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  
 increases and volume  
  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  
\[\beta = \frac{P}{{\Delta V}/V}\]
  is the bulk modulus and  
  is pressure. Rearranging this equation gives  
\[{\Delta V}/V=\frac{P}{\beta}\]
. Substitution into the thermal expansivity equation gives  
\[\frac{P}{\beta} = 3 \alpha \Delta T\]

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