Unit 2: Structure of Atom Lecture Notes

Historical Theories and the Particle Nature of Matter

• Early Indian and Greek philosophers (circa 400 B.C.) proposed that atoms are the fundamental building blocks of matter. They believed that continuous subdivision of matter would eventually reach a particle that could not be further divided.
• The word 'atom' originates from the Greek word 'a-tomio', meaning 'uncut-able' or 'non-divisible'.
• Early atomic ideas were speculative and lacked experimental verification, remaining dormant until the nineteenth century.
• John Dalton, a British school teacher, provided the first scientific basis for atomic theory in 1808. He regarded the atom as the ultimate particle of matter.
• Dalton's theory successfully explained the law of conservation of mass, the law of constant composition, and the law of multiple proportion.
• Dalton's theory failed to explain certain experimental results, such as why substances like glass or ebonite become electrically charged when rubbed with silk or fur.
• Experiments in the late 19th and early 20th centuries established that atoms are actually composed of sub-atomic particles: electrons, protons, and neutrons.

Discovery of Sub-atomic Particles

• Insights into atomic structure came from experiments involving electrical discharge through gases.
• A fundamental rule of charged particles is that like charges repel each other and unlike charges attract.
• In 1830, Michael Faraday showed that passing electricity through an electrolyte solution caused chemical reactions at electrodes, resulting in the liberation and deposition of matter, suggesting electricity has a particulate nature.
• In the mid-1850s, cathode ray discharge tubes were utilized to study electricity. These tubes are made of glass and contain two metal electrodes (cathode and anode).
• Electrical discharge through gases is only observable at very low pressures and very high voltages. Pressure is adjusted by evacuation.
• When high voltage is applied, current flows as a stream of particles from the negative electrode (cathode) to the positive electrode (anode), known as cathode rays or cathode ray particles.
• The flow is verified by using a perforated anode and coating the tube behind it with a phosphorescent material like zinc sulphide ($ZnS$). Impact on the coating produces a bright spot.
• Characteristics of Cathode Rays:
  • They travel from cathode toward anode.
  • The rays themselves are invisible but are detectable via fluorescent or phosphorescent materials (e.g., television picture tubes).
  • They travel in straight lines in the absence of electrical or magnetic fields.
  • They consist of negatively charged particles (electrons) as they behave like negative charges in electrical or magnetic fields.
  • Their characteristics do not depend on the electrode material or the nature of the gas in the tube.

Atomic Properties and Fundamental Constants

• J.J. Thomson (1897) measured the charge to mass ratio ($e/m_e$) of the electron using perpendicular electrical and magnetic fields.
• Amount of deviation depends on:
  • Magnitude of charge: Higher charge interaction leads to greater deflection.
  • Mass: Lighter particles undergo greater deflection.
  • Field strength: Deflection increases with increased voltage or magnetic field strength.
• Thomson's calculated value: $\frac{e}{m_e} = 1.758820 \times 10^{11}\,\text{C kg}^{-1}$.
• R.A. Millikan (1906-14) used the oil drop experiment to determine electron charge. He found it to be $-1.6 \times 10^{-19}\,\text{C}$. The current accepted value is $-1.602176 \times 10^{-19}\,\text{C}$.
• Mass of the electron ($m_e$) was calculated by combining these results: $m_e = 9.109382 \times 10^{-31}\,\text{kg}$.
• Discovery of Protons: Modified cathode ray tubes revealed canal rays (positively charged gaseous ions). Their mass depends on the gas used. The smallest and lightest was from hydrogen, called a proton ($1919$).
• Discovery of Neutrons: James Chadwick ($1932$) bombarded a thin sheet of beryllium with $\alpha$-particles, emitting neutral particles with mass slightly greater than protons ($1.674927 \times 10^{-27}\,\text{kg}$, compared to proton's $1.6726216 \times 10^{-27}\,\text{kg}$).

Early Atomic Models and Radioactivity

• Wilhalm R\u00f6entgen ($1895$) discovered X-rays when electrons struck materials in cathode ray tubes. They possess high penetrating power and are not deflected by fields.
• Henri Becqueral ($1896$) discovered radioactivity. Marie Curie and Rutherford furthered this work, identifying three ray types:
  • $\alpha$-rays: Helium nuclei ($He^{2+}$), high energy, two units of positive charge, four units of mass.
  • $\beta$-rays: Negatively charged particles similar to electrons.
  • $\gamma$-rays: High energy neutral radiations, most penetrating.
• J.J. Thomson's Model ($1898$): 'Plum pudding' or 'watermelon' model. Atom is a sphere of positive charge (radius $\sim 10^{-10}\,\text{m}$) with electrons embedded. It assumed uniform mass distribution.
• Rutherford's Scattering Experiment ($1909$): Bombarded a gold foil ($\sim 100\,\text{nm}$ thick) with $\alpha$-particles.
  • Observations: Most particles passed undeflected; a small fraction was deflected by small angles; a very few ($1$ in $20,000$) bounced back ($180^\circ$).
  • Conclusions: Atom is mostly empty space. Positive charge and mass are concentrated in a tiny volume (nucleus). Radius of nucleus is $\sim 10^{-15}\,\text{m}$. Analogy: If nucleus is a cricket ball, atom radius is $5\,\text{km}$.
• Rutherford's Model Postulates:
  • Positive charge/mass concentrated in the nucleus.
  • Electrons move around nucleus in circular paths (orbits), resembling the solar system.
  • Electrons and nucleus are held by electrostatic forces.

Nuclear Composition: Atomic and Mass Numbers

• Atomic number ($Z$) = Number of protons in the nucleus = number of electrons in a neutral atom.
• Mass number ($A$) = Number of protons ($Z$) + Number of neutrons ($n$). Collectively called nucleons.
• Notation: ${^{A}_{Z}X}$.
• Isobars: Atoms with the same mass number but different atomic numbers (e.g., ${^{14}_{6}C}$ and ${^{14}_{7}N}$).
• Isotopes: Atoms with the same atomic number but different mass numbers due to different neutron counts.
  • Hydrogen isotopes: Protium (${^{1}_{1}H}$, $99.985\%$), Deuterium (${^{2}_{1}D}$, $0.015\%$), Tritium (${^{3}_{1}T}$, trace amounts).
  • Chemical properties are determined by electron count (protons), so isotopes show identical chemical behavior.

Failures of Rutherford's Model

• Stability Problem: According to Maxwell's electromagnetic theory, an accelerated charged particle (electron in orbit) should emit radiation. This energy loss should cause the electron to spiral into the nucleus in $10^{-8}\,\text{s}$.
• Distribution Problem: It says nothing about how electrons are distributed or their energies.
• Stationary Electrons: If electrons were stationary, electrostatic attraction would pull them into the nucleus.

Developments Leading to the Bohr Model

• Neils Bohr improved early models by incorporating the dual character of electromagnetic (EM) radiation (wave-like and particle-like properties) and experimental atomic spectra results.
• Wave Nature of EM Radiation: Maxwell ($1870$) proposed that oscillating charged particles produce alternating electrical and magnetic fields transmitted as waves.
  • Waves do not require a medium and travel at the speed of light ($c = 2.997925 \times 10^8\,\text{m s}^{-1}$, approx $3.0 \times 10^8\,\text{m s}^{-1}$).
  • Frequency ($\nu$) and wavelength ($\lambda$) relationship: $c = \nu \lambda$.
  • Wavenumber ($\bar{\nu}$): Number of wavelengths per unit length: $\bar{\nu} = \frac{1}{\lambda}$.
• EM Spectrum Regions:
  • Radio: $\sim 10^6\,\text{Hz}$.
  • Microwave: $\sim 10^{10}\,\text{Hz}$.
  • Infrared: $\sim 10^{13}\,\text{Hz}$.
  • Visible light: $\sim 10^{15}\,\text{Hz}$.
  • Ultraviolet: $\sim 10^{16}\,\text{Hz}$.
• Black-body Radiation: Ideal bodies that absorb and emit all frequencies uniformly. Max Planck ($1900$) proposed energy is emitted/absorbed in discrete units called quanta. Equation: $E = h\nu$, where Planck's constant $h = 6.626 \times 10^{-34}\,\text{J s}$.
• Photoelectric Effect: Electrons are ejected from metals (like $K, Rb, Cs$) upon exposure to light. Einstein ($1905$) applied Planck's theory: $h\nu = h\nu_0 + \frac{1}{2}m_e v^2$. $h\nu_0$ is the work function ($W_0$).

Atomic Spectra and Bohr's Model of Hydrogen

• Emission Spectrum: Spectrum of radiation from a substance that has absorbed and then emitted energy.
• Absorption Spectrum: Like a photographic negative of emission, showing dark spaces where specific wavelengths were absorbed.
• Line Spectra (Atomic Spectra): Discrete bright lines emitted by gas-phase atoms. Elements have unique line spectra ('fingerprints').
• Hydrogen Line Series:
  • Balmer formula ($1885$): $\bar{\nu} = 109,677 \left( \frac{1}{2^2} - \frac{1}{n^2} \right)\,\text{cm}^{-1}$, where $n \geq 3$.
  • Rydberg expression: $\bar{\nu} = 109,677 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\,\text{cm}^{-1}$.
  • 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 Postulates for Hydrogen ($1913$):
  • Electrons move in circular 'stationary' orbits with fixed radius and energy.
  • Energy change occurs only during jumps between orbits: $\Delta E = E_2 - E_1 = h\nu$.
  • Angular momentum is quantized: $m_e vr = n\frac{h}{2\pi}$, where $n = 1, 2, 3...$.
• Bohr's Mathematical Results:
  • Radii: $r_n = n^2 a_0$ where $a_0 = 52.9\,\text{pm}$.
  • Energy: $E_n = -R_H \left( \frac{1}{n^2} \right)$ where $R_H = 2.18 \times 10^{-18}\,\text{J}$.
  • For hydrogen-like ions ($He^+, Li^{2+}, Be^{3+}$): $E_n = -2.18 \times 10^{-18} \left( \frac{Z^2}{n^2} \right)\,\text{J}$; $r_n = \frac{52.9(n^2)}{Z}\,\text{pm}$.
• Limitations of Bohr Model: Fails to explain multi-electron atoms, finer 'doublet' details of spectra, Zeeman effect (magnetic splitting), Stark effect (electric splitting), and chemical bonding.

Mechanics of the Microscopic World

• de Broglie Relation ($1924$): Matter has dual behavior. $\lambda = \frac{h}{mv} = \frac{h}{p}$. This was confirmed by electron diffraction.
• Heisenberg Uncertainty Principle ($1927$): Impossible to determine exactly both position and momentum of an electron simultaneously: $\Delta x \Delta p_x \geq \frac{h}{4\pi}$.
  • Implications: Rules out definite electron trajectories. Significant only for microscopic objects.
• Quantum Mechanics: Theoretical science developed by Heisenberg and Erwin Schr\u00f6dinger ($1926$) dealing with dual nature of matter. Schr\u00f6dinger Equation: $\hat{H}\psi = E\psi$ where $\hat{H}$ is the Hamiltonian operator.
• Atomic Orbital: The wave function $\psi$ for an electron. Probability density is $|\psi|^2$. Regions where $|\psi|^2$ is zero are called nodes.

Quantum Numbers and Orbital Shapes

• Principal Quantum Number ($n$): Positive integer ($1, 2, 3...$). Defines shell size and energy. Number of orbitals in a shell = $n^2$.
• Azimuthal Quantum Number ($l$): Range $0$ to $n-1$. Defines three-dimensional shape (sub-shells).
  • $l=0$ (s), $l=1$ (p), $l=2$ (d), $l=3$ (f).
• Magnetic Orbital Quantum Number ($m_l$): Range $-l$ to $+l$. Defines spatial orientation. Total orientations = $2l+1$.
• Electron Spin Quantum Number ($m_s$): Two orientations ($+1/2, -1/2$), represented by $\uparrow$ and $\downarrow$.
• Shapes:
  • s-orbitals: Spherical. Size increases with $n$. Number of radial nodes = $n-1$.
  • p-orbitals: Two lobes (dumb-bell). Designated $p_x, p_y, p_z$. Nodal plane passes through nucleus. Radial nodes = $n-2$.
  • d-orbitals: Designated $d_{xy}, d_{yz}, d_{xz}, d_{x^2-y^2}, d_{z^2}$. First four have four lobes.
  • Total nodes = $(n-1)$ [sum of angular nodes ($l$) and radial nodes ($n-l-1$)].

Rules for Filling Orbitals and Configuration

• Energies of Orbitals:
  • For hydrogen, energy depends only on $n$. Orbitals with same energy are 'degenerate'.
  • For multi-electron atoms, energy depends on $n$ and $l$. Determined by $(n+l)$ rule: Lower $(n+l)$ means lower energy. If $(n+l)$ is equal, the orbital with lower $n$ has lower energy.
• Aufbau Principle: Orbitals are filled in order of increasing energies: $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p...$
• Pauli Exclusion Principle: No two electrons in an atom can have the same four quantum numbers. An orbital can hold only two electrons with opposite spins.
• Hund's Rule of Maximum Multiplicity: Electron pairing in degenerate orbitals (p, d, f) doesn't start until each orbital is singly occupied.
• Electronic Configuration: Distribution of electrons into orbitals (e.g., Hydrogen is $1s^1$, Helium is $1s^2$).
• Exceptional Configurations: Chromium ($Z=24$) is $[Ar] 3d^5 4s^1$ and Copper ($Z=29$) is $[Ar] 3d^{10} 4s^1$. Extra stability arises from:
  • Symmetrical distribution of electrons.
  • High exchange energy: Energy released when electrons with same spin in degenerate orbitals swap positions.