Spectroscopy Cumulative Exam

Your Name________________________________

Physical Chemistry Cumulative Exam Fundamentals of Spectroscopy for Analysis of Molecular Structures and Interactions December 9, 2016 Tatyana Polenova, Principal Examiner Total points: 100 1. The absorption of radiation energy by a molecule results in the formation of an excited molecule. Given enough time, it would seem that all of the molecules in a sample would have been excited and no more absorption would occur. Yet in practice we find that the absorbance of a sample at any wavelength remains unchanged with time. Why? (10 pts) (17.5 CH)

2. List some important differences between fluorescence and phosphorescence. (10 pts) (17.43 CH)

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Spectroscopy Cumulative Exam

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3. Account for the number of lines observed in the EPR spectra of benzene and naphthalene anion radicals shown in figure below (left and right, respectively). How would you use isotopic substitution to assign the two hyperfine splitting constants in naphthalene? (10 pts) (17.41 CH)

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Spectroscopy Cumulative Exam

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4. Using the explicit formulae for spherical harmonics (see Appendix), show that the rotational transition J=0→J=2 is forbidden in microwave spectroscopy (the rigid rotator approximation) (10 pts) EX 13-12 McQ

5. The bond length in 12C14N is 117 pm and its force constant is 1630 N/m. Predict the vibrationalrotational spectrum of this molecule. (10 pts) (13-3 McQuarrie)

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Spectroscopy Cumulative Exam

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6. Given that Re=156.0 pm and k=250.0 N/m for 6Li19F, use the rigid rotator-harmonic oscillator approximation to construct to scale an energy-level diagram for the first five rotational levels in the v=0 and v=1 vibrational states. Indicate the allowed transitions in an absorption experiment, and calculate the frequencies of the first few lines in the R and P branches of the vibration-rotation spectrum of 6Li19F. (10 pts) (13-12 McQ)

7. Give the number of normal vibrational modes of O2, C2H2, CBr4, C6H6. Explain. (10 pts)(17.18 CH)

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Spectroscopy Cumulative Exam

Your Name________________________________

8. Use the particle-in-a-one-dimensional box model to calculate the longest-wavelength peak in the electronic absorption spectrum of hexatriene

. (10 pts)(17.29 CH)

9. Many aromatic hydrocarbons are colorless, but their anion and cation radicals are often strongly colored. Give a qualitative explanation for this phenomenon. (10 pts) (17.30 CH)

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Spectroscopy Cumulative Exam

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10. Short questions on NMR spectroscopy a) What is the ground state spin for 12C? Why? (2 pts)

b) What is the field strength (in Tesla) needed to generate a 13C frequency of 100 MHz? (1 pt)

c) What is a powder pattern in NMR spectroscopy? (1 pt)

6

Spectroscopy Cumulative Exam

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d) Sketch an anisotropic NMR spectral lineshape arising from a rhombic chemical shift anisotropy interaction. Label the main features of the lineshape and explain where the specific functional form arises from. (2 pts)

e) Describe the protocol for determining protein structures by NMR spectroscopy. (4 pt)

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Spectroscopy Cumulative Exam

Your Name________________________________ Appendix

I. Spherical harmonics

8

Physical Chemistry Cumulative Examination Topic: Thermodynamics November 12, 2016 University of Delaware Department of Chemistry and Biochemistry Primary Examiner: Sandeep Patel Secondary Examiner: Tatyana Polenova

This is the Fall 2016 Thermodynamics Cumulative Exam for the Physical Chemistry Division. Please answer three (4) out of the seven (7) questions to the best of your ability. Please make sure that your answer is clearly stated in the solution, and that all steps in your solution process are understandable and explicit. State all assumptions you invoke. Please leave your answers in the Blue Book. It may be helpful to first read all problems before trying to solve any. Good Luck.

⎛ ∂G ⎞ ⎜ ⎟ ⎝∂ni ⎠ P,T ,{ n

1. Prove:

j ≠i

}

⎛ ∂A ⎞ = ⎜ ⎟ ⎝∂ni ⎠V ,T ,{ n

j ≠i

}

where G is the extensive Gibbs free energy, A is the extensive Helmholtz Free Energy, ni is the number of moles of species ‘i’, V is volume, T is temperature, and P is pressure.

€                                        

2.  Derive an expression for the equilibrium constant for the following reaction involving ideal gases at constant temperature:

ν A A + ν B B ⇔ νC C + ν D D where A, B, C, D represent ideal gases and the ν are stoichiometric coefficients.

€

           

3. Consider adiabatic compression of air and helium from 1 atmosphere to 10 atm. Initial temperature is 300K. What is the final temperature after compression?

γ air =

€

Cp

C v = 1.4

; γ helium =

Cp

C v = 5.3

4. Provide an expression for the energy of a system in terms of the pair correlation or pair distribution function, g(r), where r is the magnitude of the distance between any 2 particles in the system.

5. In this problem you will deal with vapor-liquid (2-phase) equilibrium of a binary (2component) system. Consider the system acetonitrile(1)/nitromethane(2) to conform to ideal solution and ideal vapor behavior. The temperature dependence of the saturation (vapor) pressures of the two species is given by:

2945.47 t + 224.0 2972.64 = 14.2043 − t + 209.0

ln P1saturation = 14.2724 − ln P2saturation

where pressure is in units of KPa (kilopascal) and temperature in the above relations is in degrees Celsius.

€

A. For a liquid composition of x1=0.6, what are the total pressure and the composition (y1 and y2) of the vapor phase at a temperature of 75 Celsius?

B. What are the saturation temperatures for each fluid at a pressure of P = 70KPa?

6. Derive the equality:

⎛ ∂ 2 P ⎞ ⎛ ∂CV ⎞ ⎜ ⎟ = T⎜ 2 ⎟ ⎝ ∂V ⎠T ⎝ ∂T ⎠V Remember that U is a state function (hence its total differential is an exact differential).

⎛ ∂U ⎞ ⎛ ∂P ⎞ ⎟ = T⎜ ⎟ − P . ⎝ ∂V ⎠T ⎝ ∂T ⎠V

Also, the following relation € may be useful: ⎜

€

7. For a one-component, two-phase system at equilibrium, show that the locus of points where the two phases, α and β, are in equilibrium is given by the solution to the differential equation:

dµ s β v α − s α v β = dT v β − vα where

s

is molar entropy, and

€

€

€

v is molar volume.

Physical Chemistry Cumulative Examination September 9, 2016 Major Examiner: D. P. Ridge Work 100 points worth of questions: 1. (15 points) Find B as a function of time for the sequential reaction below. Assume at time t=0, A=a and B=C=0. k1 k2 A  B  C 2. (20 points) Equal volumes of two equimolar solutions of reactants A and B are mixed, and the reaction A + B  C occurs. At the end of 1 h, A is 90% reacted. How much of A will be left unreacted at the end of 2 H if the reaction is: (a) First order in A and zero order in B? (b) Zero order in A and first order in B? (c) Zero order in both A and B? (d) First order in A and one-half order in B? 3. (20 points) Find the rate law for the chain reaction below. You may use the steady state and long chain approximations as needed. Br2 Br + H2 H + Br2 H + HBr 2Br

k1

2Br

k2 k3 k4 k5

HBr + H HBr + Br H2 + Br Br2

4. (10 points) Define the quantities in the Arrhenius equation for the T dependence of the rate constant, k=Ae–Ea/RT. Draw a diagram indicating the relationship between Ea for the forward and reverse reactions and ∆Eo, the thermodynamic energy change for the reaction.

5. (10 points). Given the equation below and the Arrhenius equation (k = Aexp(EA/RT)), Define the relationship between the entropy and enthalpy of activation and the parameters A and EA (note: ∆G  is the free energy of activation).  kT k = b e − ∆G / RT h 6. (10 points). Define a linear free energy relationship and give an example. 7. (15 points). Derive the equation for the primary salt effect using transition state theory. (Hint: logγi = -0.51zi2I1/2 and I = 1/2∑ciZi2). 8. (10 points) Define adiabatic and diabatic potential energy surfaces and give an example of each. When does a reaction follow the adiabatic PES and when does it follow the diabatic PES. 9. (10 point) Make a topographical sketch of a potential energy surface with the barrier in the entrance channel for a three atom reaction: A + BC  AB + C. Label the axes in terms of the interatomic distances. Let the axis perpendicular to the paper be the energy axis. Label the transition state. Trace a typical reaction trajectory. With the barrier in the entrance channel is this reaction more likely to be endothermic or exothermic? Are the products more likely to be vibrationally hot or translationally hot? 10. (10 points) A particular surface is exposed to evaporated metal atoms. The probability that a surface site is covered by a layer N atoms deep can be calculated from PN = (n)Ne−n/N! where n is the average number of atoms per site. For what value of n are 90% of the substrate atoms covered by at least one metal atom (P0 = 0.1)? Is it possible to ignore the substrate in interpreting the chemistry on this surface if n = 2? 12. (15 points) Derive the analogue to the Lineweaver-Burke equation for the enzyme catalyzed reaction between A and B with the following mechanism. k1 A + E EA k-1 B + EA

EAB

k2 k-2 k3

EAB

Products

13. (15 points) Consider a crossed beam experiment between a beam of A molecules with mass mA and velocity vA and a beam of B molecules with mass mB and velocity vB. The beams cross at right angles. Draw a Newton diagram (i. e. a diagram in velocity space) for the system. Indicate the relative velocity v and center of mass velocity V. Indicate the elastic scattering circle for unreactive A.

Physical Chemistry Cumulative Examination May 14, 2016 Cecil Dybowski and Tatyana Polenova Work 100 points worth of questions. 1. (15 points) For the mechanism below, find expressions for [A(t)] and [B(t)]. As boundary conditions, assume that, at t = 0, the value of [A] is [A(0)] and the value of [B] = 0. 𝐴𝐴

𝐵𝐵

𝑘𝑘𝑓𝑓

→

𝑘𝑘𝑟𝑟

→

𝐵𝐵

𝐴𝐴

2. (10 points) The half-life of 14C is 5730 years. If a tree is chopped down today (2016), in what year will the wood contain 90% of the 14C it has today? 3. (20 points) The photochemical chlorination of chloroform to produce carbon tetrachloride by the reaction 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶3 + 𝐶𝐶𝐶𝐶2 → 𝐶𝐶𝐶𝐶𝐶𝐶4 + 𝐻𝐻𝐻𝐻𝐻𝐻 is thought to proceed by the following mechanism:

𝐶𝐶𝐶𝐶 ∙

+

𝐶𝐶𝐶𝐶3 𝐶𝐶 ∙

2 𝐶𝐶𝐶𝐶3 𝐶𝐶 ∙

𝐶𝐶𝐶𝐶2

𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶3

+

+

𝐶𝐶𝐶𝐶2

𝐶𝐶𝐶𝐶2

ℎν

→

𝑘𝑘2

→ 𝑘𝑘3

→

𝑘𝑘4

→

2 𝐶𝐶𝐶𝐶 ∙

𝐶𝐶𝐶𝐶3 𝐶𝐶 ∙ 𝐶𝐶𝐶𝐶𝐶𝐶4

2 𝐶𝐶𝐶𝐶𝐶𝐶4

+

+

𝐻𝐻𝐻𝐻𝐻𝐻

𝐶𝐶𝐶𝐶 ∙

Derive the steady-state rate law for the formation of carbon tetrachloride for this mechanism. Be sure to state any approximations you make. 4. (15 points) Relaxation analysis is used to determine rate constants for processes from relaxation times. For example, Pörscke, Uhlenbeck, and Martin studied the dimerization of the decanucleotide A4GCU4 to form the double stranded dinucleotide. Calling the nucleotide D and its dimer D2, the process involves these two steps: 2 𝐷𝐷

𝐷𝐷2

𝑘𝑘1

→

𝑘𝑘−1

��

𝐷𝐷2

2 𝐷𝐷

Using relaxation analysis, derive an expression for the relaxation time, τ, in terms of the rate constants in the equation, and equilibrium concentrations, [D2]eq and [D]eq. Show all work.

5. (10 points) The Arrhenius equation for the rate constant is purely empirical: 𝐸𝐸𝑎𝑎 𝑘𝑘(𝑇𝑇) = 𝐴𝐴 exp(− ) 𝑅𝑅𝑅𝑅 It often appears to fit data very well. However, most theories, such as Eyring theory, do not predict such a simple dependence on temperature. For example, one sees equations such as 𝑅𝑅𝑅𝑅 ∆𝐺𝐺 ≠ 𝑘𝑘(𝑇𝑇) = 𝑒𝑒𝑒𝑒𝑒𝑒 �− � 𝑅𝑅𝑅𝑅 ℎ𝑁𝑁0 where R is the gas constant, N0 is Avogadro’s number, h is Planck’s constant, and ∆G ≠ is the free energy of activation. Derive an equation that gives the free energy of activation in terms of the Arrhenius parameters for a reaction. Explain clearly how you arrive at this solution. 6. (10 points) Make a topographical sketch of a reaction potential energy surface for a three-atom reaction: 𝐴𝐴 + 𝐵𝐵𝐵𝐵 → 𝐴𝐴𝐴𝐴 + 𝐶𝐶 Label the two in-plane axes in terms of interatomic distances, RAB and RBC. Let the axis perpendicular to the paper be the energy axis. Label the point that corresponds to the transition state. Trace a typical reactive trajectory and a typical unreactive trajectory. 7. (15 points) The reaction of NO with Cl2 can be monitored by examining [NO(t)] or [Cl2(t)]. To the right are some data for [NO(t)] as a function of time when Cl2 is present in large excess. (a) Determine the order of reaction with respect to NO from these data. (b) From these data alone, what can you say about the order with respect to Cl2? (c) Explain how one would determine the rate constant for this reaction.

Time (s)

0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

NO Conc. 0.00049 0.00038 0.000321 0.000271 0.000239 0.000206 0.000189 0.000172 0.000152 0.000145 0.000132

8. (10 points) Assume an ideal-gas phase of reactive molecules with diameter, d, and average relative velocity 8𝑘𝑘𝐵𝐵 𝑇𝑇 𝜋𝜋𝜋𝜋 where the mass of the molecule is m, and the remainder of the symbols have their standard meanings. Give an expression for the encounter-limited rate constant for a dimerization reaction. 𝑣𝑣𝑟𝑟𝑟𝑟𝑟𝑟

=

�

9. (10 points) Consider the reaction of nitric oxide with oxygen: 2 𝑁𝑁𝑁𝑁 + 𝑂𝑂2 → 2 𝑁𝑁𝑂𝑂2 The rate law for this reaction is found to be: 𝑑𝑑[𝑁𝑁𝑁𝑁] −� � = 𝑘𝑘 [𝑁𝑁𝑁𝑁]2 [𝑂𝑂2 ] 𝑑𝑑𝑑𝑑 What is interesting about this reaction is that, experimentally, the rate constant is found to decrease with increasing temperature. Propose a reasonable mechanism that would explain both the orders and the observed negative Arrhenius activation energy.

10. (15 points) Langmuir-Hinshelwood kinetics involves reaction of molecules on a surface after adsorption. (a) Derive the Langmuir absorption isotherm for a single species, in equilibrium with a gas phase at a pressure, P, by obtaining an expression for the fractional coverage, θ. Be sure to explain any assumptions. Let the adsorption and desorption rate constants be ka and kd, respectively. (b) Do an equivalent derivation for two gas-phase species, with partial pressures, PA and PB, respectively. Let the rate constants for adsorption be kAa and kBa, respectively. Let the desorption rate constants be kAd and kBd, respectively. Find expressions for θA and θB, the fractions of sites covered by the two different molecules in terms of the rate constants and the partial pressures. (c) If the rates of adsorption and desorption are fast (i.e. the reactive step on the surface is rate-determining), write an expression for the rate of reaction.

Physical Chemistry Cumulative Exam Spectroscopy April 9, 2016

Answer 10 questions. All questions weighted equally.

1. Define the transition dipole moment in terms of the wave functions for the initial and final states and the components of the dipole moment operator. Explain briefly the connection between the transition dipole moment and spectroscopic selection rules.

2. A series of lines in the spectrum of atomic hydrogen lies at 656.46 nm, 486.27 nm, and 410.29 nm. What is the wavelength of the next line in the series? What is the ionization energy of the atom when it is in the lower state of the transitions.

3. The characteristic emission from K atoms when heated is purple and lies at 770 nm. On close inspection, the line is found to have two closely spaced components, one at 766.7 nm and the other at 770.1 nm. Account for this observation, and deduce what information you can.

4. Rotational absorption lines for 1H35Cl were found at the following wavenumbers: 83.2, 104.13, 124.73, 145.37, 165.89, 186.23, 206.60, and 226.86 cm-1. Calculate the moment of inertia and bond length of the molecule. Predict the positions of the corresponding lines in 2H35Cl.

5. Predict the shape of the nitronium ion. NO2+, from its Lewis structure and the VSEPR model. It has one Raman active vibrational mode at 1400 cm-1, two strong IR active modes at 2360 cm-1 and 540 cm-1, and one weak IR mode at 3735 cm-1. Are these data consistent with the predicted shape of the molecule? Assign the vibrational wavenumbers to the modes from which they arise. 6. The 1H35Cl molecule is quite well described by the Morse potential with De = 5.33 eV, ν=2989.7 cm-1,and xeν=52.05 cm-1. Assuming that the potential is unchanged on deuteration, predict the dissociation energies (Do) of (a)1H35Cl and (b) 2H35Cl. 7. Consider the molecule H2O. (a)To what point group does the molecule belong: (b)How many normal modes of vibration does the molecule have? (c) What are the symmetries of the normal modes of vibration for this molecule? (d)Which of the vibrational modes of this molecule are infrared active: (e)Which of the vibrational modes of this molecule are Raman active?

8. The vibrational wavenumber of the oxygen molecule in its electronic ground state is 1580 cm-1, whereas that in the first excited state (B 3Σu-), to which there is an allowed electronic transition, is 700 cm-1. Given that the separation in energy between the minima in their respective potential energy curves of these two

electronic states is 6.175 eV, what is the wavenumber of the lowest energy transition in the band of transitions originating from the v=0 vibrational state of the electronic ground state to this excited state: Ignore any rotational structure or anharmonicity. 9. The homonuclear diatomic halogen anions have 2Σu+ ground states. Each anion has excites states with term symbols 2Πg, 2Πu, and 2Σg+. To which excited states are transitions allowed from the ground state.

10. A certain molecule fluoresces at a wavelength of 400 nm with a half-life of 1.0 ns. It phosphoresces at 500 nm. If the ratio of the transition probabilities for stimulated emission for the S*  S to the TS transitions is 1.0 x 105, what is the half-life of the phosphorescent state? 11. The photoelectron spectrum of NO can be described as follows. Using He 584 pm (21.21eV) radiation there is a single strong peak at photoelectron kinetic energy 4.69 eV and a long series of 24 lines starting at 5.56 eV and ending at 2.2 eV. A shorter series of six lines begins at 12.0 eV and ends at 10.7 eV. Account for this spectrum.

12. The photoelectron spectrum of H2O was obtained with 21.21 eV He radiation. A band near 7.0 eV shows a long vibrational series with spacing 0.125 eV. The bending mode of H2O lies at 1596 cm-1. What conclusions can you drawabout the characteristics of the orbital occupied by the photoelectron.

13. The Hamiltonian for a system of two sets of non-interacting spins is given below in terms of the magnetogyric ratio, γ, the applied magnetic field B and the shielding constants for the two sets of spins, σ1 and σ2. Find the eigenvalues of the Hamiltonian, E1 and E2, for the two spin functions ψ1 = α(1)α(2) and ψ2 = α(1)β(2). 14. Predict the chemically shifted 1H peaks and the multiplet splitting of each peak you would observe in the proton nmr spectrum of diethyl ether. 15. Using the matrix representations:

Find the eigenvalues of α and β with respect to

.

Name _________________________________________________________________ Physical Chemistry Cumulative Exam Quantum Mechanics March 12, 2016 Please place a completed copy of the exam in Dr. Polenova’s mailbox by 12 noon Tatyana Polenova, Principal Examiner Total points: 100 I. Basic Concepts (40 pts total) 1. Calculate the de Broglie wavelengths for (21 pts, 7 pts each): a) an electron with a kinetic energy of 100 eV

b) a proton with a kinetic energy of 100 eV

c) an electron in the first Bohr orbit of a hydrogen atom

1

Name _________________________________________________________________ 2. (9 pts) If we locate an electron to within 20 pm, what is the uncertainty in its speed?

3. (10 pts) Plank’s distribution law (see below) gives the radiant energy density of electromagnetic radiation emitted between ν and ν+Δν. Integrate the Planck distribution over all frequencies to obtain the total energy emitted. What is the temperature dependence? Do you know whose law this is? 𝑑𝜌 𝜈, 𝑇 =

8𝜋ℎ 𝜈 ! 𝑑𝜈 𝑐 ! 𝑒 !! !! ! − 1

You will need to use the integral

! ! ! !" ! ! ! !!

!!

= !".

2

Name _________________________________________________________________ II. Quantum Mechanics of Simple Systems; Hydrogen Atom (30 pts total) 4. (10 pts) Show that the normalized wave function for a particle in a three-dimensional box with sides of length a, b, and c is 𝛹 𝑥, 𝑦, 𝑧 =

8 𝑎𝑏𝑐

! !

𝑠𝑖𝑛

𝑛! 𝜋𝑦 𝑛! 𝜋𝑥 𝑛! 𝜋𝑧 𝑠𝑖𝑛 𝑠𝑖𝑛 𝑎 𝑏 𝑐

3

Name _________________________________________________________________ 5. (10 pts) Show explicitly that Ĥ𝜓 = −

𝑚! 𝑒 ! 𝜓 8𝜀!! ℎ!

for a ground state of a hydrogen atom

6. (10 pts) Show that rotational transitions of a diatomic molecule occur in the microwave region of the far infrared region of the spectrum.

4

Name _________________________________________________________________ III. Multielectron Atoms, Bonding in Polyatomic Molecules (15 pts total, 8 + 7 pts) 7. Show that the two-term helium Hartree-Fock orbital 𝜙 𝑟 = 0.81839𝑒 !!.!!"#$! + 0.52072𝑒 !!.!"###! is normalized.

5

Name _________________________________________________________________ 7. Calculate the delocalization energy, the charge on each carbon atom, and the bond orders for the allyl radical, cation, and anion. Sketch the molecular orbitals for the allyl system.

6

Name _________________________________________________________________ IV. Physical Chemistry Seminars (15 pts total, 3 pts each) For the most recent Physical Chemistry colloquium that took place on February 15, 2016, answer the following questions: a) What was the names of the speaker?

b) What was the speaker affiliation? (Institution and Department)

c) What was the topic of the seminars and the scientific questions asked?

d) What were the techniques used to address the specific scientific questions? (list all methods that were discussed during the seminar)

e) What were the main conclusions of the colloquium lecture?

7

Name _________________________________________________________________

(T=298.15 K) H 2O + H OH

∆Hf (kJ/mol) -285.83 0 -229.83

8

º

S (J/K-mol) 69.995 0 -10.753

Name _________________________________________________________________ Essential formulae and expressions.

9

Name _________________________________________________________________ Essential formulae and expressions. (con’d)

10

Name _________________________________________________________________ 3

1 ! 1 $ 2 − r a0 n = 1, l = 0, ml = 0 ψ100 ( r ) = # & e π " a0 % 3

1 ! 1 $ 2! r $ − r 2a n = 2, l = 0, ml = 0 ψ 200 ( r ) = # & #2 − &e 0 a0 % 4 2π " a0 % " 1 n = 2, l = 1, ml = 0 ψ 210 ( r, θ , φ ) = 4 2π

3

! 1 $ 2 ! r $ − r 2a # & # & e 0 cosθ " a0 % " a0 % 3

1 ! 1 $ 2 ! r $ − r 2a0 n = 2, l = 1, ml = ±1 ψ 211 ( r, θ , φ ) = sin θ e±φ # & # &e 8 π " a0 % " a0 % 3

1 ! 1 $ 2 ! r $ − r 3a0 n = 3, l = 2, ml = 0 ψ320 ( r, θ , φ ) = 3cos2 θ −1) # & # 2 &e ( 81 6π " a0 % " a0 % π

N 2 ∫ ψ * (τ )ψ (τ ) dτ = 1;

2π

∞

* N 2 ∫ sin θ dθ ∫ d φ ∫ ψ n,l,m ( R, θ, φ ) ψn,l,ml ( R, θ, φ ) = 1 l 0

0

0

3

1 # 1 & 2 # r & − r 2a0 ψ 2 px ( r, θ , φ ) = sin θ cos φ % ( % (e 4 2π $ a0 ' $ a0 ' 3

1 # 1 & 2 # r & − r 2a0 ψ 2 py ( r, θ , φ ) = sin θ sin φ % ( % (e 4 2π $ a0 ' $ a0 ' 3

1 # 1 & 2 # r & − r 2a0 ψ 2 pz ( r, θ , φ ) = cos φ % ( % (e 4 2π $ a0 ' $ a0 ' 3

2 ψ3 px ( r, θ , φ ) = 81 2π

# 1 & 2 # r r 2 & − r 3a % ( % 6 − 2 ( e 0 sin θ cos φ $ a0 ' $ a0 a0 '

2 ψ3 py ( r, θ , φ ) = 81 2π

# 1 & 2 # r r 2 & − r 3a % ( % 6 − 2 ( e 0 sin θ sin φ $ a0 ' $ a0 a0 '

3

3

2 # 1 & 2 # r r 2 & − r 3a0 ψ3 pz ( r, θ , φ ) = cos φ % ( %6 − (e 81 2π $ a0 ' $ a0 a02 '

11

Physical Chemistry Cumulative Examination Topic: Thermodynamics February 13, 2016 University of Delaware Department of Chemistry and Biochemistry Primary Examiner: Sandeep Patel Secondary Examiner: Lars Gundlach

This is the Spring 2016 Thermodynamics Cumulative Exam for the Physical Chemistry Division. Please answer three (3) out of the six (6) questions to the best of your ability. Please make sure that your answer is clearly stated in the solution, and that all steps in your solution process are understandable and explicit. State all assumptions you invoke. Please leave your answers in the Blue Book. It may be helpful to first read all problems before trying to solve any. Good Luck. 1. The linear, 1-dimensional quantum harmonic oscillator is a fundamental model of chemical bonds as well as diatomic molecules. Consider this simple example of a quantum harmonic oscillator. The Hamiltonian is given by:

!2 ∂ 2 1 ˆ H =− + µω 2 x 2 2 2µ ∂x 2 The eigenvalues of H are:

€

⎛ 1 ⎞ E n = ⎜n + ⎟!ω ⎝ 2 ⎠

n = 0,1, 2,.....

A. What is the canonical ensemble partition function, Q(T), for this system? B. Provide the expression€ for the Helmholtz Free Energy.

C. Provide an expression for the average energy, given by E

D. What is the entropy of the oscillator?

€

= H =−

∂ (ln [Q(T )]) . ∂β

2.  For a system at constant N (particle number), volume (V), and temperature (T). The ideal gas partition function, Utotal (r N ) = 0 , is: 3N /2

Z IG (N,V,T ) =

1 ! 2π mkBT $ # & N! " h 2 %

VN

N is the number of particles, ‘m’ is the mass of each particle (all are same), ‘h’ is Planck’s constant, V is the volume, and T is temperature. There are no interactions between ideal gas particles, hence Utotal (r N ) = 0 . For a gas whose individual particles have a molecular volume ‘b’, the volume available is (V-Nb). Moreover, we will allow for any given particle to ‘feel’ an attractive ‘force’ from all other particles taken collectively as some average ‘thing’ capable of exerting a ‘force’ (this is referred to as a mean-field approximation, another very, very common approximation in the physical sciences and engineering; it allows us to simply a many-body problem into an effective single-body problem! The internal energy of this mean-field model is:

" N2 % Utotal (r N ) = −a $ ' #V & where the constant ‘a’ represents the interaction strength and (N/V) is the particle density of the system. Taking into account the particle properties ‘a’ and ‘b’, the canonical ensemble partition function becomes: 3N /2

Z(N,V,T ) =

1 ! 2π mkBT $ # & N! " h 2 %

( N 2a + (V − Nb) N exp * ) VkBT ,

2A Based on the mean-field partition function Z(N,V,T), what is the equation of state of the gas this represents?

2B. What is the Helmholtz Free Energy?

2C. What is the internal energy, U?

             

3. Physical particle adsorption onto surfaces is an often-invoked initial step in catalytic reaction mechanisms. Gas-phase particles must ‘condense’ onto the surface of the solid phase (where catalysis can occur). Adsorption does not involve true covalent (chemical) bonding, but strong non-bonding interactions (such as dispersion, van der Waals, dipoledipole, and the like); hence it is often referred to as physical adsorption. The right panel of the Figure shows particles adsorbing onto a model surface. Here we consider a simple model of adsorption of spherical particles from gas onto the top layer of a model twodimensional solid surface described as a lattice of adsorption sites; the particles adsorb up to a maximum single layer (monolayer). The lattice represents the average position of particles on the surface effectively taking into account the dynamics (i.e. average motions of adsorbed particles) so that explicit consideration of dynamic/kinetic effects on the surface is not needed. This model is the two-dimensional analogue to a constant N, V, T system (constant particles, volume, and temperature).

                                                                            3A. The surface has ‘A’ total adsorption lattice sites (‘A’ total sites where a gas-phase particle can ‘stick’ to). One particle can occupy one lattice site. If ‘N’ indistinguishable particles adsorb onto the surface lattice, what is the entropy of the state with ‘N’ adsorbed particles in the limit as N ! large, A! large (under these conditions, the surface density of particles, θ = N/A, remains constant and finite as we have seen before with bulk density when considering three-dimensional lattice models of bulk fluids)? Provide your answer with an expression that involves θ.

 

3B. The interaction energy of an adsorbing particle with the surface lattice site is ‘w’ (w < 0). For ‘N’ particles adsorbed to the surface, what is the total internal energy, U? Here we only consider contributions from the adsorption process (lattice site and adsorbing particle only), neglecting any contributions to total internal energy from interactions between solid atoms or between adsorbed particles.

3C. What is the Helmholtz Free Energy of the state with ‘N’ adsorbed particles? Provide your answer with an expression that involves θ.

3D. What is the chemical potential of the ‘N’ adsorbed particles? Provide your answer with an expression that involves θ.

3E. If the total pressure of the gas particles above the surface is ‘P’, and the gas is ideal, what is the chemical potential of the gas-phase particles at pressure ‘P’ and temperature ‘T’? You are free to define the reference state ideal gas chemical potential to be 0 at this temperature, ‘T’.

3F. Applying the idea that the ‘condensation’ of gas-phase particles onto the surface is a like a phase transition, or that the gas and the adsorbed state of particles are two phases, determine a relation that connects the pressure of the gas, ‘P’, to the number of particles adsorbed onto the surface, ‘N’, and the number of surface sites, ‘A’? Provide your answer with an expression that involves θ. The results if this analysis leads to the famous Langmuir adsorption isotherm, which relates surface coverage to pressure. Adsorption isotherms are important in the field of solid state heterogeneous catalysis, semiconductors, and electronics.

                   

€

4. Consider adiabatic compression of air and helium from 1 atmosphere to 10 atm. Initial temperature is 300K. What is the final temperature after compression?

γ air =

Cp

C v = 1.4

; γ helium =

Cp

C v = 5.3

5. For a diatomic ideal gas near room temperature, what fraction of the heat supplied is available for external work if the gas is expanded at constant pressure? At constant temperature?

6. Draw the Carnot Cycle on a graph with x-axis being volume and y-axis being pressure. The steps in a Carnot process include isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression back to the original state.

Page |1 Physical Chemistry Cumulative, November, 2015

Name:____________________________

Quantum Mechanics November 14th, 2015 Principal Examiner: L. Gundlach Second Examiner: D. Ridge

Answer question worth 100 points of the possible 140 points. Clearly indicate which questions you want graded. Otherwise I will grade the first 100 points.

Page |2 Physical Chemistry Cumulative, November, 2015 1. (20 points) The Balmer Series of the H atom is measured with a spectrometer with a resolution

𝜆𝜆 Δ𝜆𝜆

resolvable?

= 5 ∗ 105 . Up to which principal quantum number n are two neighboring lines

Page |3 Physical Chemistry Cumulative, November, 2015

1 𝑟𝑟

2. (20 points) Calculate the expectation values 〈𝑟𝑟〉 and 〈 〉 for the 1s and 2s state in a hydrogen atom. Compare the result to the Bohr radius.

Page |4 Physical Chemistry Cumulative, November, 2015 � , 𝑥𝑥��, and �𝐻𝐻 � , 𝑝𝑝̂ �. You might need the Taylor series 3. (20 points) Calculate the commutators �𝐻𝐻 2 expansion F(x) = F(x=0)+F’x+1/2 F’’ x +… . Discuss the result in terms of classical observables.

Page |5 Physical Chemistry Cumulative, November, 2015

4. (20 points) Consider the spin function for a 3 electron system. Demonstrate that 𝛼𝛼(1)𝛼𝛼(2)𝛼𝛼(3) is an eigenfunction of 𝑆𝑆̂ 2 and 𝑆𝑆̂𝑧𝑧 and determine the corresponding eigenvalues. Ladder operators 𝑆𝑆̂± = 𝑆𝑆̂𝑥𝑥 ± 𝑖𝑖𝑆𝑆̂𝑦𝑦 , and the identities 𝑆𝑆̂± = 𝑆𝑆̂1± + 𝑆𝑆̂2± + 𝑆𝑆̂3± and 𝑆𝑆̂𝑧𝑧 = 𝑆𝑆̂1𝑧𝑧 + 𝑆𝑆̂2𝑧𝑧 + 𝑆𝑆̂3𝑧𝑧 might be helpful.

Page |6 Physical Chemistry Cumulative, November, 2015 5. (25 points) The 2 dimensional harmonic oscillator in the xy plane is perturbed by the term 𝐻𝐻 ′ = 𝑎𝑎𝑎𝑎𝑎𝑎. The full Hamiltonian reads: 𝐻𝐻 =

1 �𝑝𝑝𝑥𝑥2 2𝑚𝑚

𝑘𝑘 2

+ 𝑝𝑝𝑦𝑦2 � + (𝑥𝑥 2 + 𝑦𝑦 2 ) + 𝑎𝑎𝑎𝑎𝑎𝑎

a) Show that the problem can be separate into two equations for a=0. b) List the first 3 energy levels and their degeneracy (a=0).

c) Use degenerate perturbation theory to calculate the splitting in the first degenerate state for small a. This involves solving a 2x2 secular equation.

Page |7 Physical Chemistry Cumulative, November, 2015 6. (25 points) Explain how a Hartree-Fock calculation of He is performed. Explain the requirements for the wavefunction and write down the relevant Hamilton operator and integrals without solving them. Explain how the method works.

Page |8 Physical Chemistry Cumulative, November, 2015 7. (10 points) You walk into a pizza shop. They sell three sizes of pizza: small, medium, and large. All are perfectly circular, have the same thickness, and density of toppings. The price of the large pizza equals the sum of the small and the medium: PL=PM+PS. You see a group of your friends already sitting at a table in the pizza shop with one of each size pizza In front of them and they have a perfect square box for the leftovers on the table. Each of the three pizzas is cut in perfect sixths. Now you are trying to decide if you are buying one large pizza or one small plus one medium. The price is the same. How can you figure out which choice gives the better value using only what is placed on the table.

Page |9 Physical Chemistry Cumulative, November, 2015

[Grab your reader’s attention with a great quote from the document or use this space to emphasize a key point. To place this text box anywhere on the page, just drag it.]

2𝑝𝑝𝑥𝑥 =

1

√2 𝑖𝑖

(2𝑝𝑝−1 + 2𝑝𝑝1 )

(2𝑝𝑝−1 − 2𝑝𝑝1 ) √2 2𝑝𝑝𝑧𝑧 = 2𝑝𝑝0 2𝑝𝑝𝑦𝑦 =

P a g e | 10 Physical Chemistry Cumulative, November, 2015

Spherical Polar Coordinates: 𝑥𝑥 = 𝑟𝑟 sin 𝜃𝜃 cos 𝜙𝜙 , 𝑦𝑦 = 𝑟𝑟 sin 𝜃𝜃 sin 𝜙𝜙 , 𝑧𝑧 = 𝑟𝑟 cos 𝜃𝜃 𝑧𝑧 𝑟𝑟 2 = 𝑥𝑥 2 + 𝑦𝑦 2 + 𝑧𝑧 2 , cos 𝜃𝜃 = , 𝑟𝑟

𝑑𝑑𝑑𝑑 = 𝑟𝑟 2 sin 𝜃𝜃 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑,

tan 𝜙𝜙 =

𝑦𝑦 𝑥𝑥

0 ≤ 𝑟𝑟 ≤ ∞, 0 ≤ 𝜃𝜃 ≤ 𝜋𝜋, 0 ≤ 𝜙𝜙 ≤ 2𝜋𝜋

----------------------------------------------------------------------------------------------------------------

-----------------------------------------------------------------------------------------The harmonic oscillator:

Physical Chemistry Cumulative, October, 2015

page 1

SPECTROSCOPY October, 2015 Principal Examiner: C. R. Dybowski Second Examiner: T. Polenova

Work questions worth 100 points from this examination. If you answer more than 100 points, only the first 100 points, in order, will be graded. Do NOT mark any marks on questions that you do not want graded! Question

Points

1 2 3 4 5 6 7 8 9 10 Total

SOME USEFUL DATA AND CONSTANTS Avogadro’s constant, L 6.0221367×1023 molecules Bohr magneton, µB 9.274015×10-24 J/T Bohr radius. a0 5.2917725×10-11 m Boltzmann’s constant, kB 1.380658×10-23 J/K Debye, D 3.33564×10-30 C-m Electron charge, e 1.60217733×10-19 C Electron rest mass, me 9.1093897×10-31 kg Faraday constant, 96485 C

Gas constant, R Acceleration of gravity, g Hartree, Eh Speed of light in vacuo, c Permittivity of vacuum, ε0 Planck’s constant, h Proton rest mass, mp

8.31451 J/K 9.80665 m sec-2 4.3597482×10-18 J 2.99792458×108 m sec-1 8.8541878×10-12 C2 J-1 m-1 6.620755×10-34 J sec 1.67623×10-27×10-11 kg

Physical Chemistry Cumulative, October, 2015

page 2

1. (15 points) Selection rules for dipole transitions involve evaluation of integrals. The dipole moment operator being d, the transition-moment integral is

< n | qr | k > = < n | d | k >

where q is a charge and r is the position vector describing the dipole moment. Consider the vibrational states of the harmonic oscillator, {| n >} . Knowing that these harmonic-oscillator states are proportional to Hermite polynomials, prove (Do not just state.) the general selection rule for vibrational transitions of a harmonic oscillator between states with arbitrary quantum numbers n and k.

Physical Chemistry Cumulative, October, 2015

page 3

2. (15 points) Describe each normal coordinate of CO2 in terms of the positions of the three atoms, using the coordinate system shown (with the y axis coming directly out of the page and assuming the origin at the equilibrium position of the carbon atom. (You should write each mode in terms of the deviation, qi = ri - ri,eq of each atom involved. Give the symmetry designation of each normal mode in the point group of CO2.

x z

y O2

C

O1

Physical Chemistry Cumulative, October, 2015

page 4

3. (10 points) 2,2’-diphenylpicrylhydrazyl (DPPH) is a common standard material for EPR spectroscopy. Its structure is shown in the figure. The ESR spectrum of this stable radical is shown in the adjacent figure.

NO2 N

N

O2N NO2 Explain the spectrum of DPPH in terms of its structure and its effect on the spectroscopy. Be brief but complete.

Physical Chemistry Cumulative, October, 2015

page 5

4. (10 points) In 1965, James Cooley and John Tukey made a contribution that revolutionized the way infrared spectroscopy and NMR spectroscopy are performed. What was that contribution, and why was it so important?

Physical Chemistry Cumulative, October, 2015

page 6

5. (15 points) (a) Which of the following electronic transitions are allowed? [Indicate by checking the appropriate box.] (b) For each transition, briefly explain why or why not in the space immediately below it. (c) For each allowed transition, give those molecules of the following to which the transition could apply: HCl, DCl, H2O, BF3, N2O, N2, O2. (a)

1Σ + g

↔ 1Πu

 Allowed.

 Not Allowed.

(b)

1A 1g

↔ 1B2u

 Allowed.

 Not Allowed.

(c)

1Σ

 Allowed.

 Not Allowed.

(d)

1Σ + g

 Allowed.

 Not Allowed.

↔ 1∆

↔ 1Πg

Physical Chemistry Cumulative, October, 2015

page 7

6. (10 points) Consider the NMR spectra of bromodifluoromethane at natural abundance. [In the following parts, be sure to indicate parameters (by scales or indications of some kind) on all spectra.] (a) Sketch the 1H NMR spectrum of this material.

(b) Sketch the 13C NMR spectrum obtained when high-power decoupling is applied to the protons.

Physical Chemistry Cumulative, October, 2015

page 8

7. (10 points) When gases are expanded adiabatically into a vacuum they cool to very low temperatures. I2 (Be = 0.052 cm-1) is expanded into a vacuum and the infrared spectrum shows several lines. If the line of maximum emission corresponds to molecules that start in the state with J" = 3, what is the approximate temperature after expansion, assuming all degrees of freedom are in thermal equilibrium?

Physical Chemistry Cumulative, October, 2015

page 9

8. (15 points) For each spectroscopy listed below, give the common source of electromagnetic energy used in a spectrometer. (a)

Electron spin resonance _____________________________________________________________

(b)

Infrared spectroscopy _______________________________________________________________

(c)

Visible spectroscopy_________________________________________________________________

(d)

EXAFS _________________________________________________________________________

(e)

Moessbauer spectroscopy __________________________________________________________

Physical Chemistry Cumulative, October, 2015

page 10

9. (10 points) The pure rotational Raman spectrum of oxygen gas is shown in the figure (L. C. Hoskins, J. Chem. Ed., 1975). (a) What is the selection rule for pure rotational Raman spectroscopy?

(b) From the spectrum, estimate the equilibrium rotational constant for oxygen.

(c) Why would one do Raman spectroscopy rather than infrared spectroscopy on oxygen?

Physical Chemistry Cumulative, October, 2015

page 11

10. (10 points) Who won the Nobel Prize in Chemistry for 2015? At what institutions did the scientists do the work for which they were given the Nobel Prize? For what specific work were they cited? Explain in two or three sentences why it is important. This prize was accompanied by some minor controversy. Explain in a sentence or two what the controversy was.

Physical Chemistry Cumulative Examination September 12, 2015 Major Examiner: D. P. Ridge Work 100 points worth of questions: 1. (15) For the mechanism below find A and B as a function of time. Assume that at t=0, [A]=a and [B]=0. Assume that both processes are first order and the rate constants are equal. k1 A B k2 2. (20 points) Equal volumes of two equimolar solutions of reactants A and B are mixed, and the reaction A + B  C occurs. At the end of 1 hour, A is 30% reacted. How much of A will be left unreacted at the end of 2 hours if the reactions is: (a) First order in A and zero order in B? (b) First order in B and zero order in A? (c) Zero order in both A and B? (d) First order in A and one-half order in B? 3. (20 points) Find the rate law for the chain reaction below. You may use the steady state and long chain approximations as needed. Consider H2 and C2H6 to be trace products. k1 CH3CHO

CH3 + CHO k2

CHO + CH3CHO

CO + H2 + CH3CO k3

CH3 + CH3CHO

CH4 + CH3CO k4

CH3CO

CH3 + CO k5

2CH3

C2H6

4. (10 points) Find the relationship between the rate constants and the relaxation time for the return to equilibrium for the following reaction: k1 A

B + C k2

5. (15 points). Given the equation below and the Arrhenius equation (k = Aexp(EA/RT)), Define the relationship between the entropy and enthalpy of activation and the parameters A and EA (note: ∆G  is the free energy of activation).  kT k = b e − ∆G / RT h (b)Which of the following reactions should have the largest pre-exponential factor and why?

A

k1 TS I

B

k2 TSII

C

Transition State

Frequencies

Energy Barrier

TSI TSII

3150, 2100, 2100, 570 3100, 2000, 1900, 300

1.05 eV 1.10 eV

6. (10 points) Given that ΔG = PΔV at constant T and P: (a) Find the relationship between the ln(k) (where k is the rate constant) and the volume of activation ∆Vǂ. (b) For the dimerization of cyclopentadiene in n-butyl chloride ∆Vǂ = -22.0 cm3/mol. Will the reaction rate increase or decrease with pressure? (c) At T=298K what pressure will the reaction rate differ from that at 1 atm by 10%? Assume ∆Vǂ to be independent of pressure. 6. (10 points). Define a linear free energy relationship and give an example. 8. (15 points)The decomposition of N2O (1Σ linear ground state) to N2 (1Σ ground state) and O (3P ground state) is important in atmospheric chemistry. The activation energy for the reaction is 60.6 kcal/mol, but the thermodynamic bond strength is 39.7 kcal/mol. (a)Suggest an explanation for the difference. (b)Given that the lowest singlet state of the O atom (1D) is 45.4 kcal/mol above the ground state sketch a potential energy surface for the N2O reaction and indicate the diabatic and adiabatic reaction pathways. 9. (10 point) Make a topographical sketch of a potential energy surface with the barrier in the entrance channel for a three atom reaction: A + BC  AB + C. Label the axes in terms of the interatomic distances. Let the axis perpendicular to the paper be the energy axis. Label the transition state. Trace a typical reaction trajectory. With the barrier in the entrance channel is this reaction more likely to be endothermic or exothermic? Are the products more likely to be vibrationally hot or translationally hot?

10. (10 points) A particular surface is exposed to evaporated metal atoms. The probability that a surface site is covered by a layer N atoms deep can be calculated from PN = (n)Ne−n/N! where n is the average number of atoms per site. For what value of n are 90% of the substrate atoms covered by at least one metal atom (P0 = 0.1)? Is it possible to ignore the substrate in interpreting the chemistry on this surface if n = 2? 12. (15points) Derive the analogue to the Lineweaver-Burke equation for the enzyme catalyzed reaction between A and B with the following mechanism. k1 A + E E∙A k-1 E∙A + B

k2 Products

Physical Chemistry Cumulative Examination April 11, 2015 Major Examiner: D. P. Ridge Work 100 points worth of questions: 1. (20) Find B as a function of time for the autocatalytic reaction below. Assume at time t=0, A=a and B=b. A + B



2B

2. (10 points) A 10g rock sample is found to produce radioactive decay at a rate of 1000 counts per second. Assume the decay is from 238U which has a half life of 4.5 x 109 years. What is the weight per cent concentration of 238U in the sample? 3. (20 points) Find the rate law for the chain reaction below. You may use the steady state and long chain approximations as needed. Consider H2 and C2H6 to be trace products. k1 CH3CHO

CH3 + CHO k2

CHO + CH3CHO

CO + H2 + CH3CO k3

CH3 + CH3CHO

CH4 + CH3CO k4

CH3CO

CH3 + CO k5

2CH3

C2H6

4. (10 points) Define the quantities in the Arrhenius equation for the T dependence of the rate constant, k=Ae–Ea/RT. Draw a diagram indicating the relationship between Ea for the forward and reverse reactions and ∆Eo, the thermodynamic energy change for the reaction.

5. (15 points). Given the equation below and the Arrhenius equation (k = Aexp(EA/RT)), Define the relationship between the entropy and enthalpy of activation and the parameters A and EA (note: ∆G  is the free energy of activation).  kT k = b e − ∆G / RT h (b)Which of the following reactions should have the largest pre-exponential factor and why?

A

k1 TS I

B

k2 TSII

C

Transition State

Frequencies

Energy Barrier

TSI TSII

3150, 2100, 2100, 570 3100, 2000, 1900, 300

1.05 eV 1.10 eV

6. (10 points). Define a linear free energy relationship and give an example. 7. (15 points). Derive the equation for the primary salt effect using transition state theory. (Hint: logγi = -0.51zi2I1/2 and I = 1/2∑ciZi2). 8. (15 points)The decomposition of N2O (1Σ linear ground state) to N2 (1Σ ground state) and O (3P ground state) is important in atmospheric chemistry. The activation energy for the reaction is 60.6 kcal/mol, but the thermodynamic bond strength is 39.7 kcal/mol. (a)Suggest an explanation for the difference. (b)Given that the lowest singlet state of the O atom (2D) is 45.4 kcal/mol above the ground state sketch a potential energy surface for the N2O reaction and indicate the diabatic and adiabatic reaction pathways. 9. (10 point) Make a topographical sketch of a potential energy surface with the barrier in the entrance channel for a three atom reaction: A + BC  AB + C. Label the axes in terms of the interatomic distances. Let the axis perpendicular to the paper be the energy axis. Label the transition state. Trace a typical reaction trajectory. With the barrier in the entrance channel is this reaction more likely to be endothermic or exothermic? Are the products more likely to be vibrationally hot or translationally hot? 10. (10 points) A particular surface is exposed to evaporated metal atoms. The probability that a surface site is covered by a layer N atoms deep can be calculated from

PN = (n)Ne−n/N! where n is the average number of atoms per site. For what value of n are 90% of the substrate atoms covered by at least one metal atom (P0 = 0.1)? Is it possible to ignore the substrate in interpreting the chemistry on this surface if n = 2? 12. (15points) Derive the analogue to the Lineweaver-Burke equation for the enzyme catalyzed reaction between A and B with the following mechanism. k1 A + E E∙A k-1 E∙A + B

k2 Products

Cumulative Examination in Physical Chemistry March 7, 2015 Dr. Teplyakov Quantum Mechanics This is a three-hour cumulative exam. IT IS CLOSED-BOOK. No additional materials beyond this sheet are needed or allowed to answer the questions. There are two parts in this examination. Both are closed-book. Part I covers very basic quantum mechanics problems. You must answer three out of four questions in this part. Beauty is in the eye of the beholder – choose wisely. You can draw a diagram if it helps you answer a question. Part II is based on developments in the field of chemistry and QM in particular. It is also closed-book. You must answer this question to complete the exam and have a shot at 100 points. The total will amount to 100 points, 25 points apiece. Once completed, you can place the exam in the envelop provided and slide it under the door of my office, 112 LDL. Good luck! Part I. Quantum mechanics and computational chemistry. 1) (25 points). In the computational world, there are many theoretical and computational ways to describe a model system. Compare advantages and disadvantages of HartreeFock, Møller-Plesset, and DFT models. Don’t write an essay, you can make up a table to help you answer this question. 2) (25 points). What is the main assumption of the Hartree-Fock approximation and in that context what is the significance of the Slater determinant? 3) (25 points). Within the DFT framework, atomic orbitals can be expanded in terms of Gaussian functions, which have a general form: g ijk (r ) = Nx i y j z k exp(−αr 2 ) Here x,y,z are position coordinates measured from the nucleus of an atom, i,j,k are non-negative integers, and α is an orbital exponent. Give an example of an s-type function and a p-type function generated based on this notation. You answers can contain unspecified α parameters, of course. 4) (25 points) Selecting an appropriate model for computational investigation is quite difficult. However, in many cases trial and error methods are supplemented with a comparison with experimental work. Let’s say that for a given set of models, it was found that 6-31++G** basis set is appropriate. What does it mean? Explain the meaning of every character in this notation. Part II. Nobel prizes in computational chemistry. 1) (25 points) The two most recently awarded Nobel prizes for achievements in computational chemistry were given to Arieh Warshel, Michael Levitt, and Martin Karplus (2013) and Walter Kohn and John A. Pople (1998). In a paragraph or two describe the major contributions for which these Nobel prizes were awarded.

CUMULATIVE EXAMINATION THERMODYNAMICS February, 2015 Dybowski/Polenova

SOME IMPORTANT RELATIONS:

dU = T dS - PdV + ∑i,α µ αi dnαi dH = T dS + V dP + ∑i,α µ αi dnαi dA = - S dT - P dV + ∑i,α µ αi dnαi

dG = - S dT + V dP + ∑i,α µ αi dnαi Work any 5 problems. Only the first five questions in order in the booklet will be graded, so be sure to only work five and ensure that they are the first five in the booklet. 1.

Dieterici's equation of state for a gas is:  -a  P ( V m - b) = R T exp   RT Vm

where a and b are Dieterici constants and R is the gas constant. (a) Derive explicit formulas for the dependence of the first two virial coefficients, B(T) and C(T), of a Dieterici gas on temperature in terms of the Dieterici constants, the temperature, and the gas constant. 2.

Using standard mathematical techniques and the definition of heat capacity at  ∂2 P   ∂ Cv  constant volume, show that:   = T  2  Show all work, beginning from an  ∂V T  ∂ T V obvious point. 3. One kg of iron (with a temperature-independent CP = 0.47 kJ/K-kg) at 100°C is placed into thermal contact with one kg of water (with a temperature-independent CP = 4.19 kJ/K-kg) at 0°C. Calculate the equilibrium temperature and the entropy change upon equilibration. State any assumptions you make. 5. At a second-order phase transition, the volume is continuous. Use this fact to derive a relationship for the pressure dependence of the transition temperature for a second-order phase transition. [The differential form of this relationship is known as Ehrenfest's equation, after Paul Ehrenfest.]

6. In the Handbook of Chemistry and Physics, one finds the following expression for the vapor pressure of a material: 0.05223 a log10 ( P /torr ) = b T From the table accompanying this equation, one has the following data: Substance

a

b

Argon (liquid)

6826.0

6.9605

Argon (solid)

7814.5

7.5741

Estimate the temperature and pressure at the triple point of argon and the enthalpies of sublimation, vaporization and fusion of krypton at that point. 7. The table below lists the standard free energies of formation of several hydrates of MgCl2 at 25°C. Which hydrate is the stable form in air at 25°C if the relative humidity is 80% [typical of Delaware in August]? The vapor pressure of water is 23.76 torr at 25°C. Explain in detail how you arrived at the answer. ∆fGΘ (kJ/mol)

Hydrate MgCl2

-592.33

MgCl2•H2O

-862.36

MgCl2•2H2O

-1118.5

MgCl2•4H2O

-1633.8

MgCl2•6H2O

-1278.8

8. The mean molar activity coefficients of NaCl in aqueous solution at 25°C are given in the table as a function of concentration. Conc. (mol/kg)

0.001

0.005

0.010

0.050

0.100

0.500

1.000

2.000

γ

0.966

0.929

0.904

0.823

0.778

0.682

0.685

0.671

Calculate the mean molar activity of an NaCl solution at each concentration. 9. Two important equations of statistical mechanics are the Ornstein-Zernicke equation and the Percus-Yevick equation. (a) What is being evaluated by solution of the Ornstein-Zernicke equation? (b) What did Ornstein and Zernicke posit that gave

rise to the equation? (c) What did Percus and Yevick do to attempt a solution?

Page |1 Physical Chemistry Cumulative, December, 2014

Name:____________________________

Spectroscopy December 12th, 2014 Principal Examiner: L. Gundlach Second Examiner: T. Polenova

Answer question worth 100 points of the possible 140 points. Clearly indicate which questions you want graded and which ones not. Otherwise I will grade the first 100 points.

Page |2 Physical Chemistry Cumulative, December, 2014 1. (15 points) The following 1H-NMR spectrum is measured for a substance with the formula C8H10. What compound is this? Draw the structural formula. (The right spectra are magnifications of the multiplets on the left).

P a g e  | 3  Physical Chemistry Cumulative, December, 2014    2. (5 points) What mechanisms lead to a line‐broadening in vibrational‐rotational gas phase  spectra?                    3. (25 points) Natural HCl is composed of 75% H35Cl and 25% H37Cl. The vibrational‐rotational  spectrum of natural HCl is thus composed of two line systems.  a) Give the frequency of the first line in the R‐branch for H37Cl?  b) Calculate the difference in the fundamental vibrational frequency  same force constant 



 assuming the 

481  for both molecules. (This is a small effect. Use at least 4 

decimal digits).  c) Calculate the difference in bond length for both species from the position of the first peak in  the R‐branch. (The unit are in THz and the comma is the decimal mark, i.e. the center of the  spectrum is somewhere around 86THz. Useful equations are given at the end. Use Bn=Be=B  for the rotational constant). 

          

P a g e  | 5  Physical Chemistry Cumulative, December, 2014    4. (20 points) The UV/Vis spectrum of a solution of substance A in water with a concentration of   3 ∙ 10

 is measured in a cuvette of length 7.5

.  The transmission at 250 nm is 44.75%. 

a) Calculate the extinction coefficient of substance A. (The extinction coefficient is proportional  to the logarithm of the ratio of applied light intensity to transmitted light intensity.)  b) Substance B is added to the cuvette with a concentration of 3*10

 . The transmission 

at 250 nm drops to 35%. What is the Extinction coefficient of substance B? (You can neglect  the small change in volume.)                  

 

Page |6 Physical Chemistry Cumulative, December, 2014 5. (10 points) Derive the atomic term symbols for a p5 electron configuration of equivalent electrons. What term is the groundstate.

6. (10 points) What EPR spectrum do you expect from the benzene radical anion? Draw and explain. (Pascal’s triangles can be found at the end)

P a g e  | 7  Physical Chemistry Cumulative, December, 2014    7. (10 points) The following spectrum falls out of your lab journal when you pick it up.  With what method was this spectrum most likely taken? How does this method work? What  information can be gained from the spectrum?   

              8. (10 points) For the following light sources explain how EM radiation is generated and what  properties the respective radiation has, i.e. energy, bandwidth,… (a sketch can help):  a) Deuterium lamp        b) Laser diode            c) blackbody   

 

Page |8 Physical Chemistry Cumulative, December, 2014

9. (15 points) Fill in all configurations and terms that are missing in the Grotrian diagram of He below. Indicate 3 allowed and 3 forbidden transitions between the states by solid and dashed lines respectively.

P a g e  | 9  Physical Chemistry Cumulative, December, 2014  10. (10 points) What do you need to build an FTIR spectrometer? Sketch the optical beam path.  How does the raw data look like (draw a graph)? How do we generate a spectrum from this  (draw another graph)?   

P a g e | 10 Physical Chemistry Cumulative, December, 2014 11. (10 points) A king demands a tax of 1000 gold sovereigns from each of the 10 regions of his empire. The tax collectors for each region bring him the requested bag of gold at the end of the year. An informant tells the king that one of the tax collectors is cheating and is giving coins that are consistently 10% lighter than they should be, but he does not know which collector is cheating. The king knows that each coin should weight 1 ounce exactly. How can the king identify the cheat by using a weighing device exactly once?

P a g e | 11 Physical Chemistry Cumulative, December, 2014

Peak numbers and relative intensities for hyperfine coupling of equivalent nuclei of spin I.

Relative Intensities for I = ½

Relative Intensities for I = 1

Diatomic M olecules

The H armonic Oscillator Approximation n OJ e

=

where

µ

=

0,1, 2,3,4, ...

=

mAmB mA +ms

The Anharmonic Oscillator Approximation

Table 12.1 Molecular Constants of Some Diatomic Molecules Molecule

-OJe I cm-1

Do/eV

12c 16

0 14N 160 160 iH19F 1H2H 1H 1H 1H35 Cl 1H79Br 1H1211 19F3sCl r Ci3 5Cl 9Br 19 F 12 13sCl 79Br81 Br 12 11271 I

9.756 5.080 4.511 4.476 4.430 3.754 3.056 2.616 2.475 2.19 2.152 1.971 1.5417

2TrC 2170.21 2359.61 1580.36 4138.52 3817.09 4395.24 2989.74 2649.67 2309.5 793.2 564.9 671. 384.18 323.2 214.57

X eOJe / -1 --cm

2TrC 13.461 14.456 12.073 90.069 94.958 117.91 52.05 45.21 39.73 9.9 4.0 3.0 1.465 1.07 0.6127

Re/nm

-B e I cm-1 c

0.11282 1.9314 0.1094 2.010 0.12074 1.44566 0.09171 20.939 0.07414 45.655 0.07417 60.809 0.12746 10.5909 0.1414 8.473 0.1604 6.551 0.16281 0.516508 0.2438 0.1988 0.17556 0.357165 0.232069 0.114162 0.2283 0.08091 0.2666 0.03735

-a e cm-1 c 0.01748 0.0187 0.01579 0.770 1.9928 2.993 0.3019 0.226 0.183 0.004358 0.0017 0.005214 0.000536 0.000275 0.000117

From G. Herzberg, Molecular Spectra and Molecular Structure L Spectra of Diatomic Molecules, Van Nostrand, New York, 1950.

The Rigid Rotor Approximation with Vibrational Distortion J

= 0, 1, 2, 3, 4, . . . -

J ::;; m ::;; J

gJ

The Rotation Constants Bn

h

=

8Jr 2J

where

12-2

e

= 21+1

Cumulative Examination in Physical Chemistry November 8, 2014 Dr. Teplyakov Kinetics

There are five questions in this test. You must answer 4 questions out of 5 to receive full credit. Only the first 4 that you started will be graded. Beauty is in the eye of the beholder – choose wisely. You can draw a diagram if it helps to answer a question. You are provided a copy of the most referred to surface kinetics paper by Redhead (attached) but if you prepared to this exam, you will not need it to answer most questions as it mostly applies the traditional kinetics approaches to surface processes. Good luck! 1) (20 points) Describe competitive adsorption process for two gases at constant temperature, assuming that both gases follow the Langmuirian model, by deriving the formula for the dependence of coverage of either one of the two gases on the partial pressures of these gases. State all the assumptions clearly. Formula without derivation will be awarded no credit. You can leave your answer in terms of the rate constants for adsorption and desorption for gases A and B, respectively. 2) (20 points) The following data were recorded for absorption of Kr on charcoal at 193.5 K. Use the Langmuir model to construct the adsorption isotherm and determine Vm (volume of adsorbate in the high pressure limit, corresponding to a monolayer coverage) and the equilibrium constant for adsorption/desorption. Vads (cm3/g) 5.98 7.76 10.1 12.35 16.45 18.05 21.1

P (Torr) 2.45 3.5 5.2 7.2 11.2 12.8 16.1

3) (20 points) Derive the expression for the rate of the surface-catalyzed reaction following adsorption of a compound A that transforms into 2B and the products are only formed from surface-bound B. You can assume that the temperature of the reaction is sufficiently high so that the product desorbs from the surface immediately once formed and there is not readsorption of this product to the surface.

4) (20 points) The Redhead equation (Equation (6) in the paper attached) is commonly used by surface scientists all over the world because it gives a quick estimate of the activation energy of a surface reaction based on a single peak in a temperature programmed desorption experiment. In other words, if I know the temperature of the peak, at which the reaction product is evolved from the surface and if I can control the linear heating rate of my sample, I can predict an approximate activation energy for most surface processes with rather high accuracy as long as my pre-exponential factor is on the order of 108 – 1013 s-1 (for the first order process). What is the reason for such a limitation, what is the origin of these limits, and what other limitations hinder the universal application of this useful equation? 5) (20 points) From the experimental data shown below, determine activation energy of desorption of the compound followed as accurately as you can based on Redhead analysis. The figure presents two traces of the same exact experiment for submonolayer coverage of compound A condensed on an unreactive surface (no chemical transformation, desorption is the only process) at 150 K and desorbing at 1 K/s and 8.4 K/s linear heating rates, respectively. State all the assumptions clearly. The number without explanation or derivation will be awarded no points.

Physical Chemistry Cumulative Examination, page 1

October, 2014

Quantum Chemistry C. Dybowski/T. Polenova "`Where could you keep anything so tiny?' Milo asked, trying very hard to imagine such a thing. The Mathemagician stopped what he was doing and explained simply, `Why, in a box that's so small you can't see it -- and that's kept in a drawer that's so small you can't see it, in a house so small you can't see it, on a street that's so small you can't see it, in a city that's so small you can't see it, which is part of a country that's so small you can't see it, in a world that's so small you can't see it.'" A conversation in Digitopolis from The Phantom Tollbooth by Norton Justen

Answer five questions as completely as you can. Be succinct and put the answers in order in the blue book. All five questions have equal weight. If more than five questions are answered, only the first five will be graded.

1.

The phenomenon of spin polarization of rare gases (e.g. xenon-129 and helium-3) by interaction with rubidium atoms excited with a laser is a topic of activity lately, although it traces back to work by Alfred Kastler at the École Normal Supérieure in the 1950s. (a) To begin, give the ground-state configuration of atomic Rb. (b) Out of the ground-state configuration, what term(s) arise? Explain your answer. (c) What is the first excited configuration, and what terms arise from that configuration? Explain your answer. (d) What transitions are possible between the ground term and the terms that arise from the first excited configuration? Explain clearly.

2.

The gravitational interaction between a planet (mass, m) and the Sun (mass M) is given the potential

V (r) = G

Mm r

where G is the universal gravitational constant that has the value 6.673 × 10 J m kg . The force due to this potential acts along the line directly between the two objects and r is the distance between 30 the two objects. The mass of the Sun is reported to be 1.989 × 10 kg, and the mass of the Earth is 24 reported to be 5.972 × 10 kg. (a) Assuming that the Sun is sufficiently far from Earth (compared to the distance between them) that its mass can be considered to be concentrated in a point [Okay, we know that is a stretch, but let’s do it anyway.], write the hamiltonian for the Earth subject to the gravitational potential of the Sun. Use standard symbols for the various parameters. Write Schroedinger’s equation for this situation (in three dimensions). (b) This equation can be separated. Show the various separated equations. Be sure to define all variables you use. (c) Indicate the solutions to those portions that can be solved, giving results for the wave function and energy contribution of each mode. -11

3.

-2

For ethylene, one may generate the following partial set of basis functions. (The symbols indicate the particle on which the atomic orbital is centered and its form.) ψA ∝ C12s + C22s ψC ∝ C11s - C21s

ψB ∝ C12pz - C22pz ψD ∝ C12py + C22py

H1

H3 C1

H2

Z

C2 H4

(a) What are the symmetry operations of the D2h group to which ethylene belongs? Y (b) Using A and B to designate eigenvalues with respect to the major rotation (+1 and –1, respectively), g and u for even and odd symmetry under inversion and 1 and 2 for even and odd symmetry, respectively, under the major reflection, construct the character table for this group.

Physical Chemistry Cumulative Examination, page 2 October, 2014 (c) With your character table, give designations for the symmetry-adapted orbitals above. (d) What multielectron terms arise from the product ψBψD? 4.

In quantum chemistry, one often focuses on finding stationary states of a system with a particular hamiltonian operator, H, by solving Schroedinger's time-dependent equation

H Ψ=i 

∂Ψ ∂t

(a) Discuss the meaning of a stationary state. (b) Show in an equation the form of the solution to this equation for a stationary state. (c) For a non-stationary state, what approximation is usually made to allow solution of Schroedinger’s time-dependent equation? Be specific. 5.

(a) What is the generalized Hellmann-Feynmann theorem? [Answer in an equation or in words.] (b) Use the Hellmann-Feynmann theorem to show that, for a harmonic oscillator in a state with quantum number n, +∞

∫Ψ

* n

-∞

1 2 x Ψ n dx = (n + ) hν /k 2

where k is the force constant, ν is the fundamental frequency of the oscillator and h is Planck’s constant. Show your work. 6.

In 1964, John Bell gave a theorem which reinforced the belief that quantum mechanics describes the world. State and discuss Bell’s theorem. In so doing, you will need to explain the paper by Einstein, Podolsky, and Rosen that brought his paper about. Explain the concepts of hidden variables, as well as the concept of locality and action at a distance, and how did Bell’s theorem resolves the paradox created by Einstein, Podolsky, and Rosen.

7.

An interesting quantum mechanical prediction is tunneling through a barrier. Such a model is important in numerous applications. Consider a barrier of height V0 and width a. Calculate the transmission coefficient, T, for a Coulomb wave of momentum, k, incident upon this barrier.

SOME USEFUL PHYSICAL CONSTANTS -11 -8 -1 a0 = 5.2917725 × 10 m c = 2.99792458 × 10 m s -19 -2 g = 9.80665 m s e = 1.60217733 × 10 C -34 -23 kB = 1.380658 × 10 J/K h = 6.6260755 × 10 J s -31 -27 mp = 1.672623 × 10 kg me = 9.1093897 × 10 kg -12 2 -1 -1 ε0 = 8.8541878 × 10 C J m

Physical Chemistry Cumulative Examination Topic: Thermodynamics September 6, 2014 University of Delaware Department of Chemistry and Biochemistry 034 Drake Hall Primary Examiner: Sandeep Patel Secondary Examiner: Doug Ridge

This is the Fall 2014 Thermodynamics Cumulative Exam for the Physical Chemistry Division. Please answer the five (5) questions to the best of your ability. Please make sure that your answer is clearly stated in the solution, and that all steps in your solution process are understandable and explicit. State all assumptions you invoke. Attached to the problems are items that may be of use in answering the questions. Please leave your answers in the Blue Book. Good Luck.

                               

1. A set of experiments finds that a pure substance follows the following relations:

E = cT + eo N P = aTρ 3 where a, c, and eo are constants independent of N, E, and V. E is the energy of the fluid, N the number of moles, and V the extensive volume, and ρ the system density. For this € fluid, find the underlying expression for the entropy, S(N,V,E) up to constants that are independent of E and V. You may further consider that entropy, S, must have extensive behavior.

                                                                   

€

2. It can be shown that the partition function of an ideal gas of “N” diatomic molecules in an external electric field, ε, is:

 

⎛ [q]N ⎞ Q = ⎜ ⎟ ⎝ N! ⎠

with

⎛ µε ⎞ ⎛ k T ⎞ q = C ⎜ B ⎟ sinh⎜ ⎟   ⎝ µε ⎠ ⎝ kB T ⎠

  Here, T is temperature, kB is Boltzmann’s constant, µ is the dipole moment of a single molecule, and C is a constant independent of ε. The partition function, Q, relates to the € Energy through the following equation: Helmholtz Free

 

€ ⎡ [q]N ⎤ A = −k B T ln Q = −k B T ln ⎢ ⎥   ⎣ N! ⎦

  Using this information along with the Fundamental Thermodynamic Relation for the total derivative of the Helmholtz Free energy:

 

€ dA = −SdT − PdV − (Nµ ) dε  

  where µ is the average dipole moment of a molecule in the direction of the external field, ε, show that at constant temperature and volume:

 

€

⎡ ⎛ µε ⎞ ⎛ k B T ⎞ ⎤ µ = µ⎢coth⎜ ⎟ − ⎜ ⎟ ⎥   ⎝ k B T ⎠ ⎝ µε ⎠ ⎦ ⎣

€

                                       

€

  3. For a fluid whose equation of state is given as:

P=

NRT aN 2 − 2 V − nb V

3A. What is the change in Helmholtz Free Energy for an isothermal change in volume from V1 to V 2? €

3B. What is the change in internal energy, U?

4. Two charge-neutral particles interact via a Lennard-Jones potential with parameters ε = 0.15kcal /mole; σ = 4.0angstroms . This potential is a pair potential, and as such is only a function of the separation between the two particles. 4A. Provide as high-quality a sketch as you can for this potential.

€ 4B. At what atomic separation is this potential a minimum? 4C. Discuss the force each particle experiences from the other at the separation you computed in Part B.

4D. If the two particles now also interact with a Coulomb potential (each particle has charge ‘q’ in dimensions of the unit electron charge), there will be a value of the charge, ‘q’, where the attractive Lennard-Jones interaction will be overwhelmed by the repulsive charge-charge interaction. Determine the value of the charge, ‘q’, at which the potential energy minimum vanishes. The Coulomb potential is :

UCoulomb (rij ) =

€

qiq j 4 πε o rij

5A. Consider a pure substance in this problem. Initially, in a sealed, rigid container there are equal moles of pure liquid and its vapor at initial temperature T1 and initial pressure P1. The container is heated at constant volume to a temperature T2 at which there is still liquid-vapor equilibrium. What is the expression for the final pressure? Assume ideal vapor behavior and constant molar volume of the liquid. Assume temperature independence of the heat capacities of the liquid and vapor as well as the enthalpy of vaporization.

5B. What is the liquid fraction (in molar basis) at the final pressure? (derive the expression you obtain via your analysis, not a numerical response per se).

Physical Chemistry Cumulative Examination April 12, 2014 Major Examiner: Cecil Dybowski Work 5 problems. If you work more than five, only the first five in order will be graded. Leave problems you do not work absolutely blank. 1. For the mechanism below, find the concentrations of A, B and C as a function of time. 𝐴

𝑘𝐵

�� 𝑘𝐶

𝐵

𝐴 �� 𝐶 Assume that, at t = 0, [A] = a and [B] = [C] = 0. 2. Derive the relationship between the half-life and the initial reactant concentration [A]0 for a reaction of order n, where n ≠ 1. 3. Find the relationship between the rate constants and the relaxation time for return to equilibrium after a perturbation is applied. 𝑘𝑓 𝐴  𝐵 𝑘𝑟

4. The reaction of NO with Cl2 is followed by monitoring the amount of NO left after the reaction is initiated. (a) What is the order with respect to NO? (b) What is the rate constant for this reaction ? [HINT: You may assume that chlorine is present in excess.] Time (s) 0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

NO Conc. (Arb. Units) 0.00049 0.00038 0.000321 0.000271 0.000239 0.000206 0.000189 0.000172 0.000152 0.000145 0.000132

5. In the following table, indicate which phrases in column B best describe those in A, but putting the number in the blank. Column A

Column B 1. RRKM Theory

____ Arrhenius equation ____ Belousov-Zhabotinsky reaction ____ Eley-Rideal kinetics ____ Eyring equation ____ Langmuir’s equation ____ Lindemann’s mechanism

2. 𝜃(𝑃) = 𝑏𝑃/(1 + 𝑏𝑃) 3.

1 𝑣

=

1

𝑣𝑚𝑎𝑥

+

𝐾𝑀 1 𝑣𝑚𝑥 [𝑆]0

4. ∆𝐺 𝜃 = ∆𝐻 𝜃 − ∆𝑆 𝜃

5. The product is directly involved in an equilibrium. 6. Surface reaction that involves one reactant from the gas phase. 7. The steady-state approximation

____ Lineweaver-Burk equation ____ Preequilibrium Approximation ____ Rice-Herzfeld demonstrations ____ Transition state

8. 𝑘(𝑇) = 𝐴 exp(−𝐸𝑎 /𝑅𝑇 𝑘𝑇

9. 𝑘(𝑇) = �ℎ𝐶 𝜃 � 𝑒𝑥𝑝�−∆𝐺 𝜃 /𝑅𝑇�

10. The last product-forming step is slower than the equilibrium exchange between reactants and intermediates. 11. Surface reaction between two adsorbed materials. 12. Activated complex 13. Marcus theory 14. Decay of a radioactive nucleus 15. Many organic reactions involve reactive intermediates. 16. The rate constant is calculated from the partition functions of the reactants and products. 17. Förster energy transfer 18. An example of intersystem crossing 19. Oscillating reaction

6. The decomposition of N2O (1Σ linear ground state) to N2 (1Σ ground state) and O (3P ground state) is important in atmospheric chemistry. The activation energy for the reaction is 60.6 kcal/mol, but the thermodynamic bond strength is 39.7 kcal/mol. Suggest an explanation for the difference. 7. Spectroscopic absorption is a kinetic process that has been described before. Discuss Einstein’s theory of spectroscopic absorption. In particular, discuss the relationship of his parameters and the line shape, particularly as it pertains to saturation. 8. Discuss the polymerization of ethylene in some detail, including equations where appropriate. Explain all approximations, symbols, and phrases you use. You should include information on various mechanisms, and whether or not they are significant for this particular system.

Cumulative Examination in Physical Chemistry March 8, 2014 Dr. Teplyakov Quantum Mechanics This is a three-hour cumulative exam. IT IS CLOSED-BOOK. No additional materials beyond this sheet are needed or allowed to answer the questions. You should write your answers in the blue notebooks provided. Those are the only notes that will be graded. There are two parts in this examination. Both are closed-book. Part I covers very basic quantum mechanics problems. You must answer three out of four questions in this part. Beauty is in the eye of the beholder – choose wisely. You can draw a diagram if it helps you answer a question. Part II is based on recent developments in the field of chemistry and QM in particular. It is also closed-book. You must answer this question to complete the exam and have a shot at 100 points. The total will amount to 100 points, 25 points apiece. Once completed, you can place the exam in the envelop provided and slide it under the door of my office, 112 LDL. Good luck! Part I. Quantum mechanical properties of hydrogen atoms. 1) (25 points). Compute the energy of an electron in the n = 1 Bohr orbit and the de Broglie wavelength of this n = 1 electron. You can use any reasonable units to answer this question. 2) (25 points). According to Koopmans’ theorem, what would be the ionization energy of n = 1 s electron in a hydrogen atom. Is the XPS of hydrogen common? If yes, explain how it is useful, if not, explain why. 3) (25 points). Imagine if a hydrogen atom behaved as a planetary system, with electron rotating at a constant orbit about an extremely heavy nucleus. The mass of the electron is 9.1094 x 10-31 kg, the kinetic energy of this electron is 2.1792 x 10-18 J, the radius of the circular orbit for this electron is 0.519 x 10-10 m. Assume that the nuclear mass is infinite and calculate the angular momentum of the electron in such a system. Remember that the kinetic energy of the electron is given by T =

P2 PR2 + Θ 2 , where 2m 2mR

PR is a linear momentum and PΘ is the angular momentum. 4) (25 points). a) Calculate the frequencies and wavelengths for the two Lyman spectral lines of lowest energy for hydrogen atom; b) Calculate the frequencies and wavelengths for the two Balmer spectral lines of lowest energy for hydrogen atom Part II. Recent Nobel prizes in chemistry and physics. 1) (25 points) Name and briefly describe the contributions of the scientists who received the most recent Nobel prize in chemistry. Their achievements are very much related to the topic of this cume.

Thermodynamics Cumulative Exam

Your Name________________________________

Physical Chemistry Cumulative Exam Fundamentals of Thermodynamics and Applications to Chemical Systems February 8, 2014 When done (and no later than 12 noon), place the exam into Professor Polenova’s mailbox Tatyana Polenova, Principal Examiner Total points: 100

1. Short problems (15 points, 7.5 pts each) 1a) (Multiple choice) What is the total pressure, in atmospheres, of a 10.0 L container that contains 10 moles of nitrogen gas and 10 moles of oxygen gas at 300 K?

a) 24.6 atm

b) 49.3 atm

c) 2,460 atm

d) 4,930 atm

1b) A 0.040 M solution of a monoprotic acid is 13.5% dissociated. What is the dissociation constant of the acid?

1

Thermodynamics Cumulative Exam

Your Name________________________________

2. (15 pts, 5 pts each) A. One mole of an ideal monatomic gas is compressed from 2.0 atm to 6.0 atm while being cooled from 400 K to 300 K. Calculate ∆S for the process.

B. At 303 K, the vapor pressure of benzene is 118 Torr and that of cyclohexane is 122 Torr. Calculate the vapor pressure of a solution for which χ(benzene) = 0.25 assuming ideal behavior.

C. A sample of supercooled water freezes at -10 oC. What are the signs of ∆H, ∆S, and ∆G for this process? Explain clearly. All the changes refer to the system.

2

Thermodynamics Cumulative Exam

Your Name________________________________

3. For each statement in column A, give the letter of the best match from column B. There is only one best match. (18 pts, 3 pts each) A

B

General change in internal energy for a closed system

a)

Thermodynamic definition of entropy

b) Henry’s law

Gibbs-Helmholtz equation

c) γici

.

PA=χAP A is

d)

Chemical potential of an ideal gas is

e) reaction coordinate f)

is the definition of

g) 0 m) 0 n) µ = µ0 + RT ln(P/P0) o) equilibrium constant p) ∆G0

3

Thermodynamics Cumulative Exam

Your Name________________________________

4. (18 pts) At a certain temperature, the equilibrium pressures of NO2 and N2O4 are 1.6 bar and 0.58 bar, respectively. If the volume of the container is doubled at constant temperature, what would be the partial pressures of the gases when equilibrium is re-established?

4

Thermodynamics Cumulative Exam

Your Name________________________________

5. (18 pts) A solution of equal concentrations of lactic acid and the lactate anion was found to have a pH = 3.08. What is the pKa of lactic acid?

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Thermodynamics Cumulative Exam

Your Name________________________________

6. (16 pts, 8 pts each) Diesel cycle drawn in the figure below is a thermodynamic cycle describing a heat engine and consisting of four steps: constant-pressure and constant-volume, connected by reversible adiabatic (isentropic) steps.

a) For the Diesel cycle, calculate q, w, and ∆E associated with each step.

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Thermodynamics Cumulative Exam

Your Name________________________________

b) Derive an expression of a thermal efficiency for this engine in terms of temperatures of the individual steps.

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Thermodynamics Cumulative Exam

Your Name________________________________

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Thermodynamics Cumulative Exam

Your Name________________________________

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Thermodynamics Cumulative Exam

Your Name________________________________

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Page |1 Physical Chemistry Cumulative, December, 2013

Name:____________________________

Spectroscopy December 14th, 2013 Principal Examiner: L. Gundlach Second Examiner: T. Polenova

Answer question worth 100 points of the possible 140 points. Clearly indicate which questions you want graded and which ones not. Otherwise I will grade the first 100 points.

Page |2 Physical Chemistry Cumulative, December, 2013 1. (15 points) Predict the number of chemically shifted 1H peaks and the multiplet splitting of each peak that you expect to measure for 1,1,2-trichloroethane. Draw the molecule and explain your answer.

Page |3 Physical Chemistry Cumulative, December, 2013

2. (15 points) Explain what a spin-echo experiment is, how and with what instrument it is performed and what advantage it has.

Page |4 Physical Chemistry Cumulative, December, 2013 3. (15 points) PdCl2(NH3)2 is a square-planar molecule with cis-trans isomerism. Draw the molecule, identify possible vibrational modes, and determine if they are IR active. The Pd-N and Pd-Cl modes are identified in the two IR spectra below. Which of the two is the trans spectrum?

Page |5 Physical Chemistry Cumulative, December, 2013 4. (15 points) An IR absorption spectrum of an organic compound is shown here:

Use the characteristic group frequencies below to decide whether this compound is more likely a hexene, hexane, or hexanol. Explain your answer.

Page |6 Physical Chemistry Cumulative, December, 2013 5. (10 points) Derive the ground state atomic term symbols for H, F-, and Na+.

6. (10 points) Give the order of magnitude for the energy of the transition that is probed by each of the spectroscopic techniques: THz-spectroscopy: Near edge X-ray absorption spectroscopy: NMR: Raman spectroscopy: Rotational IR spectroscopy: Fluorescence spectroscopy: Mass spectrometry:

Page |7 Physical Chemistry Cumulative, December, 2013 7. (10 points) a) The computer of your Fourier Transform (FT) IR spectrometer is broken but you can still extract the raw data that gives you the signal intensity as a function of mirror position. Which one of the two measurements has higher accuracy? Which one corresponds to a higher frequency? B B

20 100 80

10

60

20

Intensity

Intensity

40

0

-10

0 -20 -40 -60

-20 -80

-150

-100

-50

0

position [µm]

50

100

150

-100 -20

-10

0

10

20

position [µm]

b) (10 points) We want the abscisses of the spectrum after FT to be in units of frequency. Explain how the abscisse of the raw data has to be transformed before the Fourier transformation is performed. Give equations and the necessary fundamental constants.

Page |8 Physical Chemistry Cumulative, December, 2013 8. (15 points) A DFT calculation results in the following probability density distribution for the HOMO orbital and the LUMO orbital:

ℎ𝜈

LUMO

HOMO

Predict what kind of solvatochromic shift you mesure when you take a vis-absorption spectrum of the compound in heptane and in DMF.

9.

(10 points)Discuss similarities and differences between X-ray photoelectron spectroscopy and Auger electron spectroscopy. Explain the principle of operation, advantages, and disadvantages of each technique.

Page |9 Physical Chemistry Cumulative, December, 2013 10. (10 points) For a grating with 10’000 lines/cm, calculate the angle between the grating normal direction and the first order diffracted beam at 707.106 nm wavelength and normal incidence. If you don’t know the equation a drawing will help.

P a g e | 10 Physical Chemistry Cumulative, December, 2013 11. (5 points) A snail climbs up a 10-foot pole. It covers three feet each day and sleeps at night. While it sleeps it slides down the pole by one foot. When does it reach the top of the pole?

Physical Chemistry Cumulative Examination November 9, 2013 Major Examiner: D. P. Ridge Work 100 points worth of questions: 1. (15 points) For the mechanism below find A and B as a function of time. Assume that at t=0, [A]=a and [B]= 0. Assume that both processes are first order and the rate constants are equal. k1

A

B

k2

2. (10 points) Use the steady state approximation to find the rate law for the mechanism below. Under what circumstances will the reaction be zero order in O2. NO + NO N2O2 + O2

k1 k-1 k2

N2O2 2NO2

3. (10 points)Find the relationship between the rate constants and the relaxation time for return to equilibrium for the following reaction. k1 A

B+C k-1

4. (10 points). Given the equation below and the Arrhenius equation (k = Aexp(EA/RT)), Define the relationship between the entropy and enthalpy of activation and the parameters A and EA (note: ∆G  is the free energy of activation).  kT k = b e − ∆G / RT h

5. (10 points) Given that ΔG = PΔV at constant T and P: (a) Find the relationship between the ln(k) (where k is the rate constant) and the volume of activation ∆Vǂ.

(b) For the dimerization of cyclopentadiene in n-butyl chloride ∆Vǂ = -22.0 cm3/mol. Will the reaction rate increase or decrease with pressure? (c) At T=298K what pressure will the reaction rate differ from that at 1 atm by 10%? Assume ∆Vǂ to be independent of pressure. 6. (10 points). Derive the equation for the primary salt effect using transition state theory. (Hint: logγi = -0.51zi2I1/2 and I = 1/2∑ciZi2). 7. (10 points) Find the “encounter limited” rate constant given that the reactant cross section is πd2 and the average relative velocity is [8kT/πm]1/2 where d is the reactant radius, k is Boltzman’s constant, T is the temperature and m is the reactant reduced mass. Give your answer in terms of the symbols in the given expressions. What are the units of the rate constant? 8. (15 points) Consider the potential V(r) = A/r8 - B/r4. (a)

Find R, the value of r at which V(r) is a minimum for the 8-4 potential in terms of A and B.

(b)

Find the harmonic force constant f = (d2V/dr2)R for the 8-4 potential .

(c)

Sketch χ(b) for U(r) for the 8-4 potential. The scattering angle should be positive at impact parameters, b, where repulsion dominates and negative at impact parameters where attraction dominates. Indicate the rainbow angle (where χ(b) is a minimum) on your sketch.

9. (15 points)The decomposition of N2O (1Σ linear ground state) to N2 (1Σ ground state) and O (3P ground state) is important in atmospheric chemistry. The activation energy for the reaction is 60.6 kcal/mol, but the thermodynamic bond strength is 39.7 kcal/mol. (a) Suggest an explanation for the difference. (b) Given that the lowest singlet state of the O atom (1D) is 45.4 kcal/mol above the ground state sketch a potential energy surface for the N2O reaction and indicate the diabatic and adiabatic reaction pathways. 10. (10 points) Make a topographical sketch of a potential energy surface with the barrier in the entrance channel for a three atom reaction: A + BC  AB + C. Label the axes in terms of the interatomic distances. Let the axis perpendicular to the paper be the energy axis. Label the transition state. Trace a typical reaction trajectory. With the barrier in the entrance channel is this reaction more likely to be endothermic or exothermic? Are the products more likely to be vibrationally hot or translationally hot? 11. (15 points) With regard to the Langmuir isotherm:

(a) Derive the Langmuir absorption isotherm for θ, the equilibrium fraction of sites occupied by absorbate on a surface. Note that the absorption rate is proportional to the product of the pressure of adsorbate and the fraction of sites not occupied (1-θ), while the desorption rate is proportional to the fraction of sites occupied. Give your result in terms of ka and kd, the rate constants for the absorption and desorption processes and the pressure of adsorbate, P. (b) Suppose two species, A and B, adsorb on the surface with adsorption and desorption rate constants kaA, kaB, kdA and kdB. Find an expression for θ = θA + θB in terms of the rate constants and the pressures. 12. (15 points) Derive the analogue to the Lineweaver-Burke equation for the enzyme catalyzed reaction between A and B with the following mechanism. k1 A + E EA k-1 k2 B + EA

EAB k-2 k3

EAB

Products

13. (15 points) Consider a crossed beam experiment between a beam of A molecules with mass mA and velocity vA and a beam of B molecules with mass mB and velocity vB. The beams cross at right angles. Draw a Newton diagram (i. e. a diagram in velocity space) for the system. Indicate the relative velocity v and center of mass velocity V. Indicate the elastic scattering circle for unreactive A.

Name _________________________________________________________________ Physical Chemistry Cumulative Exam Quantum Mechanics of Angular Momentum October 12, 2013 Please place a completed copy of the exam in Dr. Polenova’s mailbox by 12 noon Tatyana Polenova, Principal Examiner Total points: 100

I. Basic Definitions (30 pts total) 1. Evaluate the following matrix elements (12 pts, 4 pts each): a) b) c)

2. Confirm that

and

. (8 pts total, 4 pts each)

3. (10 pts, 5 pts each) a) Confirm that the Pauli matrices below satisfy the angular momentum commutation relations when we write

and hence provide a matrix representation of angular

momentum.

b) Why does the representation correspond to s = ½ ? Hint: form the matrix representing s2 and establish its eigenvalues. II. The Coupling of Angular Momenta (25 pts total) 1. a. What angular momentum states can arise from a system with two sources of angular momentum, one with j1=1/2 and the other with j2=3/2? Specify the states. Hint: use the Clebsch-Gordan series. (10 pts) b. Construct the state with j=3/2 and mj=-1/2 for a p electron (7 pts).

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Name _________________________________________________________________ 2. (8 pts, 4 pts each) Use the vector model of angular momentum to derive the value of the angle between the vectors representing a) two α spins b) an α and a β spin in a state with S = 1 and MS = +1 and MS = 0, respectively.

III. Spin Angular Momenta and Rotation Operators (20 pts total, 10 pts each) Construct the complete basis set matrices for spin angular momentum and rotation operators for spin I=1.

IV. Physical Chemistry Seminars (25 pts total, 5 pts each) For the last two Physical Chemistry seminars that took place on October 4 and October 11, 2013, answer a set of very basic questions regarding the content of the seminars.

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Physical Chemistry Cumulative Examination Topic: Thermodynamics September 7, 2013 University of Delaware Department of Chemistry and Biochemistry 207 Brown Laboratory Primary Examiner: Sandeep Patel Secondary Examiner: Cecil Dybowski

This is the Fall 2013 Thermodynamics Cumulative Exam for the Physical Chemistry Division. Please answer the six (6) questions to the best of your ability. Please make sure that your answer is clearly stated in the solution, and that all steps in your solution process are understandable and explicit. Attached to the problems are items that may be of use in answering the questions.

1(A) (7 Points). A constant volume system is brought in contact with a thermal reservoir at temperature Tf. If the initial temperature of the system is Ti, calculate the entropy change, ∆S, of the total system (system + heat bath). Make any assumptions you feel are appropriate and state them explicitly. 1(B) (7 Points). Assume that the same temperature change is effected by successive contacts of the system with N heat reservoirs at temperatures range from Ti to Tf in increments of ∆T as follows: Ti, Ti+∆T, Ti+2∆T, Ti+3∆T, ….., Tf - ∆T, Tf. Note that N∆T=Tf-Ti. Show that in the limit as N∞ and ∆T 0, with N∆T=Tf-Ti fixed, the total entropy change (system + reservoirs) is zero. 1(C) (1 Point) What is the significance of any differences or similarities in the results of parts A and B?

2 (20 Points Total). A reversible heat engine operates between two heat reservoirs, T1 and T (T2>T1). T1 is taken to have infinite mass. The warmer reservoir consists of a finite amount of gas at constant volume and weakly temperature dependent constant volume heat capacity, Cv. After a long time, T2 = T1. 2(A). What amount of heat is extracted from the warmer reservoir during the time over which the temperature of the warmer bath equilibrates to that of the cooler bath? 2(B). What is the entropy change of the initially warmer reservoir during this period? 2(C). How much work did the engine perform over this time?

3(A) (15 Points) Take a large number, N, of localized particles in an external magnetic field, H. Each particle has spin ½. Find the number of states accessible to the system (taken to be the N particles in the field) as a function of the z-component of the total spin of the system, Ms. 3(B). (5 Points) Determine the value of Ms that is associated with the largest number of accessible states.

4 (25 Points Total) A simple model for addressing macroscopic polymer elasticity considers a onedimensional model of N polymer molecules. Each polymer molecule is linked to another in an end-to-end fashion. The angle between successive links is equally likely to be 0 or 180 degrees. See Figure 1 for a schematic of the model.

Figure 1

4(A). How many ways, g(N,m), are possible to realize a connected length of the N polymer molecules equal to L=2md where 2m is related to the net ‘directionality’ of linkages (i.e. excess of 0 degrees over 180 degrees). 4(B). For m1 and L