Atomic Structure: Quantum Numbers & Electronic Config

Atomic structure bridges classical and quantum physics, explaining how electrons are arranged in atoms. JEE tests Bohr model calculations, quantum number relationships, orbital shapes, and electronic configuration rules.

Bohr Model for Hydrogen-like Species: Applicable to H, He+, Li₂+, etc. (single electron).
Key equations: radius $r_{n}$ = 0.529 x $n^{2}$/Z angstroms.
Energy $E_{n}$ = -13.6 x $\frac{Z^{2}}{n^{2}}$ eV (or -2.18 x $10^{-18}$ x $\frac{Z^{2}}{n^{2}}$ J).
Velocity $v_{n}$ = 2.18 x $10^{6}$ x Z/n m/s.
Frequency of revolution = v/(2*$\pi$r).
Angular momentum L = nh/(2*$\pi$) (quantised).
For emission: 1/$\lambda$ = R*$Z^{2}$(1/$n1^{2}$ - 1/$n2^{2}$), where R = 1.097 x $10^{7} \; m^{-1}$ (Rydberg constant).
Energy of photon emitted: E = 13.6*$Z^{2}$(1/$n1^{2}$ - 1/$n2^{2}$) eV.

Hydrogen Spectrum Series: Lyman (n1=1, UV), Balmer (n1=2, visible), Paschen (n1=3, IR), Brackett (n1=4, IR), Pfund (n1=5, far IR). Total spectral lines from level n: n(n-1)/2. First line = shortest transition (n2 = n1+1). Last line (series limit) = longest transition (n2 = infinity).

De Broglie Wavelength: $\lambda$ = h/(mv) = h/p.
For electrons accelerated through V volts: $\lambda$ = 12.27/$\sqrt{V}$ angstroms.
For Bohr orbits: circumference = n*$\lambda$ (standing wave condition). De Broglie showed that Bohr's quantisation condition is equivalent to fitting n complete wavelengths around the orbit.

Heisenberg Uncertainty Principle: $\delta_{x}$ * $\delta_{p}$ >= h/(4*$\pi$).
Or $\delta_{x}$ * m * $\delta_{v}$ >= h/(4*$\pi$). This sets a fundamental limit on simultaneously knowing position and momentum. For macroscopic objects, the uncertainty is negligibly small. For electrons, it is significant — this is why we cannot define electron trajectories (orbits) and must use probability distributions (orbitals).

Quantum Numbers: Four numbers describe each electron: (1) Principal quantum number n (1, 2, 3, ...): determines energy level and size.
Number of orbitals in shell n = $n^{2}$.
Maximum electrons in shell n = $2n^{2}$. (2) Azimuthal quantum number l (0 to n-1): determines subshell shape and angular momentum.
l = 0 (s), 1 (p), 2 (d), 3 (f).
Number of orbitals in subshell = 2l+1. (3) Magnetic quantum number $m_{l}$ (-l to +l): determines orbital orientation in space. (4) Spin quantum number $m_{s}$ (+$\frac{1}{2}$ or -$\frac{1}{2}$): spin up or spin down. No two electrons can have all four quantum numbers identical (Pauli exclusion principle).

Orbital Shapes: s orbitals: spherical, no nodal plane, l=0.
p orbitals: dumbbell-shaped, 1 nodal plane, l=1 ($p_{x}$, $p_{y}$, $p_{z}$ along axes).
d orbitals: double-dumbbell or cloverleaf, 2 nodal planes, l=2 ($d_{xy}$, $d_{yz}$, $d_{xz}$, $d_{x2}$-y2, $d_{z2}$).
f orbitals: complex shapes, 3 nodal planes, l=3.
Radial nodes = n - l - 1.
Angular nodes = l.
Total nodes = n - 1.

Electronic Configuration Rules: (1) Aufbau principle: fill lower energy orbitals first. Order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. Use (n+l) rule: lower (n+l) fills first; if equal, lower n fills first. (2) Pauli exclusion principle: maximum 2 electrons per orbital with opposite spins. (3) Hund's rule: in degenerate orbitals, electrons occupy each singly (with parallel spins) before pairing. Half-filled and fully-filled subshells have extra stability (exchange energy).
Exceptions: Cr = [Ar]$3d^{5} \; 4s^{1}$ (not $3d^{4} \; 4s^{2}$).
Cu = [Ar]$3d^{10} \; 4s^{1}$ (not $3d^{9} \; 4s^{2}$).

Photoelectric Effect: $E_{photon}$ = h*$\nu$ = hc/$\lambda$.
Kinetic energy of ejected electron: $\text{KE}_{max}$ = h$\nu$ - $\phi$ (where $\phi$ = work function = threshold energy). Below threshold frequency $\nu_{0}$, no electrons are ejected regardless of intensity. Above $\nu_{0}$, KE increases linearly with frequency. Number of electrons ejected depends on intensity, not frequency.

The key problem-solving concept is applying quantum number rules to determine valid electron configurations and using Bohr model equations for hydrogen-like species.