# Coefficient of expansion

the coefficient of expansion is defined as linear coefficient of expansion or thermal expansion and as volumetric expansion coefficient or cubic coefficient of expansion.

## linear Wärmeausdehnungskoeffizient

the linear coefficient of expansion is a measure for the relative length variation, for example a staff of the length [itex] L< /math>, per degree of change of temperature. It is a material-specific size of, those for one homogeneous solids defined is through:

[itex] \ alpha = \ frac {1} {L} \ cdot \ frac {\ delta L} {\ delta T} [/itex]

The length variation of a staff when even heating up or cooling around the temperature difference [itex] \ delta T< /math> can be computed, by the linear coefficient of expansion [itex] \ alpha< /math> the staff material with the staff length [itex] L< /math> and the temperature difference [itex] \ delta T< /math> one multiplies:

[itex] \ Delta L = \ alpha \ cdot L \ cdot \ delta T [/itex]

## volume-specific coefficient of expansion

the volume-specific coefficient of expansion describes the relative change of the volume [itex] V< /math> with the temperature [itex] T< /math>.

It is defined through:

[itex] \ gamma= \ frac {1} {V} \ left (\ frac {\ part V} {\ part T} \ right) _p [/itex]

To isotropic solids applies [itex] \ gamma = 3 \ cdot \ alpha [/itex].

The volume-specific coefficient of expansion results, there the mass [itex] \ rho (T) \ V (T) cdot [/itex] , from the density math \ <rho> (T) /math is <temperature independent> as a function of the temperature:

[itex] \ gamma= \ frac {1} {\ rho} \ left (\ frac {\ part \ rho} {\ part T} \ right) _p [/itex]

If the coefficient of expansion is well-known as function of the temperature, then the density results out:

[itex] \ rho (T) = \ rho (T_0) \ cdot \ exp \ left (- \ int_ {T_0} ^ {T} \ gamma (T) \ cdot dT \ right) [/itex]

Here is [itex] T_0 [/itex] any temperature, e.g. [itex] T_0 [/itex] = 298.15 K = 25 °C, with that the density [itex] \ rho (T_0) [/itex] admits is.

Green iron showed that the quotient [itex] \ alpha/c_p< /math> between thermal coefficients of expansion [itex] \ the alpha< /math> and the specific thermal capacity [itex] c_p [/itex] of the temperature is independent.