Chem 302

Final Exam

May 16, 2002

Name __________

Full credit will be given to correct answers only when ALL the necessary steps are shown. DO NOT GUESS THE ANSWER.

This is a closed book and closed notes exam, and you are responsible to be sure that your exam has no missing pages ( 7 pages). You are allowed a crib sheet.

If you consider that there is not enough information to solve a problem, you have to specify the missing information and describe the problem solving procedure.

Whoever thinks a faultless

piece to see

Thinks what ne'er was, nor is,

nor e'er shall be.

- Alexander Pope,

An Essay on Criticism

If therefore angels are not composed of matter and form, as was said above, it follows that it would be impossible to have two angels of the same species

... The motion of an angel can be continuous or discontinuous as it wishes ...

And thus an angel can be at one instant in one place, and at another instant in another place, not existing at any intermediate time.

- Thonas Aquinas (1268)

Honor Statement

I have neither give nor received aid in this examination.

Full signature _______________________________

Problem 1 (25 points)

Suppose that a particle in a 1D-box is in the state psi defined by

and zero otherwise.

Note that psi is not an eigenfunction for the particle in the box.

a) Show that psi is normalized.

b) Calculate the expected value of the energy < E > .

Problem 2 (25 points)

To prove that L² commutes with each component separately

[L², L_x²] = 0

where [...] stands for the commutator and

L² = L_x² + L_y² + L_z² _, (2-1)

we have to prove first that

[L_x, L_y] = (i h/2 π) L_z (2-2)

[L_y, L_z] = (i h/2 π) L_x

[L_z, L_x] = (i h/2 π) L_y

Prove only Eq. (2-2) and then prove Eq. (2-1).

Problem 3 (25 points)

Calculate in the basis of transition state-theory, the rate constant at 300K for the reaction

H + HBr --> H₂ + Br

Use the following data:

H-Br intermolecular distance = 141.4 pm

H-Br vibrational wavenumber = 2650 cm^-1

Take the activation complex to be linear, with the following characteristic derivable from quantum mechanics:

H-H distance = 150.0 pm

H-Br distance = 142.0 pm

Center of mass is located at 283.3 pm from the first hydrogen.

Wavenumber for the symmetrical stretch = 2340 cm^-1

Wavenuber for the two degenerate bending vibrations = 460 cm^-1

Remember that I = ∑m_i r_i² where r_i is the distance from the center of mass.

Problem 4(25 points)

Consider the following proposed mechanism for the decomposition of ozone

a) Does this mechanism predicts a first order reaction during the early stages, barely any product, of the decomposition?

b)What is the pseudo-order during the latter stages, huge accumulation of product, of the reaction?

Hint: Find the dominant terms for each the initial and final stages of the reaction.

Problem 5 (25 points)

a) (15 points) The U. S. Department of Commerce estimated that the earth's human population was increasing at an average rate of 2% per year during the period 1960-1970. Considering that the earth's population in 1965 was estimated to be 3.34 billion people, calculate the time required for the population of the earth to double, and then compare it to the observed value after 35 years (2000).

b) (5 points) Predict the earth's population in the year 2515.

c) (5 points) If the total surface of this planet is approximately 1,860,000 billion square feet and 80% of this surface is covered by water calculate the surface per person in the year 2515.

Problem 6(25 points)

Discuss two implementations to the Lotka-Volterra equations that yield a more realistic ecological model.