Structure of Atom — NEET Notes

Discovery of subatomic particles

Dalton's 1808 atom was indivisible. Four decades of experiments dismantled that picture. In the 1850s, when partially evacuated cathode-ray tubes were subjected to a high voltage, a stream of particles flowed from the cathode to the anode. These cathode rays travelled in straight lines, bent toward positive plates in an electric field, and behaved identically regardless of the electrode material or the gas inside the tube — proof that the same negatively charged constituent existed in every atom. In 1897, J. J. Thomson measured the charge-to-mass ratio of the electron by balancing electric and magnetic fields against each other in a cathode-ray tube and got e/m_e = 1.758820 × 10¹¹ C kg⁻¹. In 1906–14 R. A. Millikan's oil-drop experiment pinned down the charge itself at −1.602 × 10⁻¹⁹ C, fixing the mass of the electron at 9.109 × 10⁻³¹ kg.

The discovery of the positive counterpart followed soon after. Modified cathode-ray tubes produced canal rays — positively charged ions whose mass and e/m ratio depended on the gas in the tube. The lightest such positive particle, obtained when the tube contained hydrogen, was named the proton and was characterised in 1919. The mass discrepancy between protons-plus-electrons and the actual mass of most atoms forced the conjecture of a neutral particle. James Chadwick confirmed this in 1932 by bombarding a thin sheet of beryllium with alpha particles, releasing a stream of neutral particles slightly more massive than protons. He named these neutrons.

Two side experiments from the same era proved important. Wilhelm Roentgen in 1895 found that fast electrons striking a metal target produced highly penetrating, electrically neutral radiation of very short wavelength (≈0.1 nm) — what he named X-rays. Henri Becquerel in 1896 observed that uranium salts emit radiation spontaneously, the phenomenon of radioactivity. Rutherford later resolved this radiation into α (helium nuclei, +2 charge), β (high-energy electrons) and γ (high-energy, neutral electromagnetic radiation) — and used the α-particles as projectiles in his most famous experiment.

Thomson and Rutherford atomic models

Thomson in 1898 proposed the first sub-atomic model. He pictured the atom as a sphere of radius ~10⁻¹⁰ m, with positive charge distributed uniformly throughout the volume and electrons embedded in it like seeds in a watermelon — the plum-pudding model. It accounted for overall electrical neutrality but predicted no internal structure: the mass of the atom was supposed to be spread evenly through the volume.

That picture was demolished by Rutherford's α-particle scattering experiment (Geiger, Marsden and Rutherford, 1909). A beam of α-particles from a radioactive source was directed at a gold foil ~100 nm thick, surrounded by a circular fluorescent zinc-sulphide screen. If Thomson were correct, the α-particles should have ploughed straight through with only minor angular deflections. Instead, three things happened: most α-particles passed undeflected, a small fraction was deflected through small angles, and roughly 1 in 20,000 bounced back nearly 180°. Rutherford concluded that the positive charge and almost all the mass of the atom were concentrated in an extremely small, dense region he named the nucleus. If a cricket ball represents the nucleus, the radius of the atom would be about 5 km.

Rutherford's model — nucleus at the centre, electrons revolving like planets around the sun — was a genuine advance but carried a fatal classical problem. An electron in a circular orbit is constantly being accelerated (its direction keeps changing), and Maxwell's electromagnetic theory demands that an accelerated charged particle continuously radiates energy. The orbiting electron should therefore lose energy, spiral inward, and crash into the nucleus in about 10⁻⁸ s. Atoms exist; they do not collapse. The model also says nothing about how electrons are distributed or what their energies are.

Atomic number, mass number, isotopes & isobars

The positive charge of the nucleus comes from its protons. The atomic number (Z) is the number of protons in the nucleus, which equals the number of electrons in a neutral atom — and is what fixes the chemical identity of the element. Every atom with seven protons is nitrogen, no exceptions. The mass number (A) is the total number of nucleons: A = Z + n. The composition of any nuclide is written ^A_ZX — for example ⁸⁰₃₅Br has 35 protons, 35 electrons (if neutral) and 80 − 35 = 45 neutrons.

Isotopes are atoms with the same atomic number but different mass numbers — they differ only in neutron count, and because chemical behaviour is governed by electrons, all isotopes of an element show essentially the same chemistry. Hydrogen has three: protium (¹₁H, 99.985%), deuterium (²₁H, ~0.015%) and tritium (³₁H, trace, radioactive — emits low-energy β⁻ particles, t_½ ≈ 12.33 years). Isobars are the converse: same mass number but different atomic number — for example ¹⁴₆C and ¹⁴₇N. Isoelectronic species have the same electron count but may differ in nuclear charge — Na⁺, Mg²⁺, F⁻, O²⁻ all carry ten electrons.

Developments leading to Bohr — light, photons, atomic spectra

Bohr's hydrogen model was built on three nineteenth- and early-twentieth-century developments: Maxwell's electromagnetic wave theory of light, Planck's quantisation of energy, and the observation of discrete atomic line spectra.

Maxwell (1870) showed that an accelerated charged particle emits oscillating, mutually perpendicular electric and magnetic fields — electromagnetic waves that travel through vacuum at c ≈ 3.0 × 10⁸ m s⁻¹. The relationship c = ν λ ties frequency and wavelength together. The electromagnetic spectrum runs from radio waves (~10⁶ Hz) up through microwaves, infrared, the narrow visible band (~4 × 10¹⁴ to 7.5 × 10¹⁴ Hz), ultraviolet, X-rays and γ-rays. Wave theory explained diffraction and interference but failed for four phenomena: black-body radiation, the photoelectric effect, the temperature dependence of heat capacity, and atomic line spectra.

Max Planck resolved black-body radiation in 1900 with a single radical postulate: atoms in the cavity walls emit and absorb energy not continuously but in discrete packets — quanta — whose energy is proportional to the frequency:

E = h ν   with   h = 6.626 × 10⁻³⁴ J s

Planck's quantum hypothesis (1900)

Einstein in 1905 used the same idea to explain the photoelectric effect. Light striking a clean metal surface ejects electrons only above a metal-specific threshold frequency ν₀; below it, no electrons emerge no matter how intense the light. Above ν₀, the kinetic energy of the ejected photoelectron rises linearly with the frequency of the light, while the number of electrons ejected scales with the intensity. Einstein pictured light as a stream of photons of energy hν that hand all of their energy to a single electron in one collision:

½ m_e v²  =  h ν − h ν₀  =  h ν − W₀

Einstein's photoelectric equation (1905)

This forced acceptance that light has both wave and particle properties — wave-particle duality. Atomic line spectra delivered the third clue. When an electric discharge is passed through hydrogen gas, the excited atoms emit light at a small set of discrete wavelengths rather than a continuous rainbow. Balmer (1885) found a formula for the visible lines; Rydberg (1888) generalised it. Every hydrogen line, in any series, obeys:

ν̄  =  109677 cm⁻¹ · (1/n₁² − 1/n₂²)

Rydberg formula — the wavenumber of every hydrogen line

The five series correspond to n₁ = 1, 2, 3, 4, 5 — Lyman (UV), Balmer (visible), Paschen, Brackett and Pfund (all infrared). Any theory of the atom would have to reproduce this formula.

Bohr's model for the hydrogen atom

Niels Bohr (1913) wove Planck's quantisation into Rutherford's nuclear atom and explained the hydrogen spectrum quantitatively. His model rests on four postulates and yields three concrete predictions.

The first orbit — the ground state of hydrogen — has the smallest radius and the lowest (most negative) energy. The negative sign is conventional: energy is set to zero for an electron infinitely far from the nucleus, so any bound electron has negative energy.

Bohr's model extends naturally to hydrogen-like one-electron species — He⁺, Li²⁺, Be³⁺ — by inserting the nuclear charge Z:

r_n = (n²/Z) · 52.9 pm    E_n = −2.18 × 10⁻¹⁸ · (Z²/n²) J

Bohr formulae for hydrogen-like ions

This Z-dependence is a frequent NEET test. The radius of the third Bohr orbit of Li²⁺ equals (9/3) × 52.9 = 158.7 pm — the exact NEET 2022 numerical, derived from r ∝ n²/Z.

Subtract two energies and the frequency-rule plus the Z-scaling reproduces the Rydberg formula:

ΔE  =  R_H · Z² · (1/n_i² − 1/n_f²)

Energy of a Bohr transition — emission if n_i > n_f

For hydrogen, transitions ending at n_f = 1 give the Lyman series in the UV; at n_f = 2 the Balmer series in the visible; at n_f = 3, 4, 5 the Paschen, Brackett and Pfund series in the infrared. Limitations: Bohr's model cannot account for the fine doublet structure of hydrogen lines, the spectra of any atom beyond hydrogen-like systems, the Zeeman effect (line splitting in a magnetic field) or the Stark effect (in an electric field), and it cannot explain chemical bonding.

Towards the quantum-mechanical model

Two ideas in the 1920s replaced Bohr's well-defined circular orbit with something fundamentally fuzzier. Louis de Broglie in 1924 proposed that matter, like radiation, has dual wave-particle character. Every moving particle carries a wavelength:

λ  =  h / (m v)  =  h / p

de Broglie relation — every moving mass is a wave

For a cricket ball the wavelength is astronomically smaller than any measurable length and the wave nature is invisible; for an electron (m ≈ 9.1 × 10⁻³¹ kg) the wavelength is comparable to atomic dimensions. de Broglie's prediction was confirmed when electron beams were shown to diffract — the operating principle of the electron microscope.

Werner Heisenberg in 1927 articulated a deeper consequence of duality — the uncertainty principle:

Δx · Δp_x  ≥  h / (4 π)

Heisenberg uncertainty principle (1927)

It is impossible to know the exact position and exact momentum of an electron at the same time. Locate the electron tightly (Δx small) and its velocity becomes wildly uncertain. The principle has trivial consequences for macroscopic bodies (a 1 mg object has ΔvΔx far below experimental sensitivity) but is decisive for electrons. The Bohr orbit — a perfectly defined path of an electron — violates the uncertainty principle and is therefore not physically meaningful for an electron. We can speak only of the probability of finding an electron somewhere; the well-defined orbit must be replaced by a wave function.

The quantum-mechanical model of the atom

In 1926 Erwin Schrödinger wrote down the equation that governs the electron's wave behaviour:

Ĥ ψ  =  E ψ

Time-independent Schrödinger equation — Ĥ is the Hamiltonian operator

When solved for the hydrogen atom, the equation yields a discrete set of allowed energies and a corresponding wave function ψ for each. ψ itself has no physical meaning, but |ψ|² gives the probability density of finding the electron at any point. These wave functions are called atomic orbitals. Five features of the quantum-mechanical model are worth committing to memory:

1. Electron energies are quantised — only certain specific values are allowed, falling out automatically as solutions to the Schrödinger equation.
2. Quantised energies arise from the electron's wave nature — they are not postulates.
3. Exact position and exact velocity cannot be known simultaneously (Heisenberg). The path of an electron in an atom is therefore not defined; we work with probabilities.
4. An atomic orbital is the wave function ψ for an electron in an atom. An orbital can hold at most two electrons. Many orbitals exist for a given atom.
5. |ψ|² is the probability density. Where |ψ|² is large, the electron is most likely to be found.

Each orbital in hydrogen is characterised by three quantum numbers (n, l, m_l); a fourth (m_s) is needed to fully label an electron. In a multi-electron atom the Schrödinger equation cannot be solved exactly, but approximate methods show the orbitals are similar in shape to those of hydrogen — only contracted and with energies that depend on both n and l.

Quantum numbers — addressing every electron

Four quantum numbers between them specify a unique electron in an atom. Three of them (n, l, m_l) describe the orbital; the fourth (m_s) describes the spin of the electron within that orbital. Their allowed values are tightly coupled — l depends on n, and m_l depends on l. This coupling generates almost every NEET trap in this chapter.

Two examples make the bookkeeping concrete. For n = 3 the allowed l values are 0, 1, 2 — so the third shell contains one 3s orbital (l = 0, m_l = 0), three 3p orbitals (l = 1, m_l = −1, 0, +1) and five 3d orbitals (l = 2, m_l = −2, −1, 0, +1, +2). Total orbitals = 1 + 3 + 5 = 9 = n². Total electron capacity = 2 × 9 = 18 = 2n². For l = 3 (f subshell), m_l takes seven values, giving seven f orbitals.

Shapes of s, p and d orbitals

An orbital's shape is the contour of constant |ψ|² that encloses about 90% probability of finding the electron. Different l values give different shapes.

s orbitals (l = 0) are spherically symmetric around the nucleus — the probability of finding the electron at a given distance is the same in every direction. The 1s orbital has its probability density maximum at the nucleus and decreases monotonically outward. The 2s orbital starts high, drops to zero (a radial node), rises again to a small maximum and tails off. In general an ns orbital has (n − 1) radial nodes. Size: 1s < 2s < 3s < 4s; energy increases with n.

p orbitals (l = 1) appear from the second shell onward (n ≥ 2). Each p orbital has two lobes on opposite sides of a nodal plane passing through the nucleus. Because m_l can be −1, 0 or +1, there are three p orbitals (p_x, p_y, p_z) with axes mutually perpendicular. All three are identical in size, shape and energy — only their orientation differs. The probability density on the nodal plane is zero, so a p electron is never found there. The number of radial nodes for an np orbital is n − 2; angular nodes equal l, so one angular node per p orbital.

d orbitals (l = 2) require n ≥ 3. The five d orbitals are designated d_xy, d_yz, d_xz, d_x²−y² and d_z². The first four have four lobes lying in or between the coordinate planes (the 'double dumbbell' shape); d_z² has two lobes along the z-axis plus a doughnut in the xy plane. All five 3d orbitals are equivalent in energy (degenerate). Each d orbital has two angular nodes; total nodes = n − 1 = l + (n − l − 1) where (n − l − 1) is the radial-node count.

Aufbau, Pauli, Hund — the three filling rules

Three rules determine how electrons populate the orbitals of a multi-electron atom.

Aufbau principle — German for "building up." In the ground state, electrons fill orbitals in order of increasing energy. The order is given by the (n + l) rule: lower (n + l) fills first; when two orbitals tie, the lower n fills first. The resulting sequence is

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s …

This is why potassium (Z = 19) writes [Ar] 4s¹ rather than [Ar] 3d¹: 4s (n + l = 4) lies below 3d (n + l = 5) in energy.

Pauli exclusion principle (Wolfgang Pauli, 1926): no two electrons in an atom can share the same set of four quantum numbers. Equivalently — an orbital holds at most two electrons and they must have opposite spins. The capacity of the nth shell is therefore 2n².

Hund's rule of maximum multiplicity: in degenerate orbitals (same energy, e.g. the three 2p orbitals), electrons enter singly with parallel spins before any pairing occurs. Pairing in p, d and f subshells begins only with the 4th, 6th and 8th electron respectively. Maximum unpaired electrons → maximum exchange energy → extra stability.

Electronic configurations of the elements

Writing the ground-state configuration of an element is mechanical once the three rules are in place. The notation is nl^x: the principal number, the subshell letter, and a superscript counting the electrons. Hydrogen is 1s¹. Helium is 1s² (closing the K shell). The next eight elements fill the second shell — Li (1s² 2s¹), Be (1s² 2s²), then the 2p subshell across B, C, N, O, F, Ne to give Ne = 1s² 2s² 2p⁶.

From Na onward, configurations are written using the previous noble-gas core in brackets, then the new electrons explicitly. The electrons in completely filled inner shells are core electrons; those added to the highest-n shell are valence electrons.

Two patterns recur across the table. The first is the 4s-before-3d swap at K and Ca, after which the 3d subshell fills across Sc → Zn — with the Cr and Cu exceptions noted above. The second is the (n − 1)d filling that defines the d-block transition metals, and the analogous (n − 2)f filling that defines the lanthanides and actinides.