Calculate the Separation of Energy, Frequency of Radiation, and Probability. Adv

Question

Calculate the Separation of Energy, Frequency of Radiation, and Probability. Advanced P-Chem Problem.?
To a first approximation, a PI electron in a linear polyene may be considered to be a particle in a 1-D box. Some linear polenes, of which B-Carotene is an example, are important biological co-factors that participate in processes as diverse as the absorption of solar energy in photosynthesis and protection against harmful biological oxidations. B-Carotene contains 22 conjugated Carbon atoms and the average inter-nuclear distance is 140pm. In the ground state, each level up to n=6 is occupied by two electrons.

A. Calculate the separation in energy between the ground state and the first excited state in which one electron occupies the state with n=7.

B. Calculate the frequency of the radiation required to produce a transition between these two states.

C. Calculate the total probability of finding an electron between Carbon atoms 11 and 12 in the ground state of B-Carotene.

Explanation / Answer

From a particle in a box quantum problem, the energy separation is given by En = h^2 n^2 pi^2 / (2mL^2) where h is the h-bar plank constant, n = 1,2,3,4.. , pi is pi, m is the mass of the electron, and L = 140pm. The solution for a) is simply E1 = h^2 pi^2 / (2mL^2)

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