For a long time, atoms were considered the smallest indivisible unit of matter. Experiments in the late 19th and early 20th centuries revealed that atoms are themselves made of smaller particles — electrons, protons, and neutrons.
J.J. Thomson studied cathode ray discharge tubes — a gas at low pressure sealed inside a tube with electrodes, subjected to high voltage. The rays travelling from the cathode to the anode were found to bend towards a positive plate in an electric field, showing they carry a negative charge. Thomson calculated the charge-to-mass ratio ($e/m$) of these particles, called electrons, as $1.76 \times 10^{8}\ \text{C/g}$.
Later, R.A. Millikan's oil drop experiment determined the actual charge on an electron: $e = 1.6 \times 10^{-19}\ \text{C}$. Combining this with Thomson's ratio gave the electron's mass: $9.1 \times 10^{-28}\ \text{g}$ — about 1/1837th the mass of a hydrogen atom.
Using a modified cathode ray tube with a perforated cathode, Goldstein observed rays travelling in the direction opposite to cathode rays — called canal rays or anode rays. These carried a positive charge, and the lightest such particle (obtained using hydrogen gas) was named the proton, with a mass of $1.672 \times 10^{-24}\ \text{g}$ (about 1837 times that of an electron) and a charge of $+1.6 \times 10^{-19}\ \text{C}$.
James Chadwick discovered the neutron by bombarding a beryllium sheet with alpha particles, producing a neutral particle with mass almost equal to that of a proton ($1.675 \times 10^{-24}\ \text{g}$) and no charge. This explained the mass discrepancy that protons and electrons alone couldn't account for.
| Particle | Charge | Mass | Discovered by |
|---|---|---|---|
| Electron | −1 (−1.6×10⁻¹⁹ C) | 9.1×10⁻²⁸ g | J.J. Thomson |
| Proton | +1 (+1.6×10⁻¹⁹ C) | 1.672×10⁻²⁴ g | Goldstein |
| Neutron | 0 | 1.675×10⁻²⁴ g | Chadwick |
A neutral atom also has electrons equal in number to its protons.
Isotopes — same atomic number (Z), different mass number (A); e.g., $^1_1\text{H}$, $^2_1\text{H}$, $^3_1\text{H}$. Isobars — same mass number, different atomic number; e.g., $^{40}_{18}\text{Ar}$ and $^{40}_{20}\text{Ca}$. Isotones — same number of neutrons, different atomic and mass numbers; e.g., $^{14}_{6}\text{C}$ and $^{16}_{8}\text{O}$ (both have 8 neutrons).
J.J. Thomson proposed that an atom is a sphere of uniform positive charge with electrons embedded in it, like seeds in a watermelon (or "plum pudding"). It successfully explained the overall neutrality of the atom but failed to explain later experimental results like the scattering of alpha particles.
In the famous gold foil (alpha-particle scattering) experiment, a beam of alpha particles was directed at a thin gold foil. Most particles passed straight through, some deflected at small angles, and a very small fraction bounced almost straight back.
Most of the atom is empty space (since most particles passed through undeflected). A small, dense, positively charged centre — the nucleus — repelled the few particles that came close to it, causing large-angle deflections. Electrons revolve around this nucleus in circular orbits, held by electrostatic attraction.
Rutherford's model couldn't explain why revolving electrons — which are accelerating charges and should continuously emit energy according to classical electromagnetic theory — don't spiral into the nucleus. It also failed to explain the line spectrum of hydrogen.
Light and other forms of radiant energy travel as electromagnetic waves — oscillating electric and magnetic fields perpendicular to each other and to the direction of travel. All electromagnetic radiation travels through vacuum at the same speed, $c = 3 \times 10^8\ \text{m/s}$.
where $\nu$ = frequency (Hz or s⁻¹), $\lambda$ = wavelength (m), and $\bar\nu$ = wavenumber (m⁻¹).
The complete range of electromagnetic radiation — from long-wavelength radio waves to short-wavelength gamma rays — is called the electromagnetic spectrum, and includes radio, microwave, infrared, visible, ultraviolet, X-ray, and gamma regions in increasing order of frequency (and energy).
Classical physics couldn't explain why heated bodies emit radiation of certain frequencies (black-body radiation) or why light striking a metal knocks out electrons only above a certain frequency. Max Planck resolved this by proposing that energy is emitted or absorbed in discrete packets called quanta (photons, in the case of light), not continuously.
where $h$ = Planck's constant $= 6.626 \times 10^{-34}\ \text{J s}$.
When light of sufficient frequency strikes a metal surface, electrons are ejected instantly — but only if the frequency of light exceeds a minimum value called the threshold frequency ($\nu_0$), regardless of the light's intensity. Increasing intensity beyond that threshold increases the number of electrons ejected, not their individual energy; increasing frequency increases the kinetic energy of each ejected electron.
Energy of incident photon = work function ($h\nu_0$, the minimum energy to eject an electron) + kinetic energy of the ejected electron.
When hydrogen gas is excited electrically and the emitted light is passed through a prism, it produces not a continuous rainbow but a series of sharp, separate lines — a line (atomic) spectrum. This was the key evidence Bohr's model needed to explain, which Rutherford's model could not.
where $n$ = orbit number, $Z$ = atomic number. The negative sign shows the electron is bound to the nucleus; energy becomes less negative (higher) as $n$ increases.
Calculate the radius of the second Bohr orbit of hydrogen.
$r_2 = 0.529 \times \dfrac{2^2}{1} = 0.529 \times 4 = 2.116\ \overset{\circ}{\text{A}}$
It works well only for single-electron species (H, He⁺, Li²⁺) and fails to explain the spectra of multi-electron atoms, the splitting of spectral lines in a magnetic field (Zeeman effect), or the wave nature of electrons — limitations resolved later by the quantum mechanical model.
Louis de Broglie proposed that just as light shows both wave and particle behaviour, matter (electrons, protons, and even larger objects) also has an associated wavelength.
where $p = mv$ is momentum. This wavelength is only significant (measurable) for very small masses, like electrons — for everyday objects it's far too small to detect.
It is impossible to simultaneously determine, with perfect precision, both the exact position and exact momentum of a microscopic particle like an electron. This isn't a limitation of measuring instruments — it's a fundamental property of nature at that scale, and directly follows from the wave nature of matter.
where $\Delta x$ = uncertainty in position, $\Delta p$ = uncertainty in momentum. This principle led directly to the idea of orbitals (regions of probability) replacing Bohr's fixed, well-defined orbits.
Since Bohr's fixed orbits gave way to the quantum mechanical model, an electron's position and energy are now described using four quantum numbers.
| Quantum Number | Symbol | Describes | Allowed Values |
|---|---|---|---|
| Principal | n | Shell / energy level & size of orbital | 1, 2, 3... |
| Azimuthal (angular) | l | Sub-shell & shape of orbital | 0 to (n−1) |
| Magnetic | mₗ | Orientation of orbital in space | −l to +l (including 0) |
| Spin | mₛ | Direction of electron spin | +½ or −½ |
Sub-shells corresponding to $l = 0, 1, 2, 3$ are named s, p, d, f respectively. For a given $n$, the number of orbitals in a sub-shell is $(2l+1)$, and the total number of orbitals in a shell is $n^2$.
For n = 3, list the possible values of l and the sub-shells present.
$l$ can be $0, 1, 2$ → sub-shells present are $3s, 3p, 3d$.
A node is a region where the probability of finding an electron is zero. Radial nodes $= n - l - 1$; angular nodes $= l$; total nodes $= n - 1$.
Electrons occupy orbitals in order of increasing energy — lower-energy orbitals fill first. The order generally follows the $(n+l)$ rule: orbitals with lower $(n+l)$ values fill first; if two orbitals have the same $(n+l)$ value, the one with lower $n$ fills first.
No two electrons in the same atom can have all four quantum numbers identical. This means a single orbital can hold a maximum of 2 electrons, and they must have opposite spins.
Electrons are distributed among orbitals of the same sub-shell (equal energy) singly first, with parallel spins, before any orbital receives a second electron. Pairing begins only once every orbital in that sub-shell has one electron.
The electronic configuration of an atom shows how its electrons are distributed across shells and sub-shells, written using the notation $n l^{x}$ (e.g., $1s^2\, 2s^2\, 2p^6$ for neon).
Half-filled ($p^3, d^5, f^7$) and fully-filled ($p^6, d^{10}, f^{14}$) sub-shells have extra stability due to two effects: greater symmetry of electron distribution, and greater exchange energy — the stabilising energy released when electrons with the same spin in degenerate orbitals exchange positions, which is maximum when the sub-shell is exactly half or fully filled.
Chromium (Z = 24) is $[\text{Ar}]\, 3d^5\, 4s^1$, not $3d^4\, 4s^2$ — one electron shifts from 4s to 3d to achieve a stable half-filled d sub-shell. Similarly, copper (Z = 29) is $[\text{Ar}]\, 3d^{10}\, 4s^1$, not $3d^9\, 4s^2$, to achieve a stable fully-filled d sub-shell.