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Chem 5314 homework #6 solutions
Problem 1
A molecule has an excited state 500 cm

1
above its ground state. Only consider these
two states. The excited state is doubly degenerate and the ground state is nondegenerate.
For what temperature range does the population of molecules in the excited state exceed
that of the ground state?
solution:
We can, without loss of generality, set
E
0
= 0 and
E
1
≡
Δ
E
= 500 cm

1
. Then
q
= 1 + 2
e

β
Δ
E
. We are asked for the temperature range for which
P
(1)
/P
(0)
≥
1. This
corresponds to 2
e

β
Δ
E
≥
1 or

β
Δ
E
≥ 
ln 2. Multiplying by

1 on both sides (which
reverses the inequality) and rearranging gives Δ
E
≤
k
B
T
ln 2 or
T
≥
Δ
E/
(
k
B
ln 2). Putting
in the numbers (convert units and multiple by
hc
to turn wavenumbers into energy) gives
T
≥
1039 K.
Problem 2
The vibrational frequency of I
2
is 208 cm

1
. At what temperature will the population in
the ﬁrst excited state be half of the ground state?
solution:
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 Fall '11
 Nielsen
 Physical chemistry, Mole, pH

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