What this chapter is about
This chapter explores how scientists discovered and described the internal structure of the atom. You will learn about the experiments that revealed subatomic particles—electrons, protons and neutrons—and how models of the atom evolved from Thomson's plum-pudding model through Rutherford's nuclear model to Bohr's quantised orbits and finally to the modern quantum mechanical model.
The chapter introduces you to the dual nature of matter and radiation, where particles can behave like waves and light can behave like particles. You will study how electrons are arranged in atoms using quantum numbers and orbitals, and how the shapes of s, p and d orbitals determine where electrons are most likely to be found.
After studying this chapter, you should be able to calculate wavelengths, frequencies and energies of radiation, write electronic configurations of elements using the aufbau principle, Pauli exclusion principle and Hund's rule, and explain why atoms emit light of specific colours when heated.
Key ideas
- Discovery of subatomic particles: Cathode ray experiments showed electrons exist in all matter; canal ray experiments revealed protons; Chadwick discovered neutrons in 1932.
- Rutherford's nuclear model: Most of the atom is empty space; the positive charge and nearly all the mass are concentrated in a tiny nucleus; electrons orbit this nucleus.
- Electromagnetic radiation: Light travels as waves characterised by wavelength (λ), frequency (ν) and speed (c = 3 × 10⁸ m/s), related by c = νλ.
- Quantisation of energy: Planck proposed that energy is emitted or absorbed in discrete packets called quanta; the energy of one quantum is E = hν, where h = 6.626 × 10⁻³⁴ J·s.
- Bohr's model of hydrogen: Electrons occupy fixed circular orbits with quantised angular momentum (mvr = nh/2π); they emit or absorb light only when jumping between orbits.
- de Broglie relation: All matter has wave character; the wavelength of a particle is λ = h/mv, where m is mass and v is velocity.
- Heisenberg uncertainty principle: It is impossible to know both the exact position and exact momentum of an electron simultaneously; the uncertainty is given by Δx × Δp ≥ h/4π.
- Quantum mechanical model: Electrons exist in orbitals (not orbits); each orbital is described by four quantum numbers (n, l, mₗ, mₛ) and has a characteristic shape and energy.
Formulas and facts to remember
- Formula or fact: c = νλ · Meaning: Speed of light equals frequency times wavelength.
- Formula or fact: E = hν = hc/λ · Meaning: Energy of a photon depends on its frequency.
- Formula or fact: λ = h/mv · Meaning: de Broglie wavelength of a particle with mass m and velocity v.
- Formula or fact: Δx × Δp ≥ h/4π · Meaning: Heisenberg uncertainty relation for position and momentum.
- Formula or fact: Eₙ = −13.6/n² eV · Meaning: Energy of an electron in the nth orbit of hydrogen (Bohr model).
- Formula or fact: rₙ = 0.529 × n² Å · Meaning: Radius of nth Bohr orbit for hydrogen.
- Formula or fact: Aufbau principle · Meaning: Electrons fill orbitals in order of increasing energy: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, …
- Formula or fact: Pauli exclusion principle · Meaning: No two electrons in an atom can have the same set of all four quantum numbers.
- Formula or fact: Hund's rule · Meaning: Electrons occupy degenerate orbitals singly first, with parallel spins, before pairing.
Worked examples
Example 1: Calculating photon energy
Problem: Yellow light from a sodium lamp has a wavelength of 589 nm. Calculate the energy of one photon.
Solution:
Step 1: Convert wavelength to metres. λ = 589 nm = 589 × 10⁻⁹ m
Step 2: Use E = hc/λ. E = (6.626 × 10⁻³⁴ J·s × 3 × 10⁸ m/s) / (589 × 10⁻⁹ m) E = (1.988 × 10⁻²⁵ J·m) / (5.89 × 10⁻⁷ m) E = 3.37 × 10⁻¹⁹ J
The energy of one photon of yellow sodium light is approximately 3.4 × 10⁻¹⁹ J.
Example 2: de Broglie wavelength of a moving ball
Problem: A cricket ball of mass 160 g is bowled at 144 km/h. Calculate its de Broglie wavelength.
Solution:
Step 1: Convert units. Mass m = 160 g = 0.16 kg Velocity v = 144 km/h = 144 × 1000 / 3600 m/s = 40 m/s
Step 2: Apply λ = h/mv. λ = 6.626 × 10⁻³⁴ / (0.16 × 40) λ = 6.626 × 10⁻³⁴ / 6.4 λ = 1.04 × 10⁻³⁴ m
This wavelength (about 10⁻³⁴ m) is far too small to observe, which is why we do not notice wave behaviour in everyday objects.
Example 3: Writing electronic configuration
Problem: Write the electronic configuration of iron (atomic number 26).
Solution:
Step 1: Fill orbitals in order of increasing energy using the aufbau principle. 1s² → 2s² → 2p⁶ → 3s² → 3p⁶ → 4s² → 3d⁶
Step 2: Count total electrons: 2 + 2 + 6 + 2 + 6 + 2 + 6 = 26. Correct.
The electronic configuration of Fe is 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁶ 4s², or in short form [Ar] 3d⁶ 4s².
Note: The 3d orbitals have six electrons; by Hund's rule, four are unpaired (one each in four orbitals) and two are paired in one orbital.
Common mistakes
- Confusing wavelength and frequency → Remember they are inversely related: higher frequency means shorter wavelength.
- Using wrong units in de Broglie equation → Always convert mass to kg, velocity to m/s, and use h in J·s.
- Writing 3d before 4s in ground-state configurations → Although 3d is lower in principal quantum number, 4s fills first in multi-electron atoms; fill by energy, not by n value alone.
- Thinking orbitals are circular paths → Orbitals are probability regions where electrons are likely to be found, not fixed tracks like planetary orbits.
- Violating Hund's rule when distributing electrons → Do not pair electrons in degenerate orbitals until each orbital has one electron with parallel spin.
Quick revision
- Atom has a tiny, dense, positively charged nucleus; electrons occupy space around it.
- Light is both wave (characterised by λ and ν) and particle (photon with energy E = hν).
- Electrons in atoms have wave nature; their behaviour is described by orbitals, not orbits.
- Four quantum numbers (n, l, mₗ, mₛ) fully specify an electron's state.
- Electronic configurations follow aufbau, Pauli exclusion and Hund's rule.
- Heisenberg's principle sets a fundamental limit on how precisely we can know position and momentum together.