Download MS Comprehensive/PhD Qualifying Exam in Physical Chemistry Fall

MS Comprehensive/PhD Qualifying Exam in Physical Chemistry
Fall 2015
MS comprehensive/PhD qualifying exam
Department of Chemistry
Subject: Physical Chemistry
Date: November 13, 2015
Time: Three hours
Closed book.
Instructions: There are three sections. You must use a separate blue book for each section. Write the
section number of question number of the cover of each blue book.
MS Comprehensive/PhD Qualifying Exam in Physical Chemistry
Fall 2015
1. (10 points) Energy. What has more energy, the kinetic energy of a one and one-half ton sedan at 60
m.p.h. or the chemical energy of a four-oz. (King Size) Snickers candy bar (440 Kcal)?
FYI: 1 mile = 1609.34 meters, 1 ton = 1000 kg
2. (10 points) Entropy. Which of the following two distributions, if any, has a greater entropy?
Distribution 1: p(1) = 12 , p(2) = p(3) = p(4) =
Distribution 1: p(2) = 12 , p(1) = p(3) = p(4) =
1
6
1
6
3. (15 points) Boltzmann Distribution. You have a thermodynamic system with three states. You
observe the probabilities p1 = 0.8, p2 = 0.15, p3 = 0.05 at T = 300 K. What are the energies ✏2 and ✏3
of states 2 and 3? Express your answer in units of kT , relative to the ground state.
4. Henry’s law. Henry’s law relates partial pressure (p) to the mole fraction (x) by p = KH x, where KH
is the Henry’s law constant.
(a) (10 points) Raoult’s law specifies that p = p⇤ x, where p⇤ is the vapor pressure of the pure component. Are the laws inconsistent? How do we know which law to apply to a given situation?
(b) (10 points) Divers, returning from deep dives, can get the bends from nitrogen gas bubbles in their
blood. Assume that blood is largely water. The Henry’s law constant for N2 in water at 25 C is
86,000 atm. The hydrostatic pressure is 1 atm at the surface of a body of water and increases by
approximately 1 atm for every 33 feet in depth. What is the N2 solubility at the surface versus 100
feet in depth? How does this relate to the bends?
5. Stirling Engine. A Stirling engine has the pV cycle shown in Figure 7.11 and uses and ideal gas
working fluid. The steps are quasi-static.
(a)
(b)
(c)
(d)
(10 points)
(10 points)
(10 points)
(15 points)
wtot .
How much work is done in the constant-volume segments, w12 and w34
What is the energy change around the cycle, Utot = U12 + U23 + U34 + U41 ?
What is the total work performed around the cycle, wtot = w12 + w23 + w34 + w41 ?
If T2 = 2000 K, T1 = 300 K, V2 = 1 L, V1 = 0.01L, and n = 0.1 mol of ieal gas, compute
Page 2
Questions for Qualifying Exam in Physical Chemistry
Quantum Part
Q1
Two unnormalized excited state wavefunctions of the H atom are:
!
!!
!
!!
(i)
! = 2−
(ii)
! = ! ∙ !"#$ ∙ !"#$ ∙ !
!!
!! !!
!
(a) Normalize both functions.
(b) Confirm that these two functions are mutually orthogonal.
(c) Using normalized version of the wavefunction (i), determine the expectation value of
the commutator of the position (!) and momentum operator (!! ).
(d) Do the same as in (c) for the position operator (!) and momentum operator (!! ).
Q2
Is the operator !! ! Hermitian?
Q3
Provide at least two “pure” quantum mechanical phenomena. Explain why these
phenomena cannot be explained/rationalized by Classical Mechanics.
Useful formula:
!
! ! ! !!" !" =
!
!!
! !!!
Chemical Kinetics
1. The protein rhodopsin is responsible for the absorption of light in photoreceptor
cells. Following the absorption of light, the protein goes through the complex
series of reactions shown in the figure. We assume the lifetimes shown in the
figure are reciprocals of the corresponding rate constants k1, k2, … kn at 310 K.
(a) Write the rate law for the formation of MI following a very short (10-12 s) flash
of light that converts all of the R initially present to BR.
(b) If R is initially present at 1 × 10-3 M, what is the initial rate of formation of
MI? State any approximations or assumptions that you make.
(c) Experimentally, it is found that the ratio [MII]/[MI] rapidly (within 5 × 10-3 s)
reaches a constant value of 1.0 after light absorption, even though [MI] and [MII]
individually decrease slowly with time. What is the value for k4 and k-4, the rate
constants for the reaction MI ! MII and MII ! MI, respectively?
(d) It has been proposed that the transformation MI ! MII involves large changes
in protein structure. The enthalpy change for this step was determined to be
ΔH°310 (MI ! MII) = 42 kJ/mol. Using the result of part (c), calculate ΔS°310 for
this transformation and interpret it in terms of a possible structural change.
(e) Write the rate law for the disappearance of MII.
(R)
(BR)
(LR)
(MI)
scotopsin
(MII)
(MIII)
2. A polymer is a molecule with a very high molar mass made by joining many
monomers together. In the polymerization of ethylene, the initial step is:
R2
2R! initiation
The R" species reacts with an ethylene molecule (M) to generate another radical:
R! + M
M 1!
Reaction of M1" with another monomer leads to the growth or propagation of the
polymer:
M1 ! + M
kp
M2! propagation
This step can be repeated with hundreds of monomer units. The propagation terminates
when two radicals combine:
M’! + M”!
kt
M’-M”
termination
The initiator in the polymerization of ethylene commonly is benzoyl peroxide:
[(C6H5COO)2]
2 C6H5COO!
This is a first-order reaction. The half-life of benzoyl peroxide at 100 °C is 19.8 min.
(a) Calculate the rate constant (in min-1) of the reaction.
(b) If the half-life of benzoyl peroxide is 7.30 h at 70 °C, what is the activation
energy (in kJ/mol) for the decomposition reaction?
(c) Write the rate laws for the elementary steps in the above polymerization process.
(d) What condition would favor the growth of long high-molecular-mass
polyethylene?