What is the formula used to calculate the maximum number of electrons for each shell?

The maximum number of electrons that can occupy each shell in an atom is determined by the formula (2n^2), where (n) represents the principal quantum number, which corresponds to the shell number. This formula arises from the principles of quantum mechanics, particularly the solutions to the Schrödinger equation for hydrogen-like atoms.

In detail, each shell (or energy level) can hold a certain capacity of electrons, which increases with the shell number. The principal quantum number (n) indicates the shell's level, starting with (n=1) for the first shell, (n=2) for the second, and so on. For example:

• For the first shell ((n=1)): (2(1^2) = 2) electrons.

• For the second shell ((n=2)): (2(2^2) = 8) electrons.

• For the third shell ((n=3)): (2(3^2) = 18) electrons.

This quadratic relationship captures how as one moves to higher shells, the ability to accommodate more electrons increases significantly, specifically due to the square of the shell number being multiplied by 2.