# Bohr radius of a helium ion

1. Oct 5, 2013

### leroyjenkens

1. The problem statement, all variables and given/known data
Using the Bohr model, find the atomic radius for a singly ionized He+ atom in the n =
1 (ground) state and the n = 2 (first excited) state. Then find the wavelength of the
emitted photon when an electron transitions from the n = 2 to the n = 1 state.

2. Relevant equations
$$a_0 = \frac{4∏e_0h^2}{me^2}$$

m = mass of the electron
h = h bar. Reduced Planck's constant
e = elementary charge

3. The attempt at a solution
I need some way of adding a proton and a neutron to the equation, but there's no such variable. I can't find any equation relating to the Bohr radius that isn't just constants. The Borh model can supposedly give you the radius of a helium cation, but that's the only equation I can find and it's all constants.

Last edited by a moderator: Oct 6, 2013
2. Oct 6, 2013

### Physics Monkey

The Bohr model proceeds in two steps. First, you solve for the velocity as a function of the radius assuming the Coulomb force provides the radial acceleration. Second, you impose that the angular momentum be quantized. This then determines a discrete set of allowed radii and energies.

Among these physical ingredients, how does single ionized Helium differ from Hydrogen? For example, the mass of the hydrogen atom plays no role in Bohr's model (which assumes the nucleus is infinitely anyway), so the presence of a neutron which doesn't interact electrostatically should be irrelevant. What other differences are there?

3. Oct 6, 2013

### Staff: Mentor

Don't mix LaTeX with [noparse] and [/noparse] tags. Use _ and ^. Corrected that for you.

4. Oct 6, 2013

### leroyjenkens

Helium has a greater positive charge, so the nucleus going to pull on the lone electron stronger than the hydrogen nucleus would, which will make the radius smaller.

I'm looking through the equations in this book and I have....

$$v=\frac{nh}{mr}$$

h = h bar

$$r = n^{2}a_0$$

$$v = \frac{nh}{mn^{2}a_{0}}$$

The book doesn't say what r stands for, but it's part of the angular momentum equation L = mrv, so it must be radius of the electron orbit, which is just the radius of the atom.

So the radius of the atom is r, but isn't that what the Bohr radius is supposed to be?

The formula for a0 is...
$$a_{0}=\frac{4∏ε_{0}h^{2}}{me^{2}}$$

So I'm guessing the key has to do with the e2 in the denominator, since the charges of the hydrogen and helium are different.
Thanks, was wondering why it wasn't working.

5. Oct 6, 2013

### vela

Staff Emeritus
Do what Physics Monkey suggested. Start with F=ma and $mvr=n\hbar$ to find what you need.