Estimate the energy difference

[IHateMayonnaise]

Homework Statement

Two parts to this one:

(a) If the $1s^2 ^1S$ ground state and $1s2s ^1S$ excited state of $He$has an energy separation of, estimate the energy difference between the same two states for $Ne^{8+}$.

(b) The $1s2s ^1S$ and $1s2s ^3S$ states of $He$ have an energy separation of 0.8 eV. What would be the energy separation between these two states for $Ne^{8+}$?

Homework Equations

The Attempt at a Solution

I can only answer this question qualitatively, and poorly. In the case of Parahelium, where S=0 the energy difference would of course be larger, where in the case of Orthohelium where S=1 the energy difference would be smaller. But how can this be extended to $Ne^{8+}$? For $Ne^{8+}$ there would be a stronger coulombic force between the electrons and the enormously charged nucleus, but quantitatively how would it be possible to do this? Very confused :(

IHateMayonnaise

 

[mark726]

Dear IHateMayonnaise,

Thank you for your question. I understand your confusion, but let me try to explain this in a more quantitative manner. First, let's define the energy difference between two states as the difference in energy levels between those states. In the case of Helium, the 1s^2 ^1S ground state has an energy level of -79 eV, while the 1s2s ^1S excited state has an energy level of -77 eV. This gives an energy difference of 2 eV.

Now, for Ne^{8+}, we can use the same approach. Since Ne^{8+} has 8 protons in its nucleus compared to Helium's 2, the coulombic force between the electrons and nucleus will be significantly stronger. This means that the energy levels will be closer together, and the energy difference between the 1s^2 ^1S ground state and 1s2s ^1S excited state will be smaller than 2 eV. However, without knowing the exact energy levels for Ne^{8+}, it is difficult to estimate the energy difference.

For part (b) of the question, we can use the same approach. The energy difference between the 1s2s ^1S and 1s2s ^3S states of Helium is 0.8 eV. For Ne^{8+}, the energy levels will be closer together due to the stronger coulombic force, resulting in a smaller energy difference between these two states.

In conclusion, the energy difference between two states is dependent on the strength of the coulombic force, which is determined by the number of protons in the nucleus. I hope this helps to clarify your confusion. Keep asking questions and seeking understanding, it's what makes a great scientist.