Structure of Atom

2.1 Core concepts

Discovery of the electron. J.J. Thomson (1897) constructed a cathode ray discharge tube and measured the charge-to-mass ratio of the cathode-ray particles by balancing mutually perpendicular electric and magnetic fields. The result, e/m_e = 1.758820 × 10¹¹ C kg⁻¹, was independent of both the cathode material and the residual gas, proving that the negatively charged particle now called the electron is a universal constituent of all atoms. Cathode rays travel in straight lines in field-free space, originate at the cathode, and are deflected toward the positive plate in an electric field (NCERT §2.1.1–2.1.2, pp. 30–31; Figs. 2.1, 2.2). R.A. Millikan's oil-drop experiment (1906–14) measured the charge on each electron by balancing gravity against an applied electric field on charged oil droplets; the charge was always an integer multiple of e = 1.6022 × 10⁻¹⁹ C, and combining with e/m_e gave the electron mass m_e = 9.1094 × 10⁻³¹ kg (NCERT §2.1.3, p. 31).

Proton and neutron. Modified discharge tubes with perforated cathodes produced canal rays moving toward the cathode — these were positive ions whose smallest, lightest member came from hydrogen and was identified as the proton (charge +1.602 × 10⁻¹⁹ C; mass 1.672 × 10⁻²⁷ kg). The neutron was discovered by James Chadwick (1932) by bombarding beryllium with α-particles; the resulting penetrating neutral radiation consisted of particles slightly heavier than protons (m_n = 1.674 × 10⁻²⁷ kg, q = 0) (NCERT §2.1.4 + Table 2.1, p. 32–33).

Thomson model (1898). An atom is a sphere of radius ~10⁻¹⁰ m of uniformly distributed positive charge with electrons embedded like seeds in a watermelon. Total charge is zero, mass is uniformly distributed. The model explained electrical neutrality but failed later scattering tests (NCERT §2.2.1, p. 33; Fig. 2.4).

Rutherford's nuclear atom (1911). Geiger and Marsden bombarded a very thin gold foil (~100 nm) with α-particles. Most α-particles passed undeflected; a few were deflected through small angles; and roughly one in 20,000 bounced almost straight back. Rutherford concluded that (i) most of the atom is empty space, (ii) the entire positive charge and almost all the mass are concentrated in a tiny nucleus of radius ~10⁻¹⁵ m, and (iii) electrons revolve around the nucleus in orbits like planets around the Sun (NCERT §2.2.2, p. 34; Fig. 2.5).

Atomic and mass numbers. The atomic number Z = number of protons in the nucleus = number of electrons in the neutral atom — Z defines the element. The mass number A = Z + n = total nucleons. The isotope notation is ᴬ_ZX. Isotopes have the same Z but different A (¹H, ²D, ³T; ¹²C, ¹³C, ¹⁴C; ³⁵Cl, ³⁷Cl) and share chemistry because chemistry depends on the electron count. Isobars have the same A but different Z (¹⁴₆C and ¹⁴₇N) (NCERT §2.2.3–2.2.4, p. 35).

Failure of the Rutherford model. A revolving electron is undergoing centripetal acceleration; by Maxwell's electromagnetism an accelerating charge must continuously radiate, lose energy, and spiral into the nucleus in ~10⁻⁸ s. The model also says nothing about the distribution or energies of electrons (NCERT §2.2.5, p. 36).

EM radiation as waves. Maxwell (1870) showed that oscillating charged particles produce mutually perpendicular electric and magnetic fields propagating as transverse electromagnetic waves at c = 3.0 × 10⁸ m s⁻¹ in vacuum, with no medium required. Wave parameters: frequency ν (Hz), wavelength λ (m), c = νλ, wavenumber ν̄ = 1/λ. The electromagnetic spectrum stretches from radio (~10⁶ Hz) through microwave, IR, visible (380–780 nm, ~10¹⁵ Hz), UV, X-ray to γ-ray (NCERT §2.3.1, pp. 37–39; Fig. 2.7).

Planck's quantum theory. Classical wave theory could not explain black-body radiation or the photoelectric effect. Planck (1900) proposed that energy is absorbed and emitted only in discrete quanta of E = hν, where h = 6.626 × 10⁻³⁴ J s, so that allowed energies are E = 0, hν, 2hν, … nhν. The black-body intensity now varies smoothly with temperature and explains why hot objects glow red before orange/white (NCERT §2.3.2, pp. 40–41).

Photoelectric effect. Heinrich Hertz (1887) observed that UV light ejects electrons from a metal surface. Three classical puzzles emerged: (i) emission occurs only above a threshold frequency ν₀; (ii) the kinetic energy of the ejected electron depends on frequency, not intensity; (iii) the number of electrons depends on intensity. Einstein (1905) used Planck's quanta to explain all three at once: hν = hν₀ + ½m_e v² = W₀ + KE, where W₀ = hν₀ is the work function. KE = h(ν − ν₀). This earned Einstein the 1921 Nobel Prize (NCERT §2.3.2, pp. 41–43; Fig. 2.9; eq. 2.7).

Atomic line spectra. Incandescent solids emit a continuous spectrum, but excited atoms emit at discrete wavelengths producing a line spectrum — every element has a unique fingerprint (He was discovered spectroscopically in the Sun before being found on Earth). The hydrogen line spectrum is described by Rydberg's formula ν̄ = 109,677 (1/n₁² − 1/n₂²) cm⁻¹, organised into series — Lyman (n₁ = 1, UV), Balmer (n₁ = 2, visible), Paschen (n₁ = 3, IR), Brackett (n₁ = 4, IR) and Pfund (n₁ = 5, far IR) (NCERT §2.3.3 + Table 2.3, pp. 44–46; Figs. 2.10–2.11).

Bohr's model of the atom (1913). Three postulates: (i) the electron moves only in fixed stationary circular orbits of definite radius and energy without radiating; (ii) energy is constant in an orbit but is absorbed (n_low → n_high) or emitted (n_high → n_low) only in transitions with frequency ν = ΔE/h; (iii) angular momentum is quantised: m_e v r = n h/2π. From these, the energy and radius of hydrogen-like (one-electron) species are Eₙ = −2.18 × 10⁻¹⁸ (Z²/n²) J atom⁻¹ and rₙ = 52.9 (n²/Z) pm. The Bohr radius a₀ = 52.9 pm is the radius for n = Z = 1. Negative energy means the bound electron is more stable than a free electron at rest (n = ∞, E = 0). Bohr's model elegantly reproduces the Rydberg constant from first principles (NCERT §2.4, eqs. 2.10–2.15, pp. 46–48).

Limitations of Bohr. The model fails to explain (i) the fine structure (doublets, triplets) of even hydrogen lines, (ii) the spectra of multi-electron atoms, (iii) the Zeeman effect (splitting in a magnetic field) and the Stark effect (splitting in an electric field), and (iv) the formation of chemical bonds — molecular geometry. It also violates the de Broglie/Heisenberg framework by giving the electron a definite trajectory (NCERT §2.4.2, p. 49).

Dual nature of matter — de Broglie (1924). Every moving particle has a wavelength λ = h/mv = h/p. The wave nature is significant only for microscopic objects: an electron at v = 10⁶ m s⁻¹ has λ ≈ 7.3 × 10⁻¹⁰ m (X-ray range) — diffractable by crystals (Davisson–Germer, 1927) — whereas a 1 kg cricket ball at 10 m s⁻¹ has λ = 6.6 × 10⁻³⁵ m, far below any measurable scale (NCERT §2.5.1, eq. 2.22, p. 50).

Heisenberg's uncertainty principle (1927). It is impossible to determine simultaneously and exactly both the position and the momentum of a microscopic particle: Δx · Δp ≥ h/4π, equivalently Δx · Δv ≥ h/(4π m). For an electron (m_e ≈ 9.1 × 10⁻³¹ kg) the product is sizeable and forbids talk of "the path of the electron"; for a 1 mg dust particle it is utterly negligible. The principle invalidates Bohr-style fixed orbits and motivates the probabilistic orbital description (NCERT §2.5.2, eq. 2.23, pp. 51–52).

Quantum-mechanical model. Schrödinger's wave equation Ĥψ = Eψ yields the allowed energies E and wave functions ψ (atomic orbitals) for the hydrogen atom. |ψ|² (Max Born interpretation) is the probability density of finding the electron at a point. For one-electron systems (H, He⁺, Li²⁺) orbital energy depends only on n; in multi-electron atoms it depends on both n and l because of electron-electron repulsion and shielding (NCERT §2.6, pp. 53–54).

Quantum numbers. Each orbital is specified by three quantum numbers and each electron by an additional spin quantum number. n (principal, 1, 2, 3 …) sets shell, size and (mostly) energy. l (azimuthal, 0 to n−1) sets the subshell (0 = s, 1 = p, 2 = d, 3 = f, 4 = g …) and orbital shape. mₗ (magnetic, −l to +l, 2l+1 values) gives orientation. mₛ (spin, +½ or −½, Uhlenbeck and Goudsmit 1925) is the intrinsic electron spin. The maximum number of orbitals in shell n is n² and the maximum number of electrons is 2n² (NCERT §2.6.1, pp. 54–56).

Shapes and nodes. s-orbitals are spherically symmetric; an ns orbital has (n − 1) radial nodes. p-orbitals are dumbbell-shaped along x, y or z axes (2pₓ, 2p_y, 2p_z); each has one angular nodal plane and (n − 2) radial nodes. d-orbitals come in five orientations — d_xy, d_yz, d_xz, d_{x²−y²}, d_{z²} — the first four have four-lobed shapes, while d_{z²} has two lobes plus a torus. Total nodes per orbital = n − 1; angular nodes = l; radial nodes = n − l − 1 (NCERT §2.6.2, pp. 57–59; Figs. 2.12–2.15).

Energy ordering and Aufbau. In hydrogen, orbitals of the same n are degenerate; in multi-electron atoms, energy increases with (n + l), and for tied (n + l), with smaller n first. The diagonal rule gives the filling order: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s … (NCERT §2.6.3–2.6.4, Table 2.5, pp. 60–62; Fig. 2.17). Pauli's exclusion principle — no two electrons can share all four quantum numbers (an orbital holds two electrons with opposite spins). Hund's rule — pairing in degenerate orbitals begins only after each has one electron with parallel spin. Cr (Z = 24) is [Ar] 3d⁵ 4s¹ and Cu (Z = 29) is [Ar] 3d¹⁰ 4s¹ because half-filled and filled subshells gain extra stability from symmetric distribution, low mutual shielding and large exchange energy (NCERT §2.6.4–2.6.6, pp. 62–65; Fig. 2.18).

2.2 Definitions to memorise

2.3 Diagrams / processes to remember

CUET routinely tests which figure depicts which phenomenon. The discovery sequence begins with Fig. 2.1 (cathode ray discharge tube — note the perforated anode used to make a beam), Fig. 2.2 (Thomson's e/m_e set-up showing electric and magnetic field plates with deflection points A, B, C), and Fig. 2.3 (Millikan's oil-drop apparatus with the atomiser, condenser plates and microscope viewer). Fig. 2.4 is Thomson's plum-pudding sphere and Fig. 2.5 the Geiger–Marsden gold-foil scattering set-up that overturned it — almost all α-particles pass straight through; one in 20,000 rebounds. Fig. 2.6 sketches Rutherford's miniature solar system.

The wave–particle figures begin with Fig. 2.7 — the full EM spectrum from radio (long λ, low ν) through visible (380–780 nm) to γ-rays (short λ, high ν) — and Fig. 2.8 (wave parameters: amplitude, wavelength and frequency). Fig. 2.9 is the classic photoelectric apparatus: a clean metal cathode irradiated by tunable-frequency UV; you should be able to label threshold frequency, work function and the linear KE-vs-ν Einstein plot. Fig. 2.10 contrasts emission and absorption line spectra of atomic hydrogen, and Fig. 2.11 maps Lyman, Balmer and Paschen transitions onto the H energy-level diagram — particularly useful for series-identification MCQs.

The orbital figures come next. Fig. 2.12 shows ψ(r) and ψ²(r) plots for 1s and 2s — the 2s function crosses zero once, creating a radial node. Fig. 2.13 gives boundary-surface diagrams for 1s and 2s as spheres. Fig. 2.14 shows 2pₓ, 2p_y, 2p_z dumbbells along the coordinate axes. Fig. 2.15 displays the five 3d orbitals — four "cloverleaf" lobes (d_xy, d_yz, d_xz, d_{x²−y²}) and the distinct d_{z²} with axial lobes plus an equatorial torus. Fig. 2.16 compares the energy-level diagrams of hydrogen (degenerate within n) versus multi-electron atoms ((n+l) ordering). Fig. 2.17 is the famous (n+l) diagonal mnemonic for the Aufbau filling order. Fig. 2.18 illustrates the six exchange pairs in a d⁵ configuration that account for the extra stability of half-filled subshells in Cr and Mn.

Three procedural workflows worth memorising: (1) the photoelectric pipeline — given metal work function W₀ and incident frequency ν, compute KE = h(ν − ν₀) and v_max from ½ m_e v² (NCERT Problem 2.8, p. 43); (2) the spectral-line pipeline — given n₁, n₂, use Rydberg ν̄ = 109,677(1/n₁² − 1/n₂²) cm⁻¹, then λ = 1/ν̄; (3) the H-like ion pipeline — Eₙ = −2.18 × 10⁻¹⁸ (Z²/n²) J, rₙ = 52.9 (n²/Z) pm, and ionisation energy = E_∞ − E_n.

2.4 Common confusions / NTA trap points

• Orbit vs orbital — a Bohr orbit is a fixed circular trajectory; an orbital is a 3-D probability region described by ψ. Don't treat them as synonyms (p. 56).
• Filling order vs removal order — 4s fills before 3d in the Aufbau order, but in transition-metal cations 4s electrons are removed first because once 3d is occupied, 3d sinks below 4s in energy. So Fe²⁺ is [Ar] 3d⁶, not [Ar] 3d⁴ 4s².
• Cr and Cu exceptions — half-filled (3d⁵) and filled (3d¹⁰) subshells are extra-stable; valence configurations are 3d⁵ 4s¹ and 3d¹⁰ 4s¹, not 3d⁴ 4s² and 3d⁹ 4s² (p. 64).
• Photoelectric KE depends on frequency, NOT intensity — intensity merely changes the number of electrons ejected (p. 42). Common distractor.
• Number of nodes — total nodes = n − 1; angular nodes = l; radial nodes = n − l − 1. Don't confuse "nodes" (where ψ = 0) with "nodal planes" (specific angular nodes).
• Bohr energy formula — Eₙ ∝ −Z²/n² (Z² in numerator, n² in denominator). NTA loves to swap them in distractors.
• Isotopes vs isobars vs isotones — isotopes share Z, isobars share A, isotones share neutron number n (not in CUET syllabus but tested as a distractor).
• de Broglie λ for macroscopic objects — though λ = h/mv applies to all objects, it is unobservably small for everyday bodies. Wave nature manifests only for very small m.
• Schrödinger's ψ has no physical meaning by itself — only \|ψ\|² is observable (Born interpretation, p. 54).
• (n+l) tie-break — when two orbitals have the same (n+l), the smaller n is filled first; thus 3d (n+l=5) is below 4p (n+l=5) only because 3 < 4 → no, here 3 < 4 so 3d first. But 4s (n+l=4) sits below 3d (n+l=5). Read the rule carefully.
• He emission lines — Helium was discovered in the Sun's spectrum before on Earth; it is not a hydrogen-like ion (He⁺ is), and only He⁺ obeys Bohr exactly.
• Spin quantum number values — +½ and −½, not +1 and −1 (a common slip).

2.5 Key reactions & formulas