How Does the Bohr Model Explain Energy Levels in One-Electron Atoms?

AI Thread Summary
The Bohr model effectively predicts energy levels for one-electron atoms, including hydrogen and ions like He+ and Li++. It involves calculating allowed Bohr radii, kinetic energy, and electric potential energy, leading to a formula for energy levels based on variables such as N, Z, and fundamental constants. A specific question regarding the radius of a "one-muon" atom is also posed, requiring numerical calculation based on the mass of the muon and the atomic nucleus. Participants are encouraged to demonstrate their problem-solving attempts in accordance with forum rules. The discussion emphasizes the application of the Bohr model to various atomic structures.
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Homework Statement



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?
r = Click here to preview your answer.wrong check mark
Click here for help with symbolic formatting.

(b) What is the allowed kinetic energy? Instead of substituting in your answer for part (a) you may use the variable r.
k = Click here to preview your answer.
Click here for help with symbolic formatting.

(c) What is the allowed electric potential energy? Instead of substituting in your answer for part (a) you may use the variable r.
Uel = Click here to preview your answer.
Click here for help with symbolic formatting.

(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 = Click here to preview your answer.
Click here for help with symbolic formatting.




(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 77 protons and 194 neutrons. Your answer should be numeric and in terms of meters.
r = m



The Attempt at a Solution



Tried all solutions for the first one but no luck !
 
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First answer is :

(N^2(hbar^2/((1/(4piepsilon0))e^2m)))/Z
 
Welcome to PF.

As per forum rules, you must demonstrate an attempt at solving the problem before receiving help.

Your answer for (a) looks correct.
 

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