CHEM20018 · Chemistry: Reactions And Synthesis
Chemistry of Materials: Bonding, Band Structure & Semiconductors
The materials section connects bonding forces to the electronic and optical behaviour of solids. Coulomb's law, with a dielectric constant εr that screens charges (water εr = 78.5), governs ionic interactions, while the Lennard-Jones (12-6) potential balances long-range attraction against short-range Pauli repulsion to set the equilibrium bond length. Overlapping molecular orbitals from many atoms create energy bands; the size of the band gap distinguishes metals, semiconductors and insulators and controls colour via λ ≈ 1240/Eg. Doping creates n- and p-type semiconductors, and a p–n junction is the heart of a silicon solar cell, whose optimal single-junction band gap is around 1.3 eV.
What this chapter covers
- 01Coulomb's law: U(r) = z₁z₂e²/(4πε₀εrr); dielectric εr screens the interaction (water 78.5 vs oil ~2)
- 02Force F = −dU/dr; comparison of interaction energy in vacuum vs water vs higher-charge ions
- 03Three primary bonds (ionic point-charge + Madelung, metallic electron sea, covalent MO overlap)
- 04Secondary (van der Waals) forces and their distance dependence ladder (1/r through 1/r⁶)
- 05Lennard-Jones (12-6) potential U(r) = B/r¹² − A/r⁶; equilibrium bond length r₀ = (2B/A)^(1/6)
- 06Band structure from MO overlap: valence band, conduction band, band gap Eg; metals/semiconductors/insulators
- 07Band gap ↔ colour: λ(nm) ≈ 1240/Eg(eV); doping (group V → n-type, group III → p-type)
- 08p–n junction and the silicon solar cell; optimal single-junction band gap ≈ 1.3 eV
Coulomb interaction energy in vacuum vs water, and a band-gap colour estimate
- 2 marks — correct substitution and vacuum value(a)(i) U = z₁z₂e²/(4πε₀εrr). In vacuum εr = 1, z₁z₂ = (+1)(−1) = −1, r = 0.30×10⁻⁹ m. Numerator e² = (1.602×10⁻¹⁹)² = 2.566×10⁻³⁸; denominator 4π(8.854×10⁻¹²)(1)(0.30×10⁻⁹) = 3.337×10⁻²⁰. U = −2.566×10⁻³⁸ ÷ 3.337×10⁻²⁰ = −7.69×10⁻¹⁹ J (attractive).
- 2 marks — value + the physical interpretation against k<sub>B</sub>T(a)(ii) In water divide by εr = 78.5: U = −7.69×10⁻¹⁹ ÷ 78.5 = −9.8×10⁻²¹ J. This is now comparable to thermal energy kBT ≈ 4.1×10⁻²¹ J, so water screening lets the ions separate (the salt dissolves).
- 2 marks — factor-of-4 scaling and repulsive sign(b) Two Mg²⁺ ions have z₁z₂ = (+2)(+2) = +4, four times the magnitude and positive (repulsive): U = +4 × 7.69×10⁻¹⁹ = +3.08×10⁻¹⁸ J. So the energy scales directly with the charge product and changes sign with charge signs.
- 2 marks — wavelength and colour(c) λ(nm) ≈ 1240 ÷ Eg(eV) = 1240 ÷ 2.0 = 620 nm, which lies in the orange-red region of the visible spectrum, so the emitted light is orange-red.
Key terms
- Dielectric constant (εr)
- The factor by which a medium reduces the Coulomb interaction between charges; high-εr solvents such as water (78.5) strongly screen ionic attractions and promote dissolution.
- Lennard-Jones potential
- The 12-6 model U(r) = B/r¹² − A/r⁶ balancing short-range Pauli/Born repulsion (B term) against longer-range dispersion attraction (A term), with minimum at r₀ = (2B/A)^(1/6).
- Band gap
- The energy gap Eg between the top of the valence band and the bottom of the conduction band; small for semiconductors, large for insulators, near zero for metals.
- Doping
- Adding trace impurity atoms to a semiconductor: group V donors give n-type (extra electrons), group III acceptors give p-type (extra holes).
- p–n junction
- The interface between p-type and n-type semiconductors that rectifies current and, when it absorbs light, separates charge to generate photovoltage — the basis of a silicon solar cell.
Chemistry of Materials: Bonding, Band Structure & Semiconductors FAQ
Why does table salt dissolve in water but not in oil?
Coulomb energy is screened by the dielectric constant εr. In water (εr = 78.5) the ion-pair attraction falls to roughly thermal energy (kBT), so the ions separate and dissolve. In oil (εr ≈ 2) the attraction stays far above kBT, so the lattice holds together and the salt is insoluble.
How does band gap relate to colour?
A material absorbs photons with energy at or above its band gap. Using λ(nm) ≈ 1240/Eg(eV), the gap fixes the absorption (or emission) wavelength. The colour you perceive in transmission or reflection is the complement of what is absorbed; for emission/luminescence it is the colour at that wavelength directly.
Why is there an optimal band gap for a single-junction solar cell?
It is a trade-off. A small band gap absorbs more of the solar spectrum (more photons, higher current) but gives a low output voltage; a large band gap gives a high voltage but absorbs few photons (low current). Power is current × voltage, so the product peaks at an intermediate value, around Eg ≈ 1.3 eV for a single junction.
Exam move
This section rewards quick plug-and-chug plus a few physical-intuition statements. Drill the Coulomb calculation in SI units, always tracking the charge product (magnitude and sign) and comparing the result with kBT ≈ 4×10⁻²¹ J to judge whether an interaction matters. Be able to derive r₀ = (2B/A)^(1/6) from the Lennard-Jones potential by setting dU/dr = 0. On the electronic side, memorise the band picture (metal/semiconductor/insulator by gap size), the λ ≈ 1240/Eg conversion, the doping rules (group V → n-type, group III → p-type), and the solar-cell band-gap trade-off argument. The constants are supplied in the appendix, so the marks come from clean execution and the right one-line explanations — practise the mock Section E questions to confirm both.
