If the principal quantum number (n) is 3, what is the maximum number of electrons that can occupy this shell?

The maximum number of electrons that can occupy a given shell is determined by the formula (2n^2), where (n) is the principal quantum number corresponding to that shell. When (n) is 3, this calculation becomes:

[ 2n^2 = 2(3^2) = 2 \times 9 = 18 ]

This means that the shell associated with the principal quantum number of 3 can accommodate a total of 18 electrons based on the allowed orbital configurations and the electronic structure principles.

In terms of atomic structure, the shell describes the overall energy level of the electrons around the nucleus, and the maximum number of electrons at each level is a fundamental aspect of quantum mechanics applied to electron configurations. Higher principal quantum numbers allow for more complex arrangements of orbitals, leading to increased electron capacity in these energy levels.