Chapter 2 — Structure Of Atom

Fig 1 — Educational Diagram: Historical development of atomic theory

Overview and timeline

• John Dalton (early 19th century): Dalton proposed that matter is made of indivisible atoms, each element consists of identical atoms with characteristic masses, and compounds form by combination of atoms in simple whole-number ratios. His ideas explained laws of conservation of mass, definite proportions and multiple proportions.
• J. J. Thomson (1897): Discovery of the electron by cathode-ray experiments showed atoms have internal structure. He measured charge-to-mass ratio (e/m) of the electron and proposed the 'plum pudding' model: electrons embedded in a diffuse positive charge.
• Robert A. Millikan (1909): Oil-drop experiment measured the elementary charge e, giving an absolute value for electron charge and enabling accurate electron mass from Thomson's e/m.
• Ernest Rutherford (1911): Gold foil (α-particle scattering) experiment revealed a small, dense, positively charged nucleus; most of the atom is empty space. Rutherford model: electrons orbit a compact nucleus, but classical mechanics could not explain stable orbits or line spectra.
• Niels Bohr (1913): Bohr introduced quantized orbits for the hydrogen atom. Postulates: electrons occupy fixed energy levels (En), emit/absorb photons when jumping between levels; angular momentum quantized as mvr = nħ. This explained hydrogen spectral lines (Balmer, Lyman series) and gave formulas for En and radii rn.
• de Broglie, Schrödinger & Heisenberg (1920s): quantum mechanics: de Broglie proposed wave nature of particles (λ = h/p). Schrödinger developed the wave equation (ψ), whose solutions give allowed energy levels and probability distributions (orbitals). Heisenberg formulated the uncertainty principle (Δx·Δp ≥ ħ/2). These replaced Bohr orbits by orbitals described by quantum numbers (n, l, m_l, m_s).

Key experiments and what they showed

• Cathode-ray tube: existence of electrons, charge-to-mass ratio.
• Oil-drop: absolute charge of electron.
• Alpha scattering (gold foil): tiny dense nucleus, mostly empty space.
• Spectral emission/absorption: quantized energy levels (Bohr) and later explained by wave mechanics.

From models to modern view

Dalton's indivisible atom → Thomson's internal structure (electrons) → Rutherford's nucleus → Bohr's quantized energy levels (good for H-like atoms) → quantum mechanics: electrons are described by wavefunctions (ψ), probabilities, and four quantum numbers. The modern atomic model explains chemical periodicity, bonding, spectra, magnetism and many technologies (spectroscopy, semiconductors, microscopy).

Important conceptual points for Class 11

• Atomic models evolved as new experimental data appeared; each model solved some problems but had limitations addressed by later theories.
• Bohr model is useful for simple hydrogen-like systems; quantum mechanics (Schrödinger + Pauli + spin) gives full description of multi-electron atoms.

Fig 2 — Educational Diagram: Discovery of subatomic particles

Overview: The atom was once thought indivisible. Experiments from the late 19th and early 20th centuries revealed three basic subatomic particles — electron, proton and neutron — and led to the nuclear model of the atom.

1. Discovery of electron (J. J. Thomson, 1897)

Experiment: Cathode-ray tube (CRT). A high voltage across electrodes in a partially evacuated tube produced a beam (cathode rays) that: (a) travelled in straight lines, (b) caused fluorescence on a screen, and (c) was deflected by electric and magnetic fields.

Key observations and conclusions:

• Deflection by electric/magnetic fields shows the rays are charged particles.
• All cathode rays had the same charge-to-mass ratio (e/m) independent of the cathode material, implying particles (later called electrons) are constituents of all atoms.

2. Charge and mass of electron (Millikan oil-drop experiment, 1909)

Experiment: Tiny charged oil drops were suspended between charged plates. By balancing gravitational and electric forces and using viscous drag corrections, Millikan measured the charge on many drops and found they were integer multiples of a smallest value (the elementary charge, e ≈ 1.602×10^−19 C). Combined with Thomson’s e/m, the electron mass m_e was obtained.

3. Discovery of positive particles (canal rays and hydrogen nucleus)

Experiment: Goldstein (1886) observed canal rays (anode rays) — positive ions moving opposite to cathode rays in specially designed tubes. Later studies showed one of these positive particles was the hydrogen ion (H+), a single-proton nucleus.

Conclusion: Atoms contain a positively charged particle (proton) with charge +e.

4. Nuclear model from alpha-particle scattering (Ernest Rutherford, 1909–1911)

Experiment: A beam of alpha particles was directed at a thin gold foil. Most passed through with little deflection, but a few were deflected at large angles, some even back-scattered.

Conclusions:

• Most of an atom is empty space (alpha particles pass through).
• There is a very small, massive, positively charged nucleus that causes large-angle deflections.
• The nuclear model replaced the uniform "plum-pudding" model.

5. Discovery of neutron (James Chadwick, 1932)

Experiment: Bombarding beryllium with alpha particles produced a neutral penetrating radiation which knocked protons out of paraffin. The properties of this radiation (no charge, mass similar to proton) indicated a neutral particle — the neutron.

Reaction used (example): 9Be + 4He → 12C + 1n

Summary of particle properties (qualitative):

• Electron: negative charge (−e), very small mass (m_e ≈ 9.11×10^−31 kg).
• Proton: positive charge (+e), mass ≈ 1.67×10^−27 kg.
• Neutron: neutral (0), mass ≈ 1.67×10^−27 kg (≈ proton mass).

Why these discoveries matter: They established that atoms are composite, led to the nuclear model and later quantum models, and form the basis of chemistry, nuclear physics and technologies like electronics, nuclear energy and medical imaging.

Fig 3 — Educational Diagram: Thomson's model of atom

Context: After the discovery of the electron (J. J. Thomson, 1897) Thomson proposed a model of the atom (circa 1904) to explain how negative electrons could exist inside an electrically neutral atom.

Basic idea (Plum‑pudding model): The atom is a sphere of uniform positive charge in which electrons are embedded like plums in a pudding. The positive charge is spread over the whole volume so that the total positive charge balances the total negative charge of the electrons; the atom as a whole is electrically neutral.

Assumptions:

• The atom is a sphere of radius R containing uniformly distributed positive charge +Q.
• Electrons (each of charge −e and mass m_e) are embedded at fixed positions inside this positive sphere.
• Electrostatic forces between the smeared positive charge and the electrons provide forces that hold electrons in place.
• For small displacements of an electron from equilibrium the net force is restoring (simple harmonic motion around equilibrium).

Simple electrostatic consequence (field inside a uniformly charged sphere):

For a uniformly charged sphere (total +Q, radius R), electric field at a distance r (r < R)
E(r) = (1 / (4πε0)) * (Q_enclosed / r^2)
But Q_enclosed = Q * (r^3 / R^3)
So E(r) = (1 / (4πε0)) * (Q * r / R^3)

Force on an electron & SHM behaviour:

F = −e E(r) = − (1 / (4πε0)) * (Q e / R^3) * r
This is a restoring force proportional to displacement r, so an electron displaced slightly undergoes simple harmonic motion with
ω^2 = (1 / (4πε0)) * (Q e) / (m_e R^3)
If the atom is neutral and Q = Z e (Z = total positive charge in units of e),
ω^2 = (1 / (4πε0)) * (Z e^2) / (m_e R^3)

Merits of Thomson's model:

• Explained electrical neutrality of atoms and existence of electrons as constituents.
• Provided a physical picture in which electrons could be in stable equilibrium (oscillatory motion) within the positive charge distribution.
• Was an important historical step toward more accurate models.

Limitations (why it was replaced):

• Could not explain the results of alpha‑particle scattering (Geiger–Marsden experiments) which showed a concentrated positive nucleus (Rutherford, 1911).
• Could not account for atomic spectral lines (discrete emission spectra) or chemical behavior dependent on nuclear charge and electron arrangement.
• The assumption of a continuous positive charge cloud has no experimental support; model gives incorrect scattering predictions and wrong spatial distribution of mass/charge.

Conclusion: Thomson's model (plum‑pudding) was a useful early attempt to place electrons inside atoms and to explain neutrality, but it failed quantitatively for scattering and spectral phenomena, and was superseded by Rutherford's nuclear model and later by Bohr and quantum models.

Fig 4 — Educational Diagram: Rutherford's nuclear model

Introduction: Rutherford's nuclear model (1911) replaced the plum‑pudding view of the atom and established that most of an atom's mass and all its positive charge are concentrated in a very small central nucleus, while electrons move around this nucleus in largely empty space.

Gold‑foil (alpha scattering) experiment — setup: A narrow beam of alpha (α) particles from a radioactive source was directed at a very thin gold foil. A circular fluorescent screen around the foil detected scattered α particles. Observations of scattering angles and counts led to key conclusions.

Key observations:

• Most α particles passed straight through the foil with little or no deflection (atom is mostly empty space).
• A small fraction were deflected through large angles (strong localized repulsive force).
• Very few (~1 in 20,000) were deflected back nearly 180° (center is very small and very dense).

Conclusions / Postulates of Rutherford's model:

• All positive charge and almost all mass of an atom are concentrated in a tiny central nucleus whose radius is on the order of 10^(-15) m.
• The nucleus carries a positive charge equal to Ze (Z = atomic number), where e is the elementary charge.
• Electrons move around the nucleus occupying the remaining space; most of the atom is empty space.
• Deflection of charged projectiles is due to Coulomb (electrostatic) repulsion between the positively charged nucleus and the incoming positively charged particles.

Physical picture & significance: The atom is not a uniform sphere of charge. Instead, a tiny dense nucleus (protons + neutrons, discovered later) is surrounded by electrons. Rutherford's model explained scattering data and laid the foundation for later quantum models (Bohr, wave mechanics) and nuclear physics.

Limitations: Rutherford's model could not explain the stability of orbiting electrons (classical electrodynamics predicts spiralling into the nucleus due to radiation) nor discrete atomic spectra. These problems were later resolved by Bohr's postulates and quantum mechanics.

Experimental & modern relevance: Rutherford scattering principles are used today in material analysis (Rutherford backscattering spectrometry), particle accelerators and detectors to probe nuclear and subnuclear structure, and underpin our understanding of nuclear reactions, radioactivity, and applications in medicine and energy.

Fig 5 — Educational Diagram: Bohr model of hydrogen atom

Overview: The Bohr model (Niels Bohr, 1913) explains the hydrogen atom spectra by combining classical mechanics with a quantum postulate: electrons move in certain allowed circular orbits without radiating energy and can jump between these orbits by absorbing or emitting a photon.

Bohr's postulates (concise):

• Electrons revolve in circular orbits about the nucleus under Coulomb attraction. Only certain orbits (stationary states) are allowed.
• An electron in an allowed orbit has a fixed energy and does not emit radiation.
• Angular momentum is quantized: m v r = nħ, where n = 1, 2, 3, ... (ħ = h/2π).
• Radiation is emitted/absorbed when an electron jumps between allowed orbits. The photon energy equals the energy difference: ΔE = hν.

Derivation (key steps):

• For a hydrogen-like atom (nucleus charge +Ze), equate Coulomb force to centripetal force: (1/4πε₀) (Ze e)² / r² = m v² / r.
• Use quantization m v r = nħ to eliminate v and solve for orbit radius r_n:

r_n = a₀ (n²/Z), where a₀ = 4πε₀ ħ² / (m e²) ≈ 0.529 × 10^-10 m (Bohr radius).

Energy of level n:

E_n = - (me e⁴ Z²) / (8 ε₀² h² n²) = -13.6 eV × (Z²/n²) (for hydrogen Z = 1). Ground state (n=1) E₁ = -13.6 eV.

Spectral lines (Rydberg formula):

When an electron falls from level n_i to n_f (n_i > n_f), the emitted photon has energy ΔE = E_i - E_f = hν = hc/λ. This leads to

1/λ = R_H (1/n_f² - 1/n_i²), where R_H ≈ 1.097373 × 10⁷ m^-1 (Rydberg constant for hydrogen).

Successes: Explains the hydrogen emission lines quantitatively (Balmer, Lyman, Paschen series), predicts ionization energy and radii, and works well for hydrogen-like (single-electron) ions such as He⁺, Li²⁺.

Limitations: Cannot explain fine structure, Zeeman effect, or spectra of multi-electron atoms; incompatible with the wave nature of electrons (replaced by quantum mechanics / Schrödinger model). Bohr's angular-momentum quantization is ad hoc; Schrödinger's wave mechanics gives deeper justification.

Practical notes: In CBSE Class 11 problems you'll often compute r_n, E_n, wavelengths for transitions, and use n values for series identification (e.g., Balmer: n_f = 2 produces visible lines).

Fig 6 — Educational Diagram: Atomic spectra

What are atomic spectra?
Atomic spectra are patterns of electromagnetic radiation (light) emitted or absorbed by atoms. They occur because electrons in atoms occupy discrete (quantized) energy levels; when an electron moves between levels it absorbs or emits a photon whose energy equals the difference between the two levels. Atomic spectra are characteristic of each element and are used to identify elements.

Types of spectra

• Continuous spectrum: Emitted by hot dense bodies (filament, blackbody). All wavelengths appear as a continuous band.
• Emission (line) spectrum: Bright lines on dark background. Produced by atoms in low-pressure gas discharged or excited — each bright line corresponds to a specific transition.
• Absorption spectrum: Dark lines on a continuous background. When continuous radiation passes through a cooler gas, photons of specific energies are absorbed corresponding to allowed transitions.

Bohr model explanation (for hydrogen-like atoms)
Key ideas (Bohr postulates): electrons move in certain allowed circular orbits without radiating; each orbit has quantized energy E_n; emission/absorption occurs when an electron jumps between orbits. For hydrogen, the energy of the nth level is given by E_n = -13.6 eV / n^2 (or E_n = -2.18 × 10^-18 J / n^2). The photon energy for a transition from n_i to n_f is ΔE = E_i - E_f = hν = hc/λ.

Rydberg formula (wavelengths of hydrogen spectral lines)
For hydrogen-like atoms, the wavelengths of spectral lines are given by the Rydberg formula:
1/λ = R_H (1/n_f^2 - 1/n_i^2), with n_i > n_f. Here R_H is the Rydberg constant for hydrogen ≈ 1.097373 × 10⁷ m^-1.

Important spectral series for hydrogen
- Lyman series: n_f = 1 (ultraviolet)
- Balmer series: n_f = 2 (visible; e.g., Hα = 656.3 nm)
- Paschen series: n_f = 3 (infrared)
Other series include Brackett (n_f = 4), Pfund (n_f = 5), etc.

Why spectra are useful (applications)
Identification of elements (flame tests, atomic emission spectroscopy), astrophysics (Fraunhofer lines in solar spectrum reveal elemental composition and physical conditions in stars), neon and sodium street lights, lasers (population inversion and stimulated emission), and analytical chemistry (qualitative and quantitative analysis).

Additional notes
- Absorption and emission lines for the same transitions occur at the same wavelengths (emission bright lines correspond to absorption dark lines at same λ).
- Real spectra show line broadening (Doppler, pressure, instrumental) and fine/hyperfine structures due to electron spin, relativistic effects and nuclear interactions.
- Selection rule (electric-dipole): Δl = ±1 usually governs allowed transitions (where l is orbital quantum number).

Fig 7 — Educational Diagram: Quantum nature of radiation

Introduction

The quantum nature of radiation means that electromagnetic radiation is not only a continuous wave but also exists as discrete packets of energy called photons. This idea explains phenomena that classical wave theory could not, such as blackbody radiation and the photoelectric effect.

Historical background and why quantum idea arose

• Blackbody radiation: Classical theories (Rayleigh–Jeans law) predicted infinite energy at short wavelengths (the ultraviolet catastrophe).
• Planck's hypothesis (1900): To resolve this, Planck proposed that energy exchange between matter and electromagnetic field occurs in discrete amounts. A harmonic oscillator (emitter/absorber) can have energies E = n h ν (n = 0,1,2,...), where h is Planck's constant and ν is frequency.
• Einstein and photons (1905): Einstein extended Planck's idea to light itself, proposing that light consists of quanta (photons) each with energy E = h ν. This explained the photoelectric effect and established the particle aspect of light.

Key consequences and evidences

• Photoelectric effect: Electrons are ejected from a metal when struck by light only if the light frequency exceeds a threshold ν0. The maximum kinetic energy of ejected electrons depends on frequency, not intensity. This matches Einstein's equation KEmax = hν − φ (φ = work function).
• Discrete spectra: Atoms emit light at specific wavelengths (line spectra). Each emitted photon corresponds to an energy difference between atomic energy levels ΔE = hν.
• Compton effect: X‑ray scattering from electrons shows a wavelength shift consistent with treating X‑rays as particles with momentum p = h/λ, supporting particle properties of radiation.

Summary of the physical picture

Radiation shows wave properties (interference, diffraction) and particle properties (photoelectric effect, Compton scattering). The quantum view: electromagnetic radiation is quantized into photons of energy E = hν and momentum p = h/λ. Energy exchange between light and matter occurs in integral multiples of hν.

Practical importance

Understanding the quantum nature of radiation is fundamental to devices such as photovoltaic cells, photo-detectors, LEDs and lasers, and to techniques like X‑ray scattering and spectroscopy.

Fig 8 — Educational Diagram: Wave–particle duality

Definition: Wave–particle duality is the concept that microscopic entities (light and matter) exhibit both wave-like and particle-like properties depending on the experiment and how they are observed. This idea unifies phenomena such as interference/diffraction (wave behaviour) and photoelectric/Compton effects (particle behaviour).

Historical basis: In the early 20th century Planck and Einstein treated light as quantized energy packets (photons) to explain blackbody radiation and the photoelectric effect. In 1924 Louis de Broglie proposed that material particles (electrons, neutrons, atoms) also have wave properties, characterized by a wavelength. Experiments (Davisson–Germer electron diffraction, electron double‑slit interference) later confirmed de Broglie’s hypothesis.

Key ideas and physical meaning:

• For photons: energy E and momentum p are related to frequency (ν) and wavelength (λ) by E = hν and p = h/λ (h = Planck’s constant). Photons act like particles with quantized energy and momentum, explaining photoelectric and Compton effects.
• For matter waves (de Broglie hypothesis): every particle of momentum p has an associated wavelength λ = h/p. For a non-relativistic particle p = mv, so λ = h/(mv). The wave character becomes significant when λ is comparable to the characteristic size (e.g., atomic spacing in crystals).
• Complementarity (Bohr): whether an entity shows wave or particle behaviour depends on the measurement setup. Both descriptions are needed for a full quantum picture.

Typical derivations (non-relativistic):

• De Broglie wavelength: λ = h/p. If p = mv (non-relativistic), λ = h/(mv).
• If an electron is accelerated through a potential difference V (so kinetic energy KE = eV), then p = sqrt(2m e V) and
  λ = h / sqrt(2 m e V).

When is wave nature observable? For macroscopic objects the de Broglie wavelength is extraordinarily small (practically zero), so classical particle behaviour dominates. For electrons, neutrons, atoms, or molecules at typical laboratory energies, λ can be comparable to atomic spacings and diffraction/interference experiments reveal wave properties.

Experimental confirmations (brief):

• Photoelectric effect (Einstein): light behaves like particles (photons) delivering energy hv to eject electrons; supports quantization of light.
• Compton effect: X‑ray scattering from electrons shows wavelength shift consistent with photon momentum transfer (particle nature of light).
• Davisson–Germer experiment: electrons scattered from a crystal show diffraction peaks, proving electrons have wave nature with wavelength equal to de Broglie wavelength.
• Electron double‑slit experiments: single electrons build up an interference pattern—each electron shows both particle arrival and underlying wave interference.

Limitations & remarks: De Broglie hypothesis is part of non-relativistic quantum mechanics; for very high speeds (relativistic) use relativistic relations (p = γmv, E^2 = (pc)^2 + (mc^2)^2). Quantum mechanics replaces classical trajectories with probability amplitudes (wavefunctions) whose squared magnitude gives detection probabilities.

Fig 9 — Educational Diagram: Heisenberg uncertainty principle

Definition: The Heisenberg uncertainty principle (1927) states that certain pairs of physical properties (conjugate variables) cannot be simultaneously measured with arbitrary precision. For position x and momentum p the relation is: Δx · Δp ≥ ħ/2, where ħ = h/2π. This is a fundamental property of quantum systems, not a limitation of instruments.

Why it occurs: Particles behave as waves (wave–particle duality). A well localized wave (small Δx) requires the superposition of many momentum (wavelength) components, so the spread in momentum (Δp) becomes large. Mathematically this follows from properties of Fourier transforms: a narrow function in x-space corresponds to a broad function in p-space.

Consequences and interpretation for Class 11: Electrons in atoms cannot have simultaneously exact position and momentum—this is why we describe electrons by orbitals (probability distributions), not definite classical trajectories. The uncertainty principle explains the stability of atoms (zero-point motion), quantum tunnelling, and limits on measurement at very small scales. For macroscopic objects, Δx and Δp values make the product negligible compared to ħ, so classical physics applies.

Energy–time form: ΔE · Δt ≥ ħ/2. This relates the uncertainty in energy to the time interval over which the energy measurement is made (used to explain natural linewidth of spectral lines and short-lived excited states).

Fig 10 — Educational Diagram: Schrödinger wave equation and quantum mechanics

Overview
The Schrödinger wave equation is the core mathematical statement of non-relativistic quantum mechanics. It describes how the quantum state of a particle (or system) changes in space and time. The solution is a complex function called the wavefunction, Ψ (psi), whose square modulus gives the probability density of finding the particle at a point in space.

Wavefunction and its meaning
Ψ(x,y,z,t) is the wavefunction. It can be complex. The physically measurable quantity is the probability density, given by |Ψ|² = Ψ*Ψ (Born interpretation). For a single particle the normalization condition is ∫|Ψ|² dτ = 1 over all space.

Time-dependent Schrödinger equation
The fundamental equation (time-dependent) is:
iħ ∂Ψ/∂t = −(ħ²/2m) ∇²Ψ + V(r,t) Ψ
Here ħ = h/2π, m is the particle mass, ∇² is the Laplacian operator and V(r,t) is the potential energy.

Time-independent Schrödinger equation
For a time-independent potential, solutions can be separated as Ψ(r,t) = ψ(r)·T(t). The spatial part satisfies the time-independent (stationary) equation:
−(ħ²/2m) ∇²ψ + V(r) ψ = E ψ
where E is the energy eigenvalue. These ψ(r) are stationary states (energy eigenfunctions).

Operators and expectation values
Physical observables correspond to operators. Examples:
• Momentum operator: p̂ = −iħ ∇
• Hamiltonian (energy) operator: Ĥ = −(ħ²/2m) ∇² + V(r)
Expectation value of observable A with operator Â: <A> = ∫ψ* Â ψ dτ.

Quantization and boundary conditions
Only those ψ that satisfy the Schrödinger equation and appropriate boundary conditions (finite, single-valued, continuous) are allowed. These conditions lead to discrete allowed energies (quantization). For example, a particle in a 1D infinite box of length L has energy levels E_n ∝ n² (n = 1,2,3...).

Quantum numbers (for hydrogen-like atoms)
Solving the Schrödinger equation for the hydrogen atom yields wavefunctions characterized by quantum numbers: principal (n), azimuthal (l), magnetic (m_l) and spin (m_s). These determine energy (mainly n for hydrogen), shape (l), orientation (m_l) and intrinsic spin (m_s = ±1/2).

Heisenberg uncertainty principle
Quantum mechanics implies limits on simultaneous knowledge of conjugate variables. For position x and momentum p: Δx · Δp ≥ ħ/2. This is a fundamental property, not due to measurement faults.

Nodes and probability distributions
Wavefunctions can have nodes (points or surfaces where ψ = 0). |ψ|² gives the probability distribution. For hydrogen, radial and angular nodes determine where electrons are likely or unlikely to be found. Orbitals (s, p, d...) are shapes of constant probability regions.

Why it matters (conceptual)
Schrödinger equation replaces classical trajectories with probability amplitudes. Particles show wave-like behavior (diffraction, interference) captured by Ψ and particle-like behavior in measurements (discrete detection events). This duality explains atomic spectra, chemical bonding, tunneling, and many microscopic phenomena.

Fig 11 — Educational Diagram: Quantum numbers and orbitals

Overview
Quantum numbers are a set of four numbers that describe the allowed states (orbitals) of electrons in an atom. Orbitals are regions in space where the probability of finding an electron is high. The four quantum numbers — principal (n), azimuthal or angular momentum (l), magnetic (m_l) and spin (m_s) — together uniquely identify an electron in an atom.

1. Principal quantum number (n)
- Denotes the main energy level or shell: n = 1, 2, 3, ...
- Gives approximate size and energy of the orbital. Higher n → larger orbital and typically higher energy.
- Maximum number of orbitals in a shell = n². Maximum electrons in a shell = 2n².

2. Azimuthal (angular momentum) quantum number (l)
- Determines shape of orbital and subshell type. For a given n, l = 0, 1, 2, ..., n−1.
- Subshell labels: l = 0 → s, 1 → p, 2 → d, 3 → f.
- Number of orbitals in a subshell = 2l + 1.

3. Magnetic quantum number (m_l)
- Describes orientation of an orbital in space: m_l = −l, −l+1, ..., 0, ..., +l.
- Each m_l value corresponds to one orbital within a subshell.

4. Spin quantum number (m_s)
- Represents intrinsic spin of the electron; possible values +1/2 or −1/2.
- Pauli exclusion principle: no two electrons in an atom can have the same set of all four quantum numbers.

Orbital shapes and nodes
- s orbitals (l = 0): spherical; probability density depends only on distance from nucleus.
- p orbitals (l = 1): dumbbell-shaped with a nodal plane at the nucleus; three orientations (p_x, p_y, p_z), corresponding to m_l = −1, 0, +1.
- d orbitals (l = 2): cloverleaf shapes (four lobes) for most; one (d_z2) has a doughnut/torus shape around z-axis.
- Number of angular nodes = l. Number of radial nodes = n − l − 1. Total nodes = n − 1.

Energy and degeneracy
- For hydrogen-like (single-electron) atoms, energy depends only on n: E_n = −13.6 eV * Z² / n² (Z = nuclear charge).
- In multi-electron atoms, energies also depend on l due to electron-electron interactions; subshells split (e.g., 2s and 2p differ in energy).

Rules for filling electrons
- Aufbau principle: electrons occupy the lowest available energy orbitals first (order: 1s < 2s < 2p < 3s < 3p < 4s < 3d ... with common crossovers).
- Pauli exclusion principle: max two electrons per orbital with opposite spins.
- Hund’s rule: electrons occupy degenerate orbitals singly with parallel spins before pairing.

Notation and examples
- Orbital notation: 3p, 4d, etc. Electronic configuration example: O (Z = 8) → 1s² 2s² 2p⁴.
- Calculation example: For n = 3: allowed l = 0,1,2 (3s, 3p, 3d). Orbitals = n² = 9 orbitals; maximum electrons = 2n² = 18 electrons.

Quantum mechanical picture
- Orbitals are solutions (wavefunctions ψ) of the Schrödinger equation. The observable probability density is |ψ|². Nodes are regions where ψ = 0 and |ψ|² = 0.
- The quantum numbers arise naturally from boundary conditions on these wavefunctions.

Key takeaways for Class 11
- Four quantum numbers (n, l, m_l, m_s) uniquely describe an electron.
- Orbital shapes (s, p, d) and nodes explain geometry and many chemical properties.
- Simple formulae (n², 2n², 2(2l+1), radial nodes = n−l−1) help count orbitals/electrons.

Fig 12 — Educational Diagram: Shapes of atomic orbitals

Overview

Atomic orbitals are regions in space around an atomic nucleus where the probability of finding an electron is high. The shape of an orbital is determined by the angular part of the wavefunction and depends on the quantum numbers (n, l, m_l). Orbital shapes explain atomic bonding, molecular geometry and many chemical properties.

Quantum numbers and general form

• Principal quantum number n = 1, 2, 3, ... — energy/shell.
• Azimuthal (angular) quantum number l = 0, 1, ..., n-1 — orbital type/shape: l = 0 (s), 1 (p), 2 (d), 3 (f).
• Magnetic quantum number m_l = -l, ..., 0, ..., +l — orientation (2l+1 values).

The general hydrogen-like orbital wavefunction is written as:
ψ_{n,l,m}(r, θ, φ) = R_{n,l}(r) · Y_{l}^{m}(θ, φ)

R_{n,l}(r) is the radial part (depends on r, n, l). Y_{l}^{m}(θ, φ) are spherical harmonics (determine the angular shape).

Nodes

• Total number of nodes = n − 1.
• Angular nodes = l (planes or cones where ψ = 0, e.g., p orbitals have 1 nodal plane).
• Radial nodes = n − l − 1 (spherical shells where radial part changes sign).

Shapes of common orbitals

• s-orbitals (l = 0): Spherically symmetric about the nucleus. Probability depends only on r. 1s is a single lobe with maximum at r > 0; higher ns have spherical radial nodes (shells).
• p-orbitals (l = 1): Dumbbell-shaped with two lobes of opposite phase separated by a nodal plane through the nucleus. Three orientations: p_x, p_y, p_z (m_l = −1, 0, +1).
• d-orbitals (l = 2): More complex: four of them have cloverleaf shapes (d_xy, d_yz, d_zx, d_x2−y2) and one (d_z2) has a donut (toroid) around the z-axis plus two lobes on z-axis. They have 2 angular nodes.
• f-orbitals (l = 3): Even more complex multi-lobed shapes used mainly in lanthanides and actinides.

Probability and radial distribution

Electron probability density at a point: |ψ|^2. Radial distribution function (probability of finding electron between r and r+dr): P(r) = 4π r^2 |R_{n,l}(r)|^2. Peaks in P(r) correspond to most probable radii (shells).

Why shapes matter (chemical consequences)

• Directional character of p and d orbitals explains directional bonding (e.g., π-bonds from sidewise overlap of p orbitals).
• Hybridization (sp, sp2, sp3) uses combinations of s and p orbitals to produce directional bonds (e.g., tetrahedral CH4 uses sp3 hybrids).
• d-orbitals are central to transition-metal chemistry: bonding, magnetism and color (crystal-field splitting depends on d-orbital orientations).

Simple hydrogenic examples (functional forms, up to constant normalization)

• 1s: R_{1,0}(r) ∝ e^{−r/a_0} (spherical).
• 2s: R_{2,0}(r) ∝ (2 − r/a_0) e^{−r/(2a_0)} (has one radial node where 2 − r/a_0 = 0).
• 2p angular part: Y_{1}^{m}(θ,φ) ⇒ p lobes with a nodal plane; radial part R_{2,1}(r) ∝ r e^{−r/(2a_0)}.

Summary of counting rules

• Number of orbitals in shell n = n^2.
• Maximum electrons in shell n = 2n^2.
• Magnetic orientations per subshell l = 2l + 1.

Useful classroom remarks: Emphasize that orbitals are probability distributions (not classical orbits). The shapes are solutions of the Schrödinger equation for hydrogen-like atoms and remain a good qualitative guide for multi-electron atoms.

Fig 13 — Educational Diagram: Electronic configuration

What is electronic configuration? Electronic configuration is the arrangement of electrons in the orbitals of an atom. It tells how electrons are distributed among different shells (energy levels), subshells (s, p, d, f) and orbitals according to quantum mechanics and governs chemical behavior.

Quantum numbers (brief):

• Principal quantum number n = 1,2,3,... (shell, energy, size)
• Azimuthal (angular) quantum number l = 0,...,n-1 (subshell: 0→s, 1→p, 2→d, 3→f)
• Magnetic quantum number m_l = −l,...,0,...,+l (specific orbital)
• Spin quantum number m_s = +1/2 or −1/2 (electron spin)

Rules for filling electrons:

• Aufbau principle: Electrons occupy orbitals of lowest energy first. Practically use (n + l) rule (Madelung rule); if (n + l) equal, lower n fills first.
• Pauli exclusion principle: No two electrons in an atom can have the same set of four quantum numbers — an orbital holds at most two electrons with opposite spins.
• Hund's rule: For degenerate orbitals (same energy), electrons occupy them singly with parallel spins before pairing.

Notation: Write shells/subshells with electron counts: e.g. 1s², 2s²2p⁶, etc. Use noble-gas shorthand: e.g. Fe = [Ar] 4s² 3d⁶.

Common exceptions (brief): Some transition metals show anomalous configurations for extra stability of half-filled or fully filled d subshells. Examples: Cr: [Ar] 4s¹ 3d⁵ (not 4s²3d⁴), Cu: [Ar] 4s¹ 3d¹⁰.

Why it matters: Electronic configuration explains periodic properties (atomic size, ionization energy, valency), bonding, magnetism (unpaired electrons), color and spectra (electronic transitions) and chemical reactivity.

Fig 14 — Educational Diagram: Stability concepts and exceptions (basic)

Basic idea of stability
An atom (or ion) is especially stable when its electrons occupy energetically favourable arrangements. In simple terms, completely filled shells/subshells (noble‑gas configuration) and half‑filled subshells often give extra stability. Stability rules used in Class 11 are the Aufbau principle, Pauli exclusion principle and Hund's rule, together with the (n + l) rule for orbital filling order.

Why some arrangements are more stable
1. Completely filled subshells (e.g. p6, d10) are symmetric and have lower energy because there is maximum pairing and exchange stabilization.
2. Half‑filled subshells (e.g. p3, d5) are also relatively stable because of symmetry and maximum parallel spins which increases exchange energy and lowers the total energy.
3. For transition elements the energy difference between the (n)s and (n−1)d orbitals is very small. Small energy differences allow one electron to move from s to d to produce a half‑filled or filled d subshell, giving an anomalous (more stable) configuration.

Common exceptions (basic list)
Because of the small energy gap between ns and (n−1)d orbitals, some elements do not follow the simple Aufbau filling. Typical examples (ground state electronic configurations):
- Chromium (Cr): expected [Ar] 3d4 4s2 → actual [Ar] 3d5 4s1 (half‑filled d5 is more stable)
- Copper (Cu): expected [Ar] 3d9 4s2 → actual [Ar] 3d10 4s1 (filled d10 is more stable)
- Molybdenum (Mo): [Kr] 4d5 5s1 (d5 extra stability)
- Silver (Ag): [Kr] 4d10 5s1 (d10 extra stability)
- Palladium (Pd): expected [Kr] 4d8 5s2 → actual [Kr] 4d10 5s0 (complete d10 preferred)

Why these exceptions occur (qualitative)
- Exchange energy: parallel spins in different orbitals lower the energy (important for half‑filled subshells).
- Electron‑electron repulsion and orbital penetration: moving an electron from s to d can reduce repulsion or increase stability because d orbitals combine better with the existing occupation.
- Small differences in orbital energies (ns vs (n−1)d) make such rearrangements energetically favourable.

Ions and removal of electrons
For transition metals, although neutral atoms may have ns occupied, on ionization the ns electron(s) are lost first and then d electrons, because in ions (n−1)d lies lower in energy than ns. Example: Fe: ground state [Ar] 3d6 4s2, on forming Fe2+ → configuration [Ar] 3d6 (both 4s electrons removed).

Practical/CBSE focus
Students should be able to: state and apply Aufbau, Pauli, Hund rules; use (n + l) rule; write ground‑state configurations for first‑row transition metals and recognize the common Cr/Cu type exceptions; explain qualitatively why half‑filled and fully filled subshells are extra stable.

Fig 15 — Educational Diagram: Isotopes, isobars and isotones

Overview & notation
Atoms (nuclides) are specified by their atomic number Z (number of protons) and mass number A (total number of protons + neutrons). The conventional notation is ^A_ZX, where X is the chemical symbol, A is the mass number and Z is the atomic number. The neutron number N = A − Z.

Isotopes
Definition: Isotopes are nuclides of the same element (same Z) but with different mass numbers A (different N). They have identical chemical properties (same electron configuration) but may show different physical or nuclear properties (e.g., different stability, radioactive behaviour).

Examples and key points:
- Hydrogen isotopes: ¹₁H (protium), ²₁H (deuterium), ³₁H (tritium).
- Carbon: ¹²₆C (stable), ¹⁴₆C (radioactive, used in radiocarbon dating).
- Chlorine: ³⁵₁₇Cl and ³⁷₁₇Cl (gives atomic mass ≈35.45).

Isobars
Definition: Isobars are nuclides with the same mass number A but different atomic numbers Z (different elements). Isobars can interconvert through beta decay (a neutron converts to a proton or vice versa), because A remains constant while Z changes.

Examples and key points:
- A = 40 isobars: ⁴⁰₂₀Ca, ⁴⁰₁₉K, ⁴⁰₁₈Ar.
- ¹⁴₆C and ¹⁴₇N are isobars (both A = 14).

Isotones
Definition: Isotones are nuclides that have the same neutron number N but different Z and A. Isotones often exhibit related nuclear structure effects because the neutron configuration is the same.

Examples and key points:
- N = 20 isotones: ³⁶₁₆S, ³⁷₁₇Cl, ³⁸₁₈Ar, ³⁹₁₉K, ⁴⁰₂₀Ca (all have 20 neutrons).
- Isotones are useful when studying nuclear shell effects tied to neutron number.

Practical consequences and real-life applications
- Isotopes in dating and tracing: ¹⁴C radiocarbon dating, stable isotope tracers in metabolism (deuterium).
- Medical and industrial isotopes: ^99mTc for imaging, ¹³¹I for thyroid therapy, deuterium in NMR and heavy water moderators in reactors.
- Isobars and beta decay: many beta-decay chains move along constant A (isobaric chains) until a stable isobar is reached.

Short worked example (average atomic mass)
Chlorine has two main isotopes with approximate natural abundances 75% ³⁵Cl and 25% ³⁷Cl. Average atomic mass ≈ 0.75×35 + 0.25×37 = 35.5 u (actual isotopic abundances give the standard atomic weight ≈ 35.45 u).

Summary comparison
- Isotopes: same Z, different A (vertical column on a nuclide chart).
- Isobars: same A, different Z (diagonal/vertical slice at fixed A across elements).
- Isotones: same N, different Z and A (horizontal row on a nuclide chart).

Fig 16 — Educational Diagram: Important formulae and physical constants

Overview: This topic summarizes the numerical values of physical constants and the key formulae used in the Class 11 chapter "Structure of Atom". These constants and relations connect wave and particle properties of matter and light (quantum ideas), allow calculation of atomic energy levels (Bohr model), and set limits on measurement (uncertainty principle).

Fundamental relations (conceptual)

• Energies and photons: a photon of frequency ν has energy E = hν and wavelength λ = c/ν.
• Wave–particle duality: particles (electrons) have wavelength λ = h/p (de Broglie relation).
• Atomic (Bohr) model for hydrogen-like atoms gives quantized radii, energies and velocities for electrons in allowed orbits.
• Quantum mechanics (Schrödinger) replaces definite orbits with wavefunctions ψ whose squared magnitude |ψ|^2 gives probability densities; radial probability plots show most probable electron distances.
• Measurement limit: Heisenberg uncertainty Δx·Δp ≥ ħ/2 limits simultaneous knowledge of position and momentum.

Important physical constants (useful values)

• Speed of light, c = 2.99792458 × 10^8 m·s⁻¹
• Planck's constant, h = 6.62607015 × 10⁻³⁴ J·s
• Reduced Planck constant, ħ = h/(2π) = 1.054571817 × 10⁻³⁴ J·s
• Electron mass, m_e = 9.10938356 × 10⁻³¹ kg
• Elementary charge, e = 1.602176634 × 10⁻¹⁹ C
• Bohr radius (a₀) = 5.29177210903 × 10⁻¹¹ m (most probable distance for ground-state H)
• Rydberg constant (for hydrogen), R_∞ = 1.0973731568508 × 10⁷ m⁻¹
• Rydberg energy (ionization energy of H ground state), R_y ≈ 13.605693 eV
• Permittivity of free space, ε₀ = 8.8541878128 × 10⁻¹² F·m⁻¹
• Avogadro number, N_A = 6.02214076 × 10²³ mol⁻¹
• Atomic mass unit, 1 u = 1.66053906660 × 10⁻²⁷ kg
• Fine-structure constant, α ≈ 1/137.035999 (dimensionless)

Context and usage

• Use E = hc/λ when converting photon wavelength to energy (chemistry: spectroscopy, photochemical reactions).
• Bohr expressions let you compute the radius r_n and energy E_n for H-like ions (useful for line spectra and ionization energies).
• Photoelectric equation (hf = φ + K_max) explains threshold frequency and stopping potential—applied in solar cells and photodetectors.
• de Broglie wavelength explains why electrons behave as waves in electron microscopes and diffraction experiments.

Short derivations (Bohr)

• Balance centripetal force and Coulomb attraction and quantize angular momentum (m_e v r = nħ) → r_n = a₀ n²/Z and E_n = -13.6 eV · Z²/n².
• Energy differences give emitted/absorbed photon: ΔE = E_final - E_initial = hν → 1/λ = R_∞ Z² (1/n₁² - 1/n₂²).

Notes for students: Memorize the constants and the core formulae (listed separately). Practice problems: compute wavelength of photon from n=3→n=2 transition in H (Balmer), compute de Broglie wavelength of an electron accelerated through a given potential, and use photoelectric eqn to find work function or stopping potential.

Fig 17 — Educational Diagram: Experimental evidence and landmark experiments

Overview

This topic covers the classic experiments that revealed subatomic particles and the internal structure of the atom. The experiments provide direct experimental evidence for the electron, proton, nucleus and for quantized energy levels. Together they shaped the modern atomic model (Thomson → Rutherford → Bohr).

1. Cathode Ray Experiments (J. J. Thomson)

• Setup: High‑vacuum tube with electrodes; application of electric and magnetic fields produces a beam (cathode ray) from the cathode to the anode.
• Observations: Rays are deflected by electric and magnetic fields; deflection direction independent of cathode material and identical for different gases (same particle everywhere).
• Conclusions: Existence of a universal negatively charged particle (electron). Measurement of charge to mass ratio e/m by balancing E and B fields and by measuring curvature in a magnetic field.

2. Canal Rays (Eugen Goldstein)

• Observation: Positive ions (canal rays) travel opposite to cathode rays in discharge tubes with perforated cathodes.
• Conclusion: Existence of positive particles (later identified as protons for hydrogen and positive ions of other gases).

3. Millikan Oil‑Drop Experiment

• Setup: Small charged oil drops suspended in an electric field; measurement of field required to hold droplets stationary and their terminal velocity.
• Observations & conclusions: Charges measured on many drops are integer multiples of a smallest value, identified as the elementary charge e. This gives the magnitude of electron charge.

4. Rutherford Alpha‑Scattering Experiment

• Setup: Thin gold foil bombarded with alpha particles; detection of scattering angles (most passed through, some deflected large angles, a few backscattered).
• Observations: Most alpha particles pass through with little deflection; a small fraction deflected at large angles or bounced back.
• Conclusions: Atom is mostly empty space with a very small, dense, positively charged nucleus (nuclear model). From scattering data one can estimate nuclear charge and approximate nuclear size.

5. Photoelectric Effect and Atomic Spectra

• Photoelectric effect: Light ejects electrons from metal surfaces only if frequency exceeds threshold; kinetic energy of emitted electrons depends on frequency, not intensity. This supports quantization of light (photons) and gives work function & Einstein's photoelectric equation.
• Emission spectra: Heated/energised atoms emit light in discrete lines (e.g., hydrogen Balmer series). Discrete lines imply quantized energy levels in atoms — a cornerstone for Bohr's model.

Logical flow

Thomson discovered the electron and measured e/m → suggested a diffuse positive charge with embedded electrons (plum‑pudding view). Rutherford’s scattering disproved that picture and established a tiny positive nucleus. Observed atomic spectra and photoelectric effect required quantized energy changes and photons — leading to Bohr’s model and later quantum mechanics.

Important experimental considerations

• Careful vacuum, precise field calibration and detector placement were crucial in all experiments.
• Corrections (e.g., air buoyancy in Millikan, multiple scattering in Rutherford) are needed for high accuracy.