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QCM701S - QUANTUM CHEMISTRY SPECTROSCOPY - 1ST OPP - JUNE 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF NATURAL AND APPLIED SCIENCES
QUALIFICATION: BACHELOR OF SCIENCE
QUALIFICATION CODE: 07BOSC
COURSE NAME: QUANTUM CHEMISTRY AND
MOLECULAR SPECTROSCOPY
SESSION: JUNE 2022
DURATION: 3 HOURS
LEVEL: 7
COURSE CODE: QCM701S
PAPER: THEORY
MARKS: 100
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER(S)
Prof Habauka M Kwaambwa
MODERATOR: Prof Edet F Archibong
INSTRUCTIONS
Answer ALL the SIX questions.
Write clearly and neatly
Number the answers clearly
All written work must be done in bule or black ink
No books, notes and other additional aids are allowed
Mark all answers clearly with their respective question
numbers
PERMISSIBLE MATERIALS
Non-programmable Calculators
ATTACHMENT
List of Useful Constants
THIS QUESTION PAPER CONSISTS OF 6 PAGES (Including this front page and List of Useful
Constants an attachment)
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QUESTION 1
[20]
(a) Define the terms blackbody radiation and UV catastrophe. Draw a schematic diagram of
the energy density, U(A), against wavelength, A, for the blackbody radiation at
temperatures Ti and T2 (where T1 < T2).
(5)
(b) Rayleigh-Jeans law of a blackbody radiation as function of frequency is given as:
u(y)
Under what condition would this theory agree with blackbody radiation experimental
results.
(2)
(c) The derivation by Bohr of the hydrogen atom given below.
v= a[a - - 2 } where Re = 109677.58 cm”
(i)
State the three basic considerations this equation is based on or was derived. (3)
(ii)
Calculate the wavelength, A(in nm) and ionisation energy (in eV) for the Balmer
line of the H emission. (The ni = 2 for the Balmer series).
(3)
(d) Electromagnetic radiation of wavelength 200 nm is used to irradiate gold metal.
(i)
Given that the work function of gold is 5.10 eV, determine the kinetic energy
(in Joules) and velocity of the electrons ejected.
(5)
(ii)
State briefly the effect, if any, of increasing the intensity of incident light of
wavelength 200 nm?
(2)
QUESTION 2
[14]
Consider a z-electron which is a part of a conjugated polymethine dye. Use the free-electron
molecular orbital (FEMO) method, which assumes that the z electrons are trapped in a 1-D
box of length 11.2 A to answer the following questions:
(a) Calculate the zero-point energy (in eV) of the system.
(3)
(b) Why is the zero-point energy equal to zero not feasible?
(2)
(c) Assuming the length of the chain to be 11.2 A, determine the transition caused by
excitation using the light of wavelength of 460 nm.
(6)
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(d) Determine the number of pi electrons.
(1)
(e) What is the main weakness of the FEMO model?
(2)
QUESTION 3
[22]
(a) One of the requirements for useful wave functions in Quantum Mechanics is that
they must be well-behaved. State briefly the meaning of well-behaved wave
function.
(2)
(b) The wave function, ‘Y, for an electron in the highest occupied molecular orbital of
polydiene based on 1-dimensional particle-in-a-box model is given by:
l
¥(x)=(2) 2sin( “*) forO<x<L
(i)
Plot the variation of ‘¥(x) and ¥*(x) fora particle-in-a-box for O<x<L.
(3)
(ii)
State for which values of x in terms of L is the probability of finding the
particle, ie. Y*(x), maximum in the range 0 <x <L.
(4)
(c) State using a mathematical expression what is meant in quantum theory for each of
the following:
(10)
(i)
Operator A is linear to the wave functions ‘Y, and 'Y;.
(ii)
Wave functions ¥; and ’, are not orthogonal.
(iii)
Operators A and B commute of wave function Y.
(iv)
Hermitian operator A of wave functions ‘¥; and 'Y,.
(v)
Expectation value, (a), of the observable A derived from a normalised
wave function ¥ .
(d) What are the physical meanings of commuting operators and orthogonal wave
functions in Quantum mechanics?
(3)
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QUESTION 4
[9]
(a) Show that the function ‘¥ =e" of the free particle is also an eigenfunction of the
l.inear operator, P=x =
th.
d
x
.
What
.is the expressio. n for the ei. genvalue
correspondin; g
to this eigenfunction?
(4)
(b) For circular motion in a fixed plane, the operator the Schrodinger equation is of the
form
_ he . av
2mr’ \\ do°
= EW, where m=O, +1, +2, +3, etc.
Show that ‘Y = —J—2~1ne.'"’ is an acceptable solution of the differential equation. What
is the eigenvalue expression?
(5)
QUESTION 5
[20]
(a) Which of the following molecules have a pure rotational spectrum and which ones
are IR active?
Which of the species would be:
(5)
(i)
microwave active?
(ii)
infrared (IR) active?
NH3, HCI, H2, CO2, O2, CH3Cl, C2Ha, CHa, cis-CH2Cl2, H202, trans-CH2Cl2, CS2
(b) The allowed rotational energy levels of a rigid diatomic molecule are given by
E, = BJ(J+1)
State the selection rule for the rotational energy transitions and derive the separation
between the successive spectral absorption lines in terms of the rotation constant, B.
(4)
(c) A particle on the surface of a sphere has quantum number J = 7. What is the
degeneracy of the energy level to which this state belongs to?
(2)
(d) The ro-vibrational spectrum is divided into three branches, namely, P, Q and R. What
is the approximate separation in terms B between the innermost line of the P and
second innermost line of the R branch?
(1)
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(e) For the rotation-vibration spectrum below, identify the wavenumber and transition
for the peak R(2).
(2)
100 oe Awe, ‘ phy 4} ("
90F §
80
mayen
g
K
nx
x
anne
o
I
p/eon7}
(f) The ro-vibrational spectrum of *H?*71, with peaks at 2296.40, 2322.60 and 2335.70 cm},
was recovered.
(i)
From the recovered data of the spectrum, what is approximate spacing between
the peaks?
(2)
(ii)
Deduce the moment inertia, |, 1H?2’I.
(2)
(iii) | Calculate the reduced of 1H?271.
(2)
(iv)
Evaluate the internuclear distance (in A) of 4H??7I.
(2)
Atomic masses (amu):
1H = 1.0079
127| = 126.90447
QUESTION 6
[14]
Two particles of masses 3.32 x 10°” kg and 31.5 x 10%’ kg are connected by a Hooke’s law
spring which requires force of 13.2 x 10? N to stretch it by 1.5 m.
(a) Calculate the force constant (in Nm**) of the system.
(2)
(b) What is the fundmamental vibration frequency (s*) of the system?
(6)
(c) Calculate the potential energy of the system when stretched by 1.5 m from its
equilibrium position?
(3)
(d) What is the zero point energy (based on quantum theory of simple harmonic
oscillator) of the system?
(3)
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LIST OF USEFUL CONSTANTS:
Universal Gas constant
Boltzmann’s constant,
Planck’s constant
Debye-Huckel’s constant,
Faraday’s constant
Mass of electron
Velocity of light
Avogadro’s constant
Na
1 electron volt (eV)
8.314) K? molt
1.381 x 1073 J Kt
6.626 x 1074J s
0.509 (mol dm’3)*? or mol®kg?>
96485 C molt
9.109 x 10? kg
2.998 x 10®ms?
6.022 x 107
1.602 x 10°79J