Chem 3502/5502
Physical Chemistry II (Quantum Mechanics)
3 Credits
Fall Semester 2009
Laura Gagliardi
Lecture 14, October 12, 2009
Solved Homework
We are given that the lowest microwave absorption of carbon monoxide (CO,
with
12
C and
16
O specified) is 115,271,000,000 s
−
1
(115.271 GHz). We are asked first to
compute the moment of inertia of CO.
Remember that the first absorption corresponds to the 0
→
1 rotational transition,
and the energy separation between these two levels is
Δ
E
=
h
ν
= 2
B
where
B
=
h
2
2
I
Rearranging, we have
I
=
h
2
2
B
=
h
2
h
!
=
1.0545
"
10
#
34
kg m
2
s
1
( )
2
6.6256
"
10
#
34
kg m
2
s
1
( )
115,271,000,000 s
1
( )
=
1.456
"
10
#
46
kg m
2
We may compute the reduced mass for this system from the atomic masses for
12
C and
16
O
μ =
m
12
C
m
16
O
m
12
C
+
m
16
O
=
12
!
1.66
!
10
"
27
kg
( )
15.9949
!
1.66
!
10
"
27
kg
( )
12
!
1.66
!
10
"
27
kg
( )
+
15.9949
!
1.66
!
10
"
27
kg
( )
=
1.138
!
10
"
26
kg
Recalling that
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I
= μ
R
2
where
R
is the bond length, we have
R
=
I
μ
=
1.456
!
10
"
46
kg m
2
1.138
!
10
"
26
kg
=
1.131
!
10
"
10
m
= 1.131 Å
The standard tabulated value is 1.128 Å. The difference reflects the very small correction
that is associated with CO
not
being a rigid rotator (it vibrates while it rotates).
Finally, we are asked to predict the frequency at which the 1
→
2 rotational transition
occurs. Recall that the spacing between adjacent transitions in the frequency spectrum is
precisely equal to the frequency of the 0
→
1 transition, we may estimate a value of twice
the ground state absorption, or 230,542,000,000 s
−
1
. If you refer back to the DIB website
we visited that had the rotational spectrum for CO, you will see that this is about right. In
cm
−
1
units, which are gotten simply by dividing
ν
by the speed of light
c
(which is
3
x
10
10
cm s
−
1
), this is 7.7 cm
−
1
.
The Hydrogen Atom (again)
It’s time to return to our old friend, the H atom. Bohr found a quantum
relationship useful for rationalizing the spectra of H, but it violated classical physics by
having one charged particle orbiting about another with no loss of energy through
radiation. de Broglie made the suggestion that the angular momentum quantization
invoked by Bohr was suggestive of waveparticle duality for the electron. We are finally
in a position to look at these waves more carefully.
Consider a proton and an electron bound together in some way. As these are
quantum particles, they may not be still (that would violate the uncertainty principle);
instead, they must be in motion.
Just as for a rigid rotator, we may divide the total
motion into a center of mass motion and a rotational motion about the center of mass.
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 Spring '08
 Staff
 Angular Momentum, total angular momentum

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