P214_hw_solution11 - Prepared by Girma 1 Physics 214 Spring...

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Unformatted text preview: Prepared by Girma 1 Physics 214 Spring 2006 Solution to Homework ] 11 Problem 1. In the Bohr atomic model, the electron in hydrogen atom revolves around the nucleus containing the proton. Its total energy, when moving with speed v at a distance r from the proton, is then the sum of the kinetic and the potential energies, E = 1 2 mv 2- k e 2 r , (1) where m is mass of the electron, e is charge of the electron and k = 1 / (4 ) is Coulombs constant. The Coulomb force by the proton on the electron is central and causes centripetal motion, k e 2 r 2 = mv 2 r . (2) Bohr then postulated that the angular momentum of the electron should be quan- tized as mvr = n ~ , (3) where n = 1 , 2 , 3 , and ~ = h/ 2, h being Plancks constant. Combining (2) and (3) gives quantized radii and velocities, r n = n 2 ~ 2 kme 2 and v n = ke 2 n ~ . (4) Putting these in the energy given by (1) gives quantized energies, E n =- mk 2 e 4 2 n 2 ~ 2 =- E 1 n 2 , where E 1 =- mk 2 e 4 2 ~ 2 . (5) Putting in the numbers, E 1 =- mk 2 e 4 2 ~ 2 =- 9 . 11 10- 31 kg (8 . 99 10 9 N.m 2 /C 2 ) 2 (1 . 60 10- 19 C ) 4 2(1 . 05 10- 34 J.s ) 2 =- 2 . 19 10- 18 J =- 2 . 19 10- 18 J 1 . 60 10- 19 eV 1 J =- 13 . 6 eV. (6) When the hydrogen atom makes a transition from a higher energy level E n 2 to a lower energy level E n 1 , it emits a photon of energy hf = E n 2- E n 1 = h c =- E 1 ( 1 n 2 1- 1 n 2 2 ) =...
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P214_hw_solution11 - Prepared by Girma 1 Physics 214 Spring...

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