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CHEM 332
Physical Chemistry
Spring 2014
Name:
Final Examination
Spring 2015 Students
Focus your attention on the **starred** questions and on the "problem set" questions.
1.
When the surface of a piece of Copper is irradiated with light form a Mercury arc ( =
253.7 nm), electrons are emitted with a maximum Kinetic Energy of 3.84 x 10-20 J. What is
the maximum wavelength of light (minimum Energy photon) which will eject an electron
from a Copper surface?
2.
The Schrodinger Equation for a “Free-Particle” in one-dimension has solutions of the form:
 = A
and
 = A
Explicitly show that  is an eigenfunction of the Hamiltonian operator
and is
not an eigenfunction of the momentum operator
. What implications does this
result have? (You only need to show this for one of the two solutions.)
3.
Do the following operators commute?
and
z
=
What are the implications of your result? (The second part is only worth 2 pts.)
4.
Consider the 4  electrons of 1,3-Butadiene to be constrained to a one dimensional box
0.552 nm long.
Assume the electrons fill the energy levels according to the Pauli Exclusion Principle.
What is the wavelength of the lowest energy photon to be absorbed? In what region of the
spectrum does this absorbance occur?
5.
Below is the IR Spectrum of n-Octane (CH3CH2 CH2 CH2 CH2 CH2 CH2 CH3):
It has been determined that the indicated spectral line is due to a C-H stretch. Assume that
the C-H group can be treated as an isolated diatomic system that behaves as a Simple
Harmonic Oscillator. Then, based on this spectral line, determine the Spring Constant k for
a C-H bond.
Mass Data:
mass H = 1.007825 amu
mass C = 12 amu
**6. For a Hydrogen atom,
R31 =
a) What is the behavior of R31 when r
∞; the large r limit?
R31 ~
b) What is the behavior of R31 when r 0; the small r limit? (Hint: Consider the value of
the quantum number l for this state.)
R31 ~
c) Determine, in units of ao, the position r where the node in this function occurs. (Hint:
Consider the value of R31 at the node.)
**7. Show that
and
do not commute.
**8. Write the Slater determinant for the ground state of the Li atom? Explain the importance of
this construction.
**9. Qualitatively, what is the principle difference between Valence Bond Theory and
Molecular Orbital Theory.
**10. Let  be an eigenfunction of . Show that ( + c) = (E + c), where c is any
constant. Why is this question important near the end of the course?
Important Results
Particle in a Box
=
 =
n = 1, 2, 3, …
E =
Simple Harmonic Oscillator
+ ½ k x2
=
 =
v = 0, 1, 2, ...
E = ħ (v + ½)
 =
G =
(v + ½) -
(v + ½)2
=
Particle on a Sphere
=
 
l = 0, 1, 2, ...
ml = -l, ..., +l
E = l (l + 1)
I = m r2
F = Be J(J + 1) - De J2(J + 1)2
B =
Constants
NA
= 6.022045 x 1023 entities/mole
kB
= 1.380662 x 10-23 J/K
c
= 2.99792458 x 108 m/sec
h
= 6.626176 x 10-34 J sec
RH
= 1.097373177 x 107 m-1
me
= 9.1095 x 10-31 kg
1 amu = 1.6605 x 10-24 g
CHEM 332
Physical Chemistry
Spring 2015
Problem Set 10
Reading
Atkins "Physical Chemistry, 8th Ed."
Chapter 11.
Atkins "Physical Chemsitry, 9th Ed.
Chapter 10.
Problems
1.
Use our first-order perturbation result for the ground state energy of the two electron atom
(Helium) to approximate the ground state energy for each of the following two electron
atoms. The "exact" solution for the energy is as given. Determine the percentage error for
each case. What trend do you notice? Explain.
atom
HLi+
Be2+
B3+
C4+
N5+
O6+
F7+
Ne8+
Eexact [a.u.]
-0.52759
-7.27991
-13.65556
-22.03097
-32.40624
-44.78144
-59.15659
-75.53171
-93.90680
("Exact" means a non-relativistic approximation to thirteenth order in Z-1.)
2.
Show the following Helium ground state wavefunction is normalized:
GS =
3.
1s(1)1s(2) [(1)(2) - (1)(2)]
Write the Slater Determinant for the ground state of the Beryllium atom.
4.
Under the Born-Openheimer approximation, the Valence Bond ground state wave function
for the H2 molecule can be written as:
 = [1sa(1) 1sb(2) + 1sa(2) 1sb(1)] ((1) (2) - (1) (2))
Then, variational energy is:
W =
=
where:
S =
= Overlap Integral
J =
K =
= Resonance Integral
We can estimate the energy of the H2 molecule as:
E ~ W + 1/R
where R is the internuclear distance in atomic units.
Numerical evaluation of the above integrals yields, in atomic units:
R
0.5
1.0
1.5
2.0
2.5
S
0.9603
0.8584
0.7252
0.5865
0.4583
J
-2.1876
-1.9042
-1.6771
-1.5192
-1.4127
K
-2.1019
-1.5633
-1.0382
-0.6361
-0.3668
Plot E vs. R for H2. Estimate the equilibrium internuclear distance RE for H2.
5.
According to valence bond theory, the bond between two atoms X-Y can be viewed as a
"mix" of covalent and ionic bonds:
 = N [A cov + B ion]
The exact "mix" is determined by B/A. This can be determined from measurements of the
molecule's dipole moment  and internuclear distance RE.
 =
For LiH, it is found experimentally that RE = 1.5853A and  5.882D. Determine B/A for
this molecule.