Why do electron shells have set limits? (1999)

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Date: Wed Mar 17 16:18:14 1999
Posted By: Dan Berger, Faculty Chemistry/Science, Bluffton College
Area of science: Chemistry
ID: 921203878.Ch

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Message:

Why do electron shells have set limits ?

As we know electron shells or energy levels can only contain a certain number of electrons ie 2,8,18,32,32,18,8 which match nicely to the first four quadratic numbers. We know what the individual shell configuration for each atom is. We use this information to explain the bonding between atoms, why gases are inert and it all fits very nicely into groups in the periodic table. What I would like to know is. Why does each particular shell have a set number of electrons that match the first four quadratic numbers. Doesn't this hint that there is a mathematical law that denotes why this is so, and therefore describes the nature of all matter ?

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It is true that electron shells can hold 2, 8, 18, 32, 50, ... electrons. While 2, 8, 18, 32, 50, ... are each twice a quadratic number (1, 4, 9, 16, 25, ..., n²), this is an accident as we will see.

The more complex mathematics behind what I am going to tell you can be found in textbooks (such as MacQuarrie's Quantum Chemistry), or you might check out my modest collection of computational chemistry links. Some of the sites have extensive lectures available on molecular and atomic quantum mechanics.

The quantum mechanical description of many-electron atoms is based on the Schrödinger equation for the hydrogen atom, which is exactly soluble.[*] The solution tells us that an electron in an atom is fully described by four quantum numbers, labeled n, l, m and s. These quantum numbers have particular allowed values:

• n can be any positive integer (1, 2, 3, 4, ...)
• l can be any non-negative integer up to n-1 (0, 1, 2, 3, 4, ..., n-1).
• m can be any integer from -l to l (-l, -l+1, ..., -1, 0, 1, ..., l-1, l).
• s can have only two values, ±½; these are sometimes labeled a and b.

The quantum numbers l and m can be understood in terms of subshells. Each shell (defined by n) can have n-1 types of subshell; the number of each type of subshell allowed is governed by the number of possible values of m. Thus,

• The first shell (n=1) can have only one type of subshell (l=0, referred to as an "s" subshell or orbital), and since the only allowed value of m is zero (m=l), there is only one s orbital.
• The second shell (n=2) can have two types of subshell (l=0, "s" and l=1, "p" orbitals).
  1. If l=0, m can only have the value 0; there is one s orbital.
  2. If l=1, m can have the values -1,0,1; there are three p orbitals.
• The third shell (n=3) can have three types of subshell (l=0, "s", l=1, "p" and l=2, "d"orbitals).
  1. If l=0, m can only have the value 0; there is one s orbital.
  2. If l=1, m can have the values -1,0,1; there are three p orbitals.
  3. If l=2, m can have the values -2,-1,0,1,2; there are five d orbitals.
• The fourth shell (n=4) can have four types of subshell (l=0, "s", l=1, "p", l=2, "d" and l=3, "f"orbitals).
  1. If l=0, m can only have the value 0; there is one s orbital.
  2. If l=1, m can have the values -1,0,1; there are three p orbitals.
  3. If l=2, m can have the values -2,-1,0,1,2; there are five d orbitals.
  4. If l=3, m can have the values -3,-2,-1,0,1,2,3; there are seven f orbitals.

You should get the picture.

Thus, we see that

The first shell contains one orbital.
The second shell contains four (1+3) orbitals.
The third shell contains nine (1+3+5) orbitals.
The fourth shell contains sixteen (1+3+5+7) orbitals.
...
The nth shell contains n² (1+3+5+7+...+2n-1) orbitals.

This is a quadratic sequence, but "by accident." There is no underlying mathematics which forces the sequence to be quadratic, it is merely an accident of the properties of the first three quantum numbers associated with an electron in the hydrogen atom.

All we have left is the Pauli Exclusion Principle, which says that no two electrons in the same atom can share all four quantum numbers; it is often stated as "an atomic orbital can hold a maximum of two electrons."

It is the maximum occupancy of two electrons per orbital which leads to the nice quadratic sequence: each shell 1,2,3,4,...,n can hold 2,8,18,32,...,2n² electrons.

The obvious next question is, "Why don't the orbitals fill in a logical manner, one shell at a time, instead of skipping around?" But that's another question and anoth er answer!

[*]

Schrödinger equations for many-electron atoms involve more than two interacting bodies (the nucleus counts as one, but the electrons must be considered individually). The general many-body problem has not been solved and so we must use approximate methods, based on the solution for the hydrogen atom, to describe all other atoms.

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