OLLSCOIL NA hEIREANN, CORCAIGH
THE NATIONAL UNIVERSITY OF IRELAND, CORK
COLAISTE NA hOLLSCOILE, CORCAIGH
UNIVERSITY COLLEGE, CORK
AUTUMN EXAMINATIONS 2003
CHEM3
Module CM3105 – Chemical Physics
Prof M.A. Chesters, Prof R.P. Brint, Dr A.A. Ruth,
Prof W.B. Jennings
THIS IS NOT A REAL EXAM PAPER. 2003 Questions in 2004
format
The use of non-programmable electronic calculators is permitted in this
examination. Use of programmable calculators is not permitted. Use
of any such calculators is a serious breach of discipline
For each module three questions are given as either/or choices. Students
are required to answer each question (1,2 and 3) choosing one of the given
choices. Answer all the parts of the elected option. Marks will not be
awarded for completing two choices of the same question. Three complete
answers are to be submitted in separate answer books.
1½ HOURS
CANDIDATES ARE ALLOWED 15 MINUTES READING TIME
USEFUL DATA:
Universal Gas Constant (R)
Avogadro’s Constant (NA)
Planck’s Constant (h)
Boltzmann’s Constant (k)
Speed of light (c)
Permittivity in a Vacuum (o)
Electronic Charge (e-)
1 atm
One Faraday, 1F
1J
1 eV
1 J s-1
1 bar
1 a.u.
1 fs
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
8.314 J mol-1 K-1 = 8.205×10-2 dm3 atm K-1 mol-1
8.314 m3 Pa K-1 mol-1
6.022×1023 mol-1;
6.626×10-34 J s;
1.381×10-23 J K-1 (= 0.695 cm-1 K-1);
2.998×108 m s-1;
8.854×10-12 C2 J-1 m-1;
1.602×10-19 C;
760 torr = 1.013×105 N m-2 = 1.013×105 J m-3;
96,485 C mol-1;
1 kg m2 s-2;
1.602×10-19 J;
1 Watt
1×105 Pa
1.660×10-27 kg
1×10-15 s
Answer 3 of the following 4 Questions
QUESTION ONE:
Answer all parts (i) … (iii)
(i) Outline the Hückel theory for calculating the  molecular orbitals of planar
conjugated hydrocarbons.
[8 marks]
(ii) Calculate the Hückel orbital energies and wavefunctions for the molecule
butatdiaene, C4H6.
[8 marks]
(iii) Define bond order in Hückel theory and calculate the bond orders of the two
distinct bonds in butadiene.
[4 marks]
QUESTION TWO
Answer both parts (i) and (ii).
(i) Show that for a molecular orbital  in the form of a linear combination of atomic
orbitals in a basis set, { n } ,    cn n , the best choice of the coefficients {c n } is
n
given by the secular equations:

m
( H mn   S mn )cn  0 , for all n
defining all quantities involved.
[10 marks]
(ii) List the steps involved in solving these equations using the molecule HF as an
example where appropriate.
[10 marks]
QUESTION THREE
Answer both parts (i) and (ii).
(i)
Calculate the free enthalpy G in [kJ mol-1] of one mole of molecular nitrogen, N2,
at a temperature of T=298 K and a pressure of p=1 bar. The rotational constant of
N2 is B=2.01 cm-1 (symmetry number =2), the fundamental vibration is 2330
cm-1. The zero point energy can be neglected in this question (U0=0 kJ mol-1) and
assume that the particles are non-interacting.
[16 marks]
(ii)
How many percent of G is due to the translational energy of N2?
[4 marks]
QUESTION FOUR:
Answer all parts (i), (ii) and (iii).
The rotational constant of D2 (symmetry number =2) is 30.44 cm-1. Calculate the
rotational partition function at 30 K,
(i) by explicit summation (up to 4 summands) and
[8 marks]
(ii) by the equation derived in the lecture.
[4 marks]
(iii) Comment on the two results with respect to the validity of the equation used in (ii)
and the way it was derived.
[8 marks]
OLLSCOIL NA hEIREANN, CORCAIGH
THE NATIONAL UNIVERSITY OF IRELAND, CORK
COLAISTE NA hOLLSCOILE, CORCAIGH
UNIVERSITY COLLEGE, CORK
SUMMER EXAMINATIONS 2003
CHEM3
Module CM3105 – Chemical Physics
Prof M.A. Chesters, Prof R.P. Brint, Dr A.A. Ruth,
Prof W.B. Jennings
The use of non-programmable electronic calculators is permitted in this
examination. Use of programmable calculators is not permitted. Use
of any such calculators is a serious breach of discipline
For each module three questions are given as either/or choices. Students
are required to answer each question (1,2 and 3) choosing one of the given
choices. Answer all the parts of the elected option. Marks will not be
awarded for completing two choices of the same question. Three complete
answers are to be submitted in separate answer books.
1½ HOURS
CANDIDATES ARE ALLOWED 15 MINUTES READING TIME
USEFUL DATA:
Universal Gas Constant (R)
Avogadro’s Constant (NA)
Planck’s Constant (h)
Boltzmann’s Constant (k)
Speed of light (c)
Permittivity in a Vacuum (o)
Electronic Charge (e-)
1 atm
One Faraday, 1F
1J
1 eV
1 J s-1
1 bar
1 a.u.
1 fs
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
8.314 J mol-1 K-1 = 8.205×10-2 dm3 atm K-1 mol-1
8.314 m3 Pa K-1 mol-1
6.022×1023 mol-1;
6.626×10-34 J s;
1.381×10-23 J K-1 (= 0.695 cm-1 K-1);
2.998×108 m s-1;
8.854×10-12 C2 J-1 m-1;
1.602×10-19 C;
760 torr = 1.013×105 N m-2 = 1.013×105 J m-3;
96,485 C mol-1;
1 kg m2 s-2;
1.602×10-19 J;
1 Watt
1×105 Pa
1.660×10-27 kg
1×10-15 s
QUESTION ONE:
EITHER
1(A) Answer all parts (i) … (iii)
(i) Outline the Hückel theory for calculating the  molecular orbitals of planar
conjugated hydrocarbons.
[8 marks]
(ii) For the cyclic molecule C4H4 calculate the Hückel orbital energies and show that
the ground state is predicted to be a biradical.
[8 marks]
(iii) Comment (detailed calculation not required) on the comparison of the bond
orders in C4H4 and butadiene C4H6.
[4 marks]
QUESTION ONE:
OR
1(B) Answer both parts (i) and (ii).
(i) Show that for a molecular orbital  in the form of a linear combination of atomic
orbitals in a basis set, { n } ,    cn n , the best choice of the coefficients {c n } is
n
given by the secular equations:

m
( H mn   S mn )cn  0 , for all n
defining all quantities involved.
[10 marks]
(ii) List the steps involved in solving these equations using the molecule HF as an
example where appropriate.
[10 marks]
QUESTION TWO:
EITHER
2(A) Answer all parts (i) …(v)
Explain the following terms giving examples where appropriate:
(i)
extended basis set
(ii)
valence state ionization energy
(iii) group orbital
(iv)
the Wolfberg Helmholtz approximation for Hab
(v)
variation principle
[4 marks each]
QUESTION TWO:
OR
2(B) Answer all parts (i) … (iv).
(i) Write down the Eyring equation for the rate of the reaction O + O  O2.
[2 marks]
(ii) Explain (detailed derivation not required) the transition state model (preconditions
and assumptions) that lead to Eyring’s rate equation.
[10 marks]
(iii) Estimate the rate of the oxygen reaction O + O  O2 with an activation energy
of 1 kJ mol-1 at 300 K, assuming reasonable values (order of magnitude) for the
required partition functions (leave the volume unspecified).
[6 marks]
(iv) Convert the “ad-hoc” value of the rate coefficient estimated in part (iii) to the
units [litre mol-1 s-1].
[2 marks]
QUESTION THREE:
EITHER
3(A) Answer both parts (i), (ii) and (iii).
Tellurium atoms have the following exceptionally low-lying electronically excited
states known from atomic spectroscopy: ground state (five fold degenerate), 4707
cm-1 (not degenerate), 4751 cm-1 (threefold degenerate), and 10559 cm-1 (fivefold
degenerate).
(i)
Calculate the electronic partition function, qe, for tellurium atoms at T=298 K
and at T=6000K.
[8 marks]
(ii)
What is the molar internal electronic energy (Ue) of tellurium at T=298 K?
Use:
N q
Ue  
q 
[8 marks]
(iii)
Calculate the molar electronic entropy, Se, at T=298 K. (S=U/T + k ln Q)
[4 marks]
QUESTION THREE:
OR
3(B) Answer all parts (i) … (v).
(i) Find a general expression for the ratio of the translational partition function qt of
two different molecules of different mass at the same temperature and volume.
[4 marks]
(ii) Calculate the ratio of the translational partition function qt of the deuterium
molecule, D2, to that of the hydrogen molecule, H2. Interpret the value in terms of the
energy level density in D2 and H2.
[4 marks]
(iii) Find a general expression for the ratio of the translational partition function qt of
a molecule in a constant volume at two different temperatures.
[4 marks]
(iv) At what temperature is the translational partition function of 1 dm3 molecular
hydrogen twice as large as at 273 K (volume fixed).
[4 marks]
(v) Calculate the absolute value of qt at the temperature found in part (iv) for 1 dm3
molecular hydrogen. What is the meaning of that value?
[4 marks]
OLLSCOIL NA hEIREANN, CORCAIGH
THE NATIONAL UNIVERSITY OF IRELAND, CORK
COLAISTE NA hOLLSCOILE, CORCAIGH
UNIVERSITY COLLEGE, CORK
SUMMER EXAMINATIONS 2004
CHEM3
Module CM3109 – Chemical Physics I
Professor M.A. Chesters, Professor R.P. Brint
Professor W.B. Jennings
The use of non-programmable electronic calculators is permitted in
this examination. Use of programmable calculators is not permitted.
Use of any such calculators is a serious breach of discipline
Three questions are given as either/or choices. Students are required to
answer each question (1,2 and 3) choosing one of the given choices.
Answer all the parts of the question. Marks will not be awarded for
completing two choices of the same question. Three complete answers
are to be submitted in separate answer books.
1½ HOURS
USEFUL DATA:
Universal Gas Constant (R)
Avogadro’s Constant (NA)
Planck’s Constant (h)
Boltzmann’s Constant (k)
Speed of light (c)
Permittivity in a Vacuum (o)
Electronic Charge (e-)
1 atm
One Faraday, 1F
1J
1 eV
1 J s-1
1 bar
=
=
=
=
=
=
=
=
=
=
=
=
=
=
8.314 J mol-1 K-1 = 8.205×10-2 dm3 atm K-1 mol-1
8.314 m3 Pa K-1 mol-1
6.022×1023 mol-1;
6.626×10-34 J s;
1.381×10-23 J K-1 (= 0.695 cm-1 K-1);
2.998×108 m s-1;
8.854×10-12 C2 J-1 m-1;
1.602×10-19 C;
760 torr = 1.013×105 N m-2 = 1.013×105 J m-3;
96,485 C mol-1;
1 kg m2 s-2;
1.602×10-19 J;
1 Watt
1×105 Pa
Answer 3 of the following 4 Questions
QUESTION ONE:
Answer all parts (i), (ii) and (iii)
(i) Explain the terms, system, ensemble, microstate, configuration and most probable
configuration.
(6 marks)
(ii) The number of microstates in a configuration is given by, W  N T ! /( n0 !n1!n2 !...) .
Define the quantities NT, n0, n1, n2 and show that the concept of a most probable
configuration leads to the Boltzmann Distribution.
(7 marks)
(iii) For a system, constant (N, T, V), consisting of N rare gas atoms, the ‘molecular’
partition function is the translational partition function, q =  T 3/2 where  is a
constant. Show that the molar heat capacity at constant volume, Cm,V = 12.47 J mol-1
K-1 .
(7 marks)
QUESTION TWO:
Either:
Answer both parts (i) and (ii)
(i) Derive the expression for the rotational partition function of a diatomic molecule.
(10 marks)
(ii) For the D2 molecule, rotational constant 30.43 cm at a temperature of 100 K,
calculate the rotational partition function by explicit summation and from the
expression in (i). Account for any difference between the two values.
-1
(10 marks)
Or:
Answer all parts (i), (ii) and (iii)
(i)
Given the statistical thermodynamic expression for the Helmholz Free Energy,
A = -k lnQ , derive the expression for the Gibbs Free Energy,
G  nRT ln (q / N A ) , for a molecule with molecular partition function, q.
(7 marks)
(ii) Hence, derive an expression for the equilibrium constant of the reaction,
A  B  C , in terms of the molecular partition functions of the molecules A, B and
C.
(7 marks)
(iii) Explain, briefly, why the equilibrium constant of diatomic isotope exchange
reactions are equal to 4
(6
marks)
QUESTION THREE:
Answer both parts (i) and (ii)
(i) Two atomic orbitals on two separate atoms form the basis set for a molecular
orbital. Explain the role of the following in determining the extent of interaction:
(a) symmetry of the atomic orbitals
(b) energies of the atomic orbitals
(c) overlap of the atomic orbitals
[10 marks]
(ii) Calculate the Huckel molecular orbitals of the molecule H2C=NH, given hN = 1.5
and kN = 1.0. What is the charge on the N atom?
[10 marks]
QUESTION FOUR:
Answer three of the following:
Equal marks for each part, total 20 marks.
(i) Define the Hamiltonian Integrals, Hmn , used in calculating the energy and
composition of an LCAO molecular orbitals and explain approximations used for
values of the integrals.
(1/3 of 20 marks)
(ii) The results of a Huckel calculation on cylcobutadiene are given below. Calculate
the energies of the orbitals in terms of the  and  parameters and the bond order of
the bonds.
(1/3 of 20 marks)
(iii) Identify the five types of terms that occur in the molecular Hamiltonian, using
the H2 molecule as an example and give the number of terms of each type that
contribute to the Hamiltonian of methane.
(1/3 of 20 marks)
(iv) Give the sequence of steps involved in solving the secular equations/determinant
to calculate the LCAO molecular orbitals of a molecule.
(1/3 of 20 marks)
1/2
-1/2
-1/2
1/2
1 / 21/2
-1 / 21/2
1 / 21/2
-1 / 21/2
1/2
1/2
1/2
1/2
OLLSCOIL NA hEIREANN, CORCAIGH
THE NATIONAL UNIVERSITY OF IRELAND, CORK
COLAISTE NA hOLLSCOILE, CORCAIGH
UNIVERSITY COLLEGE, CORK
SUMMER EXAMINATIONS 2004
CHEM3
Module CM3109 – Chemical Physics I
Professor M.A. Chesters, Professor R.P. Brint
Professor W.B. Jennings
The use of non-programmable electronic calculators is permitted in
this examination. Use of programmable calculators is not permitted.
Use of any such calculators is a serious breach of discipline
Three questions are given as either/or choices. Students are required to
answer each question (1,2 and 3) choosing one of the given choices.
Answer all the parts of the question. Marks will not be awarded for
completing two choices of the same question. Three complete answers
are to be submitted in separate answer books.
1½ HOURS
USEFUL DATA:
Universal Gas Constant (R)
Avogadro’s Constant (NA)
Planck’s Constant (h)
Boltzmann’s Constant (k)
Speed of light (c)
Permittivity in a Vacuum (o)
Electronic Charge (e-)
1 atm
One Faraday, 1F
1J
1 eV
1 J s-1
1 bar
=
=
=
=
=
=
=
=
=
=
=
=
=
=
8.314 J mol-1 K-1 = 8.205×10-2 dm3 atm K-1 mol-1
8.314 m3 Pa K-1 mol-1
6.022×1023 mol-1;
6.626×10-34 J s;
1.381×10-23 J K-1 (= 0.695 cm-1 K-1);
2.998×108 m s-1;
8.854×10-12 C2 J-1 m-1;
1.602×10-19 C;
760 torr = 1.013×105 N m-2 = 1.013×105 J m-3;
96,485 C mol-1;
1 kg m2 s-2;
1.602×10-19 J;
1 Watt
1×105 Pa
Answer 3 of the following 4 Questions
QUESTION ONE:
Answer both parts (i) and (ii)
(i) Derive the expression for the molecular translational partition function of a
molecule of mass m in a system of volume V at temperature T.
Use this expression to show that for a monatomic gas, Cm,V = 12. 48 J mol-1 K-1
(10 marks)
(ii) For the molecule CO, calculate the value of the molecular partition function at 298
K and 1 atmosphere pressure.
(Data: M(C) = 12.00 amu, M(O) = 16.00 amu, B = 1.931 cm-1, υe = 2214.2 cm-1,
ground state 1Σg+ )
(10 marks)
QUESTION TWO:
Either:
Answer all parts (i), (ii) and (iii)
(i) Show that Boltzmann’s expression for entropy, S =( k/NT) ln W , is compatible
with the dependence of changes in Internal Energy on temperature (U = T dS – p dV ).
(5 marks)
(ii) From expressions for U and S in terms of the Partition functions Q, derive an
expression for G in terms of the molecular partition function q.
(5 marks)
(iii) Estimate the standard entropy, S o, at 100 K of H2 (g) given that it is reasonable to
assume that only the lowest energy rotational and vibrational levels are occupied at
that temperature. From this value estimate the standard entropy of D2 (g).
(10 marks)
Or:
Answer all parts (i), (ii) and (iii)
(i)
For the simple equilibrium A ↔ B, derive an expression for the equilibrium
constant in terms of the molecular partition functions of A and B.
(10 marks)
(ii) Consider the ionisation of hydrogen atoms as an equilibrium reaction:
H ↔ H + + e–
Estimate a value of the equilibrium constant at 2000 K and 1 atmos pressure, given
that the Ionisation Energy of H is 109697 cm-1.
(10 marks)
(
QUESTION THREE:
Answer both parts (i) and (ii)
(i) Two atomic orbitals on two separate atoms form the basis set for a molecular
orbital. Explain the role of the following in determining the extent of interaction:
(a) symmetry of the atomic orbitals
(b) energies of the atomic orbitals
(c) overlap of the atomic orbitals
[8 marks]
(ii) State the approximations used in Hückel theory and calculate the composition and
orbital energies of the π molecular orbitals of the propanyl radical, C3H7 .
[12 marks]
QUESTION FOUR:
Answer two of the following:
(i) Define the Hamiltonian Integrals, Hmn , used in calculating the energy and
composition of an LCAO molecular orbitals and explain approximations used for
values of the integrals. What effect does adopting a zero-overlap approximation have
on the values?
(10 marks)
(ii) Derive the Hückel (4n+2) rule for aromatic hydrocarbons and comment on
limitations to its application.
(10 marks)
(iii) Show that the best choice of the LCAO coefficients
c , in    c 
n
n
n
n
is
given by:
 H
m
mn
 S
mn
c  0
m
for all n, defining all terms used in the process.
(10 marks)