How this instrument works
Solids stretch when warmed because atomic bonds behave like lopsided springs rather than ideal ones. An interatomic potential well is steeply repulsive where atoms crowd together and only gently attractive as they draw apart, so heat widens each atom's vibration and nudges its average position outward. Bonds obeying one perfect parabola would hold their spacing at any temperature and nothing would grow. That asymmetry, added up across the lattice, is exactly what α reports: fractional growth per kelvin, usually only parts per million.
Lavoisier and Laplace built their lever-and-microscope dilatometer in 1782 and pinned down metal dilation to precision nobody had reached before; those figures remain recognisable. Steel lands near 12 parts per million per kelvin, copper 17, aluminium 23, window glass 9. Borosilicate cookware sits at 3.3, which is why it shrugs off hot taps that crack soda-lime tumblers. Charles Édouard Guillaume went further in 1896 with Invar, an iron-nickel alloy barely stirring at 1.2 ppm/K, and took the 1920 Nobel Prize for it — awarded, unusually, for metrology. Reinforced concrete rests on one happier coincidence: rebar and its surrounding concrete stretch at nearly identical rates, so hot afternoons do not prise one from another.
ΔL = L₀·α·ΔT is first-order only, and α is itself mildly temperature-dependent — climbing as bonds soften, sinking toward zero near absolute zero, as thermodynamics demands. Over one hundred kelvin or so around room temperature a single tabulated value serves fine; over furnace ranges you want α integrated across your interval, or at least the mean value quoted for it. Two further assumptions bite harder in practice. Movement must be unobstructed, because clamped parts hold their length and develop stress instead. And α must not depend on direction, which fails for wood, graphite and many crystals, where separate coefficients belong to separate axes.
- Put your starting dimension into Original length — millimetres, centimetres, metres, kilometres or feet all work from that field's unit menu.
- Type your material's constant into Linear expansion coefficient (1/K) as a decimal: steel 1.2e-5, copper 1.7e-5, aluminium 2.3e-5, Invar 1.2e-6.
- Give Temperature change (K or °C) as a difference, never a thermometer reading — a climb from 5 °C to 55 °C is 50, not 55.
- Read Change in length, switching it to millimetres for anything you plan to machine, shim or cut a gap for.
- For cooling, make Temperature change negative; Change in length comes back negative too, meaning your part contracts by that much.
Worked example — a 10-metre steel beam in full sun
A 10 m steel beam goes in on a cold morning at 5 °C. By mid-afternoon, sunlit and dark-painted, its metal reads 55 °C. Original length = 10, Linear expansion coefficient = 1.2e-5, Temperature change = 50. Multiply straight through: ΔL = 10 × 1.2e-5 × 50 = 0.006 m, which is 6 mm.
Six millimetres across ten metres sounds ignorable right up to the moment you deny it room. Bolt both ends rigidly and those 6 mm have nowhere to travel, so stress arrives in their place: σ = 200 GPa × 1.2e-5 × 50 ≈ 120 MPa, roughly a third of what S355 structural steel yields at, and ample to bow a slender member sideways. Sliding bearings, slotted holes and toothed joints across a bridge deck exist to swallow those few millimetres rather than argue with them. Run an identical sum on a 300 m tower and a 50 K swing adds about 180 mm of height.
Questions
Does a hole in a heated plate get bigger or smaller?
Bigger. A warmed solid scales like a photographic enlargement — every dimension grows by the same fraction, empty ones included. A 20 mm bore in hot steel widens exactly as a 20 mm steel disc would. That is why a stuck jar lid surrenders to hot water, and why shrink-fit assemblies are put together by heating an outer ring until its bore clears the shaft. Picturing surrounding metal as pressing inward and squeezing a hole shut is far and away the commonest error here, and it gets the sign backwards.
Is 1/K the same as 1/°C, and what does ppm/K mean?
Identical, with no conversion between them. Both describe fractional change per degree of interval, and a kelvin step matches a Celsius degree exactly; only their zero points differ, and a difference erases any offset. Datasheets dress one number several ways — 12 ppm/K, 12 µm/(m·°C) and 1.2e-5 K⁻¹ all say the same thing. Imperial tables in 1/°F run smaller by a factor of 1.8, so steel shows up there as 6.7 ppm/°F. Confirm which scale a table uses before trusting numbers off it.
When should I use 3α instead of α?
Use 3α when your answer is a volume, α when it is a length, and 2α for an area. Each independent direction stretches by an identical fraction, and cross terms are too small to keep, so an isotropic solid's volumetric coefficient is very nearly triple its linear one: steel at 1.2e-5 K⁻¹ linear behaves as 3.6e-5 K⁻¹ by volume. Liquids get tabulated volumetrically from the start — mercury at 1.8e-4 K⁻¹, most oils around 7e-4 — so never divide those by three unless you genuinely want a column height in a narrow tube.
How much force does blocked movement produce?
Stress is the useful figure, and it ignores length entirely: σ = E·α·ΔT. Steel builds about 2.4 MPa for every kelvin denied (200 GPa × 1.2e-5), so a 50 K rise reaches 120 MPa whether your member spans 10 mm or 10 m. Force follows by multiplying stress across the section — 120 MPa over 2000 mm² is 240 kN. Continuously welded rail is laid at a chosen stress-free temperature for exactly this reason, keeping summer compression under what would buckle a track sideways into a sun kink.
Why does my measured growth disagree with this prediction?
Three usual culprits. Your part may not be free to travel: friction at supports, bolted connections or a bonded substrate restrain it, converting some movement into stress. Your temperature figure may describe air rather than metal — a sunlit steel section commonly runs 20 K above a shade reading, and dark paint widens that gap. Or α itself has drifted, since a coefficient tabulated at 20 °C understates most metals at 400 °C. Measure component temperature directly, and source a coefficient quoted across your actual interval.
Are there materials that barely move at all?
Several, and they are engineered that way. Invar, iron with 36% nickel, sits near 1.2e-6 K⁻¹ — a tenth of ordinary steel — because magnetic ordering contracts its lattice just as heat stretches it; Guillaume's alloy still lines cryogenic tanks and stiffens surveying tapes. Fused silica manages roughly 0.55e-6. Glass-ceramics cast for telescope mirror blanks are tuned near 0.02e-6 by blending crystalline and glassy phases whose responses oppose one another. Zirconium tungstate goes past neutral and genuinely contracts on heating.