Physical Chemistry II - Final
Exam Answers | CHEM 444
Chemistry
University of Delaware (UD)
13 pag.
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Circle Section Number:
10
11
80
81
CHEMISTRY 444, SPRING, 2010(10S)
Final Exam, May 21, 2010
Answer each question in the space provided; use back of page if extra space is needed. Answer questions so the grader can READILY
understand your work; only work on the exam sheet will be considered. Write answers, where appropriate, with reasonable numbers of
significant figures. You may use only the "Student Handbook," a calculator, and a straight edge.
1. (15 points) A. Starting with the Maxwell distribution of speeds,
 m(v 2 ) 
 m 3 / 2
F(v)dv = 4 π 
exp
 v 2 dv
−

2
2k
π
k
T
T



B 
B
derive the expression for the average speed given as:
DO NOT WRITE
IN THIS SPACE
p. 1________/15
p. 2________/10
p. 3________/10
p. 4________/10
p. 5________/10
p. 6________/10
p. 7________/15
p. 8________/10
p. 9________/10
p. 10_______/10
p. 11_______/15
p. 12_______/10
B. Consider Xenon gas at 298K and Pressure = 1 atmosphere. What is the mean free path
based on the Lennard-Jones diameter for a Xenon molecule?
=============
p. 13 ______/10
(Extra credit)
=============
TOTAL PTS
/135
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CHEM 444, Final, Spring, 2010, page 2
2. (10 Points) Multiple Choice. Select the best answer to the question.
1. The mean free path as determined in the context of kinetic theory
a. depends linearly on temperature
b. depends inversely on pressure
c. is the average distance a gas particle travels between successive collisions
d. all of the above
Answer: ___d__
2. Kinetic theory predicts the energy of a gas to depend on
a. pressure
b. viscosity
c. temperature
d. incident frequency
Answer: _c____
3. Perturbation-Relaxation methods such as temperature jump (T-jump) experiments are designed to probe
a. chemical kinetics for equilibrium processes where initial conditions cannot be controlled
b. fast reactions
c. rate constants
d. all of the above
Answer
___d___
4. Transition state theory invokes the concept(s) of
a. commutators
b. a maximum in the free energy profile along the reaction coordinate connecting reactant(s) and
product(s)
c. equipartition
d. slow reaction kinetics
Answer: __b_____
5. In heterogeneous catalysis, the variation of surface coverage of a solid catalyst at a given temperature
with changing pressure is called a(n)
a. phase diagram
b. T-x-y diagram
c. adsorption isotherm
d. turnover
Answer:___c___
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CHEM 444, Final, Spring, 2010, page 3
3. (10 Points) Consider the following reaction mechanism for a unimolecular reaction:
where A* is an energetic intermediate. What is the expression for the rate of production of B. Make any appropriate
assumptions for this problem.
Apply steady state approximation to determine the concentration of reactive species A*
d [ A* ]
dt
= 0 = k 2 [ A] [ A] − k−2 [ A* ] [ A] − k1 [ A* ]
2
k 2 [ A]
A
=
[ ] k [ A] + k
1
−2
*
Thus,
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CHEM 444, Final, Spring, 2010, page 4
4. (10 Points) Decomposition data for acetaldehyde is shown below as time versus acetaldehyde concentration.
Time
(seconds)
[CH3CHO]
(M)
Time
(seconds)
[CH3CHO]
(M)
0
0.2
80
0.09
20
0.153
100
0.079
40
0.124
120
0.070
60
0.104
Determine whether the reaction is first or second order, and the rate constant. Use a plot to show your work as well as any
relevant equations.
nd
-1
-1
Ans. 2 order reaction; k = 0.0771 M s
2
Rate = 0.0771 [CH3CHO]
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CHEM 444, Final, Spring, 2010, page 5
5. (10 Points) True or False. Determine whether the following statements are true or false; place a check in the appropriate
column.
Statement
True
False
1. Commutation is the term applied to describe the non-zero probability density for observing
quantum particles traveling through finite potential barriers rather than crossing over them.
x
2. Knowing the uncertainty in the position of a quantum mechanical particle, the Heisenberg
uncertainty principle gives the maximum uncertainty in the velocity.
x
3. The angular momentum component,
, commutes with
.
4. The Hermite polynomials form a complete basis and can be used to generate quantum
mechanical wavefunctions.
5. Given the following wavefunction,
x
x
, written as a linear combination of
eigenfunctions of an operator corresponding to the momentum of a quantum particle (that is,
−i
is
∂φ1
= p1φ1 ), the average value of the momentum one would obtain upon many measurements
∂x
x
.
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CHEM 444, Final, Spring, 2010, page 6
6. (10 Points) Consider the associated Legendre functions
and
. Show that they are orthogonal.
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CHEM 444, Final, Spring, 2010, page 7
7. (15 Points) Normalize the wavefunction for the Hydrogen atom
electron (n=2, l=1, m=0).
So to normalize, we compute the following integral.
*
ψ nlm
ψ nlm =
∫∫ N
2
(r /ao ) 2 e−r / a o cos2 θ r 2 sin θ dr dθ dφ = 1
π
∞
1= N
2
∫ (r /a )
2
o
e
−r / a o
2
r dr
0
∫ cos θ sinθ
2
0
2π
dθ
∫ dφ
0
1 = N 2 (4!) (ao3 )(2/3)(2π )
1 = 32π (ao3 )N 2
1 1
N=
 
4 2π  ao 
3/2
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CHEM 444, Final, Spring, 2010, page 8
8. (10 Points) Consider the spherical harmonic function,
associated with the specific values l=1 and m=0. What eigenfunction results after the following operator, called
Lˆ + , acts on
this wavefunction (again, use the specific values of l and m given)? Disregard normalization constants for the present case.
Lˆ +Ylm = Lˆ xYlm + iLˆ yYlm
Ylm = Alm Plm (θ )e imφ
l = 1,m = 0
Ylm = A10 cos θ (1) = A10 cos θ
Lˆ + (A10 cosθ ) = Lˆ x (A10 cosθ ) + iLˆ y (A10 cos θ )


∂
∂
∂
∂
= isin φ
+ cot θ cos φ (A10 cosθ ) + (i)(−i)cos φ
− cot θ sin φ (A10 cos θ )
∂θ
∂φ 
∂θ
∂φ 


= −iA sin φ sin θ − Acos φ cosθ
= −Asin θ [cos φ + isin φ ] = −Asin θe iφ
This last is just the spherical harmonic with l=1 and m value increased by one to m=1.
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CHEM 444, Final, Spring, 2010, page 9
9. (10 Points) Sketch a complete Grotrian diagram of all the terms arising from the ground state configuration of an
oxygen atom. Take the spin-orbit coupling into account and indicate the degeneracy (g) of each manifold
produced. Remember that Hund’s rules apply strictly only to the ground state but are often used to produce the order of
other states.
S = 0; L = 0
S = 0; L = 2
S = 1; L = 1
1
S (g=1)
1
S0 (1 level)
1
D (g=5)
1
D2 (1 level)
3
P0 (1 level)
3
P1 (3 levels)
3
P2 (5 levels)
3
P (g=9)
(1s2) (2s2) (2p6)
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CHEM 444, Final, Spring, 2010, page 10
10. (10 Points) For each statement on the left, provide the single appropriate response from the column on the right.
1. For infrared rotation-vibration spectroscopy, the
Q branch corresponds to
= __l_
2. MO-LCAO stands for __b__
3. A Slater determinantal form for a many-electron
wavefunction conveniently allows for the
accommodation of what particular property of the
wavefunctions describing particles such as
electrons ____n___.
4. An antibonding orbital ____k___
5. The selection rules for infrared vibrational
spectroscopy of diatomic molecules discussed in
this course are based on what model of interaction
between a molecule and electromagnetic
radiation__m____
a) an oscillating molecular magnetic moment
interacts with the electric field of incident
photon beam
b) molecular orbitals – linear combination of
atomic orbitals
c) exhibits a relative enhancement of
electronic density between bonding nuclei
d) molecular orbitals as a linear combination
of atomic occupations
e) because the perturbing Hamiltonian breaks
some symmetries
f) do not occur
g) an adsorption isotherm
h) main orbitals are low-calorie atomic orbitals
i) the method of initial rates
j) the number of electrons emitted is
proportional to the light intensity
k) exhibits a relative depletion of electronic
density between bonding nuclei
l) zero
m) an oscillating molecular electric dipole
moment interacts with the electric field of
incident radiation
n) antisymmetry
o) the kinetic energy of the emitted electrons
is linearly dependent on the light intensity
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CHEM 444, Final, Spring, 2010, page 11
11. (15 Points) Determine ground state (unless indicated otherwise) configurations, bond order, total orbital angular
momentum, total spin angular momentum, and the terms corresponding to the lowest energy arising from these
configurations (indicate the symmetry under inversion (g vs. u) only for the ground state species with Σ terms and
1
symmetry under reflection through the mirror plane σ (+ vs -) only for Σ terms of the ground state species) for the
following diatomic homonuclear molecules and ions:
Molecule
Configuration
BO =
Λ=
S=
H2- in its first
excited state
3e: (σg1s)2, (σu*1s)0 (σg2s)1,
½*(3-0) = 1.5
0
½
He2
4e: (σg1s)2, (σu*1s)2
½*(2-2) = 0
0
0
He2-
5e: (σg1s)2, (σu*1s)2 (σg2s)1
½*(3-2) = 0.5
0
½
2
O2
16e: (σg1s)2, (σu*1s)2,
(σg2s)2(σu*2s)2(σg2s)2(πu)4(πg*)2
½*(10-6) = 2
2 or 0
1 or 0
3
F2 +
17e: (σg1s)2, (σu*1s)2,
(σg2s)2(σu*2s)2(σg2s)2(πu)4(πg*)3
½*(10-7) =
1.5
1
½
Term
2
1
Σ
Σg+
Σg
Σg-
2
Π
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CHEM 444, Final, Spring, 2010, page 12
12. (10 Points). Using data provided in Table 12.1, calculate the energy of the n = 0, J = 1 to n = 2, J = 2 transition for
1 127
-1
H I, in cm , as accurately as these data allow you to.
as accurately as these data allow you to.
-1
-1
-1
-1
Data from Table 12.1. ωe = 2309.5 cm ; xeωe = 39.73 cm ; Be = 6.551 cm ; αe = 0.183 cm .
Use Equation 13.23 from the handbook, one can calculate the centrifugal distortion coefficient, Dc:
Then use Equation on page 12.5 to calculate the energy difference between the levels:
= 4351.76 cm-1
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CHEM 444, Final, Spring, 2010, page 13
13. (10 Points, extra credit) When we considered a particle in a one-dimensional box of width a with infinite-potential walls
in the class, the quantum mechanical solution for the ground state of this system had a form shown in a figure below for a =
1 Å. This solution has discontinuities (cusps) at x = 0 and x = a. However, the first postulate of quantum mechanics forbids
these discontinuities as the appropriate function has to be a solution of the Shrödinger equation, which for a particle in a box
st
nd
involves a second derivative. At the cusps, neither 1 nor 2 derivatives of the proposed solution are specified, meaning that
this function should not be appropriate. Did we make a mistake in the class? Is the quantum theory violated even for the
simplest of all the model systems we considered? Explain your answer in a few short sentences.
All we considered in the class was a model system of a particle in a box with infinite height walls. Quantum theory basically
tells us that there is something wrong with the model, which can not represent the real physical world. The problem is that
realistically an infinite potential can not exist and once we start considering realistic potentials (for example, really large but
not infinite), the appropriate wavefunction and all its derivatives become continuous, as we have seen in a problem of a
particle in a finite-height box or similar to the quantum mechanical harmonic oscillator.
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