What are the allowed radii and energy levels in the Bohr model?

[guitarguy1]

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?

(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.

Homework Equations

r=N^2(h^2)/((1/4piepsilon0)*e^2*m)

K=1/2*(1/(4piepsilon0)*e^2/r)

Uelectric=-K

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 don't 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 don't know where. Please help.

 

[guitarguy1]

Is there a reason that nobody has made an attempt to answer my question?

 

[guitarguy1]

Alright I figured out what I was doing wrong for parts B, C, and D. I got (1/2)*(Z/(4piepsilon0))*e^2/r for B), -((Z/(4piepsilon0))*e^2/r) for C) and -(1/2)*(Z/(4piepsilon0))*e^2/((N^2(hbar^2/((1/(4piepsilon0))e^2m)))/Z) for D).

I would just like some guidance now on this last part.

(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.

I think it just involves changing the mass of the, but just using the same answer that I found to part A. Please tell me if I'm heading in the right direction.

 

[gtzpanditbhai]

It is simple once you think about it. You do use the equation from part a. N=1 since the claculated energy level is below .5e-13. hbar = 1.05e-34 J*s. 1/4PiEpsilon = 8.99e9 N*m^2/c^2 and e = 1.60e-19 c. The tricky part is the mass. Since it says the mass is 207 times that of an electron, the mass would be (9.11e-31 * 207). So plug all that into the formula and divide by Z which is the number of proton and you should get an answer like xxxe-15m.

Your answer should be approximately 2.62e-15m

 

[guitarguy1]

It is simple once you think about it. You do use the equation from part a. N=1 since the claculated energy level is below .5e-13. hbar = 1.05e-34 J*s. 1/4PiEpsilon = 8.99e9 N*m^2/c^2 and e = 1.60e-19 c. The tricky part is the mass. Since it says the mass is 207 times that of an electron, the mass would be (9.11e-31 * 207). So plug all that into the formula and divide by Z which is the number of proton and you should get an answer like xxxe-15m.

Your answer should be approximately 2.62e-15m

Thanks. Yea it was simple. I had just gotten so used to putting my answers in terms of variables, I didnt notice that they had asked for a numerical value. : )