# 10.7: Benzene and Aromaticity


##### Learning Objectives
• Apply Hü​ckel theory to describing the pi bonding in cyclical conjugated system
• Identify the origin of aromaticity within Hü​ckel theory to describe extra stabilization in certain cyclical conjugated systems

The previous sections addressed the $$\pi$$ orbitals of linear conjugated system. Here we address conjugated systems of cyclic conjugated hydrocarbons with the general formula of $$C_nH_n$$ where $$n$$ is the number of carbon atoms in the ring. The molecule from this important class of organic molecule that you are most familiar with is benzene ($$C_6H_6$$) with $$n=6$$, although many other molecules exist like cyclobutadiene ($$C_4H_4$$) with $$n=4$$ (Figure 10.7.1 ).

##### Structure of Benzene

The structure of benzene is an interesting historical topic. In 1865, the German chemist Friedrich August Kekulé published a paper suggesting that the structure of benzene contained a ring of six carbon atoms with alternating single and double bonds. Within this argument, two resonance structures can be formulated.

However, X-ray diffraction shows that all six carbon-carbon bonds in benzene are of the same length, at 140 pm. The C–C bond lengths are greater than a double bond (135 pm), but shorter than a typical single bond (147 pm). This means that neither structures Figure 10.7.2 are correct and the true 'structure' of benzene is a mixture of the two. As discussed previously, that such a valence bond perspective results in a delocalization energy within a molecular orbital approach.

Aromatic systems provide the most significant applications of Hü​ckel theory. For benzene, we find the secular determinant

$\left|\begin{array}{cccccc}x&1&0&0&0&1\\1&x&1&0&0&0\\0&1&x&1&0&0\\0&0&1&x&1&0\\0&0&0&1&x&1\\1&0&0&0&1&x\end{array}\right|=0\label{31}$

with the six roots $$x=\pm2,\pm1,\pm1$$. This corresponds to the following energies (ordered from most stable to least since $$\beta < 0$$):

• $$E_1 = α + 2β$$
• $$E_2 = α + β$$
• $$E_3 = α + β$$
• $$E_4 = α − β$$
• $$E_5 = α − β$$
• $$E_6 = α − 2β$$

The two pairs of $$E=\alpha\pm\beta$$ energy levels are two-fold degenerate (Figure 10.7.3 ).

The resulting wavefunctions are below (expanded in terms of carbon $$| 2p\rangle$$ atomic orbitals).

\begin{align} | \psi_1 \rangle &= \dfrac{1}{\sqrt{6}} \left[ | 2p_{z1} \rangle+ | 2p_{z2} \rangle + | 2p_{z3} \rangle + | 2p_{z4} \rangle + | 2p_{z5} \rangle + | 2p_{z6} \rangle \right] \\ | \psi_2 \rangle &= \dfrac{1}{\sqrt{4}} \left[ | 2p_{z2} \rangle + | 2p_{z3} \rangle - | 2p_{z4} \rangle - | 2p_{z5} \rangle \right] \\ | \psi_3 \rangle &= \dfrac{1}{\sqrt{3}} \left[ | 2p_{z1} \rangle + \dfrac{1}{2}| 2p_{z2} \rangle - \dfrac{1}{2} | 2p_{z3} \rangle - | 2p_{z4} \rangle - \dfrac{1}{2} | 2p_{z5} \rangle + \dfrac{1}{2} | 2p_{z6} \rangle \right] \\ | \psi_4 \rangle &= \dfrac{1}{\sqrt{4}} \left[ | 2p_{z2} \rangle - | 2p_{z3} \rangle + | 2p_{z4} \rangle - | 2p_{z5} \rangle \right] \\ | \psi_5 \rangle &= \dfrac{1}{\sqrt{3}} \left[ | 2p_{z1} \rangle - \dfrac{1}{2}| 2p_{z2} \rangle - \dfrac{1}{2} | 2p_{z3} \rangle + | 2p_{z4} \rangle - \dfrac{1}{2} | 2p_{z5} \rangle - \dfrac{1}{2} | 2p_{z6} \rangle \right] \\ | \psi_6 \rangle &= \dfrac{1}{\sqrt{6}} \left[ | 2p_{z1} \rangle- | 2p_{z2} \rangle + | 2p_{z3} \rangle - | 2p_{z4} \rangle + | 2p_{z5} \rangle - | 2p_{z6} \rangle \right] \end{align} \nonumber

Each of the carbons in benzene contributes one electron to the $$\pi$$-bonding framework (Figure 10.7.3 ). This means that all bonding molecular orbitals are fully occupied and benzene then has an electron configuration of $$\pi_1^2 \pi_2^2 \pi_3^2$$. With the three lowest molecular orbitals occupied, the total $$\pi$$-bonding energy is

$E_{tot} (benzene)=2(\alpha+2\beta)+4(\alpha+\beta)=6\alpha+8\beta\label{32}$

Since the energy of a localized double bond is $$2(\alpha+\beta)$$, as determined from the analysis of ethylene, the delocalization energy of benzene is

$\Delta E = E_{tot} (benzene) - 3 E_{tot} (ethylene) = (6\alpha+8\beta ) - 3 \times 2(\alpha+\beta) = 2\beta \nonumber$

The experimental thermochemical value is -152 kJ mol-1.

## Aromaticity

In general, cyclic polyenes are only closed shell (i.e., each electron paired up) and extra stable for with (4n+2) $$\pi$$ electrons (n=0,1,2…). These special molecules have the highest delocalization energies and are said to be “aromatic”. For benzene this is $$2\beta$$ (Equation $$\ref{32}$$), which is the energy by which the delocalized $$\pi$$ electrons in benzene are more stable than those in three isolated double bonds.

##### Hückel's Rule

A stable, closed-shell conjugated cyclic structure is obtained for molecules with (4n+ 2) electrons with n=2, 6, 10, .... electrons.

Evidence for the enhanced thermodynamic stability of benzene was obtained from measurements of the heat released when double bonds in a six-carbon ring are hydrogenated (hydrogen is added catalytically) to give cyclohexane as a common product. In the following diagram cyclohexane represents a low-energy reference point. Addition of hydrogen to cyclohexene produces cyclohexane and releases heat amounting to 11.9 kJ mol-1. If we take this value to represent the energy cost of introducing one double bond into a six-carbon ring, we would expect a cyclohexadiene to release 23.9 kJ mol-1 on complete hydrogenation, and 1,3,5-cyclohexatriene to release 35.9 kJ mol-1. These heats of hydrogenation $$\Delta H_{hyd}$$ reflect the relative thermodynamic stability of the compounds (Figure 10.7.4 ). In practice, 1,3-cyclohexadiene is slightly more stable than expected, by about 8.1 kJ mol-1, presumably due to conjugation of the double bonds. Benzene, however, is an extraordinary 15 kJ mol-1 more stable than expected. This additonal stability is a characteristic of all aromatic compounds.