Structure of Atom

1. Discovery of Sub-atomic Particles

Dalton's atomic theory treated the atom as the smallest indivisible unit of matter. Experiments with electricity passed through gases in the late 1800s overturned this — atoms were shown to be built from smaller, charged pieces.

• Discovery of the electron: J.J. Thomson passed high voltage through a partially evacuated glass tube fitted with two electrodes. The stream of particles flowing from the cathode to the anode — cathode rays — travelled in straight lines, were deflected by electric and magnetic fields exactly as negatively charged particles would be, and behaved identically regardless of the electrode material or the gas used. Thomson concluded these particles, electrons, are a universal constituent of every atom. He measured the charge-to-mass ratio: $\dfrac{e}{m_e} = 1.758820 \times 10^{11}\text{ C kg}^{-1}$.
• Charge on the electron: R.A. Millikan's oil-drop experiment (1906–14) measured the charge directly: $e = -1.602176 \times 10^{-19}\text{ C}$. Combining this with Thomson's ratio gives the electron's mass, $m_e = 9.1094 \times 10^{-31}\text{ kg}$.
• Discovery of the proton: Using a perforated-anode cathode ray tube produced canal rays — positively charged particles whose mass depended on the gas used, unlike electrons. The lightest of these, from hydrogen gas, was named the proton.
• Discovery of the neutron: Chadwick (1932) bombarded beryllium with α-particles and detected a neutral particle of mass slightly greater than the proton's — the neutron. Its discovery explained why atomic mass exceeds what protons alone could account for.

2. Thomson & Rutherford Atomic Models

Thomson's 1898 model pictured the atom as a uniform sphere of positive charge with electrons embedded in it — often called the plum pudding model. It correctly explained overall electrical neutrality but could not survive the next experiment.

Rutherford directed a beam of α-particles at a thin gold foil. If Thomson's model were correct, the particles should have passed through with only minor deflection, since mass and charge were assumed spread evenly. Instead:

1. Most α-particles passed straight through — meaning most of the atom is empty space.
2. A small fraction deflected at moderate angles.
3. A tiny fraction (roughly 1 in 20,000) bounced almost straight back — meaning they met an extremely concentrated, positively charged mass.

From this, Rutherford proposed the nuclear model: a tiny, dense, positively charged nucleus at the centre, with electrons orbiting it at high speed, held by electrostatic attraction — much like planets orbiting the sun.

3. Atomic Number, Mass Number, Isotopes & Isobars

$$\text{Atomic number } (Z) = \text{number of protons} = \text{number of electrons (in a neutral atom)}$$

$$\text{Mass number } (A) = \text{number of protons } (Z) + \text{number of neutrons } (n)$$

A nuclide is written as $^{A}_{Z}\text{X}$. Given any two of $Z$, $A$, and neutron count, the third follows directly from $A = Z + n$.

Isotopes of an element show essentially identical chemical behaviour, because chemistry is governed by electron count (hence proton count), not neutron count.

4. Electromagnetic Radiation & Planck's Quantum Theory

Electromagnetic radiation consists of oscillating electric and magnetic fields travelling through space (or vacuum) without needing a medium. It is described by:

$$c = \nu \lambda$$

where $c$ is the speed of light ($3.0 \times 10^8\text{ m/s}$), $\nu$ is frequency (Hz), and $\lambda$ is wavelength. Wavenumber, $\bar{\nu} = 1/\lambda$, is also commonly used, especially in spectroscopy.

Classical wave theory could not explain black-body radiation, the photoelectric effect, or line spectra. Max Planck's 1900 resolution: energy is not absorbed or emitted continuously — it comes in discrete packets called quanta, each carrying energy

$$E = h\nu$$

5. Photoelectric Effect & Dual Nature of Radiation

When light strikes certain metal surfaces, electrons are ejected instantly — with no measurable time lag. Three observations classical wave theory could not explain:

• Ejection happens only above a threshold frequency $\nu_0$, specific to each metal — brighter light below $\nu_0$ ejects nothing at all.
• Above $\nu_0$, the number of electrons ejected depends on the intensity of light.
• The kinetic energy of each ejected electron depends only on frequency, not intensity.

Einstein (1905) explained this using Planck's quantum idea: light itself arrives as discrete photons, each of energy $h\nu$. A photon transfers its entire energy to one electron instantly:

$$h\nu = h\nu_0 + \dfrac{1}{2}m_e v^2$$

Here $h\nu_0$ is the work function ($W_0$) — the minimum energy needed to free an electron from that particular metal. This dual explanation (light behaving as both wave and particle depending on the experiment) is the dual nature of electromagnetic radiation.

6. Bohr's Model & Line Spectrum of Hydrogen

Bohr (1913) rescued the nuclear model by combining it with Planck's quantisation idea. His postulates:

1. Electrons move in fixed circular paths of definite energy, called orbits or stationary states — while in an orbit, energy does not change.
2. Energy is absorbed or emitted only when an electron jumps between orbits, and this happens in discrete amounts, not continuously.
3. The angular momentum of the electron is quantised: $m_e v r = n \cdot \dfrac{h}{2\pi}$, where $n = 1, 2, 3\ldots$ is the principal quantum number.

For hydrogen, the energy of the $n^{\text{th}}$ orbit is $E_n = -R_H \left(\dfrac{1}{n^2}\right)$, where $R_H = 2.18 \times 10^{-18}\text{ J}$. The negative sign shows the electron is more stable (lower energy) than a free electron at rest ($n = \infty$, $E_\infty = 0$).

When an excited electron falls from a higher orbit ($n_i$) to a lower one ($n_f$), it emits a photon of a specific frequency, producing the line spectrum of hydrogen — bright lines at only certain wavelengths, not a continuous rainbow. Different series (named after their discoverers) correspond to transitions ending at different final orbits:

7. Dual Nature of Matter & Heisenberg's Uncertainty Principle

If radiation can behave like particles, de Broglie (1924) proposed the reverse: matter should also have wave-like character. Every moving particle has an associated wavelength:

$$\lambda = \dfrac{h}{mv} = \dfrac{h}{p}$$

This is undetectable for everyday objects (their mass is far too large), but measurable for electrons — confirmed experimentally when electron beams were shown to diffract, just like waves.

Heisenberg (1927) then showed a fundamental limit on precision: it is impossible to simultaneously know both the exact position and exact momentum of an electron.

$$\Delta x \cdot \Delta p \geq \dfrac{h}{4\pi}$$

This single principle is what rules out Bohr's idea of a fixed circular "orbit" — a defined path requires knowing position and velocity together at every instant, which quantum mechanics says is simply not possible for a particle this small. This is why modern theory speaks only of the probability of finding an electron in a region, never a definite path.

8. Quantum Numbers & Shapes of Orbitals

Because a definite orbit is impossible, Schrödinger's wave equation (1926) replaced Bohr's fixed paths with orbitals — three-dimensional regions described by a wave function $\psi$, where $|\psi|^2$ gives the probability density of finding the electron at a point. Solving this equation for the hydrogen atom naturally produces three quantum numbers:

• Principal quantum number ($n$) — positive integer; determines shell size and, to a large extent, energy. Shells: $n = 1, 2, 3, 4\ldots$ correspond to K, L, M, N.
• Azimuthal quantum number ($l$) — ranges $0$ to $n-1$; determines subshell and orbital shape. $l = 0, 1, 2, 3$ correspond to s, p, d, f subshells.
• Magnetic quantum number ($m_l$) — ranges $-l$ to $+l$, giving $2l+1$ values; determines orbital orientation in space.
• Spin quantum number ($m_s$) — $+\tfrac{1}{2}$ or $-\tfrac{1}{2}$; describes the electron's intrinsic spin. No two electrons in the same orbital can share the same spin (Pauli exclusion, next section).

s orbitals are spherically symmetric at every $n$. p orbitals are dumbbell-shaped with two lobes, oriented along the x, y, or z axis ($p_x$, $p_y$, $p_z$) — three degenerate orbitals per subshell. d orbitals mostly have four lobes each, except $d_{z^2}$, which has a distinct shape — five degenerate orbitals per subshell.

9. Aufbau Principle, Pauli Exclusion & Hund's Rule

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

• Aufbau principle: electrons occupy the lowest-energy orbital available first. For multi-electron atoms, energy depends on both $n$ and $l$ together — the (n + l) rule predicts the order: lower $(n+l)$ fills first; if two subshells tie, lower $n$ fills first. This is why $4s$ (n+l = 4) fills before $3d$ (n+l = 5).
• Pauli exclusion principle: no two electrons in the same atom can share all four quantum numbers. Practically: an orbital holds at most 2 electrons, and they must have opposite spins.
• Hund's rule of maximum multiplicity: within a subshell, electrons occupy separate orbitals singly (with parallel spin) before any orbital gets a second electron. Pairing starts only once every orbital in that subshell already has one electron.

10. Electronic Configuration & Stability of Filled Subshells

The electronic configuration of an atom is the specific distribution of its electrons across orbitals, written as $1s^2\,2s^2\,2p^6\ldots$ etc. Two well-known exceptions to the straightforward Aufbau order occur because half-filled and fully-filled subshells carry extra stability (from symmetrical charge distribution and greater exchange energy):

• Chromium (Cr, Z = 24): expected $[Ar]\,3d^4\,4s^2$, actual $[Ar]\,3d^5\,4s^1$ — a half-filled $3d$ is more stable than a "nearly-half-filled" one.
• Copper (Cu, Z = 29): expected $[Ar]\,3d^9\,4s^2$, actual $[Ar]\,3d^{10}\,4s^1$ — a completely-filled $3d$ wins out the same way.

Electrons in fully-occupied inner shells are called core electrons; those in the outermost, highest-$n$ shell are valence electrons — it's the valence electrons that govern nearly all of an element's chemical behaviour.