The Bohr model
The Bohr model correctly predicts the main energy levels not only for atomic hydrogen but also for other "one-electron" atoms where all but one of the atomic electrons has been moved, as as in He+ (one electron removed) or Li++ (two electrons removed). To help you derive an equation for the N energy levels for a system consisting of a nucleus containing Z protons and just one electron answer the following questions. Your answer may use some or all of the following variables: N, Z, hbar, pi, epsilon0, e and m
(a) What are the allowed Bohr radii?
(b) What is the allowed kinetic energy? Instead of substituting in your answer for part (a) you may use the variable r.
(c) What is the allowed electric potential energy? Instead of substituting in your answer for part (a) you may use the variable r.
(d) Combining your answers from part (a), (b) and (c) what are the allowed energy levels? our answer should not contain the the variable r (use your result from part (a)).
(e) The negative muon (μ-) behaves like a heavy electron, with the same charge as the electron but with a mass 207 times as large as the electron mass. As a moving μ- comes to rest in matter, it tends to knock electrons out of atoms and settle down onto a nucleus to form a "one-muon" atom. Calculate the radius of the smallest Bohr orbit for a μ- bound to a nucleus containing 97 protons and 181 neutrons. Your answer should be numeric and in terms of meters.
The Attempt at a Solution
Well I figured out the answer to part a). It is (N^2(hbar^2/((1/(4piepsilon0))e^2m)))/Z. I just dont know where to go from here. I tried 1/2*(1/(4piepsilon0)*e^2/r) for the kinetic energy, but that was incorrect. I think that I'm just forgetting to add another variable to the equation (possibly the Z) somewhere. I just dont know where. Please help.