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The total mechanical energy of the electron is the sum of its kinetic energy and potential energy. Derive an expression for the total mechanical energy of the electron in the hydrogen atom; E = KE+U (3) Your result should be negative. The negative result means energy must be added to excite the electron to higher energy levels 'farther away' from the nucleus. In this coordinate system the total energy is zero at an infinite distance from the nucleus.

The total mechanical energy of the electron is the sum of its kinetic energy and potential energy. Derive an expression for the total mechanical energy of the electron in the hydrogen atom; E = KE+U (3) Your result should be negative. The negative result means energy must be added to excite the electron to higher energy levels 'farther away' from the nucleus. In this coordinate system the total energy is zero at an infinite distance from the nucleus.

Principles of Modern Chemistry
Principles of Modern Chemistry
8th Edition
ISBN:9781305079113
Author:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler
Publisher:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler
Chapter5: Quantum Mechanics And Atomic Structure
Section: Chapter Questions
Problem 28P: Quantum mechanics predicts that the energy of the ground state of the H atom is 13.6eV . Insight...
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Question c, d, e
Band theory, used to investigate the physics of semiconductors, involves an understanding of quantization of energy, energy levels and electron states for electrons in atoms. An early model of the atom, developed by Bohr, introduces these ideas along with the concepts of electric force, electric field, potential and kinetic energy discussed recently in class. In this assignment you will use algebra to derive general expressions for the radius and total energy of the electron in hydrogen using Bohr's model, and calculate numerical values for the electron's first four 'allowed' states. In Bohr's model of the hydrogen atom, the electron was assumed to move in stable circular orbits around the nucleus without radiating. The centripetal force, mv²/r, acting on the electron is due the electric force (Coulomb attraction) between the electron with charge (-e) and the nucleus of charge (+e), mv²/r = (1/4π) e²/r² where e = 1.602 x 10-19 C and Eo = 8.85 x 10-¹²C²/(N-m²). (1)
Transcribed Image Text:Band theory, used to investigate the physics of semiconductors, involves an understanding of quantization of energy, energy levels and electron states for electrons in atoms. An early model of the atom, developed by Bohr, introduces these ideas along with the concepts of electric force, electric field, potential and kinetic energy discussed recently in class. In this assignment you will use algebra to derive general expressions for the radius and total energy of the electron in hydrogen using Bohr's model, and calculate numerical values for the electron's first four 'allowed' states. In Bohr's model of the hydrogen atom, the electron was assumed to move in stable circular orbits around the nucleus without radiating. The centripetal force, mv²/r, acting on the electron is due the electric force (Coulomb attraction) between the electron with charge (-e) and the nucleus of charge (+e), mv²/r = (1/4π) e²/r² where e = 1.602 x 10-19 C and Eo = 8.85 x 10-¹²C²/(N-m²). (1)
c) The total mechanical energy of the electron is the sum of its kinetic energy and potential energy. Derive an expression for the total mechanical energy of the electron in the hydrogen atom; (3) Your result should be negative. The negative result means energy must be added to excite the electron to higher energy levels 'farther away' from the nucleus. In this coordinate system the total energy is zero at an infinite distance from the nucleus. E = KE + U d) To explain the quantization observed in the emission spectra of hydrogen, Bohr introduced 'quantization' to the classical model by suggesting the angular momentum of the electron was quantized. This means the angular momentum can only have discrete values. Bohr quantized the angular momentum in multiples of Planck's constant; (4) In this equation m is the mass of the electron m = 9.11 x 10-31 kg and ħh is the modified Plancks's constant h/2; where ħ = h/2π = 1.05 x 10-34 J-s = 6.58 x 10-16 eV-s. n=1-+ (-13.6 eV) mvr = nh Re-arrange the angular momentum expression given in Eqn. (4) for v and substitute into Eqn. (1) to prove the radius of the electron's orbit is given by r = n²ħ²/(e²km) where k= (1/4πeo) = 8.99 x 10° N-m²/C². This is the general equation for the allowed radii of the electron's 'orbit' in Bohr's model of hydrogen. Show all work. n = 1,2,3,.... -n=2 (-3.39 eV) n=3 (-1.51 eV) e) Next substitute r = n²ħ²/(e²km) into the expression for the total mechanical energy of the electron, Eqn. (3), to obtain E = (-1/n²) (me4/8e02 h²). This result gives the general equation for the electron's allowed energy values in Bohr's model of the hydrogen atom. Show all work.
Transcribed Image Text:c) The total mechanical energy of the electron is the sum of its kinetic energy and potential energy. Derive an expression for the total mechanical energy of the electron in the hydrogen atom; (3) Your result should be negative. The negative result means energy must be added to excite the electron to higher energy levels 'farther away' from the nucleus. In this coordinate system the total energy is zero at an infinite distance from the nucleus. E = KE + U d) To explain the quantization observed in the emission spectra of hydrogen, Bohr introduced 'quantization' to the classical model by suggesting the angular momentum of the electron was quantized. This means the angular momentum can only have discrete values. Bohr quantized the angular momentum in multiples of Planck's constant; (4) In this equation m is the mass of the electron m = 9.11 x 10-31 kg and ħh is the modified Plancks's constant h/2; where ħ = h/2π = 1.05 x 10-34 J-s = 6.58 x 10-16 eV-s. n=1-+ (-13.6 eV) mvr = nh Re-arrange the angular momentum expression given in Eqn. (4) for v and substitute into Eqn. (1) to prove the radius of the electron's orbit is given by r = n²ħ²/(e²km) where k= (1/4πeo) = 8.99 x 10° N-m²/C². This is the general equation for the allowed radii of the electron's 'orbit' in Bohr's model of hydrogen. Show all work. n = 1,2,3,.... -n=2 (-3.39 eV) n=3 (-1.51 eV) e) Next substitute r = n²ħ²/(e²km) into the expression for the total mechanical energy of the electron, Eqn. (3), to obtain E = (-1/n²) (me4/8e02 h²). This result gives the general equation for the electron's allowed energy values in Bohr's model of the hydrogen atom. Show all work.
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