Physical Chemistry , 1st ed.

as opposed to the nonaromatic aliphatic hydrocarbons. The Hückel approxi-

mation provides some clues for benzene’s distinctions.

Benzene has six carbon atoms arranged in a ring, each contributing one p

electron to the molecular orbitals. Therefore, the 6 6 determinant con-

structed using Hückel’s approximations looks like this:



 E  0 0 0  E



 E  0 0 0

0  E  0 0

0 0  E  0

0 0 0  E 

 0 0 0  E

 0 (15.26)

The only real difference between equation 15.26 and the earlier Hückel deter-

minants is the presence ofin the upper right and lower left corners. This is

because the molecule is cyclic and the first carbon atom is adjacent to the sixth

carbon atom.

Evaluating the above determinant requires solving a polynomial that is sixth

order in E(that is, the highest power ofEin the polynomial is E^6 ). Upon solv-

ing for the values ofEin terms ofand (which will not be shown here),

one finds the following values for E:

 2 ,

,

, , ,

and     2 . Two of the energies,

and   , are also doubly degener-

ate. An energy level diagram of these molecular orbitals is shown in Figure

15.21, along with the six electrons in the three lowest orbitals.

There are two points about the orbitals of benzene. First, all of the net

“bonding” orbitals (the orbitals having lower energy than the 2pelectrons in

the carbon atom, which have an energy of) are completely filled. Therefore,

the benzene molecule experiences the maximum possible decreasein overall

energy—and therefore the maximum possible increase in stability—that it

can. (Indeed, it is somewhat akin to the diatomic nitrogen molecule, which

has three pairs of electrons in bonding molecular orbitals.) Therefore, we ex-

pect that benzene should be more stable than expected, and it is. Second, con-

sider the delocalization energy. The total energy of the six electrons is

2(

 2 ) 4(

)  6 

 8 . Compare this to three units of ethylene

(the system against which all delocalization energies are compared), which for

the six electrons would have a electron energy of 6(

)  6 

 6 .

Benzene therefore has 2more of a decrease in energy, representing a delocal-

ization energy of approximately 150 kJ/mol. This is more than four times the

delocalization energy of butadiene, which at 0.472of delocalization energy

represents a decrease in energy of only 35.4 kJ/mol. Benzene is much more

stable than expected simply on the basis of having three double bonds! This

unexpected (but explainable, in terms of the Hückel approximation) stability

of benzene is given a name:aromaticity.Benzene is aromatic.The name was

derived from the pungent odors of benzene and benzene-related compounds.

It now specifically refers to the increased stability of certain cyclic -electron-

containing compounds.

Benzene is not the only aromatic compound. That is, six-membered rings

with (nominally) three alternating double bonds are not the only systems that

display the more-stable-than-expected character of aromaticity. A range of

Hückel determinants can be examined and a rule of thumb derived in terms

of maximal filling of bonding orbitals. It is found that planar cyclic mole-

cules that have 2, 6, 10, 14,...,electrons have all such electrons in lower-

energy bonding molecular orbitals and therefore are considered aromatic like

15.10 Benzene and Aromaticity 547

E 

Energy

 2





 2
Figure 15.21 Hückel theory predicts the above
arrangement for the six electrons in benzene.
The amount of additional stability in the or-
bitals of benzene is so great that it defines aro-
maticity.