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4. The n-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 A. The energy levels for the p-o-a-r are given by: n'h E, 2mr" where n 0, t1, t2, ±3, .. The six r-electrons in benzene fill the (singly degenerate) n 0 level, and the (doubly degenerate) nt1 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?

4. The n-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 A. The energy levels for the p-o-a-r are given by: n'h E, 2mr" where n 0, t1, t2, ±3, .. The six r-electrons in benzene fill the (singly degenerate) n 0 level, and the (doubly degenerate) nt1 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?

Chemistry
Chemistry
10th Edition
ISBN:9781305957404
Author:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Publisher:Steven S. Zumdahl, Susan A. Zumdahl, Donald J. DeCoste
Chapter1: Chemical Foundations
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Problem 1RQ: Define and explain the differences between the following terms. a. law and theory b. theory and...
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4. The n-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 A. The energy levels for the p-o-a-r are given by: n'h E, ,where n = 0, t1, ±2, 13, . 2m,r? The six r-electrons in benzene fill the (singly degenerate) n 0 level, and the (doubly degenerate) n=t1 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.
Transcribed Image Text:4. The n-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 A. The energy levels for the p-o-a-r are given by: n'h E, ,where n = 0, t1, ±2, 13, . 2m,r? The six r-electrons in benzene fill the (singly degenerate) n 0 level, and the (doubly degenerate) n=t1 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.
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