Mastering Atoms: Formulas, Traps & Exam Strategy | ExamiFly

The negative sign in $E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$ signifies that the electron is bound to the positive nucleus by attractive electrostatic forces. Zero potential energy is defined at infinite distance ($r \to \infty$). To liberate the electron completely from the nucleus (making it a free electron with zero kinetic energy), work equal to $+E_n$ must be supplied from an external source.

Excitation potential is the minimum potential difference required to accelerate a free electron to transition an atomic electron from its ground state ($n=1$) to a higher bound state ($n > 1$), such as $n=2$ (First Excitation Potential $= 10.2 \text{ V}$ for $H$). Ionization potential is the potential needed to knock the electron out of the atom entirely ($n=1 \to n=\infty$), which requires $13.6 \text{ V}$ for Hydrogen.

De Broglie proposed that an electron in a stationary orbit behaves like a standing matter wave along the circular path. For constructive interference (a stable standing wave), the orbit circumference must equal an integral number of de Broglie wavelengths: $2 \pi r_n = n \lambda$. Substituting $\lambda = \frac{h}{m v}$ into this relation yields $2 \pi r_n = n \frac{h}{m v}$, which simplifies directly to Bohr's quantization condition $m v r_n = \frac{n h}{2 \pi}$.