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1 12/05/18, Chem433 Final Exam Last Name or Student ID 1. (2 pts) 12. (3 pts) 2. (6 pts) 13. (3 pts) 3. (3 pts) 14. (2 pts) 4. (3 pts) 15. (3 pts) 5. (4 pts) 16. (3 pts) 6. (2 pts) 17. (15 pts) 7. (9 pts) 18. (6 pts) 8. (2 pts) 19. (14 pts) 9. (3 pts) 20. (2 pts) 10. (2 pts) 21. (3 pts) 11. (2 pts) 22. (8 pts) Honors: 19c. (4 pts) 23 (6 pts) Total: (100pts) The exam is close book close notes with mostly conceptual problems. Give a brief explanation for maximum credit. You can submit corrections to your answers online before midnight 12/05/18. If answered correctly online, you get half the face value credit for that question. If you do not use a cheat sheet (A4 page only), you get 10% extra. The extra questions for honor students will be counted as extra if you decide to answer them (only in class) 1

2 1. (2 pts) How many angular and how many radial nodal surfaces does 5g orbital have? 2. (6 pts) Identify whether a function, f(x), is an eigen function of the corresponding operator, A, and with what eigen value: a) f(x) = (cosx) 2 and A = d 2 /dx 2 b) f(x) = x 4 and A = x d/dx c) f(x,y) = exp(ax+by) and A = d 2 /dx dy 3. (3 pts) Identify dxz orbital: a) b) c) d) e) 4. (3 pts) Which of the following statements is/are wrong about Hermitian operators: a) the eigenvalues of a Hermitian operator are real numbers b) if two linear Hermitian operators have the same set of eigenfunctions then they commute c) eigenfunctions of a linear Hermitian operator with different eigen values are orthogonal to each other d) a Hermitian operator can be a complex operator e) the eigen functions of a linear Hermitian operator cannot be real 2

3 5. (4 pts) Calculate the commutator of the operators p 2 x and x 4. Express your answer through powers of x and px. 6. (2 pts) What is the degeneracy of J = 6 rotational state in CH4? 7. (9 pts) Consider an excited state electron configuration of fluorine 1s 2 2s 1 2p 5 3d 1 a) (1 pt) Identify all possible values of S for it. b) (1 pt) Identify all possible values of L for it. c) (3 pts) Identify all possible terms that can arise from this electron configuration. d) (2 pts) Based on the Hund s rules, what is the lowest energy term? What is its degeneracy? e) (2 pts) What are the possible states (levels) for the lowest energy term and their degeneracies? 3

4 8. (2 pts) The lowest energy state of para hydrogen for hydrogen molecule is characterized by: a) zero overall nuclear spin of two hydrogens, I = 0 b) overall nuclear spin of two hydrogens, I = 1 c) zero overall spin of two electrons, S = 0 d) overall spin of two electrons, S = 1 e) zero angular momentum of its rotational state, J = 0 f) angular momentum of its rotational state, J = 1 g) a, c, and e h) b, d, and f i) b, c, and f 9. (3 pts) What is the ground electron configuration of ClO ion? For clarity, sketch the orbital diagram illustrating the AOs mostly contributing to the MOs. Pay attention to the relative energies of AOs and MOs. 10. (2 pts) What is the bond order for this ground electron configuration of ClO? 11. (2 pts) Which atomic orbital(s) contribute the most to the LUMO of ClO? Explain. 12. (3 pts) Which molecule(s) will show microwave absorption? a) CH4 b) CO2 c) CH2Cl2 d) HC CH e) N C-C N f) none g) a and b h) a and c i) all of them j) there is no correct answer 4

5 13. (3 pts) Which molecule(s) will show rotational Raman spectrum? a) CH4 b) CO2 c) CH2Cl2 d) HC CH e) N C-C N f) none g) all of them h) c, d, and e i) a and c j) b, c, d, and e k) there is no correct answer 14. (2 pts) What is the difference between a microcanonical and canonical ensembles? 15. (3 pts) Calculate the probability of a macrostate {5,2,1}. Assume that the system consists of identical particles, and the particle states have identical energy. (Hint: what is the number of energy levels for each particle? What is the number of particles in the system?) 16. (3 pts) What are the probabilities of measuring l = 3 and lz (or m) = 2 for a dx2-y2 orbital? Explain or show derivations. 5

6 17. (15 pts) Consider CO2 molecule in the gas phase. a) (4 pts) How many unique fundamental vibrational frequencies does it have? What are the values for these frequencies in cm -1 (you don t need to be accurate, very crude estimate is enough): b) (3 pts) How many of them are IR- active (allowed)? You may want to illustrate each mode by displacement vectors. c) (8 pts) Sketch (or explain) its IR spectrum in the gas phase with as much details as you can. Identify the features one needs to pay attention to. I expect you to identify the allowed and forbidden IR transitions rovibronic structure with P, Q, and R branches, when appropriate, and their relative intensities pattern for each. Neglect anharmoncity and centrifugal distortion. Reminder: B (cm -1 ) = 16.86/I(amuÅ 2 ). Take the bond lengths RCO = 1.16 Å, 6

7 18. (6 pts) Choose the Slater determinants that are correctly representing the states arising from 1s 1 2p 1 electron configuration of He atom. Among those that are, identify the states that are eigen functions of the operator S 2 and label their spins and projections values of Ms. Are these functions also eigen functions of L 2? If they are, what are their eigen values (or values of L)? Hint: calculating the determinant might clarify the choices. a) b) c) d) e) f) 19. (14 pts) A wavefunction of a particle of mass m on a harmonic potential with force constant k is given by = 2 0> + 3 1>, where n> are the eigen functions of Hamiltonian, Ĥ n> = ħ (n+1/2) n>. a) (2 pts) Is this an eigen function of the Hamiltonian? b) (4 pts) Sketch the function, Hint: recall how each of the eigenfunction in the superposition looks like and add them with the corresponding weight. c) (4 pts) Is it this function normalized, i.e. is < > =1? If not, normalize it. Hint: you do not need to integrate; represent the function as a superposition of the normalized eigen functions, cn n, and evaluate the result by recalling that the eigen functions have to be orthogonal. 7

8 d) (4 pts) Calculate the expectation value for energy in this state, i.e. <E> = < Ĥ >. e) (Honors, 4 pts) Calculate the expectation value for <x(t)> = < (x,t) x (x,t)>. Hint: Remember that operator x can be expressed through the lowering and raising operators and the time dependence of an eigen function n of energy is given by exp(-ient/ħ), i.e., n (x,t) = n(x) exp(- ient/ħ) 20. (2 pts) Phosphorescence is: a) Any transition from an excited to the ground state b) Transition from an excited to the ground state that is accompanied by emission of a photon c) Any transition from the ground to an excited state d) Transition from the ground to an excited state that is accompanied by absorption of a photon e) A spin allowed transition f) A spin forbidden transition g) d and e h) b and f i) a and e 21. (3 pts) Calculate the partition function for an ideal gas with the thermal de Broglie wavelength of 16 pm at 300K, 1atm. 8

9 22. (8 pts) Sketch a high resolution proton NMR spectrum of Ethyl acetate (see structure) at low concentration in an aprotic solvent presuming that the coupling constants are identical (J = 7 Hz). Identify the approximate chemical shifts, the total (relative) integrals for each group, and the intensity pattern due to J-coupling (presuming that the latter). 9

10 23. Honors (6 pts) Consider scattering of an electron on a potential step of the height Vo originating at x = 0 and being higher than the particle s kinetic energy, E < Vo. How fast does the probability for the electron to penetrate at V the depth x into the wall, P = Poexp(- x), decline with E? o Calculate the numeric value of 1/ for E = 1 ev and Vo = 1.03 ev (1eV = J). The mass of electron is me = kg and Planck s constant, h = Js. What is different in the reflected wave when Vo is finite as compared to the infinite one? V=0 10

Last Name or Student ID

12/05/18, Chem433 Final Exam answers Last Name or Student ID 1. (2 pts) 12. (3 pts) 2. (6 pts) 13. (3 pts) 3. (3 pts) 14. (2 pts) 4. (3 pts) 15. (3 pts) 5. (4 pts) 16. (3 pts) 6. (2 pts) 17. (15 pts) 7.