Coefficient of thermal expansion

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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 expand in response to heating and contract on cooling; this response to temperature change is expressed as its coefficient of thermal expansion:

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.

1 Volumetric thermal expansion coefficient
2 Linear thermal expansion coefficient
3 Applications
4 External links
5 References

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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)

<math>

\beta=-{1\over\rho} \left({\partial\rho \over \partial T}\right)_{P}

    =\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P

</math>

where <math>\rho</math> is the density, <math>T</math> is the temperature, <math>V</math> is the volume, derivatives are taken at constant pressure <math>P</math>; <math>\beta</math> measures the fractional change in density as temperature increases at constant pressure. 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.

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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.

<math>\alpha={1\over L}{\partial L \over \partial T}</math>

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)

Mercury	60 (×10⁻⁶/°C)
BCB	42
Lead	29
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
InP	4.6
Tungsten	4.5
Glass, Pyrex	3.3
Silicon	3
Diamond	1
Quartz, fused	0.59

For isotropic materials, the linear thermal expansion coefficient is approximately 1/3 the volumetric coefficient.

<math>\beta = 3\alpha</math>

Proof:

<math>\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) = \frac{1}{L^3}\left(3L^2 \frac{\partial L}{\partial T}\right) = 3 \cdot \frac{1}{L}\frac{\partial L}{\partial T} = 3\alpha</math>

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Applications

For applications using the thermal expansion property, see bi-metal and mercury 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.

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