Topic 1
Thermal expansion
Heat something and it gets bigger. The interesting part is why, and the exam-worthy part is that holes, gaps, and cavities get bigger too — they expand exactly as if they were filled with the surrounding material.
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Why materials expand
Atoms in a solid sit in a potential energy well, vibrating about an equilibrium separation. If that well were a symmetric parabola, heating would just widen the vibration and the average separation would not change — solids would not expand at all.
Real interatomic potentials are asymmetric: steep on the close-in side (atoms repel hard when pushed together) and shallow on the far side. Add energy and the vibration widens more to the right than to the left, so the average separation creeps outward. That creep, summed over billions of bonds, is thermal expansion.
Linear expansion
For a rod, wire, rail, or any length L0, the change in length is proportional to both the original length and the temperature change:
Two things to notice. First, ΔL scales with L0 — a 1 km bridge moves a thousand times as far as a 1 m ruler for the same temperature swing, which is why long structures need expansion joints and short ones do not. Second, α values are around 10−5 K−1, so fractional changes are tiny; keep your significant figures.
| Material | α (10−6 K−1) | Note |
|---|---|---|
| Lead | 29 | Soft metals expand a lot |
| Aluminum | 24 | |
| Brass | 19 | The “expands more” half of most bimetallic strips |
| Copper | 17 | |
| Steel / iron | 12 | Matches concrete — that is why reinforced concrete works |
| Concrete | 12 | |
| Glass (ordinary) | 9 | Cracks under thermal shock |
| Pyrex / borosilicate | 3.3 | Low α ⇒ survives oven-to-counter |
| Invar (Ni–Fe alloy) | 1.2 | Engineered to barely move |
Steel and concrete have nearly identical α. If they did not, every summer–winter cycle would shear the rebar out of the concrete. That coincidence is the reason the material exists.
Area and volume expansion
Expansion is isotropic in an ordinary solid: every dimension grows by the same fractional amount. So a square of side L grows to L(1 + αΔT) on each side, and its area becomes L2(1 + αΔT)2. Expanding that square and dropping the (utterly negligible) α2 term:
The factors of 2 and 3 are just “how many dimensions are growing.” They are not extra physics, and they are worth being able to re-derive in one line rather than memorizing.
| Liquid | β (10−6 K−1) |
|---|---|
| Mercury | 182 |
| Water (at 20 °C) | 207 |
| Glycerin | 485 |
| Gasoline | 950 |
| Ethanol | 1120 |
The hole rule
This is the single most-tested idea on the page.
A hole expands as if it were made of the surrounding material. Heat a metal plate with a hole in it and the hole gets bigger, not smaller.
The reason is that expansion is a uniform scaling. Imagine photocopying the plate at 100.2% — every feature, including the empty ones, scales up. Nothing about the material “fills in” the hole.
Practical consequence: a stuck metal jar lid loosens under hot water partly because the lid’s inner diameter grows. And to fit a tight ring onto a shaft, you heat the ring — or cool the shaft.
Bimetallic strips
Bond two metals with different α face to face. Heat the pair and the high-α metal wants to be longer than the low-α one, but they are glued together — so the strip bends, curving toward the metal that expands less. Cool it below the bonding temperature and it curves the other way.
That bend is a temperature-controlled switch, which is what an old thermostat, a toaster timer, and a car turn-signal flasher all are.
Water: the exception that matters
Between 0 °C and 4 °C, water contracts as you heat it. Its density peaks at about 4 °C and falls off in both directions. Above 4 °C it behaves normally.
Two consequences worth being able to explain in words:
- Lakes freeze from the top down. As surface water cools toward 4 °C it gets denser and sinks. Below 4 °C it gets less dense, so the coldest water stays on top and freezes there. The ice then insulates the water beneath, and fish survive the winter.
- Ice floats. Solid water is about 9% less dense than liquid water (917 vs 1000 kg/m³) — far off the bottom of the graph above. Water is one of very few substances whose solid floats on its liquid.
Worked examples
1. Expansion joint on a bridge
A steel bridge span is 1.20 km long. Local temperatures run from −15 °C in winter to +40 °C in summer. How much gap must the expansion joints absorb?
Setup. αsteel = 12×10−6 K−1, L0 = 1200 m, ΔT = 40 − (−15) = 55 K.
Nearly 80 cm. Note that ΔT is a difference, so it is the same number in K or °C — no conversion needed here.
2. Does the hole get bigger?
An aluminum plate at 20 °C has a circular hole of diameter 2.000 cm. The plate is heated to 120 °C. Find the new diameter.
Key move. Treat the hole’s diameter as an ordinary length in aluminum. Do not try to reason about the metal “squeezing inward” — it does not.
Follow-up they like to ask: a steel bolt of diameter 2.002 cm will not fit at 20 °C but slides through once the plate is hot.
3. Overflow — when the container expands too
A 500 mL ordinary-glass beaker is filled to the brim with water at 20 °C, then heated to 80 °C. How much water spills?
Key move. The container expands too, so the spill is the difference of the two volume expansions, not the water’s alone.
ΔVglass = 3(9×10−6)(500)(60) = 0.81 mL Glass is quoted with α, so use β = 3α = 27×10−6 K−1.
Spill = 6.21 − 0.81 = 5.4 mL. If the question had used Pyrex (α = 3.3), the glass term would be 0.30 mL and the spill 5.9 mL — closer to the water-only answer, because low-expansion glass barely moves.
4. Conceptual: which way does the strip bend?
A strip is brass on top, invar on the bottom, bonded flat at 25 °C. Describe its shape at 5 °C.
Reason it out. Cooling, so both shrink — but brass (α = 19) shrinks much more than invar (α = 1.2). The top layer becomes the shorter one, so the strip curves upward: it bows with the brass on the inside of the curve.
The general rule: the layer on the outside of the curve is always the one that is currently longer. On heating, that is the high-α metal; on cooling, the low-α one.
Checklist
- I can state ΔL = αL0ΔT and say what each symbol is.
- I can explain in one sentence why holes get bigger.
- I can get β = 3α from scratch instead of memorizing it.
- I remember that in an overflow problem the container expands too.
- I can explain why a lake freezes from the top.