Proof of maximum no. of electrons in a shell

1. Apr 16, 2013

-ve

how do you prove that the maximum no. of electrons in the nth shell of an atom is twice of n squared (2n^2)

2. Apr 16, 2013

Staff: Mentor

Solving Schrödinger equation for a hydrogen atom.

3. Apr 16, 2013

-ve

thanks
^_^

4. Apr 17, 2013

DrDu

To be a bit more specific: The formula 2n^2 is based on the assumption that the shells are hydrogen like. There are deviations from this rule.
The factor 2 is due to the fact that each orbital can carry at most two electrons, one with spin up, the other with spin down.
So we have to explain why there are n^2 orbitals in each shell.
It is a peculiarity of the hydrogen atom that all orbitals having the same number of node surfaces have the same energy. There are radial nodes and spherical nodes. All orbitals in a given shell have n-1 nodes. The number of spherical nodes fixes whether we speak of an s, p, d, or f orbital. The number of spherical nodes is equal to the quantum number l with l=0 corresponding to s, l=1 to p etc. There are 2l+1 orbitals with the same value of l. So if e.g. n=4 the orbitals have 3 nodes. There are the following possibilities
# radial nodes #spherical nodes=l name multiplicity=2l+1
0 3 f 7
1 2 d 5
2 1 p 3
3 0 s 1

You can check that the sum of the multiplicities is 16=n^2.
In general $\sum_{l=0}^{n-1}(2l+1)=n^2$
as Kolmogorow, the father of modern statistics, realized as a 5 year old boy.

5. Apr 17, 2013

dextercioby

The only rigorous proof for 2n^2 would be to solve the SE for the atom which is mathematically impossible. Not even the helium atom admits a complete solution.

6. Apr 17, 2013

DrDu

Certainly. However, I think it is quite nice that in case of the H atom the energetic ordering of the orbitals depends only on the number of nodes (which can be traced back to the hidden SO(4) symmetry).
Is there a pedagogical way of making plausible that there are 2l+1independent spherical harmonics with given l?

7. Apr 20, 2013

Einstein Mcfly

Look at the experimental ionization potentials and note the pattern. Compare to the solution of the SE for the hydrogen atom.