Unit 2: Structure of Atom Study Notes on the Structure of the Atom

Historical Context: Indian and Greek philosophers (400 B.C.) proposed atoms as indivisible building blocks. John Dalton (1808) established the first scientific atomic theory, identifying the atom as the ultimate particle of matter to explain laws of chemical combination.
Discovery of Electrons: Michael Faraday (1830) suggested the particulate nature of electricity. In the mid-1850s, cathode ray discharge tube experiments revealed rays moving from the cathode to the anode. These rays consisted of negatively charged particles called electrons.
Charge-to-Mass Ratio: J.J. Thomson (1897) determined the $\frac{e}{m_e}$ ratio for electrons to be $1.758820 \times 10^{11}\,C\,kg^{-1}$ by observing deflections in perpendicular electric and magnetic fields.
Charge and Mass: R.A. Millikan (1906–14) used the oil drop experiment to determine electronic charge as $-1.602176 \times 10^{-19}\,C$ . The mass of an electron ( $m_e$ ) was calculated to be $9.1094 \times 10^{-31}\,kg$ .
Protons and Neutrons: Canal rays in modified discharge tubes led to the discovery of positively charged gaseous ions (protons). James Chadwick (1932) discovered neutrons by bombarding beryllium with $\alpha$ -particles, identifying them as neutral particles with a mass of $1.674927 \times 10^{-27}\,kg$ .

Thomson Model (1898): Proposed the "plum pudding" model, where positive charge is uniformly distributed in a sphere with electrons embedded in it.
Rutherford’s Scattering Experiment: Bombarding gold foil with $\alpha$ -particles showed most passed undeflected, while a few ( $1$ in $20,000$ ) bounced back.
Conclusions: Most of the atom is empty space. Positive charge and mass are concentrated in a tiny, dense core called the nucleus ( $radius \approx 10^{-15}\,m$ ). Electrons move around the nucleus in circular orbits.
Symbols: Atomic number ( $Z$ ) equals the number of protons. Mass number ( $A$ ) equals protons ( $Z$ ) plus neutrons ( $n$ ). Isotopes have the same $Z$ but different $A$ (e.g., $_{1}^{1}H$ , $_{1}^{2}D$ , $_{1}^{3}T$ ). Isobars have the same $A$ but different $Z$ .

Wave Nature: James Maxwell (1870) proposed that light consists of oscillating electric and magnetic fields. EMR travels in a vacuum at $c = 3.0 \times 10^8\,m/s$ . Relationship: $c = \nu \lambda$ .
Planck’s Quantum Theory (1900): Energy is emitted or absorbed in discrete "quanta." Formula: $E = h\nu$ , where $h = 6.626 \times 10^{-34}\,J\,s$ .
Photoelectric Effect: Einstein (1905) explained that photons striking metal surfaces eject electrons instantly if the frequency \nu > \nu_0 (threshold frequency). Equation: $h\nu = h\nu_0 + \frac{1}{2}m_e v^2$ .
Dual Nature: Radiation behaves as both a wave (interference, diffraction) and a particle (photoelectric effect, black-body radiation).

Postulates: Electrons move in fixed circular orbits (stationary states). Angular momentum is quantized: $m_e vr = \frac{nh}{2\pi}$ , where $n = 1, 2, 3...$ .
Energy and Radius: Radius of ${n}^{th}$ orbit: $r_n = n^2 a_0$ ( $a_0 = 52.9\,pm$ ). Energy: $E_n = -2.18 \times 10^{-18} \left(\frac{Z^2}{n^2}\right)\,J$ .
Hydrogen Spectrum: Spectral lines result from electronic transitions between levels. Wavenumber ( $\bar{\nu}$ ) for any transition: $\bar{\nu} = 109,677 \left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right)\,cm^{-1}$ . Series include Lyman (UV), Balmer (Visible), and Paschen (IR).
Limitations: Cannot explain multi-electron atoms, Zeeman effect (magnetic splitting), or Stark effect (electric splitting).

de Broglie Relation (1924): Matter exhibits wave-particle duality. Wavelength $\lambda = \frac{h}{mv} = \frac{h}{p}$ .
Heisenberg Uncertainty Principle (1927): It is impossible to determine simultaneously the exact position and momentum of a microscopic particle. $\Delta x \times \Delta p_x \ge \frac{h}{4\pi}$ .
Schrödinger Equation (1926): Describes the wave function ( $\psi$ ) of a system. $\hat{H}\psi = E\psi$ . $|\psi|^2$ represents the probability density of finding an electron.

Principal Quantum Number ( $n$ ): Defines shell size and energy ( $n = 1, 2, 3...$ ).
Azimuthal Quantum Number ( $l$ ): Defines orbital shape ( $0$ to $n-1$ ). Notation: $s$ , $p$ , $d$ , $f$ .
Magnetic Quantum Number ( $m_l$ ): Defines orientation in space ( $-l$ to $+l$ ).
Spin Quantum Number ( $m_s$ ): Defines electron spin direction ( $+\frac{1}{2}$ or $-\frac{1}{2}$ ).
Shapes: $s$ orbitals are spherical; $p$ orbitals are dumbbell-shaped with two lobes; $d$ orbitals are mostly double-dumbbell shaped. Nodes are regions where $|\psi|^2 = 0$.

Aufbau Principle: Orbitals are filled in order of increasing energy based on the $(n+l)$ rule.
Pauli Exclusion Principle: No two electrons can have the same four quantum numbers; an orbital holds a maximum of two electrons with opposite spins.
Hund’s Rule of Maximum Multiplicity: Pairing in degenerate orbitals starts only after each orbital is singly occupied.
Extra Stability: Half-filled ( $d^5, f^7$ ) and completely filled ( $d^{10}, f^{14}$ ) subshells possess extra stability due to symmetry and high exchange energy. Notable exceptions: Chromium ( $[Ar]3d^5 4s^1$ ) and Copper ( $[Ar]3d^{10} 4s^1$ ).