Constants Mass of electron me = 9.11 x 10-31 kg Electron charge e = -1.602 x...
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Constants Mass of electron me = 9.11 x 10-31 kg Electron charge e = -1.602 x 10-19 C Coulomb constant ke 8.988 x 10⁹ J-m-C-2 Vacuum permittivity co= 8.854 x 10-12 J-¹.m-¹.C² Planck constant h = 6.626 x 10-34 J-s, ħ = h/(2n) Speed of light c = 2.998 x 108 m-s-¹ Boltzmann constant k = 1.38 x 10-23 J.K-1 12 (a) The Bohr model is based on the assumption that the angular momentum is quantized. mir = nh. Derive the expression for energy En and radius rn of a system with e nucleus charge (Z = 1) and one electron at energy level n. Provide detailed derivation. (10 points) (b) The radius r at energy level n = 1 is defined as Bohr radius ao. Calculate ao in pm. (6 points) (c) For 1s orbital of the hydrogen atom, the quantum mechanical solution is (-1) * EXP(-1). exp ao V/1s = 1 T¹/2 where r = x² + y² + z² and an is the Bohr radius. Write down the expression for the probability of the elec- tron in the range [r, x+dr], [y, y + dy], and [z, z + dz]. In other words, P =?drdydz. (6 points) (d) Perform change of variables and convert the probability to spherical coordinate system. In other words, P = ?drd0do. (Hint: dV = drdydz and dV = r² sin 0 drdodo) (6 points) (e) Prove that the total probability, i.c., the integration over space, is normalized to 1. (6 points) Fox (Hint: the integration in spherical coordinate is ∞ f 2 drdodo) (f) Based on the expression in (d), find the radius r that corresponds to the highest probability, i.c., the most. likely r where the electron is located. How is it related to the Bohr radius ao? (6 points) Constants Mass of electron me = 9.11 x 10-31 kg Electron charge e = -1.602 x 10-19 C Coulomb constant ke 8.988 x 10⁹ J-m-C-2 Vacuum permittivity co= 8.854 x 10-12 J-¹.m-¹.C² Planck constant h = 6.626 x 10-34 J-s, ħ = h/(2n) Speed of light c = 2.998 x 108 m-s-¹ Boltzmann constant k = 1.38 x 10-23 J.K-1 12 (a) The Bohr model is based on the assumption that the angular momentum is quantized. mir = nh. Derive the expression for energy En and radius rn of a system with e nucleus charge (Z = 1) and one electron at energy level n. Provide detailed derivation. (10 points) (b) The radius r at energy level n = 1 is defined as Bohr radius ao. Calculate ao in pm. (6 points) (c) For 1s orbital of the hydrogen atom, the quantum mechanical solution is (-1) * EXP(-1). exp ao V/1s = 1 T¹/2 where r = x² + y² + z² and an is the Bohr radius. Write down the expression for the probability of the elec- tron in the range [r, x+dr], [y, y + dy], and [z, z + dz]. In other words, P =?drdydz. (6 points) (d) Perform change of variables and convert the probability to spherical coordinate system. In other words, P = ?drd0do. (Hint: dV = drdydz and dV = r² sin 0 drdodo) (6 points) (e) Prove that the total probability, i.c., the integration over space, is normalized to 1. (6 points) Fox (Hint: the integration in spherical coordinate is ∞ f 2 drdodo) (f) Based on the expression in (d), find the radius r that corresponds to the highest probability, i.c., the most. likely r where the electron is located. How is it related to the Bohr radius ao? (6 points)
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a The Bohr model states that the angular momentum of a single electron in an atom is quantized and can take on only certain discrete values The energy of the electron is determined by the quantum numb... View the full answer
Related Book For
Introduction to Probability and Statistics
ISBN: 978-1133103752
14th edition
Authors: William Mendenhall, Robert Beaver, Barbara Beaver
Posted Date:
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