8. Consider the structure of beta-carotene:

It is a linear polyene in which 21 bonds, 10 single and 11 double, alternate along a chain of 22 carbon atoms. Assume that beta-carotene can be modeled as a one-dimensional box or pipe with each of the 21 bonds having a length of 1.4x10^-10 m. Assume that there is one freely moving electron in the n=11 energy level that can be promoted to the n=12 energy level. Calculate the energy separation between the two energy levels. Calculate the frequency of radiation that can bring about the n=11 to n=12 transition.

ÎE = J

Î½ = Hz

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Bunny GreenfelderLv2

14 Oct 2019

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Physcial Chemistry:Ch 10 Introduction to Quantum Mechanics

Carotenes are molecules with alternating CâC and C=C bonds in which the electrons are delocalized across the entire alternating bond system. As such, the electrons can be approximated as being a particle-in-a-box. Lycopene is a carotene found in tomatoes and watermelon. Assuming that the alternating carbon-carbon bond system is 2.64 nm wide:

(a) What is the energy of the n = 11 level?

(b) What is the energy of the n = 12 level?

(d) If, by the Bohr frequency condition, âE = hÎ½, what frequency and wavelength of light corresponds to the transition from n = 11 to n = 12?

Answer:

(a) 1.05Ã10^â18 J (b) 1.24Ã10^â18 J (c) 1.9Ã10^â19 J (d) Î½ = 2.87Ã10¹⁴ s ^â1 , Î» = 1.05 Âµm

8.31 When Ã- carotene ( 1) is oxidized in vivo, it breaks inhalf and forms two molecules of retinal ( vitamin A), which is aprecursor to the pigment in the retina responsible for vision. Theconjugated system of retinal consists of 11 C atoms and one O atom.In the ground state of retinal, each level up to n = 6 is occupiedby two electrons. Assuming an average internuclear distance of 140pm, calculate (a) the separation in energy between the ground stateand the first excited state in which one electron occupies thestate with n = 7, and ( b) the frequency of the radiation requiredto produce a transition between these two states. ( c) Using yourresults, choose among the words in parentheses to generate a rulefor the prediction of frequency shifts in the absorption spectra oflinear polyenes: The absorption spectrum of a linear polyene shiftsto ( higher/ lower) frequency as the number of conjugated atoms (increases/ decreases).