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Structure of Atom Formulas

Planck's quantum theory, photoelectric effect, de Broglie relation, Heisenberg uncertainty principle, and quantum numbers.

8 Formulas•568+ Questions in Bank

Concept Diagram

Bohr Model & Hydrogen Spectral Series
n = 1 (Ground State: -13.6 eV)n = 2 (-3.40 eV)n = 3 (-1.51 eV)n = 4 (-0.85 eV)n = ∞ (Ionization limit: 0 eV)Lyman (UV)Balmer (Visible)Paschen (IR)Rydberg: 1/λ = R_H Z² (1/n₁² - 1/n₂²) | E_n = -13.6 Z²/n² eV | r_n = 0.529 n²/Z Å
Quantized energy levels n = 1, 2, 3... with Lyman (UV, n→1), Balmer (Visible, n→2), and Paschen (IR, n→3) emission spectral transitions.

Core Formulas & Mathematical Identities

8
01

Planck Quantum Energy & de Broglie Wavelength

E=hν=hcλ,λ=hmv=h2m(KE)E = h\nu = \frac{hc}{\lambda}, \quad \lambda = \frac{h}{mv} = \frac{h}{\sqrt{2m(KE)}}

Wave-particle duality relations.

Examiner Tip: Energy of one mole of photons (Einstein): E = N_A h ν.
02

Heisenberg's Uncertainty Principle

Δx⋅Δp≥h4π=ℏ2,Δx⋅Δv≥h4πm\Delta x \cdot \Delta p \ge \frac{h}{4\pi} = \frac{\hbar}{2}, \quad \Delta x \cdot \Delta v \ge \frac{h}{4\pi m}

Fundamental limit on simultaneous measurement of conjugate variables.

Examiner Tip: Also applies to energy and time: ΔE · Δt ≥ h / (4π).
03

Quantum Numbers & Nodal Surfaces

Radial Nodes=n−l−1,Angular Nodes=l,Total Nodes=n−1\text{Radial Nodes} = n - l - 1, \quad \text{Angular Nodes} = l, \quad \text{Total Nodes} = n - 1

Characterization of electronic orbitals (n, l, m, s).

Examiner Tip: Orbital angular momentum L = √(l(l+1)) ħ.
04

Bohr Model for Hydrogenic Atoms

rn=0.529n2Z A˚,vn=2.18×106Zn m/s,En=−13.6Z2n2 eV/atom=−2.18×10−18Z2n2 Jr_n = 0.529 \frac{n^2}{Z} \text{ \AA}, \quad v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s}, \quad E_n = -13.6 \frac{Z^2}{n^2} \text{ eV/atom} = -\frac{2.18 \times 10^{-18} Z^2}{n^2} \text{ J}

Quantized orbital radius, electron orbital velocity, and total electronic energy levels for single-electron species (H, He⁺, Li²⁺, Be³⁺).

Examiner Tip: Kinetic Energy K = -E_n = +13.6 Z²/n² eV; Potential Energy U = 2E_n = -27.2 Z²/n² eV.
05

Einstein's Photoelectric Equation

hν=Φ0+KEmax=hν0+eVs,KEmax=hc(1λ−1λ0)h\nu = \Phi_0 + KE_{\text{max}} = h\nu_0 + e V_s, \quad KE_{\text{max}} = hc\left(\frac{1}{\lambda} - \frac{1}{\lambda_0}\right)

Conservation of energy in photoelectric emission where Φ₀ = hν₀ is work function and V_s is stopping potential.

Examiner Tip: Slope of stopping potential V_s versus incident frequency ν is universal constant h/e.
06

Orbital & Spin Angular Momentum

Lorbital=l(l+1)h2π=l(l+1)ℏ,μs=n(n+2) BM(n=unpaired e−)L_{\text{orbital}} = \sqrt{l(l+1)}\frac{h}{2\pi} = \sqrt{l(l+1)}\hbar, \quad \mu_s = \sqrt{n(n+2)} \text{ BM} \quad (n = \text{unpaired } e^-)

Spatial angular momentum of atomic orbitals and spin-only magnetic moment in Bohr Magnetons.

Examiner Tip: For s-orbital (l = 0), orbital angular momentum is strictly ZERO.
07

Rydberg Formula & Hydrogen Emission Spectral Series

νˉ=1λ=RHZ2(1n12−1n22),Nlines=(n2−n1)(n2−n1+1)2\bar{\nu} = \frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right), \quad N_{\text{lines}} = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2}

Wavenumbers of emitted spectral lines. Lyman (n₁=1, UV), Balmer (n₁=2, visible), Paschen (n₁=3, IR), Brackett (n₁=4, IR), Pfund (n₁=5, far-IR).

Examiner Tip: Only the Balmer series falls in the human-visible region of electromagnetic spectrum.
08

Radial Probability Distribution Function

P(r) dr=4πr2Rn,l2(r) dr,rmax probability(1s)=a0=0.529 A˚P(r)\,dr = 4\pi r^2 R_{n,l}^2(r)\,dr, \quad r_{\text{max probability}}(1s) = a_0 = 0.529\,\text{\AA}

Probability of finding an electron in a spherical shell of radius r and thickness dr surrounding the nucleus.

Examiner Tip: At r = 0, probability density ψ² is maximum for s-orbitals, but radial probability P(r) is zero due to the 4πr² geometric factor.

Top Examiner Traps & Pitfalls

Frequently tested misconceptions in Structure of Atom

  • •Heisenberg uncertainty principle uses ≥ h / (4π), NOT h / (2π).
  • •Number of radial nodes = n - l - 1; angular nodes = l; total nodes = n - 1.

Fundamental Theory & Core Concepts

4 Topics

1Dual Nature of Light & Photoelectric Effect

  • ›Wave-particle duality: Light exhibits wave characteristics (diffraction, interference) and particulate behavior (photoelectric effect, blackbody radiation, Compton effect).
  • ›Photoelectric Observations: 1) Instantaneous emission (no time lag ~ 10⁻⁹ s). 2) Photoelectric current is directly proportional to incident light intensity. 3) Kinetic energy of photoelectrons depends solely on frequency ν, independent of intensity. 4) No emission occurs below threshold frequency ν₀ (work function Φ₀ = hν₀).
  • ›Einstein's Relation: hν = hν₀ + KE_max = hν₀ + e·V_s. Stopping potential V_s = (h/e)ν - (h/e)ν₀. The slope of V_s versus ν is universally h/e for all emitter metals.

2Bohr's Postulates & Emission Spectral Series

  • ›Postulate 1: Electrons revolve around nucleus in circular stationary orbits where centrifugal force balances Coulomb attraction: m·v² / r = k·Z·e² / r².
  • ›Postulate 2 (Quantization): Electron can only reside in orbits where its orbital angular momentum is an integral multiple of ħ = h / (2π): m·v·r = n·(h / 2π).
  • ›Postulate 3 (Energy Transitions): Radiation is emitted or absorbed only when an electron jumps between stationary states: ΔE = E₂ - E₁ = hν.
  • ›Spectral Series: Lyman (n₁=1, UV region), Balmer (n₁=2, only series in the visible spectrum), Paschen (n₁=3, near-IR), Brackett (n₁=4, IR), Pfund (n₁=5, far-IR). Total spectral lines from state n₂ down to n₁: N = (n₂ - n₁)(n₂ - n₁ + 1) / 2.

3Quantum Mechanics & Quantum Numbers

  • ›Principal Quantum Number (n): Determines shell, main energy level, and size of the orbital. Number of orbitals in n-th shell = n²; max electron capacity = 2n².
  • ›Azimuthal / Angular Momentum Quantum Number (l = 0, 1, ..., n-1): Determines subshell shape (l=0 is s, spherical; l=1 is p, dumbbell; l=2 is d, double dumbbell; l=3 is f, complex). Orbital angular momentum L = √(l(l+1)) ħ. Subshell electron capacity = 2(2l + 1).
  • ›Magnetic Quantum Number (m_l = -l, ..., 0, ..., +l): Spatial orientation of orbital in magnetic field. Total values = 2l + 1 (e.g. 3 p-orbitals: p_x, p_y, p_z; 5 d-orbitals: d_xy, d_yz, d_zx, d_x²-y², d_z²).
  • ›Spin Quantum Number (m_s = +1/2, -1/2): Intrinsic electronic spin angular momentum S = √(s(s+1)) ħ. Spin-only magnetic moment μ_s = √(n(n+2)) BM, where n is the number of unpaired electrons.
  • ›Nodal Planes: Radial nodes = n - l - 1 (spherical surfaces where ψ² = 0). Angular nodes = l (planar surfaces where ψ² = 0). Total nodes = n - 1.

4Rules for Filling Electrons & Orbital Stability

  • ›Aufbau Principle: Ground-state electrons occupy orbitals in increasing order of energy given by the (n + l) rule. If two orbitals have identical (n + l), the orbital with lower n has lower energy and is filled first.
  • ›Pauli's Exclusion Principle: No two electrons in the same atom can have all four quantum numbers identical. Consequently, each spatial orbital can hold at most two electrons, and they must have antiparallel (opposite) spins.
  • ›Hund's Rule of Maximum Multiplicity: Pairing of electrons in degenerate orbitals (orbitals with equal energy) does not occur until each orbital is singly occupied with parallel spins, maximizing exchange energy.
  • ›Extra Stability of Half-Filled & Fully-Filled Subshells: Arises from symmetrical charge distribution and maximum exchange energy (e.g., Cr has [Ar] 3d⁵ 4s¹ instead of 3d⁴ 4s²; Cu has [Ar] 3d¹⁰ 4s¹ instead of 3d⁹ 4s²).
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