# Benzene X Ray Diffraction

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})).

- X-ray Diffraction Analysis
- What Is X-ray Diffraction
- Benzene X Ray Diffraction Technique
- Benzene X Ray Diffraction Bragg S Law
- Benzene X Ray Diffraction Analysis

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.

X‐Ray Diffraction Evidence for a Cyclohexatriene Motif in the Molecular Structure of Tris(bicyclo2.1.1hexeno)benzene: Bond Alternation after the Refutation of the Mills–Nixon Theory†. X‐ray diffraction data for liquid benzene at 25°C are analyzed using scattering factors for C–H groups rather than for C and H atoms. This new approach permits the unique extraction of structure an. The structure of benzene and naphthalene in the liquid state. Journal of Structural Chemistry 1975, 15 (4), 504-509. DOI: 10.1007/BF00747186. X-Ray Diffraction.

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

## X-ray Diffraction Analysis

[left begin{array}{cccccc}x&1&0&0&0&11&x&1&0&0&00&1&x&1&0&00&0&1&x&1&00&0&0&1&x&11&0&0&0&1&xend{array}right =0label{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=alphapmbeta) energy levels are two-fold degenerate (Figure (PageIndex{3})).

The resulting wavefunctions are below (expanded in terms of carbon ( 2prangle) 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+2beta)+4(alpha+beta)=6alpha+8betalabel{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) = (6alpha+8beta ) - 3 times 2(alpha+beta) = 2beta]

The experimental thermochemical value is -152 kJ mol^{-1}.

## What Is X-ray Diffraction

## 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 ( 2beta) (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.

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

## Benzene X Ray Diffraction Technique

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.

## Benzene X Ray Diffraction Bragg S Law

## Contributors

## Benzene X Ray Diffraction Analysis

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

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