Chapter 4 — Structure Of The Atom

What is an atom?
The atom is the smallest particle of an element that can exist either alone or in combination and still retain the chemical properties of the element. Atoms are the building blocks of matter.

Historical development (brief):

• Dalton's model (early 1800s): Atoms are indivisible solid spheres; different elements have different atoms.
• Thomson's model (Plum‑pudding) (1897): Electrons embedded in a positively charged sphere.
• Rutherford's model (1911): Most alpha particles passed through gold foil; concluded that most of the atom is empty space with a tiny, dense, positively charged nucleus at the center containing most mass.
• Bohr's model (1913): Electrons move in fixed energy shells (orbits) around the nucleus; each shell has a discrete energy.
• Modern view / Quantum model: Electrons occupy regions of probability (orbitals) rather than fixed paths; nucleus contains protons and neutrons.

Constituents of an atom:

• Protons (p+): Positive charge (+1), relative mass ≈ 1 atomic mass unit (1 u). Number of protons = atomic number (Z).
• Neutrons (n0): No charge (0), relative mass ≈ 1 u. Protons + neutrons = mass number (A).
• Electrons (e−): Negative charge (−1), mass ≈ 1/1836 u. In a neutral atom, number of electrons = number of protons (Z).

Nucleus and electron distribution:
The nucleus (protons + neutrons) is extremely small compared to the whole atom but contains nearly all the mass. Electrons occupy shells (also called energy levels) around the nucleus. Shells are labeled K, L, M, N... corresponding to principal quantum numbers n = 1, 2, 3, 4....

Maximum electrons in a shell: The maximum number of electrons that can occupy the nth shell is given by 2n². So K (n=1) can hold 2, L (n=2) 8, M (n=3) 18, etc.

Atomic number and mass number:

• Atomic number Z = number of protons. It defines the element.
• Mass number A = number of protons + number of neutrons (A = Z + N).
• Isotopes are atoms of the same element (same Z) with different numbers of neutrons (different A). Example: 1H (protium), 2H (deuterium), 3H (tritium).

Key ideas to remember:

• Atoms are electrically neutral overall when electrons = protons.
• Nucleus is tiny but very massive and positively charged.
• Electrons determine chemical behavior (valency arises from outermost electrons).
• Bohr model is useful for simple shell diagrams; quantum model explains electron clouds and spectra more accurately.

Simple diagram suggestions: a labeled nucleus with p+ and n0, and shells showing electron counts (K, L, M), plus a small inset comparing sizes (nucleus vs atom).

Overview: The atom was once thought indivisible. Experiments from the late 19th and early 20th centuries showed atoms contain smaller charged particles: electrons, protons and neutrons. These discoveries changed the model of the atom from a solid sphere to a nucleus with electrons around it.

Discovery of Electrons (J.J. Thomson, 1897): Using a cathode ray tube, Thomson observed rays that (a) travel from negative to positive electrode, (b) produce fluorescence on glass, and (c) are deflected by electric and magnetic fields toward the positive plate. From experiments he measured charge-to-mass ratio (e/m) and concluded the rays are negatively charged particles—electrons—much lighter than atoms. This led to the "plum pudding" model (electrons embedded in a positive matrix).

Key observations about cathode rays:

• Deflected by E and B fields (showing they carry charge).
• Independent of cathode material (same particle in different gases and metals).
• Cause heating and fluorescence when they strike materials.

Discovery of Protons / Positive Rays: Experiments on discharge tubes with perforated cathodes showed positive rays moving opposite to cathode rays (canal rays). Work by E. Goldstein and later analyses showed these were positive ions. Rutherford’s experiments on atomic structure and later work identified the hydrogen ion (proton) as a fundamental positively charged particle (charge +e) concentrated in the nucleus.

Rutherford's Alpha Scattering Experiment (1911): Alpha particles were fired at thin gold foil. Most passed through, but some were deflected at large angles and a few even bounced back. Rutherford concluded that most of the atom is empty space, with a tiny, dense, positively charged nucleus at the center (containing protons and most of the mass) and electrons orbiting around it. This replaced the plum pudding model.

Discovery of Neutrons (James Chadwick, 1932): Bombarding beryllium with alpha particles produced a neutral radiation that could knock protons out of paraffin. The neutral particles had mass close to protons but no charge; Chadwick identified them as neutrons. Neutrons explained the extra mass in nuclei (why atomic mass > number of protons) and nuclear stability.

Summary of particle properties:

• Electron: charge = −1.602×10⁻¹⁹ C, mass ≈ 9.11×10⁻³¹ kg.
• Proton: charge = +1.602×10⁻¹⁹ C, mass ≈ 1.67×10⁻²⁷ kg.
• Neutron: charge = 0, mass ≈ 1.675×10⁻²⁷ kg.

Why these discoveries matter: They explain chemical behavior (electrons determine bonding), isotopes (same Z, different neutron number), atomic mass (sum of protons + neutrons), and enable technologies such as electron microscopes, particle accelerators, nuclear reactors and medical imaging.

Atomic Models

The study of atomic models traces how scientists pictured the atom as experimental evidence improved. Each model kept earlier successes and tried to resolve new experimental results. The main models studied in Class 9 are Dalton, Thomson, Rutherford and Bohr, followed by a brief note on the quantum (modern) view.

1. Dalton's Atomic Model (Early 19th century)

• John Dalton proposed that matter is made of tiny, indivisible particles called atoms.
• Postulates: Atoms of an element are identical in mass and properties; different elements have different atoms; atoms combine in simple whole-number ratios to form compounds; atoms are neither created nor destroyed in chemical reactions.
• Success: Explained laws of chemical combination (law of conservation of mass, law of definite proportions).
• Limitation: Atoms are not indivisible (subatomic particles exist) and atoms of an element can have isotopes (different masses).

2. Thomson's Model (Plum Pudding Model, 1897)

• J. J. Thomson discovered the electron (a negatively charged subatomic particle) using cathode ray experiments.
• Model idea: Atom is a sphere of positive charge with electrons embedded like plums in a pudding, to make the atom electrically neutral.
• Success: Introduced internal structure of atom and explained neutrality.
• Limitation: Could not explain results of scattering experiments which showed a concentrated positive charge.

3. Rutherford's Model (1911)

• Ernest Rutherford performed the gold foil (alpha scattering) experiment.
• Observations: Most alpha particles passed through gold foil undeflected; a few were deflected at large angles; a very small fraction even bounced back.
• Conclusions / Model: Atom has a tiny, dense, positively charged nucleus containing most of the mass; electrons revolve around the nucleus in mostly empty space.
• Success: Explained scattering results and introduced the nuclear atom and nucleus.
• Limitation: Classical electrodynamics predicts orbiting electrons should radiate energy and spiral into nucleus; Rutherford model could not explain atomic spectra (discrete lines).

4. Bohr's Model (1913)

• Niels Bohr combined Rutherford’s nuclear atom with early quantum ideas to explain hydrogen spectra.
• Main postulates:
  • Electrons orbit the nucleus in certain allowed circular orbits without radiating energy (stationary orbits).
  • Each allowed orbit corresponds to a fixed energy level labeled by quantum number n = 1, 2, 3, ...
  • Electrons emit or absorb energy as photons when they jump between allowed orbits. The photon energy equals the energy difference between levels: ΔE = hf.
• Success: Explained the discrete spectral lines of hydrogen (Balmer series etc.) and gave formulas for orbital radius and energy of hydrogen-like atoms.
• Limitation: Works well only for hydrogen and hydrogen-like (one-electron) ions; cannot fully explain multi-electron atoms, fine structure, or electron probability distributions.

5. Quantum Mechanical (Modern) View — brief note

• Electrons are described by wavefunctions; position is given by probability distributions (orbitals) rather than definite circular orbits.
• Explains fine details of spectra and chemical bonding. Full development requires quantum mechanics (Schrödinger equation) taught later.

Important Experiments (short)

• Cathode ray tube (Thomson): showed electrons and measured charge-to-mass ratio.
• Gold foil (Rutherford): led to discovery of nucleus by alpha-particle scattering.
• Spectral analysis (Balmer/Bohr): discrete emission lines of hydrogen that Bohr explained.

Summary

Atomic models evolved from indivisible solid spheres (Dalton) to internal structure with electrons (Thomson), to a small positive nucleus with orbiting electrons (Rutherford), to quantized energy levels for electrons (Bohr), and finally to a probabilistic wave description (quantum mechanics). Each model preserved earlier successes while resolving newer experimental results.

Setup: A source of alpha (α) particles (positive, helium nuclei) was placed in a lead box with a small hole so a narrow beam of α-particles struck a very thin gold foil. Around the foil was a movable fluorescent screen (or detector) to observe where α-particles hit. The entire arrangement was placed in a vacuum chamber to avoid collisions with air.

Observations:

• Most α-particles passed straight through the gold foil with little or no deflection.
• Some α-particles were deflected by small angles.
• A very few (roughly 1 in 20,000) were deflected by large angles; some even bounced nearly straight back toward the source.

Conclusions and interpretation:

• Since most α-particles passed through, atoms are mostly empty space.
• Large-angle deflections imply the presence of a very small, very massive, and positively charged centre in the atom — called the nucleus — capable of exerting a strong electrostatic force over a short distance.
• Nearly all the mass of an atom is concentrated in this nucleus; electrons occupy the remaining space around it (leading to the nuclear model of the atom).

Why the results required a new model: The Thomson 'plum-pudding' model predicted only tiny deflections because positive charge was thought to be spread out; it could not explain large-angle scattering. Rutherford's results required a concentrated positive charge (nucleus).

Qualitative physics behind deflection: An α-particle (charge +2e) approaching a nucleus of charge +Ze experiences a strong electrostatic repulsive force F ≈ k(2e)(Ze)/r² (Coulomb's law). If the α-particle passes far from the nucleus, the force is small and the path is almost straight; if it comes very close, the repulsive force can substantially alter its direction or reverse it.

Historical importance: Rutherford's experiment (1911) established the nuclear model, which led directly to later discoveries: quantized electron orbits (Bohr), the neutron, and modern nuclear physics.

Notes for class 9 level: Emphasize the three key experimental facts (most pass through, some deflected slightly, very few deflected strongly) and the three direct conclusions (atom mostly empty, tiny dense positive nucleus, electrons outside nucleus).

Overview: An atom consists of a tiny central nucleus (containing protons and neutrons) surrounded by electrons in the space around it. The nucleus carries almost all the mass and a positive charge; electrons carry negative charge and occupy shells or orbitals around the nucleus.

Nucleus:

• Made of protons (charge +e) and neutrons (neutral). Proton number = atomic number (Z). Proton + neutron = nucleon; total nucleons = mass number (A).
• Very small radius (~10^-15 m) but very dense. Nuclear forces (strong force) hold nucleons together, overcoming proton–proton repulsion.
• Nuclei determine isotopes: same Z but different number of neutrons (e.g., 12C, 14C).

Electrons and atomic structure:

• Electrons are negatively charged particles with very small mass compared to protons/neutrons. They occupy discrete energy levels or shells around the nucleus (K, L, M...; maximum electrons 2, 8, 18...).
• Electrons determine chemical behaviour: valence electrons (outer shell) participate in bonding, ion formation and conductivity.
• Models: Plum‑pudding (J.J. Thomson) was replaced by Rutherford’s nuclear model after alpha-scattering experiments. Bohr refined Rutherford for hydrogen-like atoms by introducing quantised orbits/energy levels (useful at Class 9 level to explain spectral lines).

Key concepts:

• Atomic number Z = number of protons = number of electrons in a neutral atom.
• Mass number A = number of protons + neutrons. Neutrons = A − Z.
• Atoms are electrically neutral overall; ions form when electrons are lost or gained.

Experimental evidence (brief): Rutherford’s gold‑foil experiment showed most alpha particles passed through thin foil (atom mostly empty space) while a few were strongly deflected or bounced back (tiny, dense, positively charged nucleus).

Why this matters: The nucleus explains radioactivity and nuclear energy (changes in nucleus). Electrons and their arrangement explain chemistry, electricity, light emission (atomic spectra) and the behaviour of matter in daily life.

Atomic number (Z): The atomic number is the number of protons in the nucleus of an atom. It uniquely identifies an element and determines its chemical properties. In a neutral atom, the number of electrons = Z.

Mass number (A): The mass number is the total number of nucleons (protons + neutrons) in the nucleus. A = Z + N, where N is the number of neutrons. Mass number is a whole number used to distinguish isotopes of an element.

Symbolic notation: An isotope is written as ^A_ZX (for example, ¹⁴₆C means Z = 6, A = 14).

Electrons and ions: For a neutral atom: electrons = Z. For an ion: electrons = Z − (positive charge) or electrons = Z + (magnitude of negative charge). Example: Ca²⁺ (Z = 20) has 18 electrons.

Isotopes and isobars: Isotopes have the same Z but different A (e.g., ¹²₆C and ¹⁴₆C). Isobars have the same A but different Z (e.g., ⁴⁰₁₈Ar and ⁴⁰₂₀Ca).

Relative atomic mass (Ar): The atomic mass shown in the periodic table is a weighted average of isotopic masses: Ar = Σ(f_i × m_i), where f_i is fractional abundance and m_i is the mass of isotope i. This explains why Ar is often not a whole number.

Why it matters: Z determines chemical behavior (bonding, reactivity); A affects mass, nuclear stability and radioactive properties (used in dating, medicine, energy).

Isotopes

Isotopes are atoms of the same element (same atomic number Z, same number of protons) that have different numbers of neutrons and therefore different mass numbers (A). They have identical chemical properties because chemical behaviour is determined by electrons/protons, but may differ in physical properties (mass, density) and nuclear stability (some are radioactive).

Notation: use the nuclear notation ^A_ZX where X is the element symbol, A is the mass number (protons + neutrons) and Z is the atomic number (protons). Example: ²₁H (deuterium), ³₁H (tritium).

Isobars

Isobars are atoms of different elements that have the same mass number A but different atomic numbers Z (different numbers of protons and neutrons). Because they are different elements, their chemical properties differ even though their total nucleon count is the same. Example: ¹⁴C (carbon, Z=6) and ¹⁴N (nitrogen, Z=7) are isobars (A = 14).

Key comparisons

• Isotopes: same Z, different N (neutrons). Example vertical family in a nuclide chart.
• Isobars: same A, different Z and N. Example lie along a constant-A line in a nuclide chart.
• Chemical behaviour: isotopes (nearly same), isobars (different).

Important notes

• Mass number A = Z + N (protons + neutrons).
• Relative atomic mass (Ar) of an element is the weighted average of masses of its naturally occurring isotopes, weighted by their abundances.
• Some isotopes are stable; others are radioactive (radioisotopes) and have practical uses in medicine, archaeology and industry.

What is electronic configuration? Electronic configuration is the arrangement of electrons of an atom in different energy levels (shells) and subshells (orbitals). It tells how many electrons occupy each shell (K, L, M, ...) and each type of orbital (s, p, d, f).

Shells and energy levels: Shells are labelled by the principal quantum number n = 1, 2, 3, ... which correspond to K, L, M, ... shells. Each shell has a maximum number of electrons given by the formula 2n². So:

• n = 1 (K): max 2 electrons
• n = 2 (L): max 8 electrons
• n = 3 (M): max 18 electrons (for class 9 we often use up to 8 for first 20 elements)

Subshells and orbitals: Each shell contains subshells labelled s, p, d, f. The s-subshell can hold 2 electrons, p can hold 6, d can hold 10, etc. A compact notation uses n followed by the subshell and a superscript showing the number of electrons, for example 1s² or 2p⁶.

Rules used to write electronic configurations (basic):

• Maximum electrons in a shell: 2n² (useful for K, L, M shells).
• Aufbau principle: Electrons fill lower energy orbitals first (order: 1s > 2s > 2p > 3s > 3p > 4s > 3d ... — for class 9, focus up to 3p/4s as needed).
• Pauli exclusion principle: An orbital can hold at most two electrons with opposite spins.
• Hund’s rule (basic): For degenerate orbitals (like 2p), electrons occupy separate orbitals with parallel spins before pairing.

How to write the electronic configuration (stepwise):

1. Find the atomic number (number of electrons for neutral atom).
2. Fill electrons into orbitals following Aufbau order and the maximum capacities.
3. Write the final configuration using nℓ^x notation.

Why it matters — Electronic configuration explains periodic properties (valency, reactivity, atomic size, ion formation), bonding, conductivity and spectral emissions because chemical behavior depends on valence (outermost) electrons.

Simple examples (in notation): 1s² means K shell filled; 2s²2p⁶ means L shell full (8 electrons).

What is valency?
Valency is the combining capacity of an element — the number of electrons an atom can lose, gain or share to attain a stable electron configuration (usually the noble gas configuration). For most elements in Class 9 (main-group elements) stability is reached by completing an octet (8 electrons) in the outermost shell; hydrogen is an exception and reaches stability with 2 electrons (duet).

What are ions?
An ion is a charged particle formed when an atom loses or gains electrons. A positively charged ion (cation) forms when an atom loses electrons. A negatively charged ion (anion) forms when an atom gains electrons.

How to determine valency (practical rules):

• Find the number of valence (outermost shell) electrons from the electronic configuration.
• Elements with 1, 2 or 3 valence electrons (typically metals in groups 1–3) tend to lose those electrons: valency = number of valence electrons (e.g., Na has 1 → valency 1; Al has 3 → valency 3).
• Elements with 5, 6 or 7 valence electrons (typically non-metals in groups 15–17) tend to gain (8 − valence electrons): valency = 8 − number of valence electrons (e.g., O has 6 → valency 2; Cl has 7 → valency 1).
• Elements with 4 valence electrons (e.g., C, Si) can either gain or lose 4 electrons; their valency is usually taken as 4.
• Hydrogen and helium follow the duet rule: H has valency 1 (seeks 1 more electron to complete 2).

Formation of ions (examples with electronic configuration)

• Na (atomic number 11): electronic configuration 2,8,1 → loses 1 electron → Na+ (2,8). Reaction: Na → Na+ + e−
• Cl (atomic number 17): electronic configuration 2,8,7 → gains 1 electron → Cl− (2,8,8). Reaction: Cl + e− → Cl−
• Mg (atomic number 12): 2,8,2 → loses 2 electrons → Mg2+ (2,8). Reaction: Mg → Mg2+ + 2e−
• O (atomic number 8): 2,6 → gains 2 electrons → O2− (2,8). Reaction: O + 2e− → O2−

How ionic compounds form
Ionic compounds form by transfer of electrons from metal atoms (which become cations) to non-metal atoms (which become anions). The charges on ions combine to give a neutral compound. Use the criss-cross method to write formulas: the magnitude of the charge on one ion becomes the subscript of the other ion (simplify if possible).

Example: Formation of NaCl
Na → Na+ + e− (metal loses 1)
Cl + e− → Cl− (non-metal gains 1)
Na+ and Cl− combine to form NaCl (one-to-one). This compound is common table salt used in food and industry.

Important points to remember

• Valency is about electrons lost or gained to reach a stable configuration.
• Cations are positive (lost electrons), anions are negative (gained electrons).
• Sum of positive and negative charges in an ionic compound = 0 (electroneutrality).
• Use electronic configurations or group number (in periodic table) to predict valency for main-group elements.

Atomic Mass Unit (amu or u)

An atomic mass unit (symbol: u or amu) is a convenient unit to express the masses of atoms and subatomic particles. It is defined as one twelfth (1/12) of the mass of an atom of carbon-12 (12C) in its ground state:

1 u = (1/12) × mass of 12C atom

Numerical value (CODATA): 1 u = 1.66053906660 × 10⁻²⁷ kg. In CBSE contexts this is often rounded to 1.66 × 10⁻²⁷ kg.

Why use the amu?

• Atomic and nuclear masses are extremely small in kg; using u keeps numerical values convenient (e.g., carbon = 12 u, oxygen ≈ 16 u).
• Based on a stable reference (carbon-12), it provides a standard for comparing masses of atoms and particles.

Relative Atomic Mass (Ar) or Atomic Mass

Relative atomic mass (often written as Ar or simply the atomic mass on the periodic table) is the weighted average mass of the atoms of an element compared to 1 u. It takes into account the masses of the isotopes of the element and their natural abundances.

Ar has no units because it is a ratio: the average mass of an atom of the element divided by 1 u.

Isotopes and the need for an average

Many elements occur as mixtures of isotopes (atoms with the same proton number but different neutron numbers). Each isotope has a slightly different mass. The relative atomic mass is the abundance-weighted mean of isotopic masses.

How to calculate Relative Atomic Mass (concept)

Convert percentage abundances to fractions (divide by 100). Multiply each isotope's mass (in u) by its fractional abundance and add the results:

Ar = Σ (isotopic mass × fractional abundance)

Comparison with Mass Number (A)

Mass number (A) is the total number of protons and neutrons in a particular nuclide (an integer). Relative atomic mass (Ar) is usually non-integer and is an average over isotopes.

Practical points

• Mass of proton ≈ 1.0073 u; neutron ≈ 1.0087 u; electron ≈ 5.5 × 10⁻⁴ u (usually negligible in atomic mass calculations).
• Values given in the periodic table are relative atomic masses (averages), e.g., Cl ≈ 35.45 u, Na ≈ 22.99 u.

Worked example (summary inside explanation)

For chlorine: isotopes 35Cl (mass ≈ 34.9689 u, abundance 75.77%) and 37Cl (mass ≈ 36.9659 u, abundance 24.23%).

Ar(Cl) = 34.9689 × 0.7577 + 36.9659 × 0.2423 ≈ 35.453 u.

Significance: Relative atomic mass lets chemists compare masses of different atoms in a simple, practical way and use these values in stoichiometric calculations.

Overview: An atom is mostly empty space. The atom's size is determined by the extent of its electron cloud and is about 10^-10 metre (1 Å). The nucleus, containing protons and neutrons, is extremely small compared to the atom, with a size of about 10^-15 metre (1 fm).

Typical values and orders of magnitude:

• Atomic radius (order): ~10^-10 m (0.1–2 Å depending on element). The Bohr radius a₀ = 5.29×10^-11 m is the typical radius of the hydrogen atom in its ground state.
• Nuclear radius (order): ~10^-15 m (1–10 fm). A single proton or neutron has a size ≈1 fm (0.84–0.88 fm for the proton charge radius).

Volume comparison: Because volume scales as radius³, the nucleus occupies about 10^-15 of the atom's volume (typical estimate: 10^-15–10^-14), i.e., almost all the atom's mass is concentrated in a tiny central region.

How sizes are measured: Rutherford alpha-particle scattering showed that most alpha particles pass through thin foil but some scatter at large angles, implying a small, dense nucleus. Electron scattering and high-energy probes (e.g., electron and proton scattering, muonic atom spectroscopy) give detailed charge distributions and measure nuclear radii.

Why nucleus is so small: Strong nuclear force binds nucleons tightly within a short range (~1–2 fm). Electrons are bound by the electromagnetic force and occupy a much larger quantum-mechanical cloud around the nucleus.

Simple physical picture: If an atom were the size of a football stadium (~100 m diameter), the nucleus would be like a small marble or pea at the center — almost undetectable compared to the stadium.

Short introduction
Bohr's model (1913) explained the hydrogen atom successfully by postulating that electrons move in fixed circular orbits with quantised angular momentum. It gave correct formulas for hydrogen spectral lines but has important limitations that required a new, modern quantum view.

Limitations of Bohr's model

• Works only for hydrogen-like atoms: Bohr's equations give correct energies only for single-electron systems (H, He+). They fail for multi-electron atoms because they ignore electron–electron interactions and shielding.
• No explanation of spectral line intensities and fine structure: The model predicts wavelengths but cannot explain the relative intensities of spectral lines, or small splittings (fine structure) seen due to electron spin and relativistic effects.
• Cannot explain Zeeman and Stark effects in detail: Splitting of spectral lines in magnetic (Zeeman) or electric (Stark) fields requires electron spin and quantum mechanics beyond Bohr’s picture.
• Fixed circular orbits contradict wave nature of electrons: Experiments (diffraction, interference) show electrons have wave characteristics (de Broglie). A particle in a fixed orbit with definite position and momentum contradicts this.
• No account of Heisenberg uncertainty principle: Bohr assumes precise electron position and momentum (orbit + speed), but quantum mechanics shows Δx and Δp cannot both be known exactly.
• Fails to explain chemical bonding and atomic shapes: Bohr model gives no idea of orbitals or probability distributions needed to explain covalent bonding, molecular shapes and hybridisation.

Why a modern view was needed
The experimental evidence (spectroscopy details, electron diffraction, atomic and molecular bonding behaviour) showed that electrons behave both as particles and waves and cannot be described by fixed orbits. The modern quantum-mechanical model replaces fixed orbits with wavefunctions (ψ) and orbitals that give probability distributions for electron positions. This model explains multi-electron atoms, fine and hyperfine structures, the Zeeman/Stark effects, chemical bonding and many more phenomena.

Key ideas of the modern view

• Wave–particle duality: Electrons have wave properties (de Broglie wavelength λ = h/p).
• Wavefunction and orbitals: Schrödinger’s equation gives ψ whose square |ψ|² is the probability density (an orbital is a region of high probability). Orbitals explain shapes (s, p, d), bonding and chemical properties.
• Uncertainty principle: Position and momentum cannot both be precisely known: Δx·Δp ≥ ħ/2.
• Quantisation from boundary conditions: Energy levels arise because only certain wavefunctions (standing waves) are allowed, not because electrons orbit like planets.

Conclusion (Class 9 level)
Bohr's model was a crucial step: it introduced quantisation and explained hydrogen spectra. But its limitations (single-electron applicability, fixed orbits, lack of wave nature and uncertainty) made a deeper quantum-mechanical model necessary. The modern view replaces definite orbits by probability clouds (orbitals), explains a wide range of atomic, optical and chemical phenomena, and is the accepted model in chemistry and physics today.