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)

Answer rating: 100% (QA)

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