Atomic structure and quantized description

Matter is built from atoms, each composed of a nucleus (protons + neutrons) surrounded by electrons in quantized energy levels.

Quantum numbers — Each electron is described by four numbers:

• $n$ — Principal ($1, 2, 3, \ldots$)
• $\ell$ — Angular momentum ($0$ to $n-1$)
• $m_\ell$ — Magnetic ($-\ell$ to $+\ell$)
• $m_s$ — Spin ($+\tfrac{1}{2}$ or $-\tfrac{1}{2}$)

Key relationships

• Amount of substance: $n = m/M$
• Avogadro's number: $N_A = 6.022\times 10^{23}\;\text{mol}^{-1}$
• Ideal gas law: $PV = nRT$

Electron configuration follows three rules:

1. Aufbau principle — Fill lowest energy orbitals first.
2. Pauli exclusion — No two electrons share all four quantum numbers.
3. Hund's rule — Maximise unpaired spins within a subshell.

Physics link: The discrete energy levels $E_n = -13.6/n^{2}$ eV (hydrogen) come from solving the Schrödinger equation for the Coulomb potential.

Common pitfall: Electron "orbits" are a cartoon: orbitals are probability clouds, and energy levels are discrete. An electron does not spiral between levels — it jumps, absorbing or emitting exactly the energy difference as light.

The Bohr Model

In the Bohr model, electrons orbit the nucleus in allowed orbits with quantized angular momentum:

$$L = n\hbar, \quad n = 1, 2, 3, \ldots$$

The energy of each orbit is:

$$E_n = -\frac{13.6}{n^2}\;\text{eV}$$

Transitions between levels emit or absorb a photon of energy $\Delta E = E_f - E_i$.

Although superseded by quantum mechanics, the Bohr model correctly predicts hydrogen's emission spectrum.