The particle in a box problem is a very simple model for electrons in conjugated

Question

The particle in a box problem is a very simple model for electrons in conjugated systems. Consider lycopene, which is responsible for the color of some fall leaves: Assuming that the color of lycopene is due to transition between an electron in the highest occupied energy state and the lowest unoccupied energy state, find the energy of this transition and calculate the color of lycopene. You can assume that each energy level can contain two electrons. The size of the "box" is generally assumed to contain 1 more bond than the conjugated pi system. A resonance stabilized CC bond is ~ 0.140 nm. How much of an effect on the color does this assumption of a slightly larger "box" have?

Explanation / Answer

For this problem the equation is

delta E = h2/8mL2 * (N+1)

where N = 2j+2 where j = No. of conjugated double bonds

j = 11 for this molecule

N = 24 Since it has been asked to take one more bond we will take N =25

L = N x l (l = length of C=C = 0.14 nm)

L = 25 x 0.14 = 3.5 nm

delta E = (6.626 x 10-34)2/8 x 9.109x10-31Kg x (3.5 x 10-9)2 * 25+1

delta E = 1.28 x 10-19J

lambda max = hc/delta E

= 6.626 x 10-34 x 3 x 108/1.28 x 10-19

= 1553 x 10-9 m or 1553 nm

SO this goes into the IR region of light

The contribution of the length of the box is not very much as the number is in nm and needs to be squared and so adding or removing 1 bond as a very small effect like for example in the case of 1 bond less the delta E would have been 1.33x 10-19 J instead of the present value.

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