A simple model for predicting ultra-violet/visible absorption spectra of conjuga

Question

A simple model for predicting ultra-violet/visible absorption spectra of conjugated polyenes is the free-electron molecular orbital model. in this model, the electrons of the conjugated pi system are free to travel the length of the conjugated carbon chain. in this model, the 1-dimensional particle-in-a-box model problem can be used to simulate the molecular orbitals of the electrons in the conjugated chain. The length of the box, L, is equal to the length of the conjugated carbon chain. L = n_cc (150 pm) The value n_cc corresponds to the number of carbon/carbon bonds in the conjugated chain. The electrons can be ordered two at a time in each energy level. in this problem, we will consider lycopene. Determine the energy of the transition from the lowest occupied to the first unoccupied energy level for lycopene. Determine the wavelength needed for this transition (lambda = he/Delta E). The experimental value for lycopene is 474 nm.

Explanation / Answer

There are 21 bonds in the conjugated bond network. This indicates that we have a box that is (21+1)*150 pm = 3300 pm = L

Furthermore, since there are 11 double bonds in the system, there must be 22 p electrons.

As these electrons enter the box, they must pair up in the energy levels, so that the lowest 11 energy levels will be filled.

The lowest energy transition will be when an electron in the 11th state is promoted to the 12th state by the absorption of a photon.

The energy of that transition is given by the difference in the two energy levels.

E12 E11 = h2/8ml2*(12^2 - 11^2)

E12 E11 = 23h2/8ml2

Where h is Planck's constant = 6.626 x 10^-34 Js

m = mass of electron = 9.109 x10^-36 Kg

L = 3300 pm = 3.3 x10^-9m

So,

E12 E11 = 23 x (6.626 x 10^-34 Js)^2 / 8 (9.109 x10^-31 Kg)( 3.3 x10^-9m)^2

              = 1.27 x 10^-19 J

E = h v = h c/

So = h c /E

        = (6.626 x 10^-34 Js)( 3 x 10^8m/s) /1.27 x 10^-19 J = 1.56 x 10^-6 m = 1560nm

Wavelength is much larger than experimental value.

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