Chemistry Reference
In-Depth Information
neigbours, lying in the same plane as the carbon atom, with a bond
angle of 120
◦
between eachpair of neighbours. Determine the formof
the sp
2
hybrid orbitals on the central carbon atom if its three nearest
neighbours all lie in the
xy
-plane, and one of the three neighbours is
along the
+
x
direction.
2.9
N
identical atoms each have a single electron at energy
E
a
. The atoms
are brought together to form an
N
-membered ring, in which each
atom interacts with its two neighbouring atoms, with an interaction
of strength
U
(
U
<
0
)
. It can be shown that the eigenstates,
ψ
(
r
)
of
n
this ring can be expressed in the form
N
e
i2
π
mn
/
N
ψ
(
r
)
=
φ
(
r
)
n
m
m
=
1
where
n
is the atomic orbital on the
m
th
atom. Show that the allowed energy levels in the
N
-membered ring
are given by
=
0, 1,
...
,
N
−
1, and
φ
(
r
)
m
E
n
=
E
a
+
2
U
cos
(
2
π
n
/
N
)
.
Given that each energy level can contain two electrons, calculate
the ground state binding energy per atom for all ring sizes between
N
. The model here is appropriate
to describe the interactions between neighbouring p
z
orbitals in sp
2
-
bonded carbon. Hence, provide two reasons why 6-membered sp
2
-
bonded carbon rings are strongly favoured (e.g. as in benzene, C
6
H
6
=
3and
N
=
8, and for
N
=∞
)
compared to other ring sizes.
2.10 Show that
λ
E
g
=
1.24
µ
meV, where
E
g
is a photon energy in electron
volts and
is its wavelength in microns. The III-V alloy InAs
x
Sb
1
−
x
has a fraction
x
of the group V sites occupied by arsenic (As) atoms,
and a fraction
λ
occupied by antimony (Sb) atoms. The energy
gap of InAs
x
Sb
1
−
x
(measured in eV) has been determined to vary
with composition at room temperature as
(
1
−
x
)
E
g
(
x
)
=
0.17
+
0.19
x
+
0.58
x
(
x
−
1
)
Determine the composition of the alloy with the lowest room tem-
perature energy gap, and hence estimate an upper limit on the
room temperature emission wavelength of conventional bulk III-V
semiconductors.
