H.O. Pierson. Handbook of carbon, graphite, diamond and fullerenes. Properties, processing and applications. 1993

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.

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

k shell

L shell

Electrons

Electrons

2sp3

2sp3

2sp3

2sp3

;;

~~~~~~~~~~

( .I .‘_‘,’

;:*,.).I . _.I, . * ;

. :: ;., :. .

: .. ...

^

..-.

).,. ‘..,..“.I

,.;: .,.‘,..‘.#.,.

.... .::.

. “: __c.: f_ . 8.:‘.

...._.;:.;;.‘?.-,~~~

.., ..~:.::...

;:‘.~.~;‘:.j~::....)

. .;;, :.:.f: ....

(_

. .

.i . ._

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.

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.

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’.

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.

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

Figure2.10.

Tetrahedral hybridizationaxesofthefoursp30rbitals.

Negativelobes

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

atom

form

bonds.

The

heavily

lopsided

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

bond

a sigma

bond.

Each

tetrahedron

of the

hybridized

carbon

atom

(shown

in Fig.

2.10)

combines

with four

other

hybridized

atoms to

form

a

three-dimensional,

entirely covalent,

lattice structure,

shown schematically

in Fig. 2.12.

From the

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).

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

carbon

tetrachloride

molecule

(CC&)

is similar.

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

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

bonds

which

will

be considered

later.

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

the paraffin,

olefin and acetylene

hydrocarbons,

and

their

derivatives.

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.

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.

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

2pz sigma

2p orbital

free delocalized