Help needed with Bohr model question

[nealskilling]

Hi,

I posted a question here a few months ago and got some help with an assignment question I was stuck on. I'm hoping someone can help me out again!

The question is - Following Bohr's model, derive an expression for the allowed energy levels of an electron bound to a nucleus of charge +Ze. At each stage, state explicitly any laws, postulates or assumptions you employ. Express your answer in terms of E1 of the first Bohr orbit in Hydrogen.

Okay, I know that the centripetal acceleration of the electron (mev2/r) is equal to the magnitude of the electrostatic force pulling the electron towards the nucleus (Ze2/4piE0r2).

I also know that the total energy of the electron is the sum of it's kinetic energy (0.5mev2) and the electrostatic potential energy (-Ze2/4piE0r) which equals -Ze2/8piE0r.

r is equal to n2a0, so that gives -Ze2/8piE0n2a0.

I'm told that E1 = e2/8piE0a0 = -13.6 eV.

I therefore thought that the expression for the energy of the electron would be -ZE1/n2.

I looked up the energy of an electron in ionized helium in the ground state and it is 54.4 eV. Using my equation I get 27.2 eV, exactly half of what the answer should be, which makes me think that the correct equation should be - Z2E1/n2.

I've spent hours looking at this and can't figure out why I'm getting ZE1 instead of Z2E1 in my final equation. If anyone out there can give me a hint, that would be very much appreciated.

Thanks in advance.

Neal.

[dynamicsolo]

r is equal to n2a0, so that gives -Ze2/8piE0n2a0.

I believe the "Bohr radius" for a hydrogenic atom is a'o = a0/Z , no?

[nealskilling]

Hi Dynamicsolo,

Just wanted to say thanks for your reply. You were correct, it should have been r = a0n2/Z. I went back over the calculation and was able to get the correct answer.

Thanks again.

Neal.

[dynamicsolo]

The issue is simply over the point of reference -- your result is correct if ao is taken to be the (first) Bohr radius for the hydrogenic atom with atomic number Z. The convention in a great many applications, however, is to express results in terms of the Bohr radius for hydrogen, 0.0529 nm.