Coefficient of thermal expansion

All materials change their size when subjected to a temperature change. The coefficient of thermal expansion describes how the size of an object changes with a change in temperature. Specifically, it measures the fractional change in volume per degree change in temperature at a constant pressure.

The volumetric thermal expansion coefficient is the most basic thermal expansion coefficient. All substances expand or contract when their temperature changes, and the expansion or contraction always occurs in all directions. Substances that expand at the same rate in any direction are called isotropic. Unlike gases or liquids, solid materials tend to keep their shape. For solids, one might only be concerned with the change along a length, or over some area. Expansion coefficients are specially defined for these cases, and they are known as the linear and area expansion coefficients. However, they all come from the volume expansion coefficient, which explains how the substance expands in any direction.

The general volumetric thermal expansion coefficient

In the general case of a gas, liquid, or solid, the coefficient of thermal expansion is given by

$\alpha _{V}={\frac {1}{V}}\,\left({\frac {\partial V}{\partial T}}\right)_{p}$

where partial derivatives must now be used, the subscript p indicating that the pressure is held constant during the expansion. In the case of a gas, the fact that the pressure is held constant is important, because the volume of a gas will vary appreciably with pressure as well as temperature. For a gas of low density this can be seen from the ideal gas law.

Isotropic materials, will by definition have the same ${\frac {}{}}\alpha _{L}$ in every direction. For an isotropic material,

${\frac {}{}}\alpha =3\alpha _{L}$.

For exactly isotropic materials, the volumetric thermal expansion coefficient is 3 times the linear expansion coefficient.

$\alpha _{V}\approx 3\alpha _{L}$

The volumetric thermal expansion coefficient for solids

The thermal expansion coefficient for a solid is a thermodynamic property of that solid. For a solid, we can ignore the effects of pressure on the material and the volumetric thermal expansion coefficient can be written :

$\alpha _{V}={\frac {1}{V}}\,{\frac {dV}{dT}}$

where V is the volume of the material, and $dV/dT$ is the rate of change of that volume with temperature.

What this basically means is that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic foot might expand to 1.02 cubic feet when the temperature is raised by 50 degrees. This is an expansion of two percent. If we had a block of steel with a volume of 2 cubic feet, then under the same conditions, it would expand to 2.04 cubic feet, again an expansion of two percent. The volumetric expansion coefficient would be two percent for 50 degrees or 0.04 percent per degree, or 0.0004 per degree.

If we already know the expansion coefficient, then we can calculate the change in volume

${\frac {\Delta V}{V}}=\alpha _{V}\Delta T$

where $\Delta V/V$ is the fractional change in volume (0.02) and $\Delta T$ is the change in temperature (50 degrees).

The above example assumes that the expansion coefficient did not change as the temperature changed by 50 degrees. This is not always true, but for small changes in temperature, it is a good approximation. If the volumetric expansion coefficient does change appreciably with temperature, then the above equation will have to be integrated:

${\frac {\Delta V}{V}}=\int _{T_{0}}^{T_{0}+50}\alpha _{V}(T)\,dT$

where $T_{0}$ is the starting temperature and $\alpha _{V}(T)$ is the volumetric expansion coefficient as a function of temperature T.

Linear thermal expansion coefficient for a solid

The linear thermal expansion coefficient relates the change in a material's linear dimensions to a change in temperature. It is the fractional change in length per degree of temperature change. Again, ignoring pressure, we may write:

$\alpha _{L}={\frac {1}{L}}\,{\frac {dL}{dT}}$

where L is the linear dimension (e.g. length) and $dL/dT$ is the rate of change of that linear dimension per unit change in temperature. Just as with the volumetric coefficient, the change in the linear dimension can be estimated as:

${\frac {\Delta L}{L}}=\alpha _{L}\Delta T$

Again, this equation works well as long as the linear expansion coefficient does not change much over the change in temperature $\Delta T$. If it does, the equation must be integrated.

For exactly isotropic materials, the linear thermal expansion coefficient is almost exactly one third the volumetric coefficient.

$\alpha _{V}\approx 3\alpha _{L}$

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). As an example, take a cube of steel that has sides of length L. The original volume will be $V=L^{3}$ and the new volume, after a temperature increase, will be

$V+\Delta V=(L+\Delta L)^{3}=L^{3}+3L^{2}\Delta L+3L\Delta L^{2}+\Delta L^{3}\approx L^{3}+3L^{2}\Delta L=V+3V{\Delta L \over L}$

We can make the substitutions $\Delta V=\alpha _{V}L^{3}\Delta T$ and, for isotropic materials, $\Delta L=\alpha _{L}L\Delta T$. We now have:

$L^{3}+L^{3}\alpha _{V}\Delta T=L^{3}+3L^{3}\alpha _{L}\Delta T+3L^{3}\alpha _{L}^{2}\Delta T^{2}+L^{3}\alpha _{L}^{3}\Delta T^{3}\approx L^{3}+3L^{3}\alpha _{L}\Delta T$

Since the volumetric and linear coefficients are defined only for extremely small temperature and dimensional changes, the last two terms can be ignored and we get the above relationship between the two coefficients. If we are trying go back and forth between volumetric and linear coefficients using larger values of $\Delta T$ then we will need to take into account the third term, and sometimes even the fourth term.

Anisotrpoic materials

Materials with anisotropic structures, such as crystals and many composites, will generally have different linear expansion coefficients ${\frac {}{}}\alpha _{L}$ in different directions. In anisotropic materials the total volumetric expansion is distributed unequally among the three axes. If the crystal symmetry is monoclinic or triclinic even the angles between these axes are subject to thermal changes. In such cases it is necessary to treat thermal expansion as a tensor that has up to six independent elements. A good way to determine the elements of the tensor is to study the expansion by powder diffraction.

Area thermal expansion coefficient for a solid

The area thermal expansion coefficient relates the change in a material's area dimensions to a change in temperature. It is the fractional change in area per degree of temperature change. Again, ignoring pressure, we may write:

$\alpha _{A}={\frac {1}{A}}\,{\frac {dA}{dT}}$

where A is some area on the object, and $dA/dT$ is the rate of change of that area per unit change in temperature. Just as with the volumetric coefficient, the change in the linear dimension can be estimated as:

${\frac {\Delta A}{A}}=\alpha _{A}\Delta T$

Again, this equation works well as long as the linear expansion coefficient does not change much over the change in temperature $\delta T$. If it does, the equation must be integrated.

For exactly isotropic materials, the area thermal expansion coefficient is 2/3 of the volumetric coefficient.

$\alpha _{A}={\frac {2}{3}}\alpha _{V}$

This ratio can be found in a way similar to that in the linear example above, noting that the area of a face on the cube is just $L^{2}$. Also, the same considerations must be made when dealing with large values of $\Delta T$

Thermal expansion coefficients for various 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/°C for hard solids to 10⁻³/°C for organic liquids. α varies with the temperature and some materials have a very high variation.

Theoretically, the coefficient of linear expansion can be approximated from the coefficient of volumetric expansion (β≈3α). However, for liquids, α is calculated through the experimental determination of β, so it is more accurate to state β here, rather than α. (The formula β≈3α is usually used for solids.)

material	coefficient of linear thermal expansion α α in 10−6/C° at 20 °C	coefficient of volumetric thermal expansion β β(≈3α) in 10−6/C° at 20 °C
Aluminium	23	69
Benzocyclobutene	42	126
Brass	19	57
Carbon steel	10.8	32.4
Concrete	12	36
Copper	17	51
Diamond	1	3
Ethanol	250~	750
Gallium(III) arsenide	5.8	17.4
Gasoline	317~	950
Glass	8.5	25.5
Glass, borosilicate	3.3	9.9
Gold	14	42
Indium phosphide	4.6	13.8
Invar	1.2	3.6
Iron	11.1	33.3
Lead	29	87
MACOR	9.3
Magnesium	26	78
Mercury	61~	182
Molybdenum	4.8	14.4
Nickel	13	39
Oak (perpendicular to the grain)	54	162
Pine (perpendicular to the grain)	34	102
Platinum	9	27
PVC	52	156
Quartz (fused)	0.59	1.77
Rubber	77	231
Sapphire (parallel to C axis, or [001])	5.3
Silicon Carbide	2.77	8.31
Silicon	3	9
Silver	18	54
Sitall	0.15	0.45
Stainless steel	17.3	51.9
Steel, depends on composition	11.0 ~ 13.0	33.0 ~ 39.0
Tungsten	4.5	13.5
Water	69~	207

See also

• Autovent
• Thermal expansion coefficients of the elements (data page)

External links

• Engineering Toolbox – List of coefficients of Linear Expansion for some common materials
• Article on how β is determined
• MatWeb: Free database of engineering properties for over 79,000 materials
• Clemson University Physics Lab: Linear Thermal Expansion
• USA NIST Website - Temperature and Dimensional Measurement workshop
• Hyperphysics: Thermal expansion
• Understanding Thermal Expansion in Ceramic Glazes