How do I calculate the length of a chemical bond?

[fizzicsdummy]

“In C6H8, the carbon atoms are connected in a chain. In a rough approximation, some electrons can be viewed as confined to a line with the length of this chain. C6H8 absorbs ultraviolet light with wavelength 357 nm, which causes an electron to change from the third lowest allowed energy to the fourth lowest allowed energy. Use this information to estimate the length in pm of a typical carbon-carbon bond in C6H8.”



I do know that radius = (n^2)(Bohr radius).



I assume I could just calculate the radius of a carbon atom and multiply it by 2 to find a rough estimate of the length of a carbon-carbon bond. However, I don't know how to do this and I don't know the importance of the energy level change described in the problem. I'd really appreciate help understanding this. An equation (if one exists) would also be appreciated. Thanks!

 

[TeethWhitener]

This is a popular question that comes up in intro quantum chem classes. The idea here is to take your conjugated 1,3,5-hexatriene molecule and approximate its valence pi electrons (very roughly) as particles in a 1D box of length L. The energy levels are
$$E = \frac{n^2h^2}{8mL}$$
and the hexatriene molecule has 6 conjugated pi electrons. Given the fact that electrons are fermions, this means the lowest 3 levels of the box are filled, so the first electronic transition possible is between the 3rd and the 4th level. You're told that the molecule absorbs light at $\lambda$ = 357nm, which you can plug into
$$E = \frac{hc}{\lambda}$$
to find the energy of the transition. You can then find the difference of energy between the 3rd and 4th box levels and solve for L to get the total length of the box. Since all 6 carbons are (very roughly) equivalent, you can divide this L by 6 to get the C-C bond length.