Chapter 2: Structure of Atom | Summary

J.J. Thomson (1898) proposed the 'plum pudding' model: a uniform positive sphere of radius ~10^-10 m with electrons embedded in it. Though it explained overall neutrality, it was disproved by Rutherford's alpha-particle scattering experiment (1909), in which a stream of high-energy alpha particles was directed at thin gold foil. Most passed through undeflected, a few deflected at small angles, and ~1 in 20,000 bounced back at nearly 180°.

Rutherford concluded that most of the atom is empty space, with a tiny, dense, positively charged nucleus at the centre. The nucleus radius (~10^-15 m) is vastly smaller than the atomic radius (~10^-10 m). Electrons orbit the nucleus in circular paths held by electrostatic attraction. However, Rutherford's model had two critical drawbacks: it could not explain atomic stability (an accelerating electron should radiate energy and spiral into the nucleus within 10^-8 s), and it gave no information about the distribution or energies of electrons.

Neils Bohr (1913) proposed four postulates for hydrogen: (i) electrons move in fixed circular orbits (stationary states) of definite energy; (ii) energy does not change within an orbit, but is absorbed when transitioning to a higher orbit and emitted when falling to a lower orbit; (iii) frequency of radiation emitted or absorbed equals ΔE/h (Bohr's frequency rule); (iv) angular momentum of the electron is quantised as integral multiples of h/2π. The radius of the nth orbit is rn = n²a₀, where a₀ = 52.9 pm, and the energy is En = −RH(1/n²) where RH = 2.18 × 10^-18 J.

Bohr's model successfully explained the line spectrum of hydrogen, including the Lyman (UV), Balmer (visible), Paschen, Brackett, and Pfund (IR) series. However, it failed to explain spectra of multi-electron atoms, fine structure of spectral lines (doublets), the Zeeman effect (splitting in magnetic fields), the Stark effect, and the ability of atoms to form molecules. These limitations arose because Bohr's model ignored the wave character of the electron and contradicted the Heisenberg uncertainty principle.

James Clerk Maxwell showed that accelerating charged particles produce electromagnetic waves with oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. Electromagnetic waves travel at c = 3.0 × 10^8 m/s in vacuum, and all types obey c = νλ. The electromagnetic spectrum spans radio waves (~10^6 Hz) through microwaves, infrared, visible (~10^15 Hz), ultraviolet, X-rays, to gamma rays (~10^24 Hz). Visible light occupies only a tiny portion of this spectrum.

Classical physics could not explain black body radiation, the photoelectric effect, or atomic line spectra. Max Planck (1900) resolved the black body problem by proposing that energy is quantised: atoms can emit or absorb energy only in discrete packets called quanta, with E = hν, where h = 6.626 × 10^-34 J·s is Planck's constant. Einstein (1905) extended this to explain the photoelectric effect: each photon of frequency ν carries energy hν, and electrons are ejected only if ν exceeds the threshold frequency ν₀. Kinetic energy of ejected electrons = hν − hν₀.

When atoms absorb energy and return to ground state, they emit radiation at discrete frequencies, producing a line emission spectrum unique to each element. Hydrogen's spectrum has several series: Lyman (UV, n₁=1), Balmer (visible, n₁=2), Paschen (IR, n₁=3), Brackett (IR, n₁=4), and Pfund (IR, n₁=5). The Rydberg formula ν̄ = 109,677(1/n₁² − 1/n₂²) cm⁻¹ describes all series, with 109,677 cm⁻¹ being the Rydberg constant.

An emission spectrum arises when excited atoms return to lower energy states, emitting specific wavelengths as bright lines. An absorption spectrum is the negative of emission: a continuous spectrum with dark lines where wavelengths are absorbed. Each element's spectrum is unique and can be used to identify elements — this is the basis of spectroscopy, through which elements like helium, rubidium, and caesium were discovered.

Louis de Broglie (1924) proposed that matter, like radiation, exhibits wave-particle duality. Every moving particle has an associated wavelength λ = h/mv = h/p. This was confirmed experimentally through electron diffraction. The wavelengths of macroscopic objects are immeasurably small due to large masses, but those of sub-atomic particles like electrons are detectable. The electron microscope exploits this wave nature, achieving magnifications of about 15 million times.

Werner Heisenberg (1927) stated that it is impossible to determine simultaneously the exact position and exact momentum (or velocity) of an electron: Δx × Δp ≥ h/4π. The more precisely the position is known, the more uncertain the momentum, and vice versa. This principle rules out definite electron trajectories and is significant only for microscopic objects. It demonstrated the fundamental failure of the Bohr model, which assumed electrons travel in well-defined circular orbits.

Quantum mechanics, developed by Heisenberg and Schrödinger (1926), accounts for the dual nature of matter and the uncertainty principle. Schrödinger's equation ĤΨ = EΨ, when solved for hydrogen, gives quantised energy states and corresponding wave functions (Ψ). These wave functions, called atomic orbitals, are characterised by three quantum numbers: principal (n), azimuthal (l), and magnetic (ml). The probability of finding an electron at any point is proportional to |Ψ|², the probability density.

The five key features of the quantum mechanical model are: (1) electron energy is quantised; (2) quantised levels arise from wave-like properties of electrons; (3) exact position and velocity cannot both be known (Heisenberg); (4) an atomic orbital is the wave function Ψ for one electron, containing at most two electrons with opposite spins; (5) |Ψ|² gives probability density. For multi-electron atoms, the Schrödinger equation cannot be solved exactly, but approximate methods show orbitals similar to hydrogen's, with energies depending on both n and l.

Each orbital is defined by three quantum numbers: n (principal, defines size and energy, n = 1, 2, 3…), l (azimuthal, defines shape, l = 0 to n−1; s, p, d, f…), and ml (magnetic, defines orientation, ml = −l to +l). A fourth quantum number ms (spin, +½ or −½) describes electron spin. The number of orbitals per shell is n², and the maximum electrons per shell is 2n². s-orbitals are spherical; p-orbitals have two lobes (dumbbell shape); d-orbitals have four lobes (or one with a ring); f-orbitals are more complex.

For hydrogen and hydrogen-like species, energy depends only on n: 1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f. In multi-electron atoms, energy depends on both n and l (s < p < d < f within a shell), due to shielding effects. The effective nuclear charge (Zeff) experienced by an electron decreases with increasing l for a given n. The (n + l) rule: the orbital with lower (n + l) has lower energy; if equal, lower n wins. Nodes (regions of zero probability) = n − 1 total; angular nodes = l; radial nodes = n − l − 1.

Electrons fill orbitals according to three rules: Aufbau principle (fill in order of increasing energy: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p…), Pauli exclusion principle (no two electrons can have the same four quantum numbers; each orbital holds maximum two electrons with opposite spins), and Hund's rule of maximum multiplicity (in degenerate orbitals, electrons fill singly with parallel spins before pairing). Core electrons occupy completely filled inner shells; valence electrons are in the outermost shell.

Exceptions to expected configurations arise when half-filled (d⁵, f⁷) or fully-filled (d¹⁰, f¹⁴) subshells provide extra stability due to symmetrical electron distribution and maximum exchange energy. For example, Cr is [Ar]3d⁵4s¹ (not 3d⁴4s²) and Cu is [Ar]3d¹⁰4s¹ (not 3d⁹4s²). Electronic configuration is fundamental to understanding chemical behaviour, reactivity, metallic character, and bond formation.

The structure of the atom evolved through a series of experimental discoveries and theoretical models. From Dalton's indivisible atom to the sophisticated quantum mechanical model, each step was driven by the inability of existing models to explain new observations. The quantum mechanical model, grounded in Schrödinger's equation, de Broglie's wave-particle duality, and Heisenberg's uncertainty principle, provides the most comprehensive and accurate description of atomic structure, defining electrons through probabilistic wave functions characterised by four quantum numbers.

For board examinations, this chapter is of central importance. Students must master the derivations and applications of Bohr's model for hydrogen (energy levels, radii, spectral lines), the Rydberg formula and hydrogen spectral series, the photoelectric effect equation, the de Broglie relation, and Heisenberg's principle. Electronic configurations using the Aufbau, Pauli, and Hund's rules — including exceptions for Cr and Cu — are directly tested. A firm understanding of quantum numbers, orbital shapes, and the (n+l) rule for energy ordering in multi-electron atoms is essential for success in both board exams and competitive entrance tests.