# Thermal expansion **Thermal expansion** is the tendency of matter to change its length, area, volume and [[Density|density]] when its [[Temperature|temperature]] changes. For most solids over moderate ranges the change in length is proportional to the change in temperature, `ΔL = α·L·ΔT`, and the coefficient of linear expansion α is a material property of the order of 10⁻⁵ per kelvin for metals: [[Aluminium|aluminium]] stretches by about 23 parts per million for every degree, [[Steel|steel]] by 12, the nickel–iron alloy [[Invar]] by about 1, and [[Fused_quartz|fused silica]] by half of one.[^spec-m15][^openstax-13] Expansion is small, but the forces behind it are not: a rail, a bridge deck or a glass dish that cannot expand freely develops a stress equal to the product of its stiffness, its coefficient and the temperature change, which is why expansion joints, [[Bimetallic_strip|bimetallic strips]] and low-expansion glasses exist. At the atomic scale expansion is a consequence of the asymmetry of the bond between neighbouring atoms: the repulsion on the short side is steeper than the attraction on the long side, so as thermal vibration grows the mean bond length walks outward. In the microsim below the reader sets T and watches that walk on an asymmetric pair potential (a Morse well of ILLUSTRATIVE depth), while a bar beside it reads out `ΔL/L = α·ΔT` for the chosen preset (Al 23, steel 12, Invar 1.2, fused silica 0.5, all ×10⁻⁶/K) and a two-layer strip curls with the curvature a mismatch in α produces.[^spec-m15] The equation answers one question: how much a part of a given material moves for a given change in temperature. On the [[Materials_science]] flagship this article serves the *Thermal expansion* section of Part III — Fundamentals › Properties, sharing the solid-thermal sim (C36) with [[Debye_model]], which supplies the vibrating lattice, and [[Thermal_conductivity_and_resistivity]], which asks how the same lattice carries heat. ## Prediction A perfectly harmonic crystal would not expand at all. If the energy of a bond stretched by x from its rest length were exactly `c·x²`, the vibration would be symmetric about x = 0 at every amplitude and the mean position would never move. Real bonds are anharmonic: writing the [[Potential_energy|potential energy]] as `U(x) = c·x² − g·x³ + …`, with the cubic term making the well shallower on the stretched side, and averaging x over the classical [[Boltzmann_distribution|Boltzmann distribution]] gives a mean displacement `⟨x⟩ = 3·g·k·T/(4·c²)`, linear in temperature.[^derived-te] Dividing by the bond length r₀ gives the coefficient, `α = 3·g·k/(4·c²·r₀)`, in terms of the [[Boltzmann_constant|Boltzmann constant]] k and two bond constants. That is the whole mechanism: expansion measures the asymmetry of the interatomic well. The sim uses the Morse form of the well, `U(r) = D·[1 − e^{−a·(r − r₀)}]²`, because it has the right shape with two parameters, a depth D and a width 1/a. Expanding it about r₀ gives c = D·a² and g = D·a³, so the classical mean stretch is `⟨x⟩ = 3·k·T/(4·D·a)` and `α = 3·k/(4·D·a·r₀)`.[^derived-te] Deeper, narrower wells (strong, stiff bonds) expand less. The sim's D and a are ILLUSTRATIVE, chosen so that the readout reproduces the preset α of each material, not fitted to any real bond; what the reader sees is the shape of the argument, the mean position of a jiggling atom sliding up the soft side of the well as T rises.[^spec-m15] *Try:* raise T and watch the mean position slide up the soft side of the well while the bar's ΔL/L climbs; switch the preset from aluminium to Invar and the bar barely moves. A quantitative prediction from first principles needs the whole phonon spectrum, because every vibrational mode shifts its frequency slightly as the crystal expands. The Grüneisen relation collects this into `β = γ·C_v·κ_T/V`, where β is the volume coefficient, C_v the [[Heat_capacity|heat capacity]] of the [[Debye_model|Debye]] lattice, κ_T the compressibility and γ the Grüneisen parameter, a dimensionless number of order 1 to 3 that measures how strongly the mode frequencies depend on volume. Because C_v is the Debye function of T/θ_D, β follows the same curve: it vanishes as T³ near absolute zero and levels off above the Debye temperature, so a coefficient quoted "at room temperature" is a plateau value for lead and copper but not for diamond. ## Contraction effects (negative expansion) Some materials shrink when heated. [[Water|Water]] between 0 °C and about 4 °C is the everyday case: its density rises as it warms from the melting point and peaks near 4 °C, after which it expands like any other liquid, which is why a lake freezes from the top and why ponds keep a liquid layer beneath the ice.[^openstax-13] Among solids, zirconium tungstate (ZrW₂O₈) contracts over the whole range from 0.3 K to its decomposition near 1,050 K, an isotropic negative expansion reported in 1996 with a coefficient near −9×10⁻⁶ per kelvin, caused by transverse vibrations of oxygen atoms in the rigid WO₄ and ZrO₆ units that pull the framework inward as they grow.[^mary1996] The Morse well of the sim cannot show contraction, because a single asymmetric bond always walks outward; negative expansion is a property of how bonds are arranged, not of the bond itself, and the sim's HUD says so.[^spec-m15] Engineers use these materials as fillers: a composite of a normal-expansion matrix and a negative-expansion filler can be tuned to zero net α. ## Factors Bond strength is the first factor. Substances with strong, stiff bonds have deep, narrow wells and small α: [[Diamond|diamond]] and fused silica near 1×10⁻⁶/K or less, against 20 to 30×10⁻⁶/K for soft metals such as aluminium and [[Lead|lead]], and higher again for [[Polymer|polymers]], whose chains are held together sideways by weak intermolecular forces.[^spec-m15][^openstax-13] The same ordering appears in melting points, and the empirical rule that α times the melting temperature is roughly constant for metals expresses the fact that both measure the same well depth. Crystal structure is the second. A cubic crystal expands equally in every direction, but a layered or chain-like one does not: [[Graphite|graphite]] expands strongly across its layers and hardly at all within them, and many low-symmetry crystals expand in one direction while contracting in another, so a single α is only defined for polycrystalline or amorphous material with no preferred orientation. [[Phase_transition|Phase transitions]] are the third: at a transition the volume can jump, and the coefficient measured across it is meaningless. [[Glass|Glasses]] and polymers show a kink in their expansion curve at the [[Glass_transition|glass transition]], where the coefficient jumps as the frozen liquid unfreezes. Magnetism is the fourth: in Invar, the 36 percent nickel–iron alloy for which Charles Édouard Guillaume received the 1920 Nobel Prize in Physics, a contraction of the lattice as the [[Ferromagnetism|ferromagnetic]] order weakens on warming toward the Curie point almost exactly cancels the ordinary lattice expansion, leaving α near 1×10⁻⁶/K at room temperature.[^nobel1920] ## Effect on density Mass is conserved when a body expands, so its density falls as its volume grows: `ρ = ρ₀/(1 + β·ΔT) ≈ ρ₀·(1 − β·ΔT)` for small changes, with β the volume coefficient.[^derived-te] For an isotropic solid β = 3α, so [[Copper|copper]] (α ≈ 17×10⁻⁶/K) loses about 0.5 percent of its density between 20 °C and 120 °C, and a warm [[Iron|iron]] casting weighs the same as a cold one but displaces more water.[^derived-te] In liquids and gases the effect is large enough to drive motion: warm fluid is lighter than the cold fluid around it and rises, which is the origin of natural [[Convection|convection]] in a heated room and in the ocean. The density curve of water, with its maximum near 4 °C, is the exception that stratifies lakes in winter.[^openstax-13] ## Coefficients The coefficient of linear expansion is defined by `α = (1/L)·(dL/dT)` and the volume coefficient by `β = (1/V)·(dV/dT)`, both in reciprocal kelvin; the volume coefficient of a liquid or gas is the only one that matters, since a fluid has no shape to keep. For an isotropic solid the two are tied, β = 3α, to first order.[^openstax-13] Both are functions of temperature, and a quoted value is an average over a stated range, usually near 20 °C. ### For various materials The table lists the sim's presets and a few comparison values from the Portal Book's table of expansion coefficients; the book rounds to two figures and gives aluminium as 25 and Invar as 0.9, where the sim's presets use the handbook values 23 and 1.2, a reminder that α depends on alloy and range.[^spec-m15][^openstax-13] | Material | Linear α (×10⁻⁶/K) | Note | |---|---|---| | Aluminium | 23 (book: 25) | sim preset | | Brass | 19 | book value | | Copper | 17 | book value | | Steel | 12 | sim preset; book "iron or steel" 12 | | Glass, ordinary | 9 | book value | | Glass, borosilicate | 3 | book value ("Pyrex") | | Invar | 1.2 (book: 0.9) | sim preset | | Fused silica | 0.5 (book "quartz": 0.4) | sim preset | | Water (volume β) | 210 | book value, near 20 °C | | Mercury (volume β) | 180 | book value | | Air and most gases at 1 atm (volume β) | 3,400 | book value, near 20 °C | Every solid in the list is below 30×10⁻⁶/K and every liquid above 100×10⁻⁶/K; gases, whose volume coefficient is set by the gas law rather than by any bond, sit ten times higher again. ## In solids A solid expands in every dimension at once, so the same α describes length, area and volume with factors of 1, 2 and 3.[^openstax-13] The three cases below are the bar, the plate and the block of the sim. ### Length For a bar of length L, `ΔL = α·L·ΔT`, valid while α·ΔT is small enough that the change in L during the change can be ignored.[^openstax-13] A 30 m steel rail warming by 40 K grows by 12×10⁻⁶ × 30 m × 40 K = 14 mm, which is why jointed track leaves gaps and continuously welded track is pre-stressed.[^derived-te] If the rail is clamped so that it cannot grow, the strain α·ΔT it is denied becomes a compressive stress `σ = Y·α·ΔT`, with Y the [[Young's_modulus|Young's modulus]]; for steel with Y ≈ 200 GPa the 40 K rise produces about 96 MPa, a substantial fraction of the [[Yield_(engineering)|yield]] stress of structural steel.[^openstax-13][^derived-te] ### Area A plate of area A grows by `ΔA = 2·α·A·ΔT`, because each of its two dimensions grows by α·ΔT and the cross term (α·ΔT)² is negligible.[^openstax-13] A hole in the plate grows in exactly the same proportion as the plate around it, as if the missing material were still there: heating a ring makes its inner diameter larger, not smaller, which is the principle of the shrink fit, where a collar heated in an oven slides over a shaft and grips it on cooling. ### Volume A block grows by `ΔV = 3·α·V·ΔT = β·V·ΔT`.[^openstax-13] For aluminium warmed by 100 K, α·ΔT = 0.0023, the exact volume factor (1.0023)³ − 1 = 0.0069 and the linear approximation 3α·ΔT = 0.0069 agree to two figures, which is why the first-order formulas are used for everything short of a furnace.[^derived-te] ## In gases A gas has no bonds to stretch; its expansion follows from the [[Ideal_gas_law|ideal gas law]], `p·V = N·k·T`.[^openstax-gas] At constant pressure V is proportional to the absolute temperature, so `β = (1/V)·(dV/dT) = 1/T`: every ideal gas has the same volume coefficient, 1/273 K⁻¹ = 3,660×10⁻⁶/K at 0 °C and 1/293 K⁻¹ = 3,410×10⁻⁶/K at 20 °C, which is the "3,400" of the table.[^derived-te] Unlike a solid's α, the gas coefficient is not a material constant at all but a statement about temperature itself. The [[Kinetic_theory_of_gases|kinetic theory]] gives the mechanism: at fixed pressure, faster molecules need more room to deliver the same momentum per unit area to the walls. ### Absolute zero computation Because V ∝ T at constant pressure, the volume of a gas measured at several temperatures and extrapolated to zero volume points at the absolute zero of temperature. In Celsius, `V = V₀·(1 + t/273.15 °C)` for a gas of volume V₀ at 0 °C, and V reaches zero at t = −273.15 °C, the same point for every gas dilute enough to obey the gas law; this is [[Charles's_law|Charles's law]] read backward, and the constant-volume gas thermometer, which uses the pressure instead of the volume, is the instrument by which the Kelvin scale was realized before the 2019 redefinition fixed k.[^openstax-12][^nist-codata] No real gas reaches zero volume, since it liquefies first, but the extrapolation from the region where the law holds is exact. ## In liquids Liquids expand more than solids and less than gases: mercury and water by a few hundred parts per million per kelvin, ethanol and gasoline by about a thousand.[^openstax-13] The mechanism is the same asymmetric intermolecular well as in a solid, but with the molecules free to rearrange, so that the loose packing of a liquid opens up faster than a lattice can. Only the volume coefficient is defined. The practical consequences are large because volumes are: 50 litres of gasoline warmed by 20 K grow by 950×10⁻⁶ × 50 L × 20 K ≈ 0.95 L, which is why fuel is sold by volume corrected to a reference temperature and why a full tank filled cold overflows in the sun.[^derived-te] ### Apparent and absolute A liquid is always measured in a container, and the container expands too. What a thermometer or a graduated vessel shows is the *apparent* expansion, the difference between the liquid's absolute volume coefficient and the container's: `β_apparent = β_liquid − 3·α_container`. For [[Mercury_(element)|mercury]] (β = 180×10⁻⁶/K) in ordinary glass (α = 9×10⁻⁶/K) the apparent coefficient is 180 − 27 = 153×10⁻⁶/K, and in borosilicate glass (α = 3×10⁻⁶/K) it is 171×10⁻⁶/K, so the same mercury climbs a Pyrex capillary faster than a soda-lime one.[^derived-te] A liquid-in-glass thermometer is calibrated against this apparent coefficient, and a volumetric flask is marked for the temperature at which it was calibrated. ## Examples and applications The bimetallic strip is thermal expansion turned into motion. Two strips of metals with different α, bonded face to face, must share one length at every temperature, so the higher-expansion layer is held short and the lower-expansion layer stretched, and the pair bends toward the low-expansion side. For two layers of equal thickness and equal elastic modulus, Timoshenko's 1925 analysis of the bimetal thermostat gives the curvature `1/R = (3/2)·(α₂ − α₁)·ΔT/t`, with t the total thickness; the general formula carries a prefactor of 6 multiplied by a geometry factor that equals 1/4 in this case.[^timoshenko1925] With the sim's presets, an aluminium layer on a steel layer (Δα = 11×10⁻⁶/K), 1 mm thick in total and warmed by 100 K, curls to a radius of 0.61 m, and a 100 mm strip deflects at its free end by about 8 mm, enough to open or close a contact.[^derived-te] This is the mechanism of the classic thermostat, the circuit-breaker trip and the dial thermometer; the sim's third panel bends a strip of the two chosen presets as the reader moves T.[^spec-m15] Where expansion must be absorbed rather than used, the engineer supplies room for it: the sliding or finger joints of a bridge deck, the loops in long steam and hot-water pipes, the sag allowed in overhead [[Electric_power_transmission|power lines]], the gap around a concrete slab. Where it must be prevented, the engineer picks materials whose coefficients match: [[Glass|glass]]-to-metal seals use alloys whose α matches the glass, and a [[Borosilicate_glass|borosilicate]] baking dish survives the oven because its α is a third of ordinary glass. The thermal-shock limit makes this quantitative: the temperature difference a brittle body can take before its surface cracks is `ΔT_c = σ_f·(1 − ν)/(E·α)`, with σ_f the fracture strength, ν the [[Poisson's_ratio|Poisson's ratio]] and E the modulus, so halving α doubles the survivable shock, which is the whole case for fused silica in laboratory ware.[^spec-m15] Invar finds its uses at the other extreme, where nothing may move: pendulum rods, surveying tapes and the frames of precision instruments.[^nobel1920] ## See also - [[Invar]] — the near-zero-expansion nickel–iron alloy among the sim's presets - [[Bimetallic_strip]] — the curl the sim's third panel shows - [[Thermal_shock]] — the ΔT_c = σ_f·(1 − ν)/(E·α) limit - [[Debye_model]] — the lattice whose heat capacity sets the shape of α(T) - [[Thermal_conductivity_and_resistivity]] - [[Heat_capacity]] - [[Ideal_gas_law]] ## References [^openstax-13]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 1, "Temperature and Heat" (pp. 17–74), §1.3 Thermal Expansion: linear, area and volume expansion, Table 1.2 thermal expansion coefficients, the density of water near 4 °C, and thermal stress (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^openstax-12]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 1, §1.2 Thermometers and Temperature Scales: the constant-volume gas thermometer and absolute zero (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^openstax-gas]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2* (OpenStax). Chapter 2, "The Kinetic Theory of Gases", §2.1 Molecular Model of an Ideal Gas: `p·V = N·k·T = n·R·T`, pp. 76 and 81. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 [^timoshenko1925]: Timoshenko, S. (1925). "Analysis of Bi-Metal Thermostats." *Journal of the Optical Society of America* 11 (3): 233–255. https://doi.org/10.1364/JOSA.11.000233 [^mary1996]: Mary, T. A.; Evans, J. S. O.; Vogt, T.; Sleight, A. W. (1996). "Negative Thermal Expansion from 0.3 to 1050 Kelvin in ZrW₂O₈." *Science* 272 (5258): 90–92. https://doi.org/10.1126/science.272.5258.90 [^nobel1920]: The Nobel Prize in Physics 1920, Charles Édouard Guillaume, "in recognition of the service he has rendered to precision measurements in Physics by his discovery of anomalies in nickel steel alloys." NobelPrize.org. https://www.nobelprize.org/prizes/physics/1920/summary/ [^nist-codata]: National Institute of Standards and Technology. "CODATA Internationally Recommended Values of the Fundamental Physical Constants." *The NIST Reference on Constants, Units, and Uncertainty.* https://physics.nist.gov/cuu/Constants/ [^derived-te]: Computed for this article from the equations on the page and the coefficients of [^spec-m15] and [^openstax-13]: the anharmonic mean displacement (classical Boltzmann average of x over U = c·x² − g·x³ to first order in g), the Morse expansion c = D·a², g = D·a³, the rail, stress, density, area, volume, gas, gasoline, apparent-expansion and bimetal numbers (Timoshenko equal-layer case, Y = 200 GPa for steel). Not printed in a Portal Book. [^spec-m15]: Matter & Energy Cluster contract, `_registry/plans/MATERIALS_SCIENCE_SECTIONS.md` row M15: sim concept (Morse pair potential, ILLUSTRATIVE depth), control T, presets α (Al 23, steel 12, Invar 1.2, fused silica 0.5 ×10⁻⁶/K), the bar readout `dL/L = alpha·dT`, the strip curl, and the second-wave thermal-shock sibling `dT_c = sigma_f·(1 − nu)/(E·alpha)`. ## External links - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 2*, the Portal Book behind this page, on the Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2 - The Wikipedia pair's *External links* section lists calculators and coefficient tables; none is a source of this page. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Thermal_expansion.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Thermal expansion* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Thermal_expansion.html" data-title="Thermal expansion"></div> *Built from `MICROSIM_GUIDE/specs/sims/Thermal_expansion.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Thermal_expansion) : [Wikitube](https://en.wikitube.io/wiki/Thermal_expansion) · pinned revision [1368853368](https://en.wikipedia.org/w/index.php?oldid=1368853368) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M15 · sim pending (matter/Thermal_expansion).*