Particle in a box is a model that is often used to look at the spectroscopy...

Particle in a box is a model that is often used to look at the spectroscopy of pi electrons in conjugated organic molecules. carotene, a precursor of retinal, a visual pigment found in the retina of the eye, has the formula given below. -carotene contains 11 conjugated double bonds which contribute 22 electrons that are delocalized along the length of the molecular chain. Each level of particle in a box has room for 2 electrons. a. The highest filled energy level is referred to as the HOMO (highest occupied molecular orbital). The quantum number that goes with this level is _____________. b. The lowest unoccupied (no electrons) is referred to as the LUMO (Lowest unoccupied molecular orbital). The quantum number for this level is ____________. c. To calculate the “length” of the box we will use the average bond length of 1.35 A for each carbon-carbon bond. The total length of the box in m is___________________ d. Find an expression for the energy difference between your LUMO and HOMO e. Using the expression in d. calculate the E for an electron moving between the HOMO and LUMO for -carotene. (be very careful about units!..no credit for calculation not done in correct units). f. What part of the electromagnetic spectrum would this transition occur in?

Solutions

Expert Solution

For particle in a box energy is given by,

where n - quantum no = 1, 2, 3, ..............

l - length of the box, m - mass of the particle

a.) Highest occupied molecular orbital (HOMO) is the highest filled energy orbital. Each orbital will be occupied by 2 electrons. There are total of 22 pi electrons in carotene. So if each orbital is occupied by 2 electrons, the n quantum no will be 11.

b.) Lowest unoccupied molecular orbital (LUMO) is the lowest unfilled energy orbital. So it will be the orbital just above HOMO. The quantum number of HOMO is 11 and so for LUMO it will be 12.

c.) There are total of 22 bonds. They are alternate double and single bond. The averagebond lenth of one carbon - carbon bond is given as 1.35 A^o . The total length of box will be 29.7 A^o ie, 29.7 * 10^-10 m.

d.)

ni>nj

e.) The energy difference between HOMO and LUMO will be

ni = 12, nj = 11

m - mass of electron = 9.1 * 10^-31 Kg

Energy difference = 0.0156 * 10^-17 J

f.) The wavelength corresponding to this energy is given by

c- velocity of light = 3 * 10⁸ m/s

queen_honey_blossom answered 1 year ago

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