Solve Bohr Atom Problem: Find Radius, Energy, Wavelenght

[Fabio010]

Find the first radius, the energy of "connection" and the wavelenght of the first risk of Lyman series of a hydrogen atom that electron has been "replaced" for a meson with negative charge with a mass 207*electron mass. Radius:

r = (h^2*n^2)/(4pi^2*e^2*k*me)

we know all constants so we just replace me for 207me and the solution is 2.6*10^-13m
in book solution the result is 2.8*10^-13m (i don't think that the difference is not significant)Energy of "connection"

E = - (ke^2)/(2r)

as we now radius, the energy is -4.4 *10^-16 J
in book is = -4.04*10^-16 JWavelenght (the problem)

E = hc/l (l = wavelenght)

l = hc/(4.4 *10^-16) = 4.5 * 10^-10 m 0.45 nm
but in book solutions l = 0.656nm

the problem is that with book results

l = hc/(4.04*10^-16) = 4.9 * 10^-10 m 0.49 nmSo i guess that i resolved it in a wrong way.PS: Sorry my english.

 

[217 MeV]

By simply replacing m_e with 207m_e, you are assuming that the meson is orbiting the centre of mass of the hydrogen proton. This assumption is valid for electron orbits as their mass is negligible for all intents and purposes, but as the proton is only ~9 times more massive than the meson, we must account for the fact that the proton and meson will orbit the proton-meson system's centre of mass.

To do this, we have to use the concept of reduced mass to reduce the problem back to a one-body problem. (the wiki below is quite good, as is the hyperphysics page).

http://en.wikipedia.org/wiki/Reduced_mass
http://hyperphysics.phy-astr.gsu.edu/hbase/orbv.html#rm

I recommend that you give those a read. If you need any more information, this concept is covered in every undergrad physics textbook I've come across.

 

[Fabio010]

ok, so i resolved my problem in wrong way.

because i can not replace the me for 207me?

 

[217 MeV]

I wouldn't say it was resolved in the wrong way, your reasoning will work well enough for any purpose where the orbiting particle has negligible mass compared to the nucleus.

The same idea applies when we model planetary orbits. Assuming that the orbiting body orbits the centre of mass of the more massive body gives us an accurate enough prediction, but for increased precision we have to take into account the fact that the massive body isn't stationary, it instead orbits the centre of mass of the entire system.

 

[Fabio010]

Ok but what i assumed, is what the book assumed because the solutions are correct.

Just, the wavelength solution is different. I don't know if the solutions are wrong, or E = hc/λ is not the right way to do it.

 

[217 MeV]

Well, your solution is incorrect at two significant figures, so I presume your book didn't make that assumption, and you may have to use reduced mass in the future.

It seems like you have been asked to find the first Lyman series wavelength for this atom, correct? You have worked out the wavelength of a photon with energy equal to that of the n=1 level for this atom. The first Lyman series wavelength will be brought about by the transition of the meson between the n=1 and n=2 energy levels. This should give you the correct answer.