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Material Type: Exam; Professor: Patel; Class: Physical Chemistry II; Subject: Chemistry and Biochemistry; University: University of Delaware; Term: Spring 2010;
Typology: Exams
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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,
3 / 2
2
2
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?
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.
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)
Time (seconds)
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
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
cos
∫∫ θ r
sin θ dr d θ d φ = 1 1 = N
( r / ao )
e
∫ 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
im φ
∂ ∂θ
∂ ∂φ
∂ ∂θ
∂ ∂φ
= − 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.
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
2
2
1
g
2
2
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.