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Structure of Atom

Ch. 4Std 11

Easy Overview

What's inside an atom? This chapter traces our understanding from Dalton's spheres to the quantum mechanical model. Rutherford's gold foil experiment proved the atom has a tiny, dense nucleus. Bohr explained the hydrogen spectrum with quantized orbits. But Bohr's model failed for multi-electron atoms. de Broglie proposed electrons have wave nature (λ = h/mv). Schrödinger's equation (ĤΨ = EΨ) treats electrons as wave functions — |Ψ|² gives probability density (orbitals, not orbits). Heisenberg's uncertainty principle (Δx·Δp ≥ h/4π) says we can't know both position and momentum precisely. Quantum numbers (n, l, m, s) define each electron's address. The Aufbau principle, Pauli exclusion, and Hund's rule determine electronic configurations. These concepts explain the periodic table, bonding, and spectroscopy.

Dalton's Atomic Theory and Its Limitations

Dalton (1808): atoms are indivisible, atoms of same element identical, compounds from different atoms, reactions rearrange atoms. Limitations: atoms divisible (electrons, protons, neutrons), isotopes exist (same element, different masses), nuclear reactions transform elements. Core idea of discrete atoms still stands.

Discovery of Subatomic Particles

Electron: J.J. Thomson (1897), cathode rays, e/m = 1.76×10¹¹ C/kg. Millikan (1909): e = 1.602×10⁻¹⁹ C. Proton: Rutherford (1911), charge +1.602×10⁻¹⁹ C, mass ~1836× electron. Neutron: Chadwick (1932), no charge, mass 1.008665 amu. Neutrons explain isotopes — same Z, different A.

Rutherford's Gold Foil Experiment

Fired α-particles at thin gold foil. Expected all through (plum pudding). Results: most through, ~1/20,000 deflected >90°, some bounced back. Conclusions: atom mostly empty, positive charge concentrated in tiny nucleus (~10⁻¹⁵ m vs atom ~10⁻¹⁰ m), electrons orbit. But classical physics says accelerating electrons should spiral into nucleus.

Bohr's Model of the Hydrogen Atom

Postulates: (1) Electrons in circular orbits. (2) Only allowed orbits where mvr = nh/2π (n=1,2,3...). (3) No radiation in stationary states. (4) ΔE = hν when jumping. Eₙ = -13.6/n² eV. rₙ = n² × 52.9 pm. Explained hydrogen spectrum: Lyman (n₁=1, UV), Balmer (n₁=2, visible), Paschen (n₁=3, IR). Failed for multi-electron atoms.

Hydrogen Spectrum — Spectral Lines Explained

Excited H emits specific wavelengths. Series: Lyman (UV, n₁=1), Balmer (visible, n₁=2), Paschen (IR, n₁=3), Brackett (far IR, n₁=4), Pfund (far IR, n₁=5). Rydberg: 1/λ = R(1/n₁² - 1/n₂²), R = 1.097×10⁷ m⁻¹. Hα = 656 nm (red), Hβ = 486 nm (blue-green). Only specific wavelengths = quantized energies.

Wave-Particle Duality — de Broglie's Hypothesis

de Broglie (1924): matter has wave properties, λ = h/mv = h/p. Macroscopic: 1 kg at 1 m/s → λ = 6.626×10⁻³⁴ m (undetectable). Electron: λ ~ 10⁻¹⁰ m (comparable to atomic spacing). Davisson-Germer (1927): electron diffraction from Ni crystal confirmed. Explains why only certain orbitals are stable (standing waves: 2πr = nλ).

Heisenberg's Uncertainty Principle

Δx·Δp ≥ h/4π. Can't simultaneously know exact position and momentum. For electrons, uncertainty is significant. If electron confined to nucleus (Δx~10⁻¹⁵ m), Δp ≥ 5×10⁻²⁰ kg·m/s → KE ~60 MeV (far exceeds binding energy — electron escapes). ΔE·Δt ≥ h/4π explains finite lifetimes of excited states.

Quantum Mechanical Model — Schrödinger Equation

ĤΨ = EΨ, where Ψ = wave function, Ĥ = Hamiltonian, E = energy. |Ψ|² = probability density — gives orbitals (3D probability clouds, not fixed orbits). Solutions yield quantized energies naturally from boundary conditions. Explains all atomic properties that Bohr couldn't.

Quantum Numbers — Address System for Electrons

n (principal): energy level and size (1,2,3...). l (azimuthal): shape — 0=s, 1=p, 2=d, 3=f (0 to n-1). m_l (magnetic): orientation (-l to +l). m_s (spin): ±½ (↑↓). Each orbital holds max 2 e⁻ with opposite spins. No two electrons can have all four QNs identical (Pauli).

Shapes of Atomic Orbitals

s (l=0): spherical, no angular nodes. p (l=1): dumbbell, three orientations (p_x, p_y, p_z). d (l=2): four cloverleaf + one d_z² with donut. f (l=3): complex, seven orientations. Total nodes = n-1. Radial nodes = n-l-1. Angular nodes = l.

Electronic Configuration — Aufbau Principle

Fill by increasing energy: 1s,2s,2p,3s,3p,4s,3d,4p,5s,4d,5p,6s,4f,5d,6p,7s,5f,6d,7p. (n+l) rule: fill by increasing n+l; for equal, lower n first. Exceptions: Cr (3d⁵4s¹), Cu (3d¹⁰4s¹) — half/full filled subshell stability. Noble gas shorthand: K = [Ar]4s¹.

Pauli Exclusion Principle and Hund's Rule

Pauli: no two electrons with same set of all four QNs. Each orbital: max 2 e⁻ with opposite spins. Hund: fill degenerate orbitals singly first, then pair. Maximizes total spin. Carbon (1s²2s²2p²): both 2p e⁻ go into different p orbitals with parallel spins (↑ ↑).

Stability of Half-Filled and Fully Filled Subshells

d⁵, d¹⁰, f⁷, f¹⁴ have extra stability (symmetrical charge distribution, exchange energy). Explains Cr ([Ar]3d⁵4s¹), Cu ([Ar]3d¹⁰4s¹), Mo ([Kr]4d⁵5s¹), Ag ([Kr]4d¹⁰5s¹). W (5d⁴6s²) doesn't follow — relativistic effects at high Z.

Moseley's Law and Atomic Number

Moseley (1913): √ν ∝ (Z-b) — X-ray frequency ∝ atomic number. First proof Z is fundamental, not atomic mass. Corrected periodic table: Co (Z=27) before Ni (Z=28) despite mass ordering. Predicted elements Z=43 (Tc), 61 (Pm), 72 (Hf), 75 (Re). Provided basis for modern periodic law.

Photoelectric Effect — Particle Nature of Light

Einstein (1905): light as photons (E = hν). No emission below threshold ν₀. KE of ejected e⁻ = hν - W₀, where W₀ = hν₀ (work function). KE depends on ν, not intensity (contradicts wave theory). Instantaneous emission. One photon → one electron. Proved particle nature of light.

Dual Behavior of Electromagnetic Radiation

Wave: interference (Young's double-slit), diffraction, polarization. Particle: photoelectric effect, Compton effect (X-ray wavelength shift Δλ = h(1-cosθ)/mₑc), blackbody radiation. Complementary descriptions. E = hν, p = h/λ, c = λν.

Energy Levels in Multi-Electron Atoms

In H, energy depends only on n. Multi-electron: repulsion splits — ns < np < nd < nf. Order: 1s<2s<2p<3s<3p<4s<3d<4p<5s<4d<5p<6s<4f<5d<6p<7s<5f<6d. 4s fills before 3d, but once occupied, 3d drops below 4s. Ions lose 4s first: Fe ([Ar]3d⁶4s²) → Fe²⁺ ([Ar]3d⁶).

Nodal Surfaces in Orbitals

Nodes = zero probability regions. Radial nodes (spherical, distance from nucleus). Angular nodes (planar/conical through nucleus). Total = n-1. Radial = n-l-1. Angular = l. Examples: 1s (0), 2s (1 radial), 2p (1 angular), 3s (2 radial), 3p (1 radial+1 angular), 3d (2 angular). More nodes = higher energy.

Limitations of Bohr's Model

Failed for multi-electron atoms. Couldn't explain fine structure, Zeeman effect (magnetic splitting), Stark effect (electric splitting). Violated uncertainty principle (precise position + momentum). Couldn't explain intensities, electron wave nature, or why only certain orbits allowed. Quantum mechanical model resolved all.

Key Points

  • •Thomson discovered electron (1897); Rutherford discovered nucleus (1911)
  • •Bohr: mvr = nh/2Ï€, Eâ‚™ = -13.6/n² eV, explained hydrogen spectrum
  • •Rydberg: 1/λ = R(1/n₁² - 1/n₂²), R = 1.097×10⁷ m⁻¹
  • •de Broglie: λ = h/mv; Davisson-Germer confirmed electron diffraction
  • •Heisenberg: Δx·Δp ≥ h/4Ï€; ΔE·Δt ≥ h/4Ï€
  • •Schrödinger: ĤΨ = EΨ; |Ψ|² = probability density
  • •QNs: n (shell), l (shape), m (orientation), s (spin)
  • •s=spherical, p=dumbbell, d=cloverleaf; nodes: total=n-1
  • •Aufbau: 1s→2s→2p→3s→3p→4s→3d→4p...
  • •Pauli: max 2 e⁻/orbital, opposite spins; Hund: fill singly first
  • •Cr and Cu exceptions: half-filled/filled subshell stability
  • •Moseley: √ν ∝ Z; atomic number is fundamental
  • •Photoelectric effect: hν = Wâ‚€ + KE; proves particle nature of light
  • •Bohr failed for multi-electron atoms; quantum model succeeds

Practice Questions

  • Describe Rutherford's gold foil experiment. Observations and conclusions?
  • State Bohr's postulates. Derive radius and energy of nth orbit in H.
  • Calculate wavelength of Hα in Balmer series (R=1.097×10⁷ m⁻¹).
  • Assign four quantum numbers for last electron in Cl (Z=17) and 3d electrons in Fe (Z=26).
  • Explain Heisenberg's uncertainty principle. Show electron can't exist inside nucleus.
  • Write configurations: Cr, Cu, Fe²⁺, S²⁻. Explain exceptions.
  • State Hund's rule. Apply to N (Z=7). How many unpaired electrons?
  • What is photoelectric effect? Einstein's equation. How does it prove particle nature?
  • Sketch 1s, 2s, 2p, 3d_xy, 3d_z² orbitals with nodal surfaces.
  • Calculate de Broglie wavelength of electron accelerated through 100 V.