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# Thermal Expansion

## Discussion

### Solids

For many solids, expansion is directly proportional to temperature change.

∆ℓ = αℓ0T

Areas expand twice as much as lengths do.

A = 2αA0T

Volumes expand three times as much as lengths do.

V = 3αV0T

applications

• buckling
• expansion gap/joint
• anti-scalding valve
• bimetallic strip, thermostat
• expansion of holes (mounting train tires)
• "What's more, the aircraft expands by 15-25 centimeters during flight because of the scorching heat created by friction with air. Designers used rollers to isolate the cabin from the body, so that stretching doesn't rip the plane apart." Helen Pearson "Concorde wings its way into retirement." Nature Physics Portal. October 2003.
• "Concorde measures 204ft in length - stretching between six and ten inches in-flight due to heating of the airframe. She is painted in a specially developed white paint to accommodate these changes and to dissipate the heat generated by supersonic flight." source
• 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

• length comparator
• push rod dilatometer (gives relative expansion, since the device itself expands)
• interferometer (highest precision method)
• x-ray diffactometer
• capacitance dilatometer
• strain gauge
• optical dilatometer (basically a digital camera)

anisotropic expansion

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

### liquids

Liquids can only expand in volume.

V = βV0T

Liquids have higher expansivities than solids.

β ~ 10−3/K, 3α ~ 10−5/K

applications

• Liquid in glass thermometer. The alcohol is colored red to look like wine.

ethyl alcohol 1120 × 10−6/K
mercury 181 × 10−6/K
glass 3(8.5 × 10−6/K) = 25.5 × 10−6/K

• Cars have coolant overflow tanks.
Coefficients of linear thermal expansion
material α (10−6/K)
alumina (αAl2O3) 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, typical 8.5
glass, ultra-low expansion 0.04
gold 14.2
invar (64% Fe, 36% Ni) 1.2
iron 11.8
nickel 13.3
plastics 40–120
plexiglas 93
platinum 8.8
plutonium 54
silicon 4.68
silver 18.9
steel, stainless 17.3
steel, structural 12
tin 22
titanium 8.5
tungsten 4.5
uranium 13.9
water, ice (0 °C) 51
wood (lumber), tangential 36
wood (lumber), axial 3.7
zinc 30.2
zirconium tungstate (ZrW2O8) −8.8
Coefficients of volume thermal expansion ☞ All values in both tables are averages for temperatures centered near 20 °C unless otherwise stated.
material β (10−6/K)
alcohol, ethyl 1120
gasoline 950
jet fuel, kerosene 990
mercury 181
water, liquid (1 °C) −50
water, liquid (4 °C) 0
water, liquid (10 °C) 88
water, liquid (20 °C) 207
water, liquid (30 °C) 303
water, liquid (40 °C) 385
water, liquid (50 °C) 457
water, liquid (60 °C) 522
water, liquid (70 °C) 582
water, liquid (80 °C) 640
water, liquid (90 °C) 695

### water

• anomalous expansion of water
• ice is less dense than water
• water is most dense at 4 °C (ρ = 999.973 kg/m3)
• applications
• frozen pipes burst
• turnover of lake water in spring

### plutonium

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], ∆ [delta], ∆′ [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 °C, 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."

### Gases

Behavior of gases is more complicated, gases will expand as much as pressure will allow. Check out the gas laws.

 PV = nRT PV = NkT