Absorption wavelengths of cyanine dyes The one-dimensional particle-in-a-box is

Question

Absorption wavelengths of cyanine dyes The one-dimensional particle-in-a-box is a ample model that can be used to describe the pi-electrons' in cyanine dyes: (CH_3)_2 N^+ = CH (CH CH) N^++ (CH_3)_2 (CH_3)_2 N^++ (CH = CH) CH N^+ (CH_3)_2 Here, indicates an electron pair The overall charge is +1. There is a counter ion as well but it is colorless Consider only what a chemist calls the 'pi electron pairs'. These are electrons that can move relatively freely along the conjugated chain, as indicated by the two resonance structures shown above. If one assumes a flat potential for these electrons, then the particle in a box model can be used to approximate their motion. Remember from general chemistry that electrons want to fill the lowest energy levels. If there is a number n of pi electron pairs, one ends up with n filled ('occupied') levels. If the n lowest levels are filled, then you got the ground state on the left below. A low energy excited state Is obtained by promoting an electron to a higher level. Calculate the energy gap Delta E between the highest occupied pi-level (highest occupied molecular orbital, or HOMO) and the lowest unoccupied level (LUMO), in order to approximate the energy needed for an electronic excitation. Use the particle-in-a-box energy formula for the etedron energy levels and calculate Delta E for the three cyanine dyes with k = 1, 2, 3 The length L of the box that you need to Use for the quantum mechanical model depends on how many units k there are between the amine groups. Use L = 2.65 + 5.3 Also, the number of pi-electron pairs depends on k. Make sure that you count them correctly. Convert your results to excitation wavelengths and compare with the lowest experimentally observed excitation wavelengths: (If your results are very different from the experimental values you have made a mistake somewhere.)

Explanation / Answer

E = n2h2 / 8mL2

The energy gap between the first excited state (n = 2) and the ground state
(n = 1) is

E = E2 - E1

E = E2 = 22h2 / 8mL2 - E1 = 12h2 / 8mL2

E = 3h2 / 8mL2

= 3 x ( 6.626 x 10-34 ) / 8 x 9.1093 x 10-31 x 2.652

= 3.88 x 10 -4

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