Bohr (1913) proposed three postulates: electrons move in circular orbits without radiating; only orbits where $L = n\hbar$ are allowed; radiation emitted/absorbed only during transitions. Results for hydrogen: $r_n = n^2 a_0$ (Bohr radius $a_0 = 0.529$ Å), $E_n = -13.6/n^2$ eV. Ground state ($n=1$): $E = -13.6$ eV. Second level ($n=2$): $E = -3.4$ eV. Third level: $-1.51$ eV. Fourth: $-0.85$ eV. Infinity: $0$ eV (free electron). For hydrogen-like ions (He⁺, Li²⁺, etc.): $E_n = -13.6Z^2/n^2$ eV. The quantization of energy levels directly explains the discrete line spectrum of hydrogen — electrons can only have specific energies, so photons of only specific frequencies are emitted or absorbed.

The Rydberg formula: $1/\lambda = R_H(1/n_1^2 - 1/n_2^2)$ where $n_2 > n_1$ (emission). $R_H = 1.097 \times 10^7$ m⁻¹ = 109677 cm⁻¹. Spectral series: Lyman ($n_1=1$): UV, 91.2–121.6 nm. First line (2→1): 121.6 nm (Lyman alpha). Balmer ($n_1=2$): UV+visible, 364.6–656.3 nm. Lines: H-α=656 nm (red), H-β=486 nm (blue-green), H-γ=434 nm (violet), H-δ=410 nm (violet). Paschen ($n_1=3$): near IR, 820–1875 nm. Brackett ($n_1=4$): IR. Pfund ($n_1=5$): far IR. Important rule: maximum wavelength (minimum energy) in any series = transition to adjacent level (e.g., 3→2 for Balmer). Series limit (minimum wavelength) = $n_2=\infty$. The Balmer series is historically important because it falls in visible range and was discovered first (Balmer, 1885) — before Bohr's theory explained it.

Energy of transition: $\Delta E = 13.6(1/n_1^2 - 1/n_2^2)$ eV. Wavelength: $\lambda = 1240/\Delta E$ (nm) where $\Delta E$ in eV. For ratio problems: since $1/\lambda \propto (1/n_1^2 - 1/n_2^2)$, taking ratios is straightforward. Number of lines when electron drops from level $n$: $n(n-1)/2$. For $n=4$: $4(3)/2 = 6$ lines. For $n=5$: $5(4)/2 = 10$ lines. Series limit wavelength: $\lambda_{limit} = n_1^2/R_H$ for series with lower level $n_1$. Lyman limit: $1/R_H = 91.2$ nm. Balmer limit: $4/R_H = 364.8$ nm. Common NEET questions: find wavelength of specific transition, identify series, calculate ratio of wavelengths, find number of spectral lines. Always draw energy level diagram and identify $n_1$ (lower) and $n_2$ (upper) correctly.

Three rules govern electron filling: Aufbau principle: electrons occupy lowest available energy orbital first. Order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p... (Madelung/diagonal rule). Pauli exclusion: maximum 2 electrons per orbital, with opposite spins. Hund rule: in degenerate orbitals (same energy), electrons fill singly with parallel spins before pairing. This minimises electron repulsion and is the ground state configuration. Important exceptions to Aufbau: Cr (Z=24): [Ar]3d⁵4s¹ (not 3d⁴4s²) — half-filled d is extra stable. Cu (Z=29): [Ar]3d¹⁰4s¹ (not 3d⁹4s²) — fully-filled d is extra stable. These exceptions are regularly asked in NEET. Similar exceptions occur for Mo, Ag, Au, Pd.

Principal quantum number $n$ (1,2,3,...): determines energy (∝ $-1/n^2$) and average distance from nucleus (∝ $n^2$). Azimuthal $l$ (0 to $n-1$): s-orbital ($l=0$, spherical), p ($l=1$, dumbbell), d ($l=2$, cloverleaf), f ($l=3$, complex). Number of orbitals: $2l+1$. Max electrons: $2(2l+1)$. Magnetic $m_l$ ($-l$ to $+l$): orientation. p has 3 orbitals ($p_x, p_y, p_z$); d has 5; f has 7. Spin $m_s$ ($±1/2$): intrinsic angular momentum. Nodes: radial nodes = $n-l-1$; angular nodes = $l$; total nodes = $n-1$. The 1s orbital has 0 nodes (spherically symmetric, maximum density at nucleus for hydrogen). 2s has 1 radial node (spherical shell of zero density). 2p has 1 angular node (nodal plane through nucleus). Understanding node patterns helps visualize orbital shapes and predict reactivity.

s-orbitals: spherically symmetric, one per subshell. 1s: smallest, most tightly bound. 2s: larger, one spherical node. Electron density has a maximum outside the node as well as at small $r$. p-orbitals: dumbbell shaped, along x, y, or z axis. Three degenerate p-orbitals ($p_x, p_y, p_z$). Nodal plane through nucleus. d-orbitals: five degenerate. $d_{xy}, d_{xz}, d_{yz}$: lobes between axes. $d_{x^2-y^2}$: lobes along x and y axes. $d_{z^2}$: unique shape with lobes along z-axis and ring (torus) in xy-plane. f-orbitals: seven degenerate, complex shapes. The directionality of p and d orbitals is responsible for directional bonding and geometry in molecules and coordination compounds. The different spatial orientations allow for σ, π, and δ bonds formed by different orbital overlaps.

Ionisation energy (IE): energy needed to remove outermost electron from gaseous atom. IE increases across period (left to right): increasing nuclear charge with same shielding → electrons held more tightly. IE decreases down group: increasing atomic radius → outer electron farther from nucleus → less attracted. Exceptions across period: Be (IE > B): 2s is more stable than 2p. N (IE > O): N has half-filled 2p (extra stability from exchange energy). Mg (IE > Al): 3s vs 3p. Successive ionisation energies: large jump when removing electron from inner shell. Used to determine valence electrons. Example: Na: 1st IE = 496 kJ/mol (easy, removes 3s). 2nd IE = 4562 kJ/mol (large jump — removing 2p electron, inner shell). Confirms Na has 1 valence electron. IE₁ < IE₂ < IE₃... always. Electron affinity: energy released when atom accepts electron. Cl has highest EA (349 kJ/mol).

Atomic radius decreases across period (left to right): increasing nuclear charge pulls electrons closer (same shell, increasing Z). Atomic radius increases down group: new shells added, shielding increases. Covalent radius: half the distance between two identical atoms bonded by single bond. Van der Waals radius: half the distance between non-bonded atoms in contact. Ionic radius: radius of ion. Cation: smaller than parent atom (electron removed, proton-to-electron ratio increases). Anion: larger than parent atom (electron added, repulsion increases radius). Isoelectronic species: same electrons, different nuclear charge. Example: N³⁻, O²⁻, F⁻, Ne, Na⁺, Mg²⁺, Al³⁺ all have 10 electrons. Radius decreases with increasing atomic number: N³⁻ > O²⁻ > F⁻ > Ne > Na⁺ > Mg²⁺ > Al³⁺. Lanthanide contraction: filling of 4f orbitals in lanthanides → poor shielding → radius contraction → 5th period d-block similar size to 4th period.