principle quantum number

Meaning: n labels the shell an electron occupies and can be any positive integer. A larger n means a larger orbital, an electron farther from the nucleus on average, and a higher energy. For a given n, the angular momentum quantum number l runs from 0 to n − 1, so a shell has n subshells, n² orbitals, and room for up to 2n² electrons. This is why valence electrons are easier to remove as n increases down a group.

Equation: En = −RH / n², where En is the electron's energy in a one-electron atom or ion such as hydrogen (J), RH is the Rydberg constant, 2.18 × 10⁻¹⁸ J, and n is the principal quantum number (unitless).

On the MCAT: Passages on hydrogen emission and absorption ask which transition releases the most energy or the shortest wavelength. Energy levels crowd together as n grows, so n = 2 → 1 releases more energy than 3 → 2. The photon energy is E = hf. Other questions ask which sets of quantum numbers are allowed, or how many orbitals or electrons a shell holds.

Common confusion: Students assume n alone fixes energy. The −1/n² formula holds for one-electron species. In multi-electron atoms, energy depends on both n and l, which is why 4s fills before 3d. Also, because En is negative, a higher n is closer to zero and therefore higher in energy.