6-11 ① If there are q pairs of carbon atoms forming the box, there are 2q delocalised electrons. Since 2 electrons can be in each state, this means that q states will be occupied. Therefore the highest energy occupied state will have quantum number n = q and the lowest energy available state that can receive a promoted electron will have n = q+1.
Since the box comprises q units, each of length j. the total length is qj. Our general expression for the energy of a state is:
So we can deduce that
You can readily see that (q+1)/q gets smaller, tending towards 1 as q gets bigger, and hence the energy gap also gets smaller, tending towards zero as q gets bigger. So as the length of the chain of conjugated double bonds increases, the particle in a box theory suggests that the energy needed to excite an electron to a higher energy state within it gets smaller. In turn, since (E = hc/λ), this means that these molecules are predicted to be able to absorb light of increasingly long wavelength as the length of the conjugated chain increases.
② For butadiene, q=2 (4 carbons). So putting the numbers into our expression from ①, we can get an actual value for the energy gap,
The photon needed to provide this energy has a wavelength λ = hc/E
= 6.63 x 10-34 x 3.00 x 108 / (9.6 x 10-19) = 2.1 x 10-7 m
Radiation of this wavelength is in the UV part of the spectrum. No absorption of visible light is predicted and so butadiene is expected to be a colourless compound.
In reality, the longest wavelength absorption maximum for butadiene is at 2.17 x 10-7m. This is in astonishingly close agreement with our extremely simple model and butadiene is, indeed colourless.
③ For lycopene, the number of conjugated double bonds is greater (q=11; (22 carbons in the conjugated system – make sure you noticed that the additional C=C bond at each end of the chain is not part of the conjugated system). This corresponds to a longer box length and, from our result in ①, we would predict that the energy gap needed to promote an electron would therefore be smaller and the wavelength of the photon needed for the excitation, correspondingly longer than in the butadiene case. This is qualitatively consistent with what is observed: lycopene absorbs visible light. In fact it absorbs principally in the blue part of the spectrum, which is why lycopene - and the tomatoes in which it’s found- are red.
If we do the same kind of calculation we did for buta-1,3-diene, this time we get:
This corresponds to an excitation wavelength of 1.3 x 10-6 m. This is actually in the infra-red part of the spectrum. So the agreement is not so good for this longer conjugated chain (though it is still within a factor of 3). Poor quantitative agreement is not really surprising because the particle in a box model is so simplistic. We are, for example, assuming the electrons have no potential energy which is clearly not realistic given all those nuclei around, to which they will be attracted and all the other electrons, by which they will be repelled.
