Chemistry 2000 (Spring 2008)
Problem Set #2: Molecular Orbital Theory and Larger Molecules
Solutions
Textbook Questions
• Answers in Solution Guide that came with text.
Additional Practice Problems
1.
The images below show the valence molecular orbitals obtained for the carbonate ion via a
semi-empirical calculation. Both side views and top views are provided, and each MO has
.. -1
.. .-1
been assigned an identifying letter.
..O..
..O
..O ..
.
..
..O..
-1
A
I
(a)
B
J
C
D
K
C
..
.O.
E
L
..
..O..
-1
F
M
C
..-1
.O
. ..
..O
..
G
N
C
..-1
.O
. ..
H
O
P
In a semi-empirical calculation, the core molecular orbitals are not calculated. Instead, a “core
potential” is used to approximate the nucleus plus core electrons. How many core MOs are
there in a carbonate ion, and why is it reasonable to approximate them as part of a core
potential?
There should be four core MOs in the carbonate ion because there are four core AOs on the
atoms comprising the carbonate ion. (one 1s orbital on the central carbon atom plus one 1s
orbital on each of the three oxygen atoms)
Because the core MOs do not participate in bonding (instead closely resembling the atomic
orbitals from which they were made), it is reasonable to approximate them as part of a core
potential.
(b)
Which of the MOs shown above are σ-MOs?
A, B, C, D, E, F, H, I, L, N, O, P
(c)
Which of the MOs shown above are π-MOs?
G, J, K, M
(d)
Which two π-MOs form a degenerate pair?
J and K
(e)
Which of the π-MOs are bonding?
G
Which are nonbonding?
J and K
Which are antibonding?
M
(f)
Use the pictured MOs to help you construct an MO diagram for the valence π orbitals only of
the carbonate ion. Remember to include the valence π electrons on your MO diagram.
3π
2π
Energy
1π
Total # pi electrons = Total # valence electrons - # electrons in σ skeleton = 24 – 18 = 6
.. .-1
..O
.
..
..O..
-1
C
..
.O.
-1 ..
..O..
C
..O ..
.. -1
..O ..
..O..
..-1
.O
. ..
..O
..
C
..-1
.O
. ..
..O
..
C
..
.O
.
σ skeleton
2.
The connectivity of the allyl cation (C3H5+) is shown:
(a)
Draw the resonance structures for the allyl cation.
H
H
C
C
H
+
H
C
H
H
and
H
C
C
H
C
+
H
H
(b)
Draw a molecular orbital diagram for the π bonds only of the allyl cation.
Be sure to: (i) show the relative energies of the molecular orbitals,
(ii) draw a picture of each molecular orbital,
(iii) label each molecular orbital as bonding, nonbonding or antibonding,
(iv) include electrons on your diagram.
(c)
Calculate the average C–C bond order based on your MO diagram.
σ C-C bond order = 1
π C-C bond order = 2 bonding electrons ÷ 2 C-C links = 0.5
2 electrons per bond
Total C-C bond order = σ C-C bond order + π C-C bond order = 1 + 0.5 = 1.5
(d)
Based on your MO diagram, would adding two more π electrons change the average C–C bond
order? Why or why not?
No. The next two electrons go in a nonbonding π MO, and electrons in a nonbonding orbital
don’t affect bond order.
3.
The central C–C bond in butane (CH3CH2CH2CH3) is longer than the central C–C bond in
butadiene (CH2CHCHCH2). Use molecular orbital theory to explain why this is the case.
4.
(a) Explain the difference between a metal, a semiconductor, and an insulator.
Because of the extended structures of metals, semiconductors, and insulators bands (a
collection of molecular orbitals that lie very close in energy) are formed. For a metal there is no
band gap between the filled and the unoccupied bands, i.e., valence and conduction bands. A
semiconductor has a small and an insulator has a large band gap. These differences in the band
gap result in a metal being a electric conductor, an insulator exhibiting no electric conduction,
and a semiconductor exhibiting electric conduction with increasing temperature (where
electrons can be thermally excited to the conduction band).
(b) How can you increase the conductivity of a semiconductor?
The conductivity of a semiconductor can be increase by doping the semiconductor with an
element that has an additional electron with an energy level that is just below the conduction
band (n-type doping) or with an element that has one electron less than the semiconductor with
an energy level that is just above the valence band (p-type doping).