Atomic Structure revision notes

• Cathode Rays (Electrons): Stream of negatively charged particles moving from cathode to anode in a low-pressure discharge tube. They travel in straight lines in the absence of fields and are deflected towards the positive pole by electric/magnetic fields. Their properties do not depend on the material of the electrodes or the nature of the gas. JEE Tip This established electrons as a fundamental constituent of all matter.
• Canal Rays (Positive Ions): Positively charged gaseous ions produced in modified cathode ray tubes. Their mass and charge-to-mass ratio depend heavily on the nature of the gas. JEE Tip Unlike electrons, canal rays are just ionized gas atoms.

• X-Rays: Discovered by Wilhelm Roentgen (1895) when electrons struck a dense metal target. They are un-deflected by electric/magnetic fields, have high penetrating power, and possess very short wavelengths ($\sim 0.1 \text{ nm}$).
• Radioactivity: Discovered by Henri Becquerel. Elements emit three kinds of rays: $\alpha$-particles ($He^{2+}$ nuclei) with the least penetration, $\beta$-rays (fast electrons) with 100 times more penetration, and $\gamma$-rays (high-energy neutral EMR) with 1000 times more penetration than $\alpha$-particles,.

• Thomson Model (1898): "Plum pudding" or "watermelon" model. Atom is a uniform sphere (radius $\sim 10^{-10} \text{ m}$) of positive charge with electrons embedded to give a stable electrostatic arrangement. It assumed mass was evenly distributed but failed to explain scattering experiments.

• Electromagnetic Radiation (EMR): Oscillating electric and magnetic fields produced by accelerating charged particles. Fields are perpendicular to each other and to the direction of propagation, and they do not require a medium.
• Electromagnetic Spectrum: Ordered by increasing frequency: Radio $\rightarrow$ Microwave $\rightarrow$ Infrared (IR) $\rightarrow$ Visible $\rightarrow$ Ultraviolet (UV) $\rightarrow$ X-rays $\rightarrow$ Gamma rays.
• Visible Light: Wavelength ranges from 400 nm (violet) to 750 nm (red); frequency ranges from $7.5 \times 10^{14} \text{ Hz}$ to $4.0 \times 10^{14} \text{ Hz}$,.
• Black Body Radiation: An ideal black body is a perfect absorber and radiator of energy. The amount of light emitted (intensity) and its spectral distribution depend only on the temperature. As temperature increases, the maxima of the intensity-wavelength curve shifts to a shorter wavelength. Wave theory failed to explain this curve.
• Planck’s Quantum Theory (1900): Energy is emitted/absorbed discontinuously in discrete "chunks" called quanta. The energy of a quantum is directly proportional to its frequency ($E=h\nu$).

• Einstein's Explanation (1905): Light consists of particles (photons). A photon collides with an electron, transferring its full energy instantaneously,.
• Dual Behaviour of EMR: Light exhibits both wave-like properties (diffraction, interference) and particle-like properties (black body radiation, photoelectric effect).

• Emission Spectrum: Produced when excited atoms emit radiation as they drop to a lower energy state. Appears as bright lines on a dark background,.
• Absorption Spectrum: Like a "photographic negative" of an emission spectrum. White light passed through a sample leaves dark gaps in a continuous bright spectrum corresponding to absorbed wavelengths,.

de Broglie's Hypothesis (1924)

Matter, like radiation, has dual behaviour. Every object in motion has an associated wave,.

• Macroscopic objects have undetectable wavelengths ($\approx 10^{-34} \text{ m}$) due to large mass.
• Sub-atomic particles (electrons) have measurable wavelengths.

Heisenberg Uncertainty Principle (1927)

It is impossible to simultaneously determine the exact position and exact momentum of an electron.

• Consequence: It rules out the existence of definite trajectories or "Bohr orbits".
• Macroscopic Limits: The uncertainty product for a milligram-sized object is infinitesimally small ($\approx 10^{-28} \text{ m}^2 \text{ s}^{-1}$) making it practically insignificant for large objects.

• Schrödinger Wave Equation: Incorporates wave-particle duality. Solved for the hydrogen atom, it yields quantized energy states and wave functions ($\psi$).
• Wave Function ($\psi$) & Probability Density: $\psi$ (atomic orbital) has no physical meaning. $|\psi|^2$ gives the probability density of finding the electron at a specific point in space. An orbital is mathematically defined by $\psi$.

• Effective Nuclear Charge ($Z_{eff}$) and Shielding Geometry: Inner electrons shield outer electrons. Shielding power depends heavily on orbital shape: $s > p > d > f$. Consequently, for a given $n$, the $Z_{eff}$ experienced is $s > p > d > f$, making $s$ most tightly bound.
• Aufbau Principle: Orbitals are filled in order of increasing energy based on the $(n+l)$ rule.
• Pauli Exclusion Principle: No two electrons in an atom can have the same set of all four quantum numbers. Max 2 electrons per orbital with opposite spins.
• Hund’s Rule of Maximum Multiplicity: In degenerate orbitals, pairing does not take place until each orbital is singly occupied with parallel spins.

• $(n+l)$ Rule: The orbital with the lower $(n+l)$ value has lower energy and is filled first. If $(n+l)$ values are equal, the lower $n$ fills first.
• Millikan's Oil Drop Principle: The magnitude of electrical charge on oil droplets is always an integral multiple of the fundamental electrical charge ($q = ne$).
• Bohr's Frequency Rule: $\nu = \Delta E / h$. Radiation is absorbed/emitted only when a transition occurs between two stationary states.
• Conservation of Energy in Photoelectric Effect: Energy of incident photon = Work Function + Kinetic Energy of ejected electron.

• Velocity of Light: $c = \nu \lambda$ (where $c = 3.0 \times 10^8 \text{ m s}^{-1}$).
• Wavenumber: $\bar{\nu} = \frac{1}{\lambda}$.
• Planck's Equation: $E = h\nu = \frac{hc}{\lambda}$ (where $h = 6.626 \times 10^{-34} \text{ J s}$).
• Photoelectric Effect: $h\nu = h\nu_0 + \frac{1}{2}m_ev^2$.
• Rydberg Formula: $\bar{\nu} = 109,677 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \text{ cm}^{-1}$.
• Bohr Angular Momentum: $m_e v r = \frac{nh}{2\pi}$.
• Bohr Radius: $r_n = 52.9 \left( \frac{n^2}{Z} \right) \text{ pm}$,.
• Bohr Energy: $E_n = -2.18 \times 10^{-18} \left( \frac{Z^2}{n^2} \right) \text{ J atom}^{-1}$,.
• de Broglie Wavelength: $\lambda = \frac{h}{p} = \frac{h}{mv}$.
• Heisenberg Uncertainty Principle: $\Delta x \cdot \Delta p \ge \frac{h}{4\pi} \implies \Delta x \cdot m\Delta v \ge \frac{h}{4\pi}$.

• Shielding Effect / Penetration Power: $s > p > d > f$ (for a given $n$). Because $s$ is spherical, it spends more time close to the nucleus, shielding outer electrons best.
• Energy of Orbitals (Multi-electron): $s < p < d < f$ (for a given $n$).
• Effective Nuclear Charge ($Z_{eff}$): Decreases as azimuthal quantum number $l$ increases within the same shell ($Z_{eff}$ for $s > p > d > f$).

• Calculations involving the Rydberg Equation: Calculating wavelength/frequency for specific transitions and comparing them across hydrogen-like species where the $Z^2$ factor must be included.
• Photoelectric Effect Numericals: Utilizing Einstein's equation to find work function, threshold frequency, or maximum kinetic energy.
• Identifying Valid Sets of Quantum Numbers: Rules for $n, l, m_l, m_s$ to identify impossible states.
• de Broglie Wavelength linked with Kinetic Energy: Combining $K.E. = \frac{1}{2}mv^2 = qV$ with $\lambda = \frac{h}{p}$.
• Graphs of Probability Density: Matching $|\psi|^2$ vs $r$ graphs to specific orbitals by calculating expected radial nodes.
• Exchange Energy & Exceptional Configurations: Assessing stability logic for $Cr$, $Cu$, and counting unpaired electrons.