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# 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 coefficient of thermal expansion is used:

• in linear thermal expansion
• in area thermal expansion
• in volumetric thermal expansion

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).

## Thermal expansion coefficient

The thermal expansion coefficient is a thermodynamic property of a substance given by Incropera & DeWitt (p. 537).

It relates the change in temperature to the change in a material's linear dimensions. It is the fractional change in length per degree of temperature change. $\alpha={1\over L_0}{\partial L \over \partial T}$

where $L_0\$ is the original length, $L\$ the new length, and $T\$ the temperature.

## Linear thermal expansion ${\Delta L \over L_0} = \alpha_L \Delta T$

The linear thermal expansion is the one-dimensional length change with temperature.

## Area thermal expansion

The change in area with temperature can be written: ${\Delta A \over A_0} = \alpha_A \Delta T$

For exactly isotropic materials, the area thermal expansion coefficient is very closely approximated as twice the linear coefficient. $\alpha_A\cong 2\alpha_L$ ${\Delta A \over A_0} = 2 \alpha_L\Delta T$

## Volumetric thermal expansion

The change in volume with temperature can be written: ${\Delta V \over V_0} = \alpha_V \Delta T$

The volumetric thermal expansion coefficient can be written $\alpha_V =\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p=-{1\over\rho} \left(\frac{\partial \rho}{\partial T}\right)_{p}$

where $T\$ is the temperature, $V\$ is the volume, $\rho\$ is the density, derivatives are taken at constant pressure $p\$; $\beta\$ measures the fractional change in density as temperature increases at constant pressure.

Proof: $\alpha_V =\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p=\frac{\rho}{m}\left(\frac{\partial V}{\partial \rho}\right)_p\left(\frac{\partial \rho}{\partial T}\right)_p=\frac{\rho}{m}(-\frac{m}{\rho^2})\left(\frac{\partial \rho}{\partial T}\right)_p=-{1\over\rho} \left(\frac{\partial \rho}{\partial T}\right)_p$

where $m\$ is the mass.

For exactly isotropic materials, the volumetric thermal expansion coefficient is very closely approximated as three times the linear coefficient. $\alpha_V\cong 3\alpha_L$ ${\Delta V \over V_0} = 3 \alpha \Delta T$

Proof: $\alpha_V = \frac{1}{V} \frac{\partial V}{\partial T} = \frac{1}{L^3} \frac{\partial L^3}{\partial T} = \frac{1}{L^3}\left(\frac{\partial L^3}{\partial L} \cdot \frac{\partial L}{\partial T}\right) = \frac{1}{L^3}\left(3L^2 \frac{\partial L}{\partial T}\right) = 3 \cdot \frac{1}{L}\frac{\partial L}{\partial T} = 3\alpha_L$

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 (i.e., 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: $({L + }{\Delta L})^3 = {L^3 + 3L^2}{\Delta L} + {3L}{\Delta L}^2 + {\Delta L}^3 \,$

Note that this equation contains the main term, $3L^2\$, but also shows a secondary term that scales as $3L{\Delta L}^2 = {3L^3}{\alpha}^2{\Delta T}^2\,$, 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, ${\Delta L}^3\$ is vanishingly small, and is almost universally ignored. In anisotropic materials the total volumetric expansion is distributed unequally among the three axes.

## Thermal expansion coefficients for some common materials

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. The range for α is from 10-7 for hard solids to 10-3 for organic liquids. α varies with the temperature and some materials have a very high variation. 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)

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 11.1
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
Invar 1.2
Diamond 1
Quartz, fused 0.59

## 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 0.6x10-6 range. These alloys are useful in aerospace applications where wide temperature swings may occur.

## References

1. ^ Incropera, Frank P.; DeWitt, David P. (August 9 2001). Fundamentals of Heat and Mass Transfer, 5th Edition, Wiley. ISBN 0-471-38650-2.
2. ^ Turcotte, Donald L.; Schubert, Gerald (2002). Geodynamics, 2nd Edition, Cambridge. ISBN 0-521-66624-4.