# 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.

### Volumetric thermal expansion coefficient

The volumetric thermal expansion coefficient (sometimes simply thermal expansion coefficient) is a thermodynamic property of a substance given by

[itex]\alpha=-{1\over\rho} \left({\partial\rho \over \partial T}\right)_{P}

= -{1\over V} \left({\partial V \over \partial T}\right)_{P}[itex]

where [itex]\rho[itex] is the density, [itex]T[itex] is the temperature, [itex]V[itex] is the volume, derivatives are taken at constant pressure [itex]P[itex]; [itex]\alpha[itex] 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.

### Linear thermal expansion coefficient

The expression for linear thermal expansion coefficient

[itex]\beta = {\alpha \over 3}[itex]

The expansion and contraction of material must be considered when designing large structures, when using tape or chain to measure distances for land surveys, and when designing molds for casting hot material. Engineers usually employ the linear thermal expansion coefficient, which is the fractional change in length of a bar per degree of temperature change. It is typically measured in parts per million per Celsius degree. Some values for common materials:

 Aluminum 25 Brass 19 Copper 17 Glass 9 Glass, Pyrex 4 Gold 14 Iron 12 Platinum 9 Quartz, fused 0.59 Silicon 3 Steel 13 Tungsten 4.5

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

## Applications

For applications using the thermal expansion property, see bi-metal and mercury thermometer de:Ausdehnungskoeffizient nl:Uitzettingscoëfficiënt

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