Given: The electron of mass me (9.10939 E-31 kg) and charge -Q (e = -1.60218 E-1

Question

Given: The electron of mass me (9.10939 E-31 kg) and charge -Q (e = -1.60218 E-19 Coulomb (Coul)) is attracted toward a proton of charge Q (+e) with Force of F=KQ^2/r^2 (K is 8.987 E9 Jm/Coul^2, Coulombs constant), with a force exactly balanced by the electron's centrifugal force F= mV^2/r ; where r is the radius of the H atom. (note, E-3 means 10 to the power of -3). The electron's kinetic energy KE = mV^2/2 and velocity V is so fast that the electron has relativistic properties given by De Broglie's wave equation, so that the wavelength (? or lambda) is calculated as ? lambda = h/mV , where h is Planck's constant quantum (6.626080 E-34 J sec)., Furthermore to eliminate destructive electron-wave interference, we assume that n x lambda ? = ?D = 2?r , which indicates that an integer n multiple of wavelength is equal to the circumference of the H atom = pi (3.14159) x Diameter of H atom circle; D =2r.   This H atom model allows the exact calculation of all parameters.

Find: Calculate the formulas for, and the exact values of the following parameters, using terms of the fundamental constants of the universe m, h, e, K, in order to find: H atom radius r (nm), electron velocity V (m/s, and % light-speed of C = 3.0 E8 m/s), kinetic energy KE of electrons in orbitals n = 1, 2, 3, 4, 5 ---, and the value of Rydberg's constant (J). Finally, find the wavelength lambda of light photons of energy E = hC/lambda ?, which are equivalent to the energy difference (?E) between orbitals of higher energy with n = 5, 4, or 3, relative to n= 2, and also relative to n= 1. Compare these theoretically calculated wavelengths from Bohr's H atom model with the observed emission-spectrum of wavelengths of electromagnetic light actually found for Hydrogen.

Explanation / Answer

In order for an electron to stay in an orbit around an atomic nucleus, the Coulomb force between the negative charged electron and the positively charged nucleus must be balanced with a centripetal force, so:



Where, me is the mass of the electron, v is the velocity, r is the distance between the nucleus and the electron, e is the charge of either the electron or the nucleus and k is the coulomb constant. You can find information about some physical constants like k (Coulomb constant) or like me (electron mass) in our appendix.

Therefore, the speed of the electron is related to its orbit radius, r, by the formula:



Electric potential energy of an electron with a distance r from the nucleus of an atom is defined as . There is a minus sign in the formula. This minus represents that the electron is bonded to the nucleus. When this value approaches to zero, the electron escapes. The total energy of an electron is the sum of its KE (kinetic energy)

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