Exploring the Bohr Model: Allowed Radii, Kinetic & Electric Potential Energy

In summary, the Bohr model correctly predicts the main energy levels for one-electron atoms, including those with removed electrons. To derive an equation for the energy levels of a system with a nucleus containing Z protons and one electron, the allowed Bohr radii, kinetic energy, and electric potential energy must be calculated. The resulting equation is -(1/2)*(Z/(4piepsilon0))*e^2/((N^2(hbar^2/((1/(4piepsilon0))e^2m)))/Z). For the last part, the mass of the negative muon must be taken into account, which is 207 times that of an electron. The calculated answer for the smallest Bohr orbit for a
  • #1
guitarguy1
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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?

(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.
 
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  • #2


Is there a reason that nobody has made an attempt to answer my question?
 
  • #3


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.
 
  • #4


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
 
  • #5


gtzpanditbhai said:
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. : )
 

1. What is the Bohr Model?

The Bohr Model is a simplified model of the atom proposed by Danish physicist Niels Bohr in 1913. It describes the structure of an atom as a small, positively charged nucleus surrounded by orbiting electrons at specific energy levels, also known as shells or energy levels.

2. What are allowed radii in the Bohr Model?

Allowed radii in the Bohr Model refer to the specific distances at which electrons can orbit around the nucleus without losing energy. These radii are determined by the energy levels of the electrons, which are quantized in the Bohr Model.

3. What is kinetic energy in the Bohr Model?

Kinetic energy in the Bohr Model refers to the energy that an electron possesses due to its motion around the nucleus. The kinetic energy of an electron depends on its speed and distance from the nucleus.

4. What is electric potential energy in the Bohr Model?

Electric potential energy in the Bohr Model refers to the energy that an electron possesses due to its position in relation to the positively charged nucleus. The closer an electron is to the nucleus, the higher its electric potential energy.

5. How is the Bohr Model useful in understanding atomic structure?

The Bohr Model is useful in understanding atomic structure because it provides a simple visual representation of the structure of an atom and explains the behavior of electrons in an atom. It also helps explain the relationship between energy levels and electron orbitals, and how electrons gain or lose energy when they transition between energy levels.

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