Download Chemistry 110

Chemistry 327: Applied Physical Chemistry
These questions are assigned as homework. They are in addition to the problems I’ve
assigned in previous mailings and handouts but whereas those question sets were
designed to lead you through a specific topic- these are independent and require you to
use some of the models we’ve developed. Please feel free to work together, and for God’s
sake ask questions! p.s. Don’t expect to finish all of these things in a single night!
Question 1
The wave properties of neutrons are frequently exploited to probe the crystal
structures of solids. This is done by shooting beams of neutrons and watching how
diffract through the lattice structures of those crystals. What is the velocity of a neutron
that would be useful to probe the structure of a NaCl crystal which has a lattice spacing
of about 2.5 x 10-10 m. The mass of a neutron is 1.68 x10-27 kg.
Question 2:
Write an (un normalized) wavefunction for a free neutron with the velocity (in
one dimension) you found in question 1. What information do you have about the
momentum of this particle? What information do you have about its position? How
would you improve information about the position of the neutron and at what cost?
Question 3
What is the interpretation of the complex conjugate of the wavefunction you
found in question 3? Prove it
Question 4
What is the minimum peak width of a UV fluoresence signal originating from an
excited state with a lifetime of 1.75 x10-12 seconds? (Use the energy-time uncertainty
relationship)
Question 5
Calculate the commutator pˆ , x  for the free particle wavefunction
k(x)=Akeikx What is the physical significance and consequence of this relationship?
Question 6

What is the zero-point energy of the 2-D Particle in a square box of dimensions
LxL? How does this zero point energy change with the dimensions of the box and the
mass of the particle?
Question 7
What is the ultimate source of the zero point energy and energy quantization in
any physical system?
Question 8
Write the Hamiltonian operator for the 3-D Particle in a box. What is the zero
point energy of the 3-D Particle in a box? What is the degeneracy of this ground state?
What is the energy of the first excited state of the 3-D PIB and its degeneracy? What is
the relationship between the degereracy of a system and it symmetry?
Question 9
Suppose you have an electron (H+) stuck between two negatively charged, parallel
cell membranes that are 6.7 nm apart (assume that electrical potential between the
membranes is essentially zero while and jumps very quickly to a huge value, i.e. infinity,
at the membrane itself).. What is zero point energy of the electron? What is the energy
of the lowest excited state of the electron? What wavelength of light would be used to
probe this excitation?
Question 10
Consider 3D box of dimensions L1 x L2 x L3 (L1=L2 =1.5 L2). Write its
Hamiltonian. What is the zero point energy and degeneracy of the ground state
wavefunction? What about the first excited state?
What can you say about the relationship between energetic degeneracy and coordinate
systems?
Question 11
How does the zero point energy of a harmonic oscillator change with the force
constant, k? How does it change with the mass of the particle?
Question 12
What is the zero point energy for the vibration of a hydrogen atom tethered to a
metal surface? Assume that the force constant, k, for this hydride bond is 312 N/m.
Assume also that the mass of the metal surface can be regarded as infinite.
Question 13
Estimate the energy required to excite the vibration of the tethered hydrogen atom
in the question above to its first excited state. What is the wavelength of the photon
required to effect this transition? What is this energy in terms of cm-1? What is the
energy required to effect the transition from the first excited vibrational state to the
second excited vibrational state?
Question 14
Calculate the probability that the hydrogen attached to the metal surface in
questions 12 and 13 will “tunnel” into the surface of the metal. That is, calculate the
probability that the hydrogen will be at a value of x that would put it within the metal
surface. Assuming that the metal-hydrogen bond length is 0.75 Å, this would be the total
probability that the hydrogen is between x = -0.75 Å and -∞.
Question 15
Repeat questions 12,13, and 14 for the 2H isotope, deuterium.
Question 16
Sketch the ground state, first excited state, and second excited state wavefunctions
of the quantum mechanical harmonic oscillator. How many nodes are in the 4th excited
state of the harmonis oscillator?
Question 17
Calculate the expectation values of position <x> and momentum <p>, where x is
understood as the of the bond length from the equillibrium bond distance and p is the
momentum of the vibrational motion, for the ground state and the 1stexcited states of the
harmonic oscillator. How does this value change with increasing vibrational quantum
number v?
Question 18
Calculate the expectation value of the square of position <x2> and momentum
<p2>, for the ground state and the 1st and 2nd, excited states of the harmonic oscillator.
How does this value change with increasing vibrational quantum number v? How does
this result compare with that from the previous question does this make sense. How does
this correspond the ‘classical’ limit?
Question 19
Calculate the zero point vibrations for each of the following diatomic gasses
having their respective force constants:
F2(k=1238 N/m), Cl2 (k=1175 N/m), Br2(k=1076 N/m).
Question 20
The of the force constant in both 1HF and 2HF (DF) is 1845 N/m. What are the
zero point vibrational energies of each? What wavelengths of light are needed to excite
each to their respective first excited states?
The next 3 questions consider deal with the molecule HCl. For the purposes of these
problems, we do not consider the molecule to undergo a H-Cl stretching vibration and
therefore the bond length between the two atoms is fixed at 1.3 Å. The atomic mass of the
Cl atom is 35.00 g/mol and that of the H atom is 1.00 g/mol. (hint: at some point you
may need to convert to kg per atom)
Question 21|
Determine the reduced mass, , and moment of inertia, I, of the HCl molecule.
Question 22|
Assume that the HCl molecule is free to rotate in only 2 dimensions, i.e. it is
restricted to the x,y plane. One way to think of this is that the HCl molecule is restricted
to a crystal surface and can rotate on this surface but cannot ‘tip up’ into +z direction.
The surface does not otherwise affect the HCl molecule. This is very much like a ‘particle
on a ring.’
a) What is the lowest value of angular momentum possible for the HCl molecule on this
surface? (Remember, angular momentum is a vector quantity so it has both a magnitude
and a direction) What is the rotational kinetic energy of this state? What is its
degeneracy? Write the wavefunction or wavefunctions (if degenerate) for this
‘rotational’ wavefunction. (make sure that they are normalized)
b) What is the angular momentum of the first excited state of the 2D HCl molecule?
What is the rotational kinetic energy of this state? What is its degeneracy? Write the
wavefunction or wavefunctions (if degenerate) for this rotational wavefunction.
c) What is the angular momentum of the second excited state of the 2D HCl molecule?
What is the rotational kinetic energy of this state? What is its degeneracy? Write the
wavefunction or wavefunctions (if degenerate) for this rotational wavefunction.
d) What is the energetic difference between the ground and first excited rotational states?
What is the energetic difference between the first and second excited rotational states?
e) Calculate the wavelengths of the photons required to effect each of the transitions
described in part d.
f) Why are these rotational energies quantized in the first place?
Question 23|
Free at last, our intrepid HCl molecule is liberated from the crystalline surface. It
is now free to rotate in 3 dimensions. This is very much like a ‘particle on a ball.’
a) What is the lowest value of angular momentum possible for the freely rotating HCl
molecule? What are the possible ‘orientations’ of this angular momentum vector along
the z-axis? What is the rotational kinetic energy of this state? What is its degeneracy?
Write the wavefunction or wavefunctions (if degenerate) for this ‘rotational’
wavefunction. (make sure that they are normalized)
b) What is the angular momentum of the first excited state of the 3D HCl molecule?
What are the possible ‘orientations’ of this angular momentum vector along the z-axis?
What is the rotational kinetic energy of this state? What is its degeneracy? Write the
wavefunction or wavefunctions (if degenerate) for this rotational wavefunction.
c) What is the angular momentum of the second excited state of the 3D HCl molecule?
What are the possible ‘orientations’ of this angular momentum vector along the z-axis?
What is the rotational kinetic energy of this state? What is its degeneracy? Write the
wavefunction or wavefunctions (if degenerate) for this rotational wavefunction.
d) What is the energetic difference between the ground and first excited rotational states?
What is the energetic difference between the first and second excited rotational states?
e) Calculate the wavelengths of the photons required to effect each of the transitions
described in part d.
f) Why are these rotational energies quantized in the first place?
Question 24|
Electrons behave as if they were tiny ‘particle on a ring’ because they have their
very own intrinsic angular momentum that we call spin. This kind of angular
momentum, however, is quite different than orbital angular momentum, for instance,
because it can occur in half-odd integer multiples of , i.e 1/2 , 3/2. etc. Make a
drawing that explains how the particle on a ring molecule can be used to describe this
type of weird angular momentum.
Question 25
Why are 4n+2 pi electrons required for Huckel aromaticity?
Question 26:
Estimate the wavelength of light necessary to excite an electron from the HOMO
to the LUMO of benzene. The distance from the center of the ring to any carbon is 1.4 Å.
Question:
Why can we picture the ‘localized’  molecular orbitals of benzene like we do.
What information do we lose about those  electrons when we localize the particles on
the ring? What information, if any, do we lose about the energy of these electrons.
Explain.