Structure of Atom

The atom is not indivisible — it is built of three fundamental particles. Each was found through a specific experiment, and examiners like to test which scientist, which apparatus and which conclusion go together.

Electron — J.J. Thomson, using a cathode ray discharge tube at low pressure and high voltage. Cathode rays travel from cathode to anode in straight lines, are deflected by electric and magnetic fields toward the positive plate, and their properties (charge/mass ratio, e/m = 1.758820 × 10¹¹ C kg⁻¹) do not depend on the gas or electrode material — so the electron is a universal constituent of matter. Millikan's oil drop experiment later gave the charge on the electron, 1.6 × 10⁻¹⁹ C, which combined with Thomson's e/m gives the mass of the electron, 9.1 × 10⁻³¹ kg.

Proton — Goldstein, using a perforated cathode in a discharge tube, observed rays travelling opposite to cathode rays: anode rays (canal rays). Unlike cathode rays, their charge-to-mass ratio depends on the gas used, and the smallest, lightest positive particle obtained (from hydrogen gas) is the proton, charge +1.6 × 10⁻¹⁹ C, mass 1.67 × 10⁻²⁷ kg (about 1837 times the electron's mass).

Neutron — Chadwick, by bombarding beryllium with α-particles: ⁹Be + ⁴He → ¹²C + neutron. The neutron carries no charge and has a mass almost equal to the proton's (1.675 × 10⁻²⁷ kg).

Thomson's model (plum pudding / watermelon model): the atom is a uniform sphere of positive charge with electrons embedded in it like seeds in a watermelon, the positive and negative charges being equal so the atom is neutral overall. It explained the overall neutrality of the atom but could not explain the results of Rutherford's scattering experiment.

Rutherford's α-particle scattering experiment: a beam of fast α-particles was directed at a thin gold foil (about 100 nm thick), surrounded by a circular fluorescent zinc sulphide screen. Observations: (i) most α-particles passed through the foil undeflected; (ii) a small fraction were deflected by small angles; (iii) a very small fraction (about 1 in 20,000) bounced back at angles close to 180°.

Rutherford concluded: (a) most of the atom is empty space, since most particles pass straight through; (b) a small region of the atom is very densely positively charged and contains almost all the mass — the nucleus, since only a heavy, concentrated positive charge could repel a fast, massive α-particle straight back; (c) the electrons revolve around this nucleus in circular paths (orbits), the size of the atom (~10⁻¹⁰ m) being about 10,000 times the size of the nucleus (~10⁻¹⁵ m).

Drawback: an electron moving in a circular orbit is accelerating, and an accelerating charged particle must radiate energy (electromagnetic theory). It should therefore spiral into the nucleus, making the atom unstable — but atoms are stable. This is the drawback that Bohr's model was built to fix.

Before Bohr's model, two lines of evidence — the nature of light and the atomic spectrum of hydrogen — had to come together. Electromagnetic radiation, in the wave picture (Maxwell), consists of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation, and it does not need a medium to travel (unlike sound). It is characterised by wavelength (λ), frequency (ν) and velocity, related by c = νλ, where c is the speed of light in vacuum, 3 × 10⁸ m s⁻¹. Wavenumber (𝜈̄), the number of wavelengths per unit length, is defined as 𝜈̄ = 1/λ, with SI unit m⁻¹ (commonly cm⁻¹). The electromagnetic spectrum ranges, in order of increasing frequency (decreasing wavelength), from radio waves through microwaves, infrared, visible light, ultraviolet, X-rays to γ-rays.

Visible light (400–750 nm) is only a tiny slice of this spectrum. The wave picture explained phenomena like interference and diffraction well, but it could not explain three observations: black body radiation, the photoelectric effect, and the variation of heat capacity of solids with temperature. These needed a particle picture of light.

Black body radiation could not be explained by wave theory (it predicted radiation to increase indefinitely with frequency — the 'ultraviolet catastrophe'). Max Planck resolved this by proposing that energy is emitted or absorbed by atoms/molecules only in discrete packets called quanta, and for light, a quantum is called a photon. The energy of a quantum is proportional to its frequency: E = hν, where h is Planck's constant, 6.626 × 10⁻³⁴ J s. Energy is always an integral multiple of hν (E = nhν); it cannot be emitted or absorbed continuously.

The photoelectric effect (Hertz, Lenz): when light of sufficiently high frequency strikes a metal surface, electrons are ejected instantly (photoelectrons). Wave theory predicted that intensity, not frequency, should control ejection — but experiment showed: (i) electrons are ejected only above a minimum frequency, the threshold frequency (ν₀), which is characteristic of the metal, regardless of intensity; (ii) the kinetic energy of the ejected electrons increases linearly with frequency of incident light, not with intensity; (iii) intensity only increases the number of photoelectrons (the photoelectric current), not their individual kinetic energy; (iv) emission is instantaneous, with no time lag.

Einstein explained this using Planck's quantum idea: each photon of energy hν transfers all its energy to one electron. Part of this energy (h ν₀, the work function, W₀) is used to overcome the attraction holding the electron in the metal; the rest becomes the electron's kinetic energy.

When white light is passed through a prism, it splits into a continuous rainbow — a continuous spectrum, because all wavelengths of visible light are present. But when atoms of an element are excited (by heat or an electric discharge) and the light they emit is passed through a prism, only specific, sharp coloured lines appear against a dark background, at fixed wavelengths — this is a line (atomic) emission spectrum, and it is unique to each element, like a fingerprint. A line absorption spectrum is obtained when white light is passed through a vapour of the element first — dark lines appear at exactly the wavelengths that would have been emitted, on the continuous background.

The hydrogen spectrum is the simplest and most important. Its emission lines fall into series named after their discoverers — Lyman (UV, lines end at n = 1), Balmer (visible, n = 2), Paschen, Brackett and Pfund (all infrared, n = 3, 4, 5). Each line's wavenumber fits the empirical Rydberg formula. This was the single biggest clue that electrons in an atom exist only in certain fixed, discrete energy levels — not at any arbitrary energy — because only transitions between fixed levels can produce a few sharp lines instead of a continuous smear. Bohr's model was built precisely to explain why this is so.

Bohr combined Rutherford's nuclear model with Planck's quantum idea to fix the stability problem, using three postulates:

1. The electron moves in circular orbits of fixed energy called stationary states (or orbits), and while in a stationary state it does not radiate energy, despite being in circular motion.
2. The angular momentum of the electron is quantised: it can only take values that are integral multiples of h/2π — mvr = nh/2π, where n = 1, 2, 3, ... (the principal quantum number).
3. Energy is absorbed or emitted only when the electron jumps from one stationary state to another, and the energy difference equals the energy of the photon absorbed or emitted: ΔE = hν = E₂ − E₁.

Derivation of radius: for an electron of mass m, charge −e, orbiting a nucleus of charge +Ze at radius r with speed v, the electrostatic force of attraction supplies the centripetal force: kZe²/r² = mv²/r (k = 1/4πε₀). Combining this with the quantisation condition mvr = nh/2π and eliminating v gives r_n = n²h²ε₀/(πmZe²), which numerically works out to r_n = 0.529 (n²/Z) Å.

Derivation of energy: total energy = kinetic + potential = ½mv² − kZe²/r. Using mv² = kZe²/r from the force balance, this simplifies to E = −kZe²/2r. Substituting r_n gives E_n = −2.18 × 10⁻¹⁸ (Z²/n²) J = −13.6 (Z²/n²) eV per electron. The negative sign shows the electron is bound to the nucleus (energy is taken as zero when the electron is completely free, at n = ∞).

Bohr's model explains the hydrogen line spectrum exactly: when an electron falls from a higher orbit n₂ to a lower orbit n₁, the emitted photon's energy equals E_n2 − E_n1, and working this out reproduces the Rydberg formula with R_H calculated from fundamental constants — a major triumph.

Bohr's model was a huge step forward but it has clear limits, and examiners like a crisp list:

• It explains the spectrum of hydrogen and hydrogen-like species (He⁺, Li²⁺, single-electron ions) very well, but it fails for multi-electron atoms — it could not account for their more complex spectra.
• It could not explain the splitting of spectral lines in a magnetic field (Zeeman effect) or in an electric field (Stark effect).
• It could not explain the finer details (fine structure) of the hydrogen spectrum seen with better resolution.
• It treats the electron only as a particle in a fixed circular orbit and ignores its wave nature (de Broglie), and it violates Heisenberg's uncertainty principle by assuming both a definite position (on a fixed orbit) and a definite momentum can be known simultaneously.

These gaps needed two more ideas — the dual nature of matter and the uncertainty principle — before a fully correct (quantum mechanical) model of the atom could be built.

Louis de Broglie proposed, in 1924, that if light (normally thought of as a wave) can behave like particles (photons, as in the photoelectric effect), then matter (normally thought of as particles) should also show wave-like behaviour. He derived a relation for the wavelength associated with any moving particle by combining Einstein's E = mc² with Planck's E = hν: the wavelength of a particle of mass m moving with velocity v is λ = h/mv.

This dual behaviour is significant only for very small, fast-moving particles like electrons — for macroscopic objects (a cricket ball, a car), the associated wavelength works out to be immeasurably small because of the large mass in the denominator, so wave behaviour is never observed for everyday objects. De Broglie's hypothesis was confirmed experimentally by the Davisson–Germer experiment, in which a beam of electrons was diffracted by a nickel crystal exactly as X-rays would be, proving electrons show wave-like diffraction.

De Broglie waves are the reason the quantum mechanical model replaces Bohr's fixed circular orbits with regions of high probability (orbitals) — an electron's exact path cannot be traced the way a planet's can.

Werner Heisenberg stated, in 1927, that it is impossible to determine simultaneously and with perfect accuracy both the exact position and the exact momentum (or velocity) of a microscopic moving particle such as an electron. The uncertainties in position (Δx) and momentum (Δp) are related by Δx · Δp ≥ h/4π.

This is a fundamental limitation of nature, not a flaw in measurement technique — measuring an electron's position (say, using a photon of light) necessarily disturbs its momentum, because the electron is so light that even a single photon striking it changes its velocity appreciably. The principle is significant only for very small particles with very small mass, like electrons; for macroscopic objects the uncertainty, calculated from the same formula, is far too small to be measurable (because h is extremely small compared to the momentum of everyday objects), so classical mechanics works perfectly well for them.

The direct consequence for atomic structure is enormous: since an electron's exact position and momentum cannot both be known, the idea of a well-defined orbit (a fixed path, as Bohr assumed) has to be abandoned. It is replaced by the idea of an orbital — a three-dimensional region around the nucleus where the probability of finding the electron is high. This is exactly the shift that leads into the quantum mechanical model.

The quantum mechanical model treats the electron as a wave and describes it using a mathematical function ψ (psi), the wave function, obtained by solving the Schrödinger wave equation. ψ itself has no direct physical meaning, but ψ² (or |ψ|²) at a point gives the probability density of finding the electron at that point — this is the modern replacement for a 'fixed orbit'. An orbital is the three-dimensional region in space around the nucleus where the probability of finding an electron is maximum (typically taken as 90–95%). This is fundamentally different from a Bohr orbit, which was a fixed, well-defined circular path.

Each orbital, and the electron in it, is completely described by a set of four quantum numbers:

• Principal quantum number (n) = 1, 2, 3, ... — decides the size and energy of the shell; the shell with a given n is denoted K, L, M, N for n = 1, 2, 3, 4.
• Azimuthal (angular momentum) quantum number (l) = 0 to (n − 1) for a given n — decides the shape of the subshell/orbital: l = 0, 1, 2, 3 are called s, p, d, f respectively.
• Magnetic quantum number (m_l) = −l to +l (including 0), giving (2l + 1) values — decides the orientation of the orbital in space.
• Spin quantum number (m_s) = +½ or −½ — describes the spin (clockwise or anticlockwise) of the electron about its own axis; it was not part of Schrödinger's original equation but was added to explain fine spectral detail.

For a given shell n: the number of subshells = n; the number of orbitals = n²; the maximum number of electrons = 2n². For a given subshell l, the number of orbitals = (2l + 1), so s has 1 orbital (2 electrons max), p has 3 (6 electrons max), d has 5 (10 electrons max), f has 7 (14 electrons max).

s-orbitals (l = 0) are spherically symmetric — probability of finding the electron is the same in all directions at a given distance from the nucleus; 1s, 2s, 3s all differ only in size (2s, 3s are bigger and have a spherical node inside them).

p-orbitals (l = 1) have a dumb-bell shape with two lobes on either side of the nucleus and a node (zero probability) at the nucleus itself. There are three degenerate p-orbitals per shell (p_x, p_y, p_z), oriented along the three perpendicular axes, and all three have identical energy and shape but different orientation.

d-orbitals (l = 2) are more complex: four of the five (d_xy, d_yz, d_xz, d_x²−y²) have a four-lobed 'double dumb-bell' (cloverleaf) shape, while d_z² has two lobes along the z-axis plus a torus (doughnut) of electron density around the middle.

Nodes are regions where the probability of finding the electron is zero. There are two types: radial nodes (spherical shells within the orbital), given by n − l − 1, and angular nodes (planes or cones through the nucleus), given by l. Total nodes = n − 1.

Energy ordering in a multi-electron atom is not simply by n; it follows the (n + l) rule: lower (n + l) means lower energy; if two subshells have equal (n + l), the one with lower n has lower energy. This is why 4s (n + l = 4) fills before 3d (n + l = 5), even though n = 3 < 4 — a very frequently tested point.

Three rules govern how electrons fill orbitals in the ground state of an atom:

Aufbau principle: orbitals are filled in order of increasing energy, i.e., increasing (n + l); for equal (n + l), the orbital with lower n fills first. The resulting standard filling order is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d ...

Pauli exclusion principle: no two electrons in the same atom can have all four quantum numbers identical. The practical consequence is that any orbital can hold a maximum of two electrons, and those two must have opposite spins (m_s = +½ and −½).

Hund's rule of maximum multiplicity: electron pairing in orbitals of the same subshell (degenerate orbitals, e.g. the three p orbitals) does not start until each orbital of that subshell has received one electron, and these singly-occupying electrons all have parallel (same-direction) spin. This gives the atom maximum stability.

Exceptions to Aufbau — the extra stability of half-filled and fully-filled subshells: chromium (Z = 24) is [Ar] 3d⁵ 4s¹, not the 'expected' [Ar] 3d⁴ 4s²; copper (Z = 29) is [Ar] 3d¹⁰ 4s¹, not [Ar] 3d⁹ 4s². This happens because exactly half-filled (d⁵) and fully-filled (d¹⁰) subshells have extra stability, arising from (i) symmetrical distribution of electrons and (ii) greater exchange energy between electrons of parallel spin in degenerate orbitals — an electron moves from 4s to 3d to achieve this more stable arrangement.

Electronic configuration is written using the notation nl^x (e.g. 1s² 2s² 2p⁶...), or in shorthand using the previous noble gas core in square brackets, e.g. sodium (Z = 11) = [Ne] 3s¹.