the lower level
calculated · eV
ΔE
Photon energy
calculated · eV
λ
Wavelength
calculated · nm
rₗ
Radius of the lower orbit
worked out (not typed) · nm
Formulas
Eᵤ
=
−13.6
×
Z
2
nᵤ
2
−6.044
=
−13.6
×
2
2
3
2
Eₗ
=
−13.6
×
Z
2
nₗ
2
−13.6
=
−13.6
×
2
2
2
2
ΔE = Eᵤ − Eₗ
7.556 = −6.044 − (−13.6)
λ
=
1,240
ΔE
164.1
=
1,240
7.556
rₗ
=
0.0529
×
nₗ
2
Z
0.1058
=
0.0529
×
2
2
2
Assumptions
•
One electron only around a nucleus of charge Z (H, He⁺, Li²⁺, …); Z = 1 is hydrogen.
•
The nucleus is held still (no reduced-mass correction), so Eₙ = −13.6Z² ÷ n² eV and rₙ = a₀n² ÷ Z, with a₀ = 0.0529 nm.
•
A drop from nᵤ to nₗ gives off one photon carrying the gap: λ = hc ÷ ΔE, with hc = 1240 eV·nm.
Step-by-step
WE KNOW
Z = 2, nᵤ = 3, nₗ = 2
FIND
energy of the upper level (Eᵤ), energy of the lower level (Eₗ), photon energy (ΔE), wavelength (λ), radius of the lower orbit (rₗ)
Step 1 · Find energy of the upper level (Eᵤ)
Use Eᵤ = −13.6 × Z² ÷ nᵤ²
Hydrogen’s
−13.6
eV
n
2
for
the
upper
level
nᵤ,
times
Z
2
for
the
larger
charge.
Eᵤ
=
−13.6
×
2
2
3
2
Eᵤ
=
−13.6
×
4
9
Eᵤ
=
−
54.4
9
Eᵤ = −6.044 eV
Step 2 · Find energy of the lower level (Eₗ)
Use Eₗ = −13.6 × Z² ÷ nₗ²
Hydrogen’s
−13.6
eV
n
2
for
the
lower
level
nₗ,
times
Z
2
for
the
larger
charge.
Eₗ
=
−13.6
×
2
2
2
2
Eₗ
=
−13.6
×
4
4
Eₗ
=
−
54.4
4
Eₗ = −13.6 eV
Step 3 · Find photon energy (ΔE)
Use ΔE = Eᵤ − Eₗ
The
photon
carries
the
gap
between
the
two
levels:
13.6
Z
²(
1
nₗ
2
−
1
nᵤ
2
).
ΔE = −6.044 − (−13.6)
ΔE = 7.556 eV
Step 4 · Find wavelength (λ)
Use λ = 1,240 ÷ ΔE
hc
=
1240
eV·nm
over
the
photon’s
energy
in
eV
gives
λ
in
nm.
λ
=
1,240
7.556
λ
=
164.1
nm
→
ultraviolet:
shorter
than
violet
light.
Step 5 · Find radius of the lower orbit (rₗ)
Use rₗ = 0.0529 × nₗ² ÷ Z
The
orbit
grows
as
n
2
and
shrinks
as
the
charge
Z
pulls
harder.
rₗ
=
0.0529
×
2
2
2
rₗ
=
0.0529
×
4
2
rₗ
=
0.2116
2
rₗ = 0.1058 nm
Check
Put the numbers back in. Both sides should match.
−6.044
=
−13.6
×
2
2
3
2
✓
−13.6
=
−13.6
×
2
2
2
2
✓
7.556 = −6.044 − (−13.6) ✓
164.1
=
1,240
7.556
✓
0.1058
=
0.0529
×
2
2
2
✓
Topic check
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