## Thermal Expansivity and Bulk Modulus

Generally, when you heat a substance, it expands. The expanding substance can exert of force on its surroundings. It may seem obvious that there is a relationship between the thermal expansivity of a substance and the bulk modulus
If the temperature of a solid with thermal volume expansivity
$\alpha$
increases and volume
$V$
increases by
$\Delta T$
then the volume increases by
$\Delta 3 \alpha V \Delta T \rightarrow \Delta V /V= 3 \alpha \Delta T$
.
The bulk modulus equation is
$\beta = \frac{P}{{\Delta V}/V}$
where
$\beta$
is the bulk modulus and
$P$
is pressure. Rearranging this equation gives
${\Delta V}/V=\frac{P}{\beta}$
. Substitution into the thermal expansivity equation gives
$\frac{P}{\beta} = 3 \alpha \Delta T$
.