Atomic Structure

The structure of the atom is the foundation of all chemistry. The PMDC MDCAT 2026 syllabus expects you to know the discovery of subatomic particles, Planck's quantum theory, the four quantum numbers, the shapes of s/p/d orbitals, the rules for electronic configuration, and the line spectrum of hydrogen. This is one of the most heavily tested chapters — expect 3–4 MCQs.

PMC Table of Specifications. This chapter spans six PMDC subtopics — Discovery of Proton, Planck's Quantum Theory, Spectrum of Hydrogen, Quantum Numbers, Shapes of Orbitals, and Electronic Configuration.

Discovery of Proton

Goldstein's discharge tube experiment (1886) revealed positive rays travelling in the opposite direction to cathode rays. Using a perforated cathode, he observed positively charged particles streaming through the holes — canal rays or positive rays.

The lightest positive particle is produced when the discharge tube contains hydrogen — this is the proton (named by Rutherford, 1919).
Charge: +1.602 × 10⁻¹⁹ C (equal in magnitude to the electron's charge).
Mass: 1.6726 × 10⁻²⁷ kg ≈ 1837 times the mass of an electron.
Charge/mass ratio (e/m) of the proton is the largest for any positive ion of a given gas, because hydrogen has the smallest mass.

Rutherford's gold-foil experiment (1911)

Bombarded thin gold foil with α-particles. Most passed straight through, a few deflected, very few bounced back. Conclusions: the atom is mostly empty space; the positive charge and almost all the mass are concentrated in a tiny dense nucleus; electrons orbit the nucleus.

Planck's Quantum Theory

Max Planck (1900) proposed that energy is emitted or absorbed not continuously but in discrete packets called quanta. For light, each quantum is a photon.

Key equations

Energy of one photon: E = hν, where h = Planck's constant = 6.626 × 10⁻³⁴ J·s and ν is frequency in Hz.

Speed of light relation: c = νλ — so E = hc/λ.

Energy of n photons: E = nhν.

This quantisation explained black-body radiation, the photoelectric effect (Einstein, 1905) and ultimately the line spectra of atoms.

Spectrum of Hydrogen

When a sample of hydrogen gas is excited by an electric discharge, it emits a series of discrete lines — not a continuous spectrum. Niels Bohr (1913) explained this by postulating that the electron occupies fixed circular orbits with quantised energy E_n = −13.6/n² eV, and that emission/absorption occurs only when the electron jumps between these orbits.

Spectral series

Lyman series — transitions ending at n = 1; in the UV region.
Balmer series — transitions ending at n = 2; in the visible region (the lines Balmer himself observed).
Paschen series — transitions ending at n = 3; in the infrared.
Brackett series — ending at n = 4; far IR.
Pfund series — ending at n = 5; far IR.

Rydberg formula

1/λ = R_H (1/n₁² − 1/n₂²), where R_H = 1.097 × 10⁷ m⁻¹, n₁ < n₂. n₁ is the lower (final) level for emission; n₂ the upper (initial). Gives the wavelength of every line in the H spectrum.

Common trap. Bohr's model only works for one-electron systems (H, He⁺, Li²⁺). It fails for multi-electron atoms because it ignores electron–electron repulsion. The exam favourite "Bohr explains the spectrum of sodium" is FALSE.

Quantum Numbers

Four quantum numbers fully specify the state of every electron in an atom. Two electrons in the same atom can never have the same set of all four (Pauli's principle).

The four quantum numbers

Principal quantum number (n): Determines the main energy level / shell. Allowed values: 1, 2, 3, …. Larger n means higher energy and larger orbital. Maximum number of electrons in shell n = 2n².
Azimuthal / angular momentum quantum number (l): Determines the subshell shape. Allowed values: 0, 1, …, (n − 1). Subshell letters: l = 0 → s, l = 1 → p, l = 2 → d, l = 3 → f. Number of subshells in shell n = n.
Magnetic quantum number (m or m_l): Determines the orbital orientation in space. Allowed values: −l, −l+1, …, 0, …, +l. Number of orbitals in subshell l = (2l + 1).
Spin quantum number (s or m_s): Spin direction of electron. Allowed values: +½ or −½. Each orbital holds at most 2 electrons with opposite spin.

Counts to memorise

Quantum numbers — values, meaning, electron capacity
Number	Symbol	Allowed values	Tells us	Capacity rule
Principal	n	1, 2, 3, …	Energy level / shell size	Max electrons in shell = 2n²
Azimuthal	l	0 … (n − 1)	Subshell shape (s/p/d/f)	Number of subshells in shell = n
Magnetic	m_l	−l … 0 … +l	Orbital orientation in space	Orbitals per subshell = (2l + 1)
Spin	m_s	+½ or −½	Electron spin direction	Max 2 e⁻ per orbital (Pauli)

Subshells — capacity at a glance
Subshell	l	Orbitals (2l + 1)	Max electrons
s	0	1	2
p	1	3	6
d	2	5	10
f	3	7	14

Total in shell n = 2n² → 2, 8, 18, 32, …

Shapes of Orbitals

An orbital is a region of space where the probability of finding an electron is high (about 90–95 percent). Each subshell has a characteristic shape determined by l.

s orbitals (l = 0)

Spherically symmetric about the nucleus. 1s has no node; 2s has one radial node; 3s has two radial nodes. All s orbitals are non-directional.

p orbitals (l = 1)

Dumb-bell shaped, with a nodal plane through the nucleus. Three orientations along x, y, z axes — p_x, p_y, p_z. Maximum electron density along the axis; zero at the nucleus.

d orbitals (l = 2)

Five orbitals, mostly four-lobed (d_xy, d_xz, d_yz, d_x²−y²) with d_z² shaped like a dumb-bell with a torus around the middle. Only encountered from the third shell upward.

Electronic Configuration

The arrangement of electrons across orbitals follows three rules.

The three filling rules

Aufbau principle: Electrons fill orbitals in order of increasing energy: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p …. The (n + l) rule predicts the order — lower (n + l) fills first; for ties, the lower n fills first.
Pauli's exclusion principle: No two electrons in an atom can have the same set of all four quantum numbers. Consequence: an orbital holds at most two electrons with opposite spin.
Hund's rule of maximum multiplicity: When electrons fill degenerate (equal-energy) orbitals (p, d, f), they occupy them singly first with parallel spins before any pairing. This minimises electron repulsion.

Configurations of the first 30 elements

H (1): 1s¹ · He (2): 1s².
Li (3): [He] 2s¹ · Be (4): [He] 2s².
B (5): [He] 2s² 2p¹ · C (6): 2s² 2p² · N (7): 2s² 2p³ · O (8): 2s² 2p⁴ · F (9): 2s² 2p⁵ · Ne (10): 2s² 2p⁶.
Na (11): [Ne] 3s¹ · Mg (12): 3s² · Al (13): 3s² 3p¹ · Si (14): 3p² · P (15): 3p³ · S (16): 3p⁴ · Cl (17): 3p⁵ · Ar (18): 3p⁶.
K (19): [Ar] 4s¹ · Ca (20): [Ar] 4s².
Sc (21): [Ar] 3d¹ 4s² · Ti (22): 3d² 4s² · V (23): 3d³ 4s².
Cr (24): [Ar] 3d⁵ 4s¹ (anomaly — half-filled d gives extra stability).
Mn (25): 3d⁵ 4s² · Fe (26): 3d⁶ 4s² · Co (27): 3d⁷ 4s² · Ni (28): 3d⁸ 4s².
Cu (29): [Ar] 3d¹⁰ 4s¹ (anomaly — full d gives extra stability).
Zn (30): [Ar] 3d¹⁰ 4s².

Mnemonic for Aufbau order. Use the diagonal rule: write 1s; 2s 2p; 3s 3p 3d; 4s 4p 4d 4f; … then draw arrows diagonally from upper-right to lower-left. The result — 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, … — is the order of filling.

Worked MCQs

Five MCQs that capture the high-yield testing patterns for this chapter. Read the explanation even when you get the answer right — that's where the deeper concept lives.

Q1. Which series of the hydrogen spectrum lies in the visible region?

Lyman
Balmer
Paschen
Brackett

Transitions ending at n = 2 produce the Balmer series, whose lines (656, 486, 434, 410 nm) fall in the visible spectrum. Lyman (n → 1) is UV; Paschen (n → 3) is IR.

Q2. The maximum number of electrons that can be accommodated in a subshell with l = 2 is:

l = 2 corresponds to a d subshell, which has (2l + 1) = 5 orbitals. Each orbital holds 2 electrons (Pauli), giving 5 × 2 = 10.

Q3. The ground-state electronic configuration of chromium (Z = 24) is:

[Ar] 3d⁴ 4s²
[Ar] 3d⁵ 4s¹
[Ar] 3d⁶ 4s⁰
[Ar] 4s² 4p⁴

Chromium is one of the two famous Aufbau anomalies. A half-filled 3d⁵ set (one electron in each d orbital) is more stable than the predicted 3d⁴ 4s² — an electron promotes from 4s to 3d to give 3d⁵ 4s¹.

Q4. The principle that states "no two electrons in an atom can have the same set of all four quantum numbers" is:

Aufbau principle
Hund's rule
Pauli's exclusion principle
Heisenberg's uncertainty principle

Pauli's exclusion principle limits each orbital to two electrons with opposite spin. Hund deals with degenerate orbital filling; Aufbau with the order of filling; Heisenberg with the impossibility of measuring position and momentum simultaneously.

Q5. The energy of a photon of green light (λ = 500 nm) is approximately:

3.97 × 10⁻³⁰ J
3.97 × 10⁻¹⁹ J
6.63 × 10⁻³⁴ J
3.0 × 10⁸ J

E = hc/λ = (6.63 × 10⁻³⁴)(3 × 10⁸) / (500 × 10⁻⁹) = 3.97 × 10⁻¹⁹ J. This is the order of magnitude of single-photon energies for visible light (a few eV).

Quick Recap

Proton discovered by Goldstein via canal rays; lightest in hydrogen discharge.
Planck: E = hν; energy is quantised into photons.
H spectrum series — Lyman (UV, n → 1), Balmer (visible, n → 2), Paschen (IR, n → 3).
Rydberg: 1/λ = R_H(1/n₁² − 1/n₂²).
Quantum numbers: n (shell), l (shape), m_l (orientation), m_s (spin).
Subshell capacities: s = 2, p = 6, d = 10, f = 14; shell n = 2n².
Filling rules: Aufbau (lowest energy first), Pauli (max 2 e/orbital, opposite spin), Hund (parallel spins fill degenerate orbitals first).
Cr and Cu break the rules — half/full d-shell stability.

Test yourself. Take a timed practice test or browse topic-wise MCQs to lock these concepts in.