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Chemistry 305 Midterm II – November 25, 2010
Answer all questions on the foolscap sheets provided. Be sure to include the
question number. Each question (1-5) is worth 20 points, although parts of questions are
not necessarily of equal value. Equations and other useful info are attached.
1. (a) Which of the following molecules may show a pure rotational (microwave)
absorption spectrum: H2, CH4, CH3Cl, HCl.
CH3Cl, HCl
(b) Which of the following molecules may show a pure vibrational (infrared) absorption
spectrum: CH4, CH3CH3, CH3Cl, N2.
CH4, CH3CH3, CH3Cl
(c) Which of the following molecules may show a rotational Raman spectrum:
H2, NO, N2O, CH4
H2, NO, N2O
(d) Say you wish to study the anti-Stokes portion of the Raman spectrum of a sample,
but you are having difficulty because the peaks barely rise above the baseline. What
could you do to the sample to increase the intensity of the spectrum? Explain.
Heat the sample. Most molecules do not change frequency upon scattering
(Rayleigh). Even fewer molecules undergo anti-Stokes scattering, as these lines
originate from lower populated excited states. By heating the sample, we increase
the initial population of these states, increasing the number which can undergo antiStokes scattering.
2. (a) The microwave spectrum of 79Br19F exhibits a series of absorptions 0.714 cm 1 apart.
Using the masses 79Br = 78.92 au and 19F = 19.00 au, calculate re.
1.76×10−10 m
(b) Calculate the value (in wavenumbers) for the centrifugal distortion constant D for
1 35
H Cl (re = 1.2746 Å). Use the isotopic masses 1H = 1.0078 au and 35Cl = 34.9689 au.
5.32×10−4 cm−1
(exact answer varies depending on any intermediate rounding; 5 sig figs OK)
3. (a) Assuming an accurate enough spectrometer, when we take the IR spectrum of a
diatomic molecule such as HCl, we see rotational structure on top of the vibration, which
splits the spectrum into two branches, designated P and R. To a first approximation, the
~
transitions within the branches are separated by 2 B . However, the rotational constants
of the upper and lower vibrational states are not equal, since r increases with v. Taking
the expression for wavenumber of an anharmonic oscillator and applying it to:
(i) R branch
J 1, J J 1
~
~ ~ (1 2 ~ ) ( B
R
e
e
~
B )J
~
(B
~
B )J
(ii) P branch
J
~
P
1,
J
J
1
~ ~
~ (1 2 ~ ) ( B
B )J
e
e
~
(B
~
B )J
2
2
2
Sketch the vibration-rotation spectrum of HCl, ignoring isotope effects. The sketch only
needs to be qualitative (don’t go too far with that- maybe semi-quantitative?), but it
should be as informative as possible.
R-branch (closer)
P-branch (spread out)
Ignore isotope effects and exact wavenumbers above, but sketch should be close
to this.
(b) Predict the wavenumbers of rotational Raman transitions for J = 0, 1, 2, 3 for 14N2, for
which B = 1.99 cm 1, when it is exposed to monochromatic 336.732 nm laser radiation.
Stokes: 29685.27, 29677.31, 29669.35, 29661.39 cm−1 for J = 0, 1, 2, 3
anti-Stokes: 29709.15, 29717.11 cm−1 for J = 2, 3
4. (a) For a diatomic molecule, considering only the v=0 to v=1 transition, and assuming
equal rotational constants in the upper and lower vibrational states,
S J ,v
2 Bm cm 1 where m has a positive value for ∆J= +1 and negative for
spec
0
∆J= −1, and 0
e 1 2 e . What is the wavenumber of the line in the R branch of
HI originating from the rotational state with J=2. Use the information in Table 3-3
below, and B = 6.511 cm−1.
2 e e (2309.5 2 39.73) cm 1 2230.0 cm 1
0
e
By comparison with the R branch equation given in question 3, m = J + 1, so
2230.0 cm 1 2(6.511 cm 1 )(3) 2269.1 cm 1
R (2)
(b) To be fully correct, electron spin must be included in the wavefunction. Our first
impulse might be to write the following four spin functions for two-electron systems:
3
However, the last two functions are invalid. Why?
They violate the principle of indistinguishability of particles.
(c) For each of the following nuclei, label them as being fermions or bosons:
1
H, 16O, 35Cl
For nuclei, Fermions = odd number of protons + neutrons (mass number),
Bosons = even number, thus 1H, 35Cl are fermions, 16O is boson
5.
(a) Calculate energies in cm 1 of the fundamental and the first overtone frequency of
CO using the anharmonic constants in Table 3-3 below. Compare these values to those
predicted by the harmonic oscillator using = 2143.3 cm 1.
Anharmonic: 2143.29 and 4259.65 cm−1
Harmonic: 2143.3 and 4286.6 cm−1
(b) For the vibration-rotation spectra of polyatomic molecules, we need a second
rotational quantum number, K. For symmetric top molecules, assuming negligible
centrifugal distortion, we have term values (energy levels) given by
2
1
1
1
Gvib
v
0, 1, 2, 3...
e
e v
e cm
2
2
Frot
( A B) K 2
BJ J 1
K
S J ,v
Gvib
J , ( J 1), ( J
1
v
2
Frot
cm 1 J
e
0,1, 2...
2)...... J
e
1
v
2
2
e
BJ ( J 1) ( A B) K 2 cm 1
Derive an equation for the spectral transitions (∆SJ,v) which will be given by the
selection rules Δv = 1, ΔJ = 1, ΔK = ─1. Start from the initial states, and use primes
to indicate final states. Substituting v′= v+1, J′ = J+1, K′ = K−1
S J ,v
S
3
v
2
S
1
v
2
e
2
e
e
e
e
e
3
v
2
e
1
v
2
2
2) ( A B)( K 1) 2
e
+B( J 1)( J
e
+BJ ( J 1) ( A B) K 2
2
(v 1) 2 B( J 1) ( A B)(1 2 K )
4
EQUATIONS, etc.
J=
– Branch
-2
O
NA = 6.022×1023
-1
P
0
Q
1
R
2
S
h = 6.626×10−34 J s
c = 2.998×1010 cm/s
Boltzmann factor
Population of state of energy level E
e
E / k BT
2
Be
; Erot
2I
Be J ( J 1)
For a diatomic, I
r2
h
B
2
8
mi ri 2
In
i
cI
mA mB
mA mB
where
Vibration of diatomic anharmonic oscillator
2 e (1 3 e )
0
e (1 2 e )
1
Ev,J
h
e
1/ 2
h
e
1/ 2
e
2
3 e (1 4 e )
E vib
hBe J ( J 1) hDJ J 2 ( J 1) 2
h
2
E rot
1/ 2 J ( J 1)
interaction
4 B3
D
2
e
For pure rotation spectra
2 BJ
Rotational Raman
J 2
J
where
i
i
2 B(2 J 3)
J 2
where
is incident radiation
S J ,v
1 Å = 10−10 m
BJ J 1
1
v
2
1 nm = 10−9 m
J
e
e
1
v
2
i
i
2 B(2 J 1)
is incident radiation
2
1 cm = 10−2 m
e
cm
1
1 Joule = 1 kg m2 s−2
5