Thermal Expansion



equation   solids
Δℓ  =  0 αΔT   linear expansion
ΔA  =  A0 2αΔT   areal (or superficial) expansion
ΔV  =  V0 3αΔT   volumetric (or cubical) expansion
Equations of Thermal Expansion

Thermal expansion is a small, but not always insignificant effect. Typical coefficients are measured in parts per million per kelvin (10−6/K). That means your typical classroom meter stick never varies in length by more than a 100 µm in its entire lifetime — probably never more than 10 µm while students are using it.


measurement techniques

anisotropic expansion

Some materials expand differently in different directions, notably graphite and wood (lumber).


ΔV = βV0ΔT

coolant overflow tanks in cars

Liquids have higher expansivities than solids

β ~ 10−3/K, 3α ~ 10−5/K

That's why a liquid in glass themometer works

ethyl aclohol 1120 × 10−6/K, mercury 181 × 10−6/K

The alcohol is colred red to look like wine.

glass 3(8.5 × 10−6/K) = 25.5 × 10−6/K

equation   liquids
ΔV  =  V0 βΔT   volumetric (or cubical) expansion
Equations of Thermal Expansion



Check out the gas laws

Behavior of gases is more complicated, gases will expand as much as pressure will allow

equation   gases
PV  =  nRT   ideal gas law
Equations of Thermal Expansion
material α (10−6/K)
alumina (α‑Al2O3) 5.30
aluminium 23.1
barium ferrite 10
brass 20.3
carbon, diamond 1.18
carbon, graphite ∥ 6.5
carbon, graphite ⊥ 0.5
chromium 4.9
concrete 8 ~ 12
copper 16.65
epoxy 55
germanium 6.1
glass 8.5
gold 14.2
invar (64% Fe, 36% Ni) 1.2
iron 11.8
lead 28.9
nickel 13.3
plastics 40 ~ 120
platinum 8.8
plutonium 54
silicon 4.68
silver 18.9
solder, lead-tin 25
steel, stainless 17.3
steel, structural 12
tin 22
titanium 8.5
tungsten 4.5
uranium 13.9
water, ice (0 ℃) 51
wood (lumber), tangential 36
wood (lumber), radial 26
wood (lumber), axial 3.7
zinc 30.2
zirconium tungstate (ZrW2O8) −8.8
Coefficients of linear thermal expansion
material β (10−6/K)
alcohol, ethyl 1120
gasoline 950
jet fuel, kerosene 990
mercury 181
water, liquid (1 ℃) −50
water, liquid (4 ℃) 0
water, liquid (10 ℃) 88
water, liquid (20 ℃) 207
water, liquid (30 ℃) 303
water, liquid (40 ℃) 385
water, liquid (50 ℃) 457
water, liquid (60 ℃) 522
water, liquid (70 ℃) 582
water, liquid (80 ℃) 640
water, liquid (90 ℃) 695
Coefficients of volume thermal expansion Note: All values in both tables are averages for temperatures centered near 20 ℃ unless otherwise stated.





Plutonium undergoes more phase transitions at ordinary pressures than any other element. As plutonium is heated it transforms through six different crystal structures before melting — α [alpha], β [beta], γ [gamma], Δ [delta], Δ′ [delta prime], and ε [epsilon]. Physical properties like density and thermal expansion vary significantly from phase to phase making it one of the more difficult metals to machine and work. The metallurgy of plutonium is best left to the experts.

Notes form LLNL that must be paraphrased. "One of plutonium's unique physical properties is that the pure metal exhibits six solid-state phase transformations before reaching its liquid state, passing from alpha, beta, gamma, delta, delta-prime, to epsilon. Large volume expansions and contractions occur between the stable room-temperature alpha phase and the element's liquid state. Another unusual feature is that unalloyed plutonium melts at a relatively low temperature, approximately 640 ℃, to yield a liquid of higher density than the solid from which it melts. In addition, the elastic properties of the delta face-centered cubic (fcc) phase of plutonium are highly directional (anisotropic). That is, the elasticity of the metal varies widely along different crystallographic directions by as much as a factor of six to seven."