A particle in a box model can be applied in a simplistic manner to the considera

Question

A particle in a box model can be applied in a simplistic manner to the consideration of mobile pi electrons in a conjugated system consider vitamin A2 with 12 pi electrons. If these electrons fill the first 6 pi particle in a box energy levels by pairing spin up/ spin down then the absorption of a photon of light can occur from n=6 to the vacant n=7 level. If we consider the box containing the pi electrons to have infinite potentenian beyond the ~1210 pm length of the conjugated chain and zero potential inside: a. what are the energies of the n=1-7 levels? b. what is the energy difference between n=6 and n=7? c. what does this very approximate calculation compare with experimental values?

Explanation / Answer

a) Only three emission lines are possible......no b) Photons with the highest energy are emitted in a transition from level n = 4 to level n = 3.... no , not much difference between 4&3 c) Photons with the highest frequency are emitted in a transition from level n = 1 to level n = 4...........no, this absorbs energy d) The emission line having the longest wavelength corresponds to a transition from level n = 3 to level n = 2..... no, not much difference from 3 to 2 e) The emission line having the shortest wavelength corresponds to a transition from level n = 4 to level n = 1 short wavelenghts = hi energy yes The energy difference between the energy levels with principal quantum numbers n = 6 and n = 2 may be calculated from a form of the Rydberg equation e = R.(1/n1² - 1/n2²) with R = 2.1799 x 10^(-18) J, giving e = 4.844 x 10^(-19) J. The energy difference is of course exactly the same from one atom to another, so the quantity of hydrogen present is not relevant. If the intention was to estimate the total amount of energy radiated by 6.022 x 10^23 hydrogen atoms undergoing that transition, then you simply multiply by that number (Avogadro's number), but the result - about 290kJ - has no physical significance as it is impossible to separate out those atoms from others in different states, and they are being continuously re-excited into a variety of different excited energy levels.

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