Bohr radius of a helium ion

[leroyjenkens]

Homework Statement

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.

Homework 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

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.

 

[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?

 

[Borek]

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

 

[leroyjenkens]

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?

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

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

a₀ = Bohr radius

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

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.

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

Thanks, was wondering why it wasn't working.

 

[vela]

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