SME
Physical Chemistry I: Thermodynamics & Kinetics · Problem type
Rotational partition function and the most populated level
Use this for “N₂ has B = 1.998 cm⁻¹. Find its rotational partition function and its most populated rotational level at 500 K.”
Diagram
Lines at ν̃ = 2B(J + 1), 2B = 2 × 1.931 = 3.86 cm⁻¹ apart.
Heights
follow
the
lower
level’s
share,
(2J
+
1)
e
−1.4388BJ(J + 1) ÷ T
at
298.2
K:
the
strongest
line
starts
at
J
=
7.
This is an example. Type your own values to start a new problem.
qᵣₒₜ
=
T (K)
×
1.4388
cm·K
×
B (cm⁻¹)
θᵣ
Rotational temperature
calculated · K
Jₘₐₓ
Most populated level
calculated
Formulas
θᵣ
=
1.4388
×
B
2.778 = 1.4388 × 1.931
qᵣₒₜ
=
T
σ
× θᵣ
107.3
=
298.2
1 × 2.778
Jₘₐₓ
=
√(
T
2 × θᵣ
)
−
0.5
6.825
=
√(
298.2
2 × 2.778
)
−
0.5
Assumptions
•
A linear molecule at T ≫ θᵣ, the high-temperature limit: the sum over J turns into an integral, q = T ÷ (σθᵣ).
•
The symmetry number σ is 2 for a symmetric molecule (N₂, CO₂) and 1 otherwise (HCl, CO).
•
θᵣ = hcB ÷ k = 1.4388B, with hc ÷ k = 1.4388 cm·K.
•
Level J holds a share proportional to (2J + 1)e−J(J + 1)θᵣ/T; its slope is 0 at Jₘₐₓ.
Step-by-step
WE KNOW
B = 1.931 cm⁻¹, T = 298.15 K, σ = 1
FIND
rotational temperature (θᵣ), rotational partition function (qᵣₒₜ), most populated level (Jₘₐₓ)
Step 1 · Find rotational temperature (θᵣ)
Use θᵣ = 1.4388 × B
hc
k
=
1.4388
cm·K
turns
the
rotational
constant
into
a
temperature.
θᵣ = 1.4388 × 1.931
θᵣ = 2.778 K
Step 2 · Find rotational partition function (qᵣₒₜ)
Use qᵣₒₜ = T ÷ (σ × θᵣ)
Count
the
levels
reached
at
T
,
then
divide
by
σ
:
turning
a
symmetric
molecule
end
for
end
gives
no
new
state.
qᵣₒₜ
=
298.2
1 × 2.778
qᵣₒₜ = 107.3
Step 3 · Find most populated level (Jₘₐₓ)
Use Jₘₐₓ = √(T ÷ (2 × θᵣ)) − 0.5
Set
the
slope
of
(2J
+
1)
e
−J(J + 1)θᵣ/
T
to
0
and
solve
for
J.
Jₘₐₓ
=
√(
298.15
2 × 2.7783228
)
−
0.5
Jₘₐₓ
=
√(
298.15
5.557
)
−
0.5
Jₘₐₓ
=
√
53.65
−
0.5
Jₘₐₓ = 7.325 − 0.5
Jₘₐₓ = 6.825 → The nearest whole level, J = 7, holds the most molecules: the tallest line in the spectrum starts there.
Check
Put the numbers back in. Both sides should match.
2.778 = 1.4388 × 1.931 ✓
107.3
=
298.2
1 × 2.778
✓
6.825
=
√(
298.15
2 × 2.7783228
)
−
0.5
✓
Topic check
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