Discovery of Subatomic Particles (Cathode Rays, Canal Rays and Rutherford Scattering)
The indivisible atom posited by Dalton was disproven at the turn of the 20th century through cathode-ray discharge tube experiments:
- Discovery of Electron (J.J. Thomson, 1897): Cathode rays consist of negatively charged particles with charge-to-mass ratio e/m = 1.758820 × 10¹¹ C/kg. Charge was determined by Millikan's Oil Drop experiment (e = -1.6022 × 10⁻¹⁹ C, mass m_e = 9.10938 × 10⁻³¹ kg).
- Discovery of Proton (E. Goldstein, 1886): Canal rays (anode rays) produced positively charged hydrogen ions (e/m depends on the residual gas in the discharge tube).
- Discovery of Neutron (James Chadwick, 1932): Bombardment of Beryllium sheets with α-particles yielded neutral particles (⁹₄Be + ⁴₂α → ¹²₆C + ¹₀n) with mass m_n = 1.67493 × 10⁻²⁷ kg.
- Rutherford's α-Particle Scattering Experiment (1911): Most of the atom is empty space; positive charge and mass are concentrated in a tiny dense Nucleus (r_nucleus and asymp; 10⁻¹⁵ m, r_atom and asymp; 10⁻¹⁰ m).
Review the fundamentals from our previous chapter in Class 11 Chemistry Chapter 1 Notes or plan your revision blocks with our study time planner guide.
Dual Nature of Electromagnetic Radiation and Planck's Quantum Theory
Light exhibits both wave and particle characteristics:
- Wave Nature (James Clerk Maxwell, 1870): Radiation propagates as oscillating perpendicular electric and magnetic fields traveling at c = 3.0 × 10⁸ m/s, with c = ν · λ and wavenumber ν̄ = 1 / λ.
- Planck's Quantum Theory (Max Planck, 1900): Atoms and molecules emit or absorb energy only in discrete packets called Quanta (or Photons for light):
Where Planck's Constant h = 6.62607 × 10⁻³⁴ J·s.
Photoelectric Effect and Hydrogen Emission Spectrum (Rydberg Formula)
Photoelectric Effect (H. Hertz, 1887 and Albert Einstein, 1905): When light of frequency higher than a threshold frequency ν₀ strikes a metal surface, electrons are ejected instantaneously:
Hydrogen Emission Spectrum (Rydberg Formula, Johannes Rydberg, 1890):
Where Rydberg constant R_H = 109,677 cm⁻¹ = 1.09677 × 10⁷ m⁻¹.
| Spectral Series | Lower Orbit (n₁) | Upper Orbit (n₂) | Spectral Region |
|---|---|---|---|
| Lyman Series | n₁ = 1 | n₂ = 2, 3, 4, ... | Ultraviolet (UV) |
| Balmer Series | n₁ = 2 | n₂ = 3, 4, 5, ... | Visible |
| Paschen Series | n₁ = 3 | n₂ = 4, 5, 6, ... | Infrared (Near IR) |
| Brackett Series | n₁ = 4 | n₂ = 5, 6, 7, ... | Infrared (Mid IR) |
| Pfund Series | n₁ = 5 | n₂ = 6, 7, 8, ... | Infrared (Far IR) |
Bohr's Atomic Model for Hydrogen and Postulates (Radii and Energy Equations)
In 1913, Niels Bohr formulated his quantized model for hydrogen and hydrogen-like species (He⁺, Li²⁺, Be³⁺):
- Electrons revolve only in fixed non-radiating orbits where angular momentum is quantized: m_e · v · r = (n · h) / (2π) (where n = 1, 2, 3...).
- Bohr Orbit Radius:
r_n = 0.529 × (n² / Z) Å = 52.9 × (n² / Z) pm - Bohr Orbit Energy:
E_n = -2.18 × 10⁻¹⁸ × (Z² / n²) J/atom = -13.6 × (Z² / n²) eV/atom - Velocity of Electron:
v_n = 2.18 × 10⁶ × (Z / n) m/s
Dual Nature of Matter: de Broglie Wavelength Equation
In 1924, Louis de Broglie proposed that microscopic particles like electrons have dual wave-particle properties:
Heisenberg's Uncertainty Principle
In 1927, Werner Heisenberg stated that it is physically impossible to simultaneously measure the exact position and momentum of an electron:
This renders the concept of fixed Bohr circular orbits obsolete and gives rise to three-dimensional probability orbitals.
Quantum Mechanical Model of Atom: Wavefunctions (ψ) and Probability Density (ψ²)
Erwin Schrödinger developed the fundamental wave equation: Ĥψ = Eψ.
- ψ (Wavefunction): Has no physical meaning by itself.
- ψ² (Probability Density): Represents the probability of finding an electron in a unit volume around the nucleus.
- Nodes: Regions where probability density ψ² = 0.
- Radial Nodes:
n - l - 1 - Angular Nodes:
l - Total Nodes:
(n - l - 1) + l = n - 1
- Radial Nodes:
The Four Quantum Numbers (n, l, m, s) and Orbital Shapes (s, p, d, f)
| Quantum Number | Symbol | Permitted Values | Physical Significance |
|---|---|---|---|
| Principal | n | 1, 2, 3, 4, ... | Main shell, size, and major energy level |
| Azimuthal (Orbital Angular Momentum) | l | 0 to (n - 1) | Subshell shape: s(0) spherical, p(1) dumbbell, d(2) double-dumbbell, f(3) complex |
| Magnetic | m_l | -l to +l (2l + 1 total values) | 3D spatial orientation of the orbital in space |
| Spin | m_s | +1/2, -1/2 | Intrinsic electron spin (clockwise / counter-clockwise) |
Electronic Configuration Rules: Aufbau, Pauli and Hund's Rule (Cr and Cu Anomalies)
- Aufbau Principle: Electrons occupy orbitals in order of increasing energy based on the (n + l) rule:
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d... - Pauli's Exclusion Principle: No two electrons in the same atom can have the exact same set of all four quantum numbers (n, l, m_l, m_s). An orbital can hold a maximum of 2 electrons with opposite spins.
- Hund's Rule of Maximum Multiplicity: Pairing of electrons in degenerate orbitals (p, d, f) does not occur until each orbital is singly occupied with parallel spins.
- Exceptional Configurations:
- Chromium (Z = 24): [Ar] 3d⁵ 4s¹ (instead of 3d⁴ 4s²)
- Copper (Z = 29): [Ar] 3d¹⁰ 4s¹ (instead of 3d⁹ 4s²)
- Reason: Half-filled (d⁵) and fully-filled (d¹⁰) orbitals provide symmetrical electron distribution and maximum exchange energy.
Step-by-Step Solved Exemplar Numericals for Board, JEE and NEET
Problem 1: de Broglie Wavelength of an Accelerated Electron
Calculate the de Broglie wavelength of an electron moving with a velocity of 2.05 × 10⁷ m/s (m_e = 9.11 × 10⁻³¹ kg).
Calculation: λ = (6.626 × 10⁻³⁴ J·s) / (9.11 × 10⁻³¹ kg × 2.05 × 10⁷ m/s) = (6.626 × 10⁻³⁴) / (1.868 × 10⁻²³) = 3.55 × 10⁻¹¹ m = 0.0355 nm.
Problem 2: Heisenberg Uncertainty in Position of a Fast Particle
A microscope using suitable photons is employed to locate an electron in an atom within a distance of 0.1 Å (10⁻¹¹ m). What is the uncertainty involved in the measurement of its velocity?
Calculation: Δv = (6.626 × 10⁻³⁴) / [4 × 3.1416 × (9.11 × 10⁻³¹) × 10⁻¹¹] = (6.626 × 10⁻³⁴) / (1.145 × 10⁻⁴⁰) = 5.79 × 10⁶ m/s.
Frequently Asked Questions (FAQs)
The four quantum numbers are: (1) Principal Quantum Number (n) designating main shell size and energy; (2) Azimuthal Quantum Number (l) defining orbital shape and subshell (s, p, d, f); (3) Magnetic Quantum Number (m_l) describing 3D spatial orientation; and (4) Spin Quantum Number (m_s = +1/2 or -1/2) describing intrinsic electron spin.
Chromium (Cr: [Ar] 3d⁵ 4s¹) and Copper (Cu: [Ar] 3d¹⁰ 4s¹) exhibit anomalous configurations instead of 3d⁴ 4s² and 3d⁹ 4s² because half-filled (d⁵) and fully-filled (d¹⁰) d-subshells possess extraordinary stability due to symmetrical electron distribution and maximum exchange energy.
de Broglie proposed that matter exhibits wave-particle duality: λ = h / p = h / (mv) = h / √(2mE), where h is Planck's constant (6.626 × 10⁻³⁴ J·s), m is mass, v is velocity, and E is kinetic energy.
It states that it is impossible to determine simultaneously and precisely both the position (x) and momentum (p) of a microscopic particle like an electron: Δx × Δp ≥ h / (4π).
Einstein's photoelectric equation is: hν = hν₀ + (1/2)m_e v_max², where hν is incident photon energy, hν₀ is the work function (W₀ = threshold energy required to eject an electron), and (1/2)m_e v_max² is the maximum kinetic energy of the emitted photoelectron.
An Orbit (Bohr model) is a circular planar 2D path around the nucleus where an electron revolves, violating Heisenberg's principle. An Orbital (Quantum mechanical model) is a 3D region in space around the nucleus where the probability of finding an electron is maximum (≥90%).