SME
Physical Chemistry II: Quantum & Spectroscopy · Problem type
Population ratio of two levels (Boltzmann)
Use this for “Two nondegenerate levels are 1000 cm⁻¹ apart. Find the ratio of molecules in the upper level to the lower one at 298 K.”
Chart
This is an example. Type your own values to start a new problem.
N₂
÷
N₁
=
g₂
g₁
×
e
−1.4388
×
÷
=
N₂ ÷ N₁
x₁
Share in the lower level
worked out (not typed) · %
x₂
Share in the upper level
worked out (not typed) · %
Formulas
N₂
N₁
=
(
g₂
g₁
)
×
e
−1.4388 × ΔE ÷
T
2.709
=
(
3
1
)
×
e
−1.4388 × 21.16 ÷ 298
x₁
=
100
1 + N₂ ÷ N₁
26.96
=
100
1 + 2.709
x₂
=
100
×
N₂ ÷ N₁
1 + N₂ ÷ N₁
73.04
=
100
×
2.709
1 + 2.709
Assumptions
•
The molecules are at equilibrium at temperature T, so each state holds a share proportional to e−E ÷ kT.
•
A level with degeneracy g has g states at the same energy, so it holds g times as many: N₂ ÷ N₁ = (g₂ ÷ g₁)e−ΔE ÷ kT.
•
ΔE is in wavenumbers, so ΔE ÷ kT = 1.4388ΔE ÷ T, with hc ÷ k = 1.4388 cm·K.
•
The bars share the molecules between these two levels alone.
Step-by-step
WE KNOW
g₁ = 1, g₂ = 3, ΔE = 21.16 cm⁻¹, T = 298 K
FIND
molecules up per molecule down (N₂ ÷ N₁), share in the lower level (x₁), share in the upper level (x₂)
Step 1 · Find molecules up per molecule down (N₂ ÷ N₁)
Use N₂ ÷ N₁ = (g₂ ÷ g₁) × e−1.4388 × ΔE ÷ T
The Boltzmann factor for the gap, scaled by how many states each level has.
N₂
N₁
=
(
3
1
)
×
e
−1.4388 × 21.16 ÷ 298
N₂
N₁
=
3
×
e
−30.45 ÷ 298
N₂
N₁
=
3
×
e
−0.1022
N₂
N₁
=
3
×
0.9028
N₂
N₁
=
2.709
→
The
upper
level
holds
more:
its
degeneracy
outweighs
the
Boltzmann
factor.
Step 2 · Find share in the lower level (x₁)
Use x₁ = 100 ÷ (1 + N₂ ÷ N₁)
For
each
molecule
in
the
lower
level
there
are
N₂
N₁
in
the
upper
one.
x₁
=
100
1 + 2.709
x₁
=
100
3.709
x₁
=
26.96%
Step 3 · Find share in the upper level (x₂)
Use x₂ = 100 × (N₂ ÷ N₁) ÷ (1 + N₂ ÷ N₁)
The upper level’s part of the same total.
x₂
=
100
×
2.709
1 + 2.709
x₂
=
100
×
2.709
3.709
x₂
=
270.9
3.709
x₂
=
73.04%
Check
Put the numbers back in. Both sides should match.
2.709
=
(
3
1
)
×
e
−1.4388 × 21.16 ÷ 298
✓
26.96
=
100
1 + 2.709
✓
73.04
=
100
×
2.709
1 + 2.709
✓
Topic check
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