10.5.P3
Prerequisites:
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Many biological molecules give color to objects by absorbing light of particular colors and reflecting other light. Some notable examples are beta-carotene (orange -- found in carrots and egg yolks), lycopene (red -- found in tomatoes), and chlorophyll (green -- found in many plants). The absorption spectra of beta-carotene is shown in the figure at the right. It absorbs strongly in the blue -- at about 450 nm -- so it leaves the red, yellow, and orange light that hits it to reflect back.*
* The figure is from a very old paper -- 1942 -- so the units given are Ångstroms (Å), not the SI units of nanometers (nm). 10 Å = 1 nm.)
The part of the molecule that is primarily responsible for the absorption of visible light hitting the molecule is called a chromophore. For beta-carotene, the molecular structure is shown below, and the chromophore is indicated in red on the chemical formula shown below that.
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Source: F. Zscheile et al., Plant Physiology, 17(3), 331-346 (1942)
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The chromophore consists of a long string of double bonds. A reasonable starting point might be to model the energy levels as if the electrons just could run back and forth freely along the molecule. The quantum physics tells us (see the reading Quantum string for a full derivation) if that's the case, the energy levels can be gotten assuming the electron waves are like the normal modes of an oscillating string. This model of the electron states gives the energy of the n-th level of a string of length L as
To model the excited states of the system we have to know a bit of chemistry. The electrons in the chromophore fill up the energy levels. You can only put two electrons in each state. There are 22 in the string of double bonds, so in the ground state all energy levels up to n=11 are filled.
Therefore, the first energy level that in the "ground state" is not occupied by electrons is n=12.
A. The first excited state of the molecule is defined as the state that is closest to the ground state in energy. It involves bringing an electron into the lowest available energy level of n=12. From which energy state should the electron be excited if we want to excite the lowest energy state above the ground state?
B. According to the data shown at the top, this chromophore (beta carotene) does a good job absorbing photons of wavelength lambda = 450 nm. What is the energy of this photon in eV?
C. Use the vibrating string model of the electrons in the chromophore to estimate the size of the chromophore, L. The actual length is about 3 nm. Is the model giving you a reasonable result?
Joe Redish 4/20/13
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