4. The t-electron system in benzene may be approximated as a particle-on-a-ring with a radius of r, which is also the length of the C-C bonds in the ring. This bond length is approximately 1.39 Á. The energy levels for the p-0-a-r are given by: n'h? E, where n = 0, ±1, ±2, ±3, ... %3D 2m r The six t-electrons in benzene fill the (singly degenerate) n30 level, and the (doubly degenerate) n=±1 level. The first excited state, then corresponds to exciting an electron from n-1 to n=2. a) Using the given bond length, calculate the energy change for the benzene molecule as an electron is excited from n=1 to n=2. b) Calculate the wavelength of light that benzene would need to absorb to undergo that energy change. Compare that with the experimental value of 268 nm. c) In what portion of the electromagnetic spectrum is this wavelength? d) Using the experimental wavelength from part b above, calculate the effective bond length in benzene. e) Is the particle-on-a-ring approximation valid? Give a concise justification for your thoughts on this.

4. The t-electron system in benzene may be approximated as a particle-on-a-ring with a radius of r, which is also the length of the C-C bonds in the ring. This bond length is approximately 1.39 Á. The energy levels for the p-0-a-r are given by: n'h? E, where n = 0, ±1, ±2, ±3, ... %3D 2m r The six t-electrons in benzene fill the (singly degenerate) n30 level, and the (doubly degenerate) n=±1 level. The first excited state, then corresponds to exciting an electron from n-1 to n=2. a) Using the given bond length, calculate the energy change for the benzene molecule as an electron is excited from n=1 to n=2. b) Calculate the wavelength of light that benzene would need to absorb to undergo that energy change. Compare that with the experimental value of 268 nm. c) In what portion of the electromagnetic spectrum is this wavelength? d) Using the experimental wavelength from part b above, calculate the effective bond length in benzene. e) Is the particle-on-a-ring approximation valid? Give a concise justification for your thoughts on this.

Principles of Modern Chemistry

8th Edition

ISBN:9781305079113

Author:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Publisher:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Chapter4: Introduction To Quantum Mechanics

Section: Chapter Questions

Problem 39P: Chapter 3 introduced the concept of a double bond between carbon atoms, represented by C=C , with a...

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