Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof of maximum no. of electrons in a shell

  1. Apr 16, 2013 #1

    -ve

    User Avatar

    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. jcsd
  3. Apr 16, 2013 #2

    Borek

    User Avatar

    Staff: Mentor

    Solving Schrödinger equation for a hydrogen atom.
     
  4. Apr 16, 2013 #3

    -ve

    User Avatar

    thanks
    ^_^
     
  5. Apr 17, 2013 #4

    DrDu

    User Avatar
    Science Advisor

    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.
     
  6. Apr 17, 2013 #5

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
  7. Apr 17, 2013 #6

    DrDu

    User Avatar
    Science Advisor

    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?
     
  8. Apr 20, 2013 #7
    Look at the experimental ionization potentials and note the pattern. Compare to the solution of the SE for the hydrogen atom.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof of maximum no. of electrons in a shell
  1. Electron shells (Replies: 7)

  2. Electron Sub-Shells (Replies: 2)

Loading...