The maximum number of electrons that can occupy the shell having a principal quantum number of 3 is

The maximum number of electrons that can occupy any electron shell is given by the formula 2n², where n is the principal quantum number of that shell.

For the third shell (n = 3): maximum electrons = 2 × 3² = 2 × 9 = 18.

To understand why, the third shell contains three sub-shells: 3s, 3p, and 3d. The 3s sub-shell holds a maximum of 2 electrons. The 3p sub-shell holds a maximum of 6 electrons. The 3d sub-shell holds a maximum of 10 electrons. Adding these up: 2 + 6 + 10 = 18 electrons total.

For comparison, the first shell (n = 1) holds 2 × 1² = 2 electrons, and the second shell (n = 2) holds 2 × 2² = 8 electrons. The formula 2n² is a quick way to find the electron capacity of any shell without counting each sub-shell separately.