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the particle on a ring is a useful model for the motion of electrons around the porphine ring (4). the conjugated macrocycle that forms the structural basis of the heme group and the chlorophylls. We may treat the group as a circular ring of radius 440pm, with 20 electrons in the conjugation system moving along the perimeter of the ring. As in exercise 9.26, assume that in the ground state of the molecule quantized each level is occupied by two electrons. (A) Calculate the energy and angular momentum of an electron in the highest occupied level.(B) Calculate the frequency of radiation that can induce a transition between the highest occupied and lowest unoccupied levels.

User Neodymium
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Final answer:

The angular momentum for the highest occupied state of an electron on a ring model with 20 electrons is L = 10h/2π. The energy levels are proportional to 1/n², and the frequency of radiation needed for the transition between levels can be calculated as ν = ΔE/h.

Step-by-step explanation:

To answer the question about the energy and angular momentum of an electron in the highest occupied level of a particle on a ring model and the frequency of radiation required to induce a transition between the highest occupied and lowest unoccupied levels, we utilize the concepts of quantization of angular momentum and energy levels.

In this model, the quantized angular momentum L for an electron in an orbit is given by Bohr's formula L = n(h/2π), where n is a non-negative integer, h is Planck's constant and π is pi. Using this, the angular momentum for the highest occupied state with 20 electrons (given that each level is occupied by two electrons), would be for n = 10, resulting in L = 10h/2π.

The energy levels for a particle on a circular orbit are proportional to 1/n², based on Bohr's theory. The transition energy is the difference between these levels, and the frequency (ν) of the emitted or absorbed radiation can be determined using the equation ν = ΔE/h, where ΔE is the energy difference between the initial and final states and h is Planck's constant. The frequency indicates the type of electromagnetic radiation needed for the electron transition. As the question does not provide explicit values for mass or velocity, we cannot calculate precise values for energy and frequency.

According to Hund's rule and the Heisenberg uncertainty principle, the ground state has the lowest energy possible, and the exact momentum and position of an electron cannot be simultaneously determined with precision.

User Evan Kennedy
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