26 Carbon, Graphite, Diamond, and Fullerenes
S Orbital
P Orbital
Figure 2.6. Schematic representation of the “s”and “p”orbitals.
with its spin |
uncoupled |
|
from |
the |
|
other |
electrons. |
This |
alteration |
occurs |
as |
|||||||||||
a result |
of the formation |
of hybridatomic |
orbitals, |
in which |
the |
arrangement |
|
|||||||||||||||
of the |
electrons |
of the |
I_shell |
of the |
atom in the ground |
state |
is modified |
as |
||||||||||||||
one of the 2s electron |
is promoted |
(or lifted) |
to the higher |
orbital |
2p as shown |
|||||||||||||||||
in Fig. 2.7. |
These |
new |
orbitals |
are |
called |
hybrids |
since they |
|
combine |
the |
||||||||||||
2s and the 2p orbitals. |
They |
are labeled |
sp3 since |
they |
are formed |
from |
one |
|||||||||||||||
s orbital |
and three |
p orbitals. |
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In this |
hybrid |
sp3 state, |
the |
carbon |
atom has four 2sp3 orbitals, |
instead |
|||||||||||||||
of two 2s and two 2p of the ground-state |
atom and the valence |
state |
is raised |
|||||||||||||||||||
from |
two to four. |
|
The |
calculated |
|
sp3 electron-density |
contour |
is shown |
in |
|||||||||||||
Fig. 2.8 and a graphic |
visualization |
of the orbital, in the |
shape |
of an electron |
||||||||||||||||||
cloud, |
is shown |
in Fig. 2.9.n2) |
This |
orbital |
is asymmetric, |
with |
most |
of |
it |
|||||||||||||
concentrated |
on one |
side and with |
a small |
tail on the |
opposite |
|
side. |
|
|
The Element Carbon |
27 |
Carbon Atom Ground State
k shell |
L shell |
Electrons |
Electrons |
2Px *P, 2Pz
SP3
Hybridization
|
|
2sp3 |
2sp3 |
|
2sp3 |
|
2sp3 |
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;; |
~~~~~~~~~~ |
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( .I .‘_‘,’ |
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;:*,.).I . _.I, . * ; |
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. :: ;., :. . |
: .. ... |
^ |
||
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..-. |
).,. ‘..,..“.I |
,.;: .,.‘,..‘.#.,. |
|||||||
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.... .::. |
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. “: __c.: f_ . 8.:‘. |
...._.;:.;;.‘?.-,~~~ |
||||||
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.., ..~:.::... |
;:‘.~.~;‘:.j~::....) |
. .;;, :.:.f: .... |
(_ |
||||||
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. . |
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.i . ._ |
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|
Figure 2.7. The sp3 hybridization |
|
of carbon orbitals. |
Shaded |
electrons |
are valence |
||||||
electrons |
(divalent |
for ground state, |
tetravalent |
for |
hybrid |
state). |
Arrow indicates |
||||
direction |
of electron |
spin. |
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As |
shown in |
Figs. 2.8 and 2.9 (and in following |
related |
figures), |
the |
|||||
lobes |
are labeled |
either |
+ or -. |
These signs |
refer to the sign |
of the wave |
||||
function |
and not to any |
positive |
or negative |
charges |
since |
an electron |
is |
|||
always |
|
negatively |
charged. When an orbital |
is separated |
by a node, |
the |
||||
signs |
are opposite. |
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28 Carbon, Graphite, Diamond, and Fullerenes
I |
Nodal Surface |
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\ |
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\\ |
0.3 |
0.1 |
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-0. |
‘\\ |
p;4 |
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0.2\ |
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-0.1 |
‘\ |
- /I |
” |
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-0.4,// |
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I’ |
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: |
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PbBohr Radius (a,) |
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Figure 2.8. Electron |
density |
contour |
of sp3 orbitaLI”] |
Figure 2.9. Cloud representation of sp3 hybrid orbitaLI’*]
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The Element Carbon |
29 |
|||||||||
A |
graphic |
visualization |
|
of |
the formation |
of |
the |
sp3 |
hybridization |
is |
|||||||||||||
shown |
in Fig. |
2.10. |
|
The |
four |
hybrid |
sp3 orbitals |
(known |
as tetragonal |
||||||||||||||
hybrids) |
|
have |
identical |
shape |
but |
different |
spatial |
orientation. |
Connecting |
||||||||||||||
the end |
|
points |
of these |
vectors |
(orientation |
of maximum |
probability) |
forms |
|||||||||||||||
a regular |
tetrahedron |
|
(i.e., |
a solid |
with four |
plane |
faces) |
with |
equal |
angles |
|||||||||||||
to each |
|
other |
of 109” 28’. |
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The energy |
required |
to accomplish |
the sp3 hybridization |
and |
raise |
the |
|||||||||||||||||
carbon |
atom from the |
ground |
state to the |
corresponding |
valence |
state V, |
|||||||||||||||||
is 230 |
kJ mol-‘. |
This |
hybridization |
is possible |
only |
because |
the |
required |
|||||||||||||||
energy |
is more than |
compensated |
|
by the energy |
decrease |
associated |
with |
||||||||||||||||
forming |
|
bonds |
with |
other |
atoms. |
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The hybridized |
atom |
is now |
ready |
to form |
a set |
of bonds with |
other |
||||||||||||||||
carbon |
atoms. |
It should |
be stressed that these |
hybrid |
orbitals |
(and indeed |
|||||||||||||||||
all hybrid |
orbitals) |
are formed |
only |
in the |
bonding |
process |
|
with other |
atoms |
||||||||||||||
and are |
not representative |
|
of an actual |
structure |
|
of a free |
|
carbon |
atom.[131 |
Axis below
plane of page
Figure2.10. |
Tetrahedral hybridizationaxesofthefoursp30rbitals. |
Negativelobes |
omitted for clarity.
30 Carbon, Graphite, Diamond, and Fullerenes
3.3The Carbon Covalent sp3 Bond
|
As mentioned |
above, |
carbon |
|
bonding |
is covalent |
and in the case of the |
||||||||||||||||||||||||||
sp3 bonding, |
the |
atoms |
share |
a |
|
pair |
|
of |
electrons. |
The |
four |
sp3 valence |
|||||||||||||||||||||
electrons |
of the hybrid carbon |
atom, together |
with the small size of the atom, |
||||||||||||||||||||||||||||||
result |
|
in strong covalent |
bonds, |
since |
four |
of the six |
electrons |
of the |
carbon |
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atom |
form |
bonds. |
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The |
heavily |
lopsided |
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configuration |
|
of the |
sp3 orbital |
allows |
a substan- |
|||||||||||||||||||||||
tial overlap |
and |
a strong |
bond |
when |
the |
atom |
combines |
with |
a sp3 orbital |
||||||||||||||||||||||||
from |
another |
carbon |
atom |
since |
the |
concentration |
|
of |
these |
bonding |
|||||||||||||||||||||||
electrons |
between |
the |
nuclei |
|
minimizes |
|
the |
nuclear |
repulsion |
and |
|
maxi- |
|||||||||||||||||||||
mizes |
the attractive |
forces |
between |
themselves |
|
and both |
nuclei. |
This |
bond |
||||||||||||||||||||||||
formation |
is |
illustrated |
in |
Fig. |
2.11. |
|
By |
convention, |
a |
directional |
|
(or |
|||||||||||||||||||||
stereospecific) |
orbital such |
as the sp3 is called |
a sigma (a) orbital, |
and the |
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bond |
a sigma |
bond. |
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Each |
tetrahedron |
of the |
hybridized |
carbon |
atom |
(shown |
in Fig. |
2.10) |
||||||||||||||||||||||||
combines |
with four |
other |
hybridized |
|
atoms to |
form |
a |
three-dimensional, |
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entirely covalent, |
lattice structure, |
shown schematically |
in Fig. 2.12. |
From the |
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geometrical |
standpoint, |
the |
carbon nucleus |
|
can be considered |
as the |
center |
||||||||||||||||||||||||||
of a cube |
with |
each |
of the |
four |
orbitals |
|
pointing |
to four |
alternating |
corners |
of |
||||||||||||||||||||||
the cube. |
This structure |
is the |
basis |
of the |
diamond |
crystal (see Ch. |
11). |
|
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|
A similar |
tetrahedral |
|
bonding |
arrangement |
is also found |
in the |
|
meth- |
||||||||||||||||||||||||
ane molecule |
where |
the hybridized |
|
carbon |
atom is bonded to four hydrogen |
||||||||||||||||||||||||||||
atoms. |
Four molecular |
orbitals |
are formed |
by combining |
each |
of the carbon |
|||||||||||||||||||||||||||
sp3 orbitals |
with |
the |
orbital |
of the |
|
attached |
hydrogen |
atom (Fig. |
2.13). |
The |
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carbon |
tetrachloride |
molecule |
(CC&) |
|
is similar. |
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The |
tetragonal |
angle |
|
of 109”28’ |
of the sigma-bond |
molecules |
must |
be |
||||||||||||||||||||||||
considered |
as a time-averaged |
value |
|
since |
it changes |
continuously |
|
as the |
|||||||||||||||||||||||||
result |
|
of thermal |
vibrations. |
|
The |
|
sigma-bond |
energy |
and the |
bond |
length |
||||||||||||||||||||||
will vary |
depending |
on |
the kind |
|
of |
atom |
which |
is attached |
to the |
carbon |
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atom. |
Table |
2.7 shows |
the |
bond |
energy |
and |
the |
bond |
length |
of various |
|||||||||||||||||||||||
carbon |
couples. |
The bond |
energy |
is the energy |
required |
to break one |
mole |
||||||||||||||||||||||||||
of bonds. |
|
An identical |
amount |
|
of |
energy |
is |
released |
when |
|
the |
bond |
is |
||||||||||||||||||||
formed. |
Included |
are the |
double |
|
and triple |
carbon |
bonds |
and |
other |
carbon |
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bonds |
which |
will |
be considered |
later. |
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The |
sp3 bonds |
listed |
|
in Table |
2.7 |
are found |
in all aliphatic |
compounds |
||||||||||||||||||||||||
which |
|
are organic |
compounds |
with an open-ended |
chain structure |
and include |
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the paraffin, |
olefin and acetylene |
|
hydrocarbons, |
|
and |
their |
derivatives. |
|
|
|
The Element Carbon 31
Figure 2.11. The sp3 hybrid orbital bonding (sigma bond) showing covalent bonding.
Figure |
2.12. Three-dimensional |
representation of sp3 covalent bonding (diamond |
|
structure). Shaded regionsareregionsof |
highelectronprobabilitieswherecovalent |
||
bonding |
occurs. |
|
|
32 Carbon, Graphite, Diamond, and Fullerenes
Figure 2.13. Three-dimensional representation of the methane molecule (CH,) with sigma (sp3) bonding. Shaded regions are regions of high electron probabilities where covalentbonding occurs.t41
Table 2.7. Carbon-Couples |
Bond Energies and Lengths |
|
||
|
|
|
|
Bond |
|
Hybrid |
Approximate |
bond energy* |
length |
Bond |
type |
kJ/mole |
kcal/mole |
nm |
c-c |
sps |
370 |
88 |
0.154 |
c=c |
sps |
680 |
162 |
0.13 |
c=c |
sp |
890 |
213 |
0.12 |
C-H |
sp3 |
435 |
104 |
0.109 |
C-Cl |
sps |
340 |
81 |
0.18 |
C-N |
sp3 |
305 |
73 |
0.15 |
c-o |
sps |
360 |
86 |
0.14 |
* Energy required to break one mole of bonds (Avogadro’s number)
The Element Carbon |
33 |
4.0 THE TRIGONAL sp2 AND DIGONAL sp CARBON BONDS
4.1The Trigonal sp2 Orbital
In |
addition |
to the |
sp3-tetragonal |
|
hybrid orbital |
reviewed |
in |
Sec. |
3 |
|||||||||||
above, |
two |
other orbitals complete the |
series |
of electronic |
building |
blocks |
||||||||||||||
of all carbon |
allotropes |
and compounds: |
the |
sp2 and the sp orbitals. |
|
|
||||||||||||||
Whereas |
the sp3 |
orbital |
is the |
key to |
diamond |
and |
aliphatic |
|
com- |
|||||||||||
pounds, the sp2 (or trigonal) |
orbital is the |
basis |
of all graphitic |
structures |
and |
|||||||||||||||
aromatic |
compounds. |
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The |
mechanism of the sp2 hybridization |
is somewhat |
different |
from |
||||||||||||||||
that of the sp3 hybridization. |
The arrangement |
of the electrons |
of the L shell |
|||||||||||||||||
of the atom |
in the ground |
state |
is modified |
as one |
of the |
2s |
electrons |
is |
||||||||||||
promoted |
and combined |
with |
two |
of the 2p orbitals |
(hence |
the designation |
|
|||||||||||||
sp2), to form three sp2 orbitals |
and an unhybridized |
free |
(or delocalized) |
p |
||||||||||||||||
orbital |
electron |
as shown |
in Fig. 2.14. |
The valence |
state |
is now |
four |
(V4). |
||||||||||||
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|
Carbon |
Atom |
Ground |
State |
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k shell |
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L shell |
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Electron |
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Electrons |
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1 |
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Hybridization
1
1 |
Is |
1 2sp3 1 2sp3 1 2SP2 ( 2P 1 |
Figure2.14. Thesp’hybridizationofcarbonorbitals. Shadedelectronsarevalence electrons (divalent for ground state and tetravalent for hybrid state).
34 |
Carbon, |
Graphite, |
Diamond, |
and |
Fullerenes |
|
|
|
||||||
The |
calculated |
electron-density |
contour |
of the |
sp2 orbital is similar |
in |
||||||||
shape |
to |
that |
of the |
sp3 orbital |
shown in |
Figs. |
2.8 |
and 2.9. |
These three |
|||||
identical |
sp2 orbitals |
are in the same |
plane |
and their orientation |
of maximum |
|||||||||
probability |
forms |
a 120° |
angle |
from |
each |
other |
as shown in Fig. 2.15. |
|
||||||
The |
fourth |
orbital, |
i.e., the delocalized |
non-hybridized |
p electron, |
is |
||||||||
directed perpendicularly |
to the |
plane |
of the three |
sp2 orbitals |
and becomes |
|||||||||
available |
to form |
the |
subsidiary |
pi (n) bond |
with |
other atoms. |
|
Figure 2.15. Planar section of the sp* hybrid orbitals of the carbon atom.
4.2The Carbon Covalent sp2 Bond
Like the sp3 bond, the sp bond is covalent. |
It is a strong bond, |
because |
||||||||
of the three s$ |
valence |
electrons and the small |
size of the atom. |
|
|
|
||||
The |
lopsided configuration |
of the spz orbital |
allows a substantial |
overlap |
||||||
with other |
s$ |
orbitals. |
|
This overlap is similar |
to the sp3 overlap illustrated |
in |
||||
Fig. 2.11, |
except that |
it is more |
pronounced, |
with a shorter bond |
length and |
|||||
higher bond energy, |
as shown |
in Table 2.7. |
Like the sp3 orbital, |
the s$ |
is |
|||||
directional |
and is called |
a sigma |
(a) orbital, and the bond a sigma |
bond. |
|
The Element Carbon 35
Each sps-hybridized carbon atom combines with three other sps-hybrid-
ized atoms to form a series of hexagonal structures, all located in parallel
planes |
as shown schematically in Fig. 2.16. The fourth valency, that is, the free |
|
delocalized electron, is oriented perpendicularto this plane |
as illustrated in Fig. |
|
2.17. |
Unlike the sigma (a) orbital, it is non-symmetrical |
and is called by |
convention a pi (n) orbital. It is available to form a subsidiary pi (I-C)bond.
6.71 A
• Carbon atom
Figure 2.16. Three-dimensional schematic of the graphite structure.
2pz sigma |
2p orbital |
|
free delocalized |
||
|
orbitals
Figure 2.17. Schematic of the sp* hybridized structure of graphite showing the sigma bonds and the 2pfreeelectrons (above and belowthe sigma orbit& plane).