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NAME: CHEMISTRY 444, SPRING, 2010(10S)
Circle Section Number: 10 11 80 81 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,
F(v) dv = 4 π
m
2 π kB T
3 / 2
exp −
m ( v
2
2 kB T
v
2
dv
derive the expression for the average speed given as: 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?
DO NOT WRITE
IN THIS SPACE
p. 1________/ p. 2________/ p. 3________/ p. 4________/ p. 5________/ p. 6________/ p. 7________/ p. 8________/ p. 9________/ p. 10_______/ p. 11_______/ p. 12_______/ ============= p. 13 ______/ (Extra credit) ============= TOTAL PTS /
2. (10 Points) Multiple Choice. Select the best answer to the question.
- 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
- Kinetic theory predicts the energy of a gas to depend on a. pressure b. viscosity c. temperature d. incident frequency Answer: _c____
- 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
- 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_____
- 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
4. (10 Points) Decomposition data for acetaldehyde is shown below as time versus acetaldehyde concentration. 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. Ans. 2nd^ order reaction; k = 0.0771 M-^1 s-^1 Rate = 0.0771 [CH3CHO]^2 Time (seconds)
[CH3CHO]
(M)
Time (seconds)
[CH3CHO]
(M)
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
- 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
- Knowing the uncertainty in the position of a quantum mechanical particle, the Heisenberg uncertainty principle gives the maximum uncertainty in the velocity. x
- The angular momentum component, , commutes with. (^) x
- The Hermite polynomials form a complete basis and can be used to generate quantum mechanical wavefunctions. x
- Given the following wavefunction, , written as a linear combination of eigenfunctions of an operator corresponding to the momentum of a quantum particle (that is,
− i
∂ x
= p 1 φ 1 ), the average value of the momentum one would obtain upon many measurements
is. x
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
( r / ao )
e
− r / ao
cos
∫∫ θ r
sin θ dr d θ d φ = 1 1 = N
( r / ao )
e
− r / ao
∫ r
dr cos
θ
∫ sin^ θ^ d θ^ d φ
∫ 1 = N
(4!) ( ao
)(2/3)(2 π ) 1 = 32 π ( ao
) N
N = 1 4 2 π 1 ao
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
L^ ˆ
- ,^ acts on this wavefunction (again, use the specific values of l and m given)? Disregard normalization constants for the present case.
L^ ˆ
+ Ylm =^
L ˆ
xYlm +^ i^
L ˆ
yYlm
Ylm = Alm Plm ( θ) e
im φ
l = 1 , m = 0
Ylm = A 10 cos θ ( 1 ) = A 10 cos θ
L^ ˆ
+ ( A 10 cos^ θ)^ =^
L ˆ
x ( A 10 cos^ θ)^ +^ i^
L ˆ
y ( A 10 cos^ θ)
= i sin φ
∂ ∂θ
+ cot θ cos φ
∂ ∂φ
( A 10 cos^ θ)^ +^ ( i )(− i )^ cos^ φ^
∂ ∂θ
− cot θ sin φ
∂ ∂φ
( A 10 cos^ θ)
= − i A sin φ sin θ − A cos φ cos θ
= − A sin θ (^) [cos φ + i sin φ] = − A sin θ e i φ This last is just the spherical harmonic with l=1 and m value increased by one to m=1.
10. ( 10 Points) For each statement on the left, provide the single appropriate response from the column on the right.
- For infrared rotation-vibration spectroscopy, the Q branch corresponds to = _l
- MO-LCAO stands for b
- 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___.
- An antibonding orbital ____k___
- 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
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 symmetry under reflection through the mirror plane σ (+ vs - ) only for 1 Σ terms of the ground state species ) for the following diatomic homonuclear molecules and ions: Molecule Configuration BO = Λ = S = Term
H 2
in its first
excited state
3e: (σg1s)^2 , (σu1s)^0 (σg2s)^1 , ½(3-0) = 1.5 0 ½ 2 Σ
He 2 4e: (σg1s)
2
, (σu*1s)
2
½*(2-2) = 0 0 0 1 Σg
He 2
g1s)
u*1s)
g2s)
1
g
O 2
16e: (σg1s)^2 , (σu*1s)^2 ,
(σg2s)^2 (σu2s)^2 (σg2s)^2 (πu)^4 (πg)^2
½*(10-6) = 2 2 or 0 1 or 0 3 Σg
F 2
+ 17e: (σg1s)
2
, (σu*1s)
2
(σg2s)^2 (σu2s)^2 (σg2s)^2 (πu)^4 (πg)^3
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 involves a second derivative. At the cusps, neither 1st^ nor 2nd^ 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.
