Distance of electron from nucleus

dinospamoni

1. Homework Statement

a) Consider a hydrogenic atom consisting of a carbon-12
nucleus and a single electron. What is the wavelength of the
photon emitted when this atom drops from n = 5 to the n = 2
state?

b) According to the Bohr theory, what is the distance
between the electron and the nucleus in the n = 2 state of the
hydrogenic carbon-12 atom?

2. Homework Equations

$r_n = n^2 \times a_0$

where $a_0$ is the bohr radius

3. The Attempt at a Solution

I found the answer to part A no problem, but b is giving me trouble.

I know that the equation for the radius for excited states is for a hydrogen atom. WIth a carbon nucleus I expect it to be much smaller that the bohr radius, but I can't figure out how to account for the change in nucleus size.

Any ideas?

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voko

What are the postulates of Bohr's model?

dinospamoni

Bohr's Model says that angular momentum is quantized, or only has certain values, so

$\vec{l} = \vec{r} \times \vec{p}$

simplified, it is

$l = rmv = n\hbar$

Right direction?

voko

That is not incorrect, but that does not seem to help much. What is the shape of orbits in Bohr's model, and what holds electrons in place?

dinospamoni

Ok so the electron has energy of -122.4 eV = -1.96*10-17J

The total energy of the electron is:
$E = KE + U E = \frac{1}{2} m_e v^{2} + \frac{k q^2}{r}$

And by F = ma

$\frac{m_{e} v^{2}}{r} = \frac{k q^{2}}{r^2}$

I solved for v2

and found this expression for r:

$r = \frac{2 k q^{2}}{k q^{2}-2 E}$

However I found this radius to be 1.18 * 10-11m which is wrong

voko

The equation you termed F = ma is not entirely correct, as you neglected the charge of the nucleus.

Then you could use the angular momentum rule.

dinospamoni

Ah I see.

so the force equation would be

$\frac{m_{e} v^{2}}{r} = \frac{k 6 q^{2}}{r^2}$?

voko

Ah I see.

so the force equation would be

$\frac{m_{e} v^{2}}{r} = \frac{k 6 q^{2}}{r^2}$?
Yes.

"Distance of electron from nucleus"

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