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

Historical Context and the Discovery of Sub-atomic Particles

• Early Views on Atoms (400 B.C.): Early Indian and Greek philosophers proposed that atoms are the fundamental building blocks of matter. They believed continued subdivision of matter would ultimately yield atoms, which are indivisible.
• Etymology: The word ‘atom’ is derived from the Greek word ‘a-tomio’, meaning ‘uncut-able’ or ‘non-divisible’.
• Dalton’s Atomic Theory (1808): John Dalton, a British school teacher, proposed the first scientific basis for atomic theory. He regarded the atom as the ultimate particle of matter.     * Successes: Explained the law of conservation of mass, the law of constant composition, and the law of multiple proportions.     * Failures: Could not explain electrical phenomena, such as glass or ebonite becoming charged when rubbed with silk or fur.
• Shift in Concept: Experimental observations in the late 19th and early 20th centuries established that atoms are made of sub-atomic particles (electrons, protons, and neutrons).

Discovery of the Electron and Cathode Ray Experiments

• Michael Faraday (1830): Showed that passing electricity through an electrolyte solution caused chemical reactions resulting in the deposition of matter at electrodes, suggesting the particulate nature of electricity.
• Cathode Ray Discharge Tubes (mid-1850s): Glass tubes with two metal electrodes (cathode and anode). Electrical discharge through gases is observed only at very low pressures and high voltages.     * Process: Applying high voltage causes a stream of particles to move from the negative electrode (cathode) to the positive electrode (anode).     * Detection: By making a hole in the anode and coating the tube with phosphorescent zinc sulphide ($ZnS$), rays striking the coating produce a bright spot.
• Properties of Cathode Rays:     1. They start from the cathode and move toward the anode.     2. They are invisible but detectable through fluorescent or phosphorescent materials (e.g., television picture tubes).     3. In the absence of fields, they travel in straight lines.     4. In magnetic or electrical fields, they behave like negatively charged particles.     5. Their characteristics do not depend on the electrode material or the gas used, establishing electrons as basic constituents of all atoms.
• Charge to Mass Ratio ($e/m_e$): Determined by J.J. Thomson in 1897 using a cathode ray tube with perpendicular electrical and magnetic fields.     * Factors influencing deflection:         i. Magnitude of charge: Higher charge results in greater interaction and deflection.         ii. Mass of particle: Lighter particles undergo greater deflection.         iii. Field strength: Deflection increases with voltage or magnetic field strength.     * Result: $\frac{e}{m_e} = 1.758820 \times 10^{11}\,C\,kg^{-1}$.
• Charge on the Electron ($e$): R.A. Millikan (1906-1914) used the Oil Drop Experiment.     * Mechanism: Oil droplets entered an electrical condenser through an atomizer. Air was ionized by X-rays. Droplets acquired charge by colliding with gaseous ions. Motion was controlled by adjusting voltage.     * Quantization of Charge: Determined that charge $q$ is always an integral multiple of a fundamental charge: $q = ne$, where $n = 1, 2, 3...$.     * Accepted Value: $e = -1.602176 \times 10^{-19}\,C$.
• Mass of Electron ($m_e$): Calculated by combining Thomson and Millikan’s results.     * $m_e = 9.109382 \times 10^{-31}\,kg$.

Discovery of Protons, Neutrons, and Radioactivity

• Canal Rays (Protons): Modified cathode ray tubes revealed rays carrying positive particles.     * Characteristics: Mass depends on the gas nature; charge-to-mass ratio depends on the gas; some carry multiples of fundamental charge; behavior in fields is opposite to electrons.     * Proton: The smallest and lightest positive ion, obtained from hydrogen, characterized in 1919.
• Neutrons: Discovered by James Chadwick (1932) by bombarding a thin Beryllium sheet with $\alpha$-particles. These are electrically neutral particles with a mass slightly greater than protons.
• Properties Summary:     * Electron ($e$): Mass $\approx 9.11 \times 10^{-31}\,kg$; Relative charge $-1$.     * Proton ($p$): Mass $\approx 1.673 \times 10^{-27}\,kg$; Relative charge $+1$.     * Neutron ($n$): Mass $\approx 1.675 \times 10^{-27}\,kg$; Relative charge $0$.
• Radioactivity: Henri Becquerel (1896) discovered some elements emit radiation spontaneously. This field was developed by Marie Curie, Pierre Curie, Rutherford, and Frederick Soddy.     * $\alpha$-rays: High-energy particles with two units of positive charge and four units of atomic mass (Helium nuclei).     * $\beta$-rays: Negatively charged particles similar to electrons.     * $\gamma$-rays: High-energy neutral radiations like X-rays.     * Penetrating Power: $\alpha < \beta (100\times) < \gamma (1000\times)$.

Early Atomic Models and Rutherford Scattering

• Thomson Model of Atom (1898): Spherical shape (radius $\approx 10^{-10}\,m$) with uniform positive charge. Electrons are embedded like seeds in a watermelon or plums in a pudding (Plum Pudding Model). It explained overall neutrality but failed later experimental results.
• Rutherford’s scattering experiment: Bombarded gold foil ($\approx 100\,nm$ thickness) with $\alpha$-particles.     * Observations: Most passed undeflected; small fraction deflected by small angles; very few ($\approx 1$ in 20,000) bounced back (180° deflection).     * Conclusions:         1. Most space in the atom is empty.         2. Positive charge and mass are concentrated in a tiny volume (nucleus).         3. Nucleus radius ($10^{-15}\,m$) is tiny compared to atom radius ($10^{-10}\,m$). Analogy: If nucleus is a cricket ball, atom radius is $5\,km$.
• Rutherford’s Nuclear Model: Nucleus at center; electrons move in circular paths (orbits) at high speed. Atoms resemble solar systems. Held together by electrostatic forces.     * Drawbacks: Cannot explain stability. According to Maxwell’s electromagnetic theory, accelerated charged particles (revolving electrons) must radiate energy. This would cause orbits to shrink, and electrons would spiral into the nucleus in $10^{-8}\,s$. It also says nothing about electron distribution or energies.

Atomic Number, Mass Number, and Isotopes

• Atomic Number ($Z$): Number of protons in the nucleus. In a neutral atom, $Z = \text{number of electrons}$.
• Mass Number ($A$): Total number of nucleons (protons + neutrons).     * $A = Z + n$
• Notation: ${}^{A}_{Z}X$
• Isobars: Atoms with the same mass number ($A$) but different atomic numbers ($Z$). Example: ${}^{14}_{6}C$ and ${}^{14}_{7}N$.
• Isotopes: Atoms with the same atomic number but different mass numbers.     * Hydrogen Isotopes: Protium (${}^{1}_{1}H$), Deuterium (${}^{2}_{1}H$ or $D$), Tritium (${}^{3}_{1}H$ or $T$).     * Chemical Behavior: Controlled by the number of electrons (protons). Neutrons have little effect; thus, isotopes show identical chemical behavior.

Wave Nature of Electromagnetic Radiation

• James Clerk Maxwell (1870): Proposed that accelerating charged particles produce alternating electrical and magnetic fields transmitted as electromagnetic waves.
• Properties of EM Radiation:     1. Oscillating electric and magnetic field components are perpendicular to each other and to the direction of propagation.     2. Does not require a medium; can travel in a vacuum.     3. Travel at the speed of light: $c \approx 3.0 \times 10^8\,m/s$ ($2.997925 \times 10^8\,m/s$).
• EM Spectrum: Range of frequencies/wavelengths (Radio, Micro, Infrared, Visible, UV, X-ray, Gamma).     * Visible region: $\approx 10^{15}\,Hz$.
• Mathematical Relations:     * $c = \nu \lambda$     * Wavenumber ($\bar{\nu}$): $\bar{\nu} = \frac{1}{\lambda}$. Units: $m^{-1}$, commonly $cm^{-1}$.

Particle Nature of Radiation: Planck’s Quantum Theory

• Inadequacies of Wave Theory: Could not explain Black-body radiation, Photoelectric effect, Heat capacity variations of solids, or atomic line spectra.
• Black-Body Radiation: An ideal body that absorbs and emits all frequencies. Max Planck (1900) proposed that energy is absorbed or emitted in discrete chunks called quantum.     * Equation: $E = h\nu$     * Planck’s Constant ($h$): $6.626 \times 10^{-34}\,J\,s$.
• Photoelectric Effect: Ejection of electrons when light hits a metal surface (H. Hertz, 1887).     * Observations: No time lag; number of electrons $\propto$ light intensity; kinetic energy $\propto$ frequency of light.     * Threshold Frequency ($\nu_0$): Minimum frequency required for ejection.     * Einstein’s Explanation (1905): Light consists of photons. Conservation of energy requires:     * $h\nu = h\nu_0 + \frac{1}{2} m_e v^2$     * Work function ($W_0$) is $h\nu_0$.
• Dual Nature: Light exhibits both wave-like (interference, diffraction) and particle-like properties.

Atomic Spectra and the Bohr Model

• Spectra Type:     * Continuous Spectrum: White light spread through a prism (e.g., rainbow).     * Emission Spectrum: Radiation emitted by excited atoms.     * Line Spectrum: Atomic emission at specific wavelengths with dark spaces (fingerprints for elements).
• Hydrogen Line Spectrum:     * Balmer Series Formula (1885): $\bar{\nu} = 109,677 \left( \frac{1}{2^2} - \frac{1}{n^2} \right)\,cm^{-1}$ where $n \ge 3$.     * Rydberg Formula: $\bar{\nu} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$, where $R_H = 109,677\,cm^{-1}$.     * Series: Lyman ($n_1=1$, UV), Balmer ($n_1=2$, Visible), Paschen ($n_1=3$, IR), Brackett ($n_1=4$, IR), Pfund ($n_1=5$, IR).
• Bohr’s Model Postulates (1913):     1. Electrons revolve in circular orbits (stationary states) with fixed radii and energy.     2. Energy changes only during transitions between orbits ($\Delta E = h\nu$).     3. Angular momentum is quantized: $m_e vr = \frac{nh}{2\pi}$ ($n = 1, 2, 3...$).
• Bohr Model Quantities for Hydrogen:     * Radius ($r_n$): $r_n = n^2 a_0$ where $a_0 = 52.9\,pm$.     * Energy ($E_n$): $E_n = -R_H \left( \frac{1}{n^2} \right)$ where $R_H = 2.18 \times 10^{-18}\,J$.     * Hydrogen-like ions ($He^+$, $Li^{2+}$): $E_n = -2.18 \times 10^{-18} \left( \frac{Z^2}{n^2} \right)\,J$; $r_n = \frac{52.9 \times n^2}{Z}\,pm$.
• Limitations of Bohr Model: Cannot explain spectra of multi-electron atoms, Zeeman effect (splitting in magnetic field), Stark effect (splitting in electric field), or chemical bonding.

Quantum Mechanical Model of Atom

• Dual Behavior of Matter: Louis de Broglie (1924) proposed that matter enjoys dual nature: $\lambda = \frac{h}{mv}$. Significant for microscopic objects.
• Heisenberg Uncertainty Principle (1927): Impossible to determine simultaneously the exact position and momentum of a subatomic particle: $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$. Rules out definite trajectories.
• Quantum Mechanics: Developed by Heisenberg and Schrödinger (1926).     * Schrödinger Equation: $\hat{H} \psi = E \psi$. $\psi$ is the wave function.     * Probability Density ($|\psi|^2$): Probability of finding an electron at a point.
• Quantum Numbers:     1. Principal ($n$): Shell ($K, L, M...$). Size and energy of orbital. Total orbitals = $n^2$.     2. Azimuthal/Orbital Angular Momentum ($l$): Subshell shape. Values $0$ to $n-1$. ($s=0, p=1, d=2, f=3$).     3. Magnetic ($m_l$): Spatial orientation. Values $-l$ to $+l$. Total orientations = $2l+1$.     4. Spin ($m_s$): Electron spin orientation ($+\frac{1}{2}, -\frac{1}{2}$.)
• Orbital Shapes:     * s-orbitals: Spherical. Size increases with $n$. Number of nodes = $n-1$.     * p-orbitals: Two lobes (dumbbell) along x, y, or z axes. One angular node.     * d-orbitals: Five orientations ($d_{xy}, d_{yz}, d_{xz}, d_{x^2-y^2}, d_{z^2}$). Two angular nodes.
• Energies of Orbitals:     * In H-atom, energy depends only on $n$ ($2s=2p$).     * In multi-electron atoms, energy depends on $n$ and $l$. Rule: lower $(n+l)$ has lower energy. If $(n+l)$ is equal, the lower $n$ has lower energy.
• Rules for Filling Orbitals:     * Aufbau Principle: Fill in order of increasing energy ($1s < 2s < 2p < 3s ...$).     * Pauli Exclusion Principle: Maximum two electrons per orbital with opposite spins. No two electrons have same 4 quantum numbers.     * Hund’s Rule of Maximum Multiplicity: No pairing in degenerate orbitals until each is singly occupied with parallel spins.
• Electronic Configuration Exceptions: Chromium ($[Ar] 3d^5 4s^1$) and Copper ($[Ar] 3d^{10} 4s^1$) occur because half-filled and full-filled subshells have extra stability due to symmetry and high exchange energy.

Questions & Discussion

• Problem 2.1: Calculate the number of protons, neutrons, and electrons in ${}^{80}_{35}Br$.     * Solution: $Z=35$, $A=80$. Protons = $35$, Electrons = $35$, Neutrons = $80 - 35 = 45$.
• Problem 2.2: A species has $18$ electrons, $16$ protons, and $16$ neutrons. Assign symbol.     * Solution: Protons = $16$ means Sulphur ($S$). Electrons $18 > 16$, so it is a $-2$ anion. Mass number = $32$. Symbol is ${}^{32}_{16}S^{2-}$.
• Problem 2.6: Energy of one mole of photons with $\nu = 5 \times 10^{14}\,Hz$.     * Solution: $E = h\nu = (6.626 \times 10^{-34} \,J\,s) \times (5 \times 10^{14}\,s^{-1}) = 3.313 \times 10^{-19}\,J$. One mole = $E \times N_A = 199.51\,kJ\,mol^{-1}$.
• Problem 2.16: Golf ball ($40\,g$) at $45\,m/s$ measured within $2\%$. Uncertainty in position?     * Solution: $\Delta v = 45 \times 0.02 = 0.9\,m/s$. $\Delta x \ge \frac{h}{4\pi m \Delta v} = 1.46 \times 10^{-33}\,m$.