ENG045_SL1

ENG045_SL1 - DATE DUE ENG45 Solution to homework set#1...

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DATE: September 30, 2011 DUE: October 7, 2011 ENG45, Solution to homework set #1. 1. The Bohr Atom. This is the most primitive model for why the atom is stable and for how atomic energies are quantized. It is built on a planetary type motion of an electron cir- cularly orbiting a (Hydrogen) nucleus. Based on realizations from the photoelectric effect and special relativity, where photon energy E ph = and photon momentum is p = h/λ [ h is Planck’s constant, ν is the frequency (in Hz) and λ is the wave- length. The speed of light is c = νλ ], the electron momentum is associated with a wave length, and the quantization of orbits is accomplished. The results are: r n = r B n 2 , r B = ¯ h 2 4 πε 0 m e e 2 (1) v n = v B 1 n , v B = ¯ h m e r B (2) E n = - E B 1 n 2 , E B = 1 2 e 2 4 πε 0 r B (3) where n = 1 , 2 , ··· is the (primary) quantum number, ¯ h = h/ 2 π , ε 0 is the vacuum permittivity, m e is the mass of the electron 1 , and e is the proton charge. The three cal- culated constants are the Bohr radius r B , the Bohr velocity v B , and the Bohr Energy E B . a. Calculate r B , v B , and E B as given by Eqs. (1-3) as well as with the revision given by the footnote. Find the differences between the values. Express E B in both Joules and eV. Looking up the natural constants we get: r B = 0 . 529178 ˚ A (4) v B = 2 . 18769 × 10 6 m/s (5) E B = 2 . 179905 × 10 - 18 J = 13 . 606 eV (6) If we use the reduced mass, which is μ = (1 + 5 . 45 × 10 - 6 ) - 1 m e , we get: ˜ r B = 0 . 529466 ˚ A (7) ˜ v B = v B (8)
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ENG045_SL1 - DATE DUE ENG45 Solution to homework set#1...

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