15. Volume (Cubical) expansion

The amount by which the volume of a material increases when the temperature is raised to one degree is called the coefficient of volumetric expansion.

The coefficient of volumetric expansion can be represented by the symbol $\mathrm{\gamma }$ (gamma).

Volume expansion

Consider a material with the volume of $V_1$ at $t_1^o$ \(c\) and the material is heated upto $t_2^o$ \(c\). Now, the volume of the material is $V_2$.

Let $\mathrm{\gamma }$ be the coefficient of cubical expansion.

We know that,

The change in volume \(ΔV\) is proportional to original volume $V_1$, rise in temperature \(ΔT\), and material type.

$\begin{array}{l}\mathrm{\Delta }V=V_2-V_1\mathit{and}\\ \\ \mathrm{\Delta }T=t_2-t_1\\ \\ \mathrm{\Delta }V\propto V_1\mathrm{\Delta }T\\ \\ \mathrm{\Delta }V=\mathrm{\gamma }{V}_1\mathrm{\Delta }T\end{array}$

Rearranging the above equation,

$\begin{array}{l}\mathrm{\gamma }\hspace{0.17em}=\frac{V_2-V_1}{V_1\left(t_2-t_1\right)}\\ \mathit{Where},\\ V_1\mathit{and}{V}_2\mathit{are}\mathit{the}\mathit{volumes}\mathit{of}\mathit{the}\mathit{material}\mathit{before}\mathit{and}\mathit{after}\mathit{heating}\mathit{respectively}.\\ t_1\mathit{and}{t}_2\mathit{are}\mathit{the}\mathit{temperatures}\mathit{of}\mathit{the}\mathit{material}\mathit{before}\mathit{and}\mathit{after}\mathit{heating}\mathit{respectively}.\\ \mathrm{\gamma }\mathit{is}\mathit{the}\mathit{volumetric}\mathit{coefficient}\mathit{of}\mathit{expansion}.\end{array}$

$6\mathrm{\alpha }=4\mathrm{\beta }=2\mathrm{\gamma }$