Why does a gold isotope, with an atomic number of 79, require more neutrons to be stable than a sodium isotope, with an atomic number of 11?

Here's a simplified explanation.

The protons in the nucleus repel each other. The neutrons provide the "glue" that holds the nucleus together and prevents it from flying apart.

The "glue" is the strong nuclear force. It is strong but extremely short range. It falls off extremely rapidly as the p-n distance increases.

A neon atom has 10 protons. There are three stable isotopes, with 10, 11, and 12 neutrons.

With fewer than 10 protons, the glue is not strong enough to hold the nucleus together.

If there are more than 12 neutrons, the average p-n distance is great enough that the glue has again become too weak.

Gold has one stable isotope. It contains 79 protons and 118 neutrons.

If there are fewer than 118 neutrons, the proton repulsions will be too strong for the strong force. If there are more than 118 neutrons, the average p-n distance will be large enough that the glue will again be too weak to hold the nucleus toge ther.