Atomic Structure revision notes

Key Concepts & Definitions

Atom:

Derived from Greek 'a-tomio' (uncut-able). Regarded as the ultimate, indivisible particle of matter by Dalton (1808), which was later proven wrong,.

Atomic Number (Z):

Number of protons in the nucleus, which is exactly equal to the number of electrons in a neutral atom.

Mass Number (A):

Total number of nucleons (protons ZZZ + neutrons nnn).

Isotopes:

Atoms with identical atomic number (ZZZ) but different mass numbers (AAA) due to a different number of neutrons (e.g., Protium 11H^1_1H11​H, Deuterium 12H^2_1H12​H, Tritium 13H^3_1H13​H).

Isobars:

Atoms with the same mass number (AAA) but different atomic numbers (ZZZ) (e.g., 614C^{14}_6C614​C and 714N^{14}_7N714​N).

Chemical Identity:

Chemical properties are controlled by the number of electrons (and protons). Because neutrons have very little effect on chemical behavior, all isotopes of a given element show identical chemical properties.

Discovery of Sub-atomic Particles, Radioactivity & X-Rays

• 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.
• Fundamental Particles:
  • Electron (e): Discovered by J.J. Thomson. Mass = $9.1094 \times 10^{-31} \text{ kg}$. Charge = $-1.602176 \times 10^{-19} \text{ C}$. Specific charge ($e/m_e$) = $1.758820 \times 10^{11} \text{ C kg}^{-1}$.
  • Proton (p): Smallest/lightest positive ion obtained from hydrogen gas. Charge = $+1.602176 \times 10^{-19} \text{ C}$, Mass = $1.6726 \times 10^{-27} \text{ kg}$.
  • Neutron (n): Discovered by Chadwick (1932) by bombarding Beryllium with $\alpha$-particles. Electrically neutral with a mass of $1.6749 \times 10^{-27} \text{ kg}$ (slightly heavier than a proton),.
• 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,.

Early Atomic Models

• 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.
• Rutherford’s Nuclear Model: Based on $\alpha$-particle ($He^{2+}$) scattering on a 100 nm gold foil.
  • Observations: Most passed undeflected, few deflected by small angles, and very few (1 in 20,000) bounced back (180°),.
  • Conclusions: Most of the atom is empty space. Positive charge and mass are concentrated in a tiny central volume called the nucleus (radius $\sim 10^{-15} \text{ m}$),. Electrons revolve in circular orbits (planetary model).

Electromagnetic Radiation & Planck's Quantum Theory

• 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 ($E=h\nu$).

Photoelectric Effect & Dual Nature of Light

• Photoelectric Effect (Hertz, 1887): Ejection of electrons when light strikes a metal surface.
  • No time lag between light striking and electron ejection.
  • Number of ejected electrons $\propto$ intensity/brightness of light.
  • Kinetic energy of ejected electrons $\propto$ frequency of light.
  • Occurs ONLY if incident frequency $\nu > \nu_0$ (Threshold Frequency).
• 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).

Atomic Spectra & Bohr's Model

• 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,.
• Hydrogen Line Spectrum:
  • Lyman Series ($n_1=1$): Ultraviolet.
  • Balmer Series ($n_1=2$): Visible.
  • Paschen ($n_1=3$), Brackett ($n_1=4$), Pfund ($n_1=5$) Series: Infrared.
• Bohr’s Model for Hydrogen (1913):
  • Electrons move in concentric circular paths called stationary states or orbits.
  • Quantization of Angular Momentum: Electrons only occupy orbits where angular momentum $m_evr = n\frac{h}{2\pi}$.
  • Transition occurs when energy is absorbed/emitted in discrete amounts: $\Delta E = E_{f} - E_{i} = h\nu$,.

Dual Nature of Matter & Heisenberg's Uncertainty Principle

• 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.

Quantum Mechanical Model & Quantum Numbers

• 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 $\psi$.
• Quantum Numbers:
  1. Principal Quantum Number ($n$): Determines shell, major contributor to energy and size. Max electrons per shell = $2n^2$. Number of orbitals = $n^2$.
  2. Azimuthal/Orbital Angular Momentum ($l$): Determines subshell and 3D shape. Values from $0$ to $(n-1)$. ($l=0 \rightarrow s$, $l=1 \rightarrow p$, $l=2 \rightarrow d$, $l=3 \rightarrow f$).
  3. Magnetic Orbital Quantum Number ($m_l$): Determines spatial orientation. Values from $-l$ to $+l$ (total $2l+1$ values).
  4. Electron Spin Quantum Number ($m_s$): Intrinsic spin. Values $+\frac{1}{2} (\uparrow)$ or $-\frac{1}{2} (\downarrow)$,.
• Shapes of Orbitals & Nodes:
  • Nodes: Regions where probability density $|\psi|^2$ reduces to zero.
  • s-orbitals: Spherically symmetric.
  • p-orbitals: Two lobes with a nodal plane between them. $p_x, p_y, p_z$ are mutually perpendicular.
  • d-orbitals: Four have double-dumbbell shape ($d_{xy}, d_{yz}, d_{xz}, d_{x^2-y^2}$) and one is dumbbell with a doughnut/collar ($d_{z^2}$).

Electronic Configuration & Stability

• 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.
• Stability of Half-Filled and Fully-Filled Subshells: Such configurations possess extra stability due to:
  1. Symmetrical distribution of electrons.
  2. Maximum Exchange Energy: Electrons with parallel spins in degenerate orbitals exchange positions. More exchanges = more energy released = more stability.

Important Rules, Laws & Principles

• $(n+l)$ 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.

Formulae & Equations

• 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: $\Delta x \cdot \Delta p \ge \frac{h}{4\pi} \implies \Delta x \cdot m\Delta v \ge \frac{h}{4\pi}$.
• Calculation of Nodes:
  • Total Nodes = $(n - 1)$.
  • Radial Nodes = $(n - l - 1)$.
  • Angular Nodes = $l$.

⚠️ EXCEPTIONS & ANOMALIES

• Exception 1: Stability of the Atom vs. Classical Electrodynamics
  • Anomaly: According to Maxwell's classical theory, an accelerating charged particle must emit continuous electromagnetic radiation. An electron orbiting a nucleus in Rutherford's model should rapidly spiral into the nucleus in $10^{-8} \text{ s}$.
  • Reality: Atoms are exceptionally stable; Bohr resolved this by introducing "stationary states" where classical electromagnetism is suspended,.
• Exception 2: The Photoelectric Threshold Anomaly
  • Anomaly: Classical wave theory predicted that a highly intense (bright) light of any frequency should eventually transfer enough energy to eject an electron.
  • Reality: A highly intense red light ($\approx 4.5 \times 10^{14} \text{ Hz}$) can shine on potassium for hours without ejecting a single electron, but a very weak yellow light ($\approx 5.1 \times 10^{14} \text{ Hz}$) ejects electrons instantly because it exceeds the threshold frequency ($\nu_0$).
• Exception 3: Degeneracy in Hydrogen vs. Multi-Electron Atoms
  • Anomaly: In a multi-electron atom, energies depend on both $n$ and $l$ (e.g., $1s < 2s < 2p < 3s$).
  • Reality: In Hydrogen (and $He^+$, $Li^{2+}$), orbital energy is determined solely by the principal quantum number $n$. Therefore, the $2s$ and $2p$ orbitals are degenerate, and $3s = 3p = 3d$.
• Exception 4: Exceptional Electronic Configurations of Cr and Cu
  • Anomaly: Expected Cr: $[Ar] 3d^4 4s^2$ and Cu: $[Ar] 3d^9 4s^2$,.
  • Reality: They adopt $[Ar] 3d^5 4s^1$ and $[Ar] 3d^{10} 4s^1$ respectively.
  • Why: Extra stability is achieved by symmetrical electron distribution and maximum exchange energy.
• Exception 5: Bohr Model's Failure with Fine Spectral Lines
  • Anomaly: Bohr's model perfectly predicts the primary lines of the Hydrogen spectrum.
  • Reality: It entirely fails to explain the fine structure (closely spaced doublets/triplets) seen in advanced spectroscopy. It also fails to explain Zeeman (magnetic) and Stark (electric) effect splittings.
• Exception 6: Independence of Specific Charge in Cathode vs. Canal Rays
  • Anomaly: The $e/m$ ratio for cathode rays (electrons) is universal and completely independent of the gas or electrode material.
  • Reality: The charge-to-mass ratio for canal rays (positive ions) varies wildly depending entirely on the specific gas present in the tube.

Trends & Comparisons

• 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 $s > p > d > f$).

Previous Year JEE Topics

• 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.

Memory Aids & JEE Traps

• → [JEE TIP] Trap 1 - Photoelectric vs Brightness: Increasing the intensity (brightness) of incident light increases the number of photoelectrons ejected. Kinetic energy depends only on the frequency of the incident light.
• → [JEE TIP] Trap 2 - Hydrogen Orbital Energy: In hydrogen and hydrogen-like single-electron species, orbital energy depends solely on the principal quantum number ($n$). Therefore, $3s = 3p = 3d$.
• → [JEE TIP] Trap 3 - Node Calculation: The number of radial nodes is $(n - l - 1)$, and the number of angular nodes is $l$. Do not confuse these with the total number of nodes, which is $(n - 1)$.
• → [JEE TIP] Trap 4 - Canal Ray Identity: Canal rays are positively charged gaseous ions, which vary depending on the gas in the tube. They are only considered protons if the gas used is pure hydrogen.
• → [JEE TIP] Trap 5 - Bohr Energy Scaling: As the atomic number ($Z$) increases for hydrogen-like species ($He^+$, $Li^{2+}$), the energy becomes more negative ($E_n \propto -Z^2/n^2$), meaning the electron is more tightly bound.
• → [JEE TIP] Trap 6 - Wave-Particle Boundary: Boundary surface diagrams do not show a strict physical region where the electron is contained 100% of the time. They are arbitrary contours enclosing a region of $\sim 90\%$ probability.
• → [JEE TIP] Trap 7 - Balmer Series Visibility: Only the Balmer series ($n_1 = 2$) falls in the visible spectrum. The Lyman series is UV, and Paschen/Brackett/Pfund are Infrared.
• → [JEE TIP] Trap 8 - Spin Quantum Origin: Only $n, l,$ and $m_l$ arise naturally from the Schrödinger wave equation. The spin quantum number ($m_s$) was introduced empirically by Uhlenbeck and Goudsmit to explain closely spaced doublet lines.
• → [JEE TIP] Trap 9 - Uncertainty Principle Applicability: The uncertainty principle makes it impossible to calculate exact trajectories, but this uncertainty is entirely negligible for macroscopic objects. It only has physical significance for microscopic particles,.
• → [JEE TIP] Trap 10 - Atomic Radius vs. Nucleus Size: The volume of the nucleus is negligibly small compared to the atom. The atomic radius is $\sim 10^{-10} \text{ m}$ while the nuclear radius is $\sim 10^{-15} \text{ m}$.