Ch120a: Nature of the Chemical Bond - Fall 2016
Problem Set 4
TA: Shane Flynn ([email protected])
Problem 1: Bond Energy of H2
A
To understand the interactions between hydrogen atoms one could use the Morse potential, [ V (R) = D(1 − e−β(R−r0 ) )2 ].
For a hydrogen molecule, D = 109 kcal/mol, B=1.93Å−1 , and r0 = 0.74 Å. Plot this potential curve (you must use some
”real’ plotting software ie mathematica, python, etc and include the source code).
B
Taylor Expansions are a very useful tool for understanding the nature of a physical system. Although this is not a math class
I think it is an essential skill, therefore I would like for you to do this problem as explicit as possible (ie state all assumptions
and show the algebra for each step an answer only will receive NO credit).
Show that if R ≈ r0 this potential can be approximated as a harmonic potential. What is the value of you spring constant
(k)? Based on this approximation, what is the zero point energy at the ground state (we generally use kcal/mol for units in
electronic structure).
HINT: The ground state energy of the Morse Potential is given by E(0) = 12 ~ω.
What is D0 of H2 ?
C
Assume H2 can be described by the ideal gas equation, find D298K of H2 .
HINT: Recall the bond dissociation energy is also called the enthalpy in thermodynamics, and we know the Cp of an ideal
gas linear molecule .
Problem 2: Ziegler-Natta Catalysis
Valence bonds to metals can be s-like or d-like, which influences whether pericyclic reactions involving those bonds are allowed
or forbidden. Consider the Ziegler-Natta polymerization reaction shown below:
A
Draw the transition states for the concerted polymerization reaction, illustrate its orbitals, and determine the number of
bonds in the transition state. Draw the transition state! Is this reaction symmetry allowed or forbidden?
B
In this reaction, why is the Cl2 HTi-CHCH2 Cl not an observed product (why is the Cl not transferred as the H was)? Use
orbital illustrations to explain.
C
If you tried to use the active catalyst CdH+ instead of CL3 TiH, should you expect the reaction to proceed? (Explain, you
should be able to use the paterns in A and B to explain this trend).
1
D
What about the active catalyst PdH+ instead of CL3 TiH, should you expect the reaction to proceed now? (Explain)
E
Should Cr-H or Cr-H+ have a higher barrier for H-D exchange in the reaction:
M - H + D2 → M - D + H - D
F
Now consider the reaction:
M - CH3 + D2 → M - D + CH3 - D
How would the barrier change from the correcponding exchange in Section E?
Problem 3: Surface Energy
We will analyze the surface energy of some cubic crystals, focusing on the following crystal planes: (100), (110), and (111).
Assume the lattice parameter is a (or set it = 1 for convenience), and the bond energy between two atoms is .
Comment: Please make a table where the y axis has your 3 different surfaces for the 3 different cubic structures (9 total
entries) and your x axis has the Atoms/surface, atoms/surface area, broken bonds/atom, energy/surface area.
A
For a simple cubic structure (SC), calculate the surface-atom density for the (100), (110), and (111) planes.
B
How many bonds were broken during the cleavage of each plane?
C
What is the surface energy of each of these planes? (i.e., what is the total energy of the broken bonds per atom multiplied
by its surface-atom density?)
HINT: Cleaving a solid generates two surfaces, so each surface is assumed to get half of the total bond energy.
D
Repeat parts a through c. with the body centered cubic (BCC) structure.
E
Repeat parts a through c with the face centered cubic (FCC) structure.
F
What correlations do you observe between surface atom density and surface energy in each of these structures? Can you
make a general statement about surface atom density and its relative energy?
Problem 4: Graphene
A
Draw the structure of graphene. Denote the boundaries of the unit cell. How many carbon atoms are in each unit cell of
graphene?
B
The C-C bond length is 1.34Å in ethylene, 1.40Å in benzene, and 1.54Å in ethane. What is the average C-C bond order in
graphene? What is the trend in bond length versus bond order?
2
C
A graphene sheet may be regularly cleaved in two patterns, which we’ll call “armchair” and “zig-zag”. For each structure
draw the dangling bonds. Will these dangling bonds be left open, or is there some way for them to interact in an energetically
favorable manner (if so draw how)?
D
The average C-C bond energy in graphene is 111.7 kcal/mol. The interaction of dangling bonds, if there is any, has a strength
of 20 kcal/mol due to strain. Using the numbers given in this problem, what is the edge energy for each edge? Make your
units in kcal/mol Å. When a graphene sheet is cleaved, do you expect the edges formed to be armchair, zig-zag, or a mixture?
3