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Encyclopedia > Coefficient of thermal expansion
Material Properties
Specific heat $c=frac{T}{N}left(frac{partial S}{partial T}right)$
Compressibility $beta=-frac{1}{V}left(frac{partial V}{partial P}right)$
Thermal expansion $alpha=frac{1}{V}left(frac{partial V}{partial T}right)$
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During heat transfer, the energy that is stored in the intermolecular bonds between atoms changes. When the stored energy increases, so does the length of the molecular bond. As a result, solids typically* expand in response to heating and contract on cooling; this response to temperature change is expressed as its coefficient of thermal expansion: The thermodynamic properties of materials are intensive thermodynamic parameters which are specific to a given material. ... Specific heat capacity, also known simply as specific heat (Symbol: C or c) is the measure of the heat energy required to raise the temperature of a specific quantity of a substance (thus, the name â€œspecificâ€ heat) by certain amount, usually one kelvin. ... Fluid Dynamics Compressibility (physics) is a measure of the relative volume change of fluid or solid as a response to a pressure (or mean stress) change: . For a gas the magnitude of the compressibility depends strongly on whether the process is adiabatic or isothermal, while this difference is small in... In physics, thermal expansion is the tendency of matter to increase in volume or pressure when heated. ...

The coefficient of thermal expansion is used in two ways:

• as a volumetric thermal expansion coefficient
• as a linear thermal expansion coefficient

These characteristics are closely related. The volumetric thermal expansion coefficient can be measured for all substances of condensed matter (liquids and solid state). The linear thermal expansion can only be measured in the solid state and is common in engineering applications.

* Some substances have a negative expansion coefficient, and will expand when cooled (e.g. freezing water).

## Contents

### Volumetric thermal expansion coefficient

The volumetric thermal expansion coefficient (sometimes simply thermal expansion coefficient) is a thermodynamic property of a substance given by (Incropera, 2001 p537) Here is a partial list of thermodynamic properties of fluids: temperature [K] density [kg/m3] specific heat at constant pressure [J/kg·K] specific heat at constant volume [J/kg·K] dynamic viscosity [N/m²s] kinematic viscosity [m²/s] thermal conductivity [W/m·K] thermal diffusivity [m²/s] volumetric...

$beta =frac{1}{V}left(frac{partial V}{partial T}right)_P=-{1overrho} left(frac{partial rho}{partial T}right)_{P}$

where T is the temperature, V is the volume, ρ is the density, derivatives are taken at constant pressure P; β measures the fractional change in density as temperature increases at constant pressure.

Proof:

$beta =frac{1}{V}left(frac{partial V}{partial T}right)_P=frac{rho}{m}left(frac{partial V}{partial rho}right)_Pleft(frac{partial rho}{partial T}right)_P=frac{rho}{m}(-frac{m}{rho^2})left(frac{partial rho}{partial T}right)_P=-{1overrho} left(frac{partial rho}{partial T}right)_P$

where m is the mass.

The expansion of a crystalline material occurs only when the force field of the crystal deviates from a perfect quadratic. If the force field is perfectly parabolic, no expansion will occur.

### Linear thermal expansion coefficient

The linear thermal expansion coefficient relates the change in temperature to the change in a material's linear dimensions. It is the fractional change in length of a bar per degree of temperature change.

$alpha={1over L}{partial L over partial T}$

The expansion and contraction of material must be considered when designing large structures, when using tape or chain to measure distances for land surveys, when designing molds for casting hot material, and in other engineering applications when large changes in dimension due to temperature are expected. Some values for common materials, given in parts per million per Celsius degree: (NOTE: This can also be in kelvins as the changes in temperature are a 1:1 ratio) Celsius relates to the Celsius or centrigrade temperature scale. ... The Kelvin scale is a thermodynamic (absolute) temperature scale where absolute zeroâ€”the lowest possible temperature where nothing could be colder and no heat energy remains in a substanceâ€”is defined as zero kelvin (0 K). ...

coefficient of linear thermal expansion α
material α in 10-6/K at 20 °C
Mercury 60
BCB 42
Aluminum 23
Brass 19
Stainless steel 17.3
Copper 17
Gold 14
Nickel 13
Concrete 12
Iron or Steel 12
Carbon steel 10.8
Platinum 9
Glass 8.5
GaAs 5.8
Indium Phosphide 4.6
Tungsten 4.5
Glass, Pyrex 3.3
Silicon 3
Diamond 1
Quartz, fused 0.59

For exactly isotropic materials, the linear thermal expansion coefficient is very closely approximated as one-third the volumetric coefficient. General Name, Symbol, Number mercury, Hg, 80 Chemical series transition metals Group, Period, Block 12, 6, d Appearance silvery Atomic mass 200. ... Benzocyclobutene (BCB) is an aromatic chemical compound frequently used to create photosensitive polymers. ... General Name, Symbol, Number lead, Pb, 82 Chemical series poor metals Group, Period, Block 14, 6, p Appearance bluish white Atomic mass 207. ... Aluminum is a soft and lightweight metal with a dull silvery appearance, due to a thin layer of oxidation that forms quickly when it is exposed to air. ... Brass is the term used for alloys of copper and zinc; the proportions of zinc and copper can be varied to create a range of brasses each with unique properties[1]. Note that in comparison bronze is principally an alloy of copper and tin. ... The 630 foot high, stainless-clad (type 304) Gateway Arch defines St. ... General Name, Symbol, Number copper, Cu, 29 Chemical series transition metals Group, Period, Block 11, 4, d Appearance copper, metallic Atomic mass 63. ... General Name, Symbol, Number gold, Au, 79 Chemical series transition metals Group, Period, Block 11, 6, d Appearance metallic yellow Atomic mass 196. ... General Name, Symbol, Number nickel, Ni, 28 Chemical series transition metals Group, Period, Block 10, 4, d Appearance lustrous, metallic Atomic mass 58. ... Concrete being poured, raked and vibrated into place in residential construction in Toronto, Ontario, Canada. ... General Name, Symbol, Number iron, Fe, 26 Chemical series transition metals Group, Period, Block 8, 4, d Appearance lustrous metallic with a grayish tinge Atomic mass 55. ... The old steel cable of a colliery winding tower Steel is an alloy whose major component is iron, with carbon content between 0. ... Carbon steel is a metal alloy, a combination of two elements, iron and carbon, where other elements are present in quantities too small to affect the properties. ... General Name, Symbol, Number platinum, Pt, 78 Chemical series transition metals Group, Period, Block 10, 6, d Appearance grayish white Atomic mass 195. ... Glass can be made transparent and flat, or into other shapes and colors as shown in this sphere from the Verrerie of Brehat in Brittany. ... Gallium arsenide (GaAs) is a chemical compound composed of gallium and arsenic. ... Indium phosphide (InP) is a semiconductor composed of indium and phosphorus. ... General Name, Symbol, Number tungsten, W, 74 Chemical series transition metals Group, Period, Block 6, 6, d Appearance grayish white, lustrous Atomic mass 183. ... Pyrex is a brand name of borosilicate glass introduced by Corning Glass Works in 1915. ... General Name, Symbol, Number silicon, Si, 14 Chemical series metalloids Group, Period, Block 14, 3, p Appearance as coarse powder, dark gray with bluish tinge Atomic mass 28. ... This article is about the gemstone. ... Quartz is one of the most common minerals in the Earths continental crust. ... Isotropy (the opposite of anisotropy) is the property of being independent of direction. ...

$betacong 3alpha$

Proof:

$beta = frac{1}{V} frac{partial V}{partial T} = frac{1}{L^3} frac{partial L^3}{partial T} = frac{1}{L^3}left(frac{partial L^3}{partial L} cdot frac{partial L}{partial T}right) congfrac{1}{L^3}left(3L^2 frac{partial L}{partial T}right) = 3 cdot frac{1}{L}frac{partial L}{partial T} = 3alpha$

This ratio arises because volume is composed of three mutually orthogonal directions. Thus, in an isotropic material, one-third of the volumetric expansion is in a single axis (a very close approximation for small differential changes). Note that the partial derivative of volume with respect to length as shown in the above equation is exact, however, in practice it is important to note that the differential change in volume is only valid for small changes in volume (ie the expression is not linear). As the change in temperature increases, and as the value for the linear coefficient of thermal expansion increases, the error in this formula also increases. For non-negligible changes in volume: In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

$({L + }{Delta L})^3 = {L^3 + 3L^2}{Delta L} + {3L}{Delta L}^2 + {Delta L}^3 ,$

Note that this equation contains the main term, 3L2, but also shows a secondary term that scales as 3LΔL2 = 3L3α2ΔT2, which shows that a large change in temperature can overshadow a small value for the linear coefficient of thermal expansion. Although the coefficient of linear thermal expansion can be quite small, when combined with a large change in temperature the differential change in length can become large enough that this factor needs to be considered. The last term, ΔL3 is vanishingly small, and is almost universally ignored. In anisotropic materials the total volumetric expansion is distributed unequally among the three axes. This article is being considered for deletion in accordance with Wikipedias deletion policy. ...

For applications using the thermal expansion property, see bi-metal and mercury thermometer thermocouple and Peltier_Seebeck effect. ... Close up of a maximum thermometer. ...

Thermal expansion is also used in mechanical applications to fit parts over one another, e.g. a bushing can be fitted over a shaft by making its inner diameter slightly smaller than the diameter of the shaft, then heating it until it fits over the shaft, and allowing it to cool after it has been pushed over the shaft, thus achieving a 'shrink fit'

There exist some alloys with a very small CTE, used in applications that demand very small changes in physical dimension over a range of temperatures. One of these is Invar 36, with a coefficient in the 0.0000016 range. These alloys are useful in aerospace applications where wide temperature swings may occur. Invar, also called FeNi36, is an alloy of iron (64%) and nickel (36%) with some carbon and chromium. ...

## References

• Incropera, Frank P.; David P. DeWitt (August 9 2001). Fundamentals of Heat and Mass Transfer, 5th Edition, Wiley. ISBN 0-471-38650-2.

Results from FactBites:

 Thermal expansion (1201 words) The coefficient of thermal expansion is generally defined as the fractional increase in length per unit rise in temperature. The former is related to the slope of the tangent to the length – temperature plot, while the latter is governed by the slope of the chord between two points on this curve. Determination of the thermal expansion coefficient requires the measurement of two physical quantities, displacement and temperature, for a sample of the material that is undergoing an appropriate thermal cycle.
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