Distance of electron from nucleus

  • #1
dinospamoni
92
0

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?

Homework Equations



[itex]r_n = n^2 \times a_0[/itex]

where [itex]a_0[/itex] is the bohr radius

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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  • #2
What are the postulates of Bohr's model?
 
  • #3
Bohr's Model says that angular momentum is quantized, or only has certain values, so

[itex]\vec{l} = \vec{r} \times \vec{p} [/itex]

simplified, it is

[itex]l = rmv = n\hbar[/itex]

Right direction?
 
  • #4
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?
 
  • #5
Ok so the electron has energy of -122.4 eV = -1.96*10-17J

The total energy of the electron is:
[itex] E = KE + U

E = \frac{1}{2} m_e v^{2} + \frac{k q^2}{r}[/itex]

And by F = ma

[itex] \frac{m_{e} v^{2}}{r} = \frac{k q^{2}}{r^2} [/itex]

I solved for v2

and found this expression for r:

[itex] r = \frac{2 k q^{2}}{k q^{2}-2 E} [/itex]

However I found this radius to be 1.18 * 10-11m which is wrong
 
  • #6
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.
 
  • #7
Ah I see.

so the force equation would be

[itex]\frac{m_{e} v^{2}}{r} = \frac{k 6 q^{2}}{r^2}[/itex]?
 
  • #8
dinospamoni said:
Ah I see.

so the force equation would be

[itex]\frac{m_{e} v^{2}}{r} = \frac{k 6 q^{2}}{r^2}[/itex]?

Yes.
 
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