Filling the shells

How many electrons in each shell?

Two scientists, Bohr and Bury, gave the rules for how the electrons are arranged in the shells.

1. The most electrons that a shell can hold is $2n^2$, where $n$ is the number of the shell. So:
   • K shell ($n = 1$): $2 \times 1^2 = 2$ electrons.
   • L shell ($n = 2$): $2 \times 2^2 = 8$ electrons.
   • M shell ($n = 3$): $2 \times 3^2 = 18$ electrons.
   • N shell ($n = 4$): $2 \times 4^2 = 32$ electrons.
2. The outermost shell of an atom can hold no more than $8$ electrons.
3. The shells are filled step by step. A shell takes electrons only after the shell inside it is full.

The first twenty elements

For the first twenty elements, we can use a simple scheme. Put $2$ electrons in the first shell, up to $8$ in the second and up to $8$ in the third. The rest go in the fourth. The numbers written one after the other, such as $2, 8, 1$, are the electron arrangement of the atom.

You should find that a new shell begins only when the one inside it is full: at $Z = 3$, $Z = 11$ and $Z = 19$.

A worked example

Write the arrangement of phosphorus, which has $15$ electrons.

• K shell: $2$. Electrons left: $13$.
• L shell: $8$. Electrons left: $5$.
• M shell: the $5$ that are left.

So the arrangement is $2, 8, 5$.

Practise with new numbers

Try it in Python

This program fills the shells for the first twenty elements.

Remember these

• Shell $n$ holds at most $2n^2$ electrons. K: $2$, L: $8$, M: $18$, N: $32$.
• The outermost shell holds at most $8$.
• Fill the inner shells first. Up to calcium: $2, 8, 8, 2$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many electrons can the L shell hold at most?

2. Write the electron arrangement of chlorine, which has $17$ electrons.

3. Why is $2, 10$ not a correct arrangement for the atom of neon?

4. Potassium has $19$ electrons. Why does its last electron go into the N shell, not the M shell?