# 10.7: Benzene and Aromaticity

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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 \(\PageIndex{1}\)).

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 \(\PageIndex{2}\) 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 Hückel 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 \(\PageIndex{3}\)).

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

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

Each of the carbons in benzene contributes one electron to the \(\pi\)-bonding framework (Figure \(\PageIndex{3}\)). 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

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

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 \(\PageIndex{4}\)). 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.

* Figure \(\PageIndex{4}\):* Experimental evidence for aromatic stabilization energy in benenze from the heat of hydrogenation. Energies are in kcal/mol. Image used with permission (CC-SA-BY-NC; William Resuch);

## Contributors

William Reusch, Professor Emeritus (Michigan State U.), Virtual Textbook of Organic Chemistry

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