### Theory:

Cubical expansions:
When we heat a solid, if there is an increase in the volume of the body, this is called Cubical expansion. It is also known as Volumetric 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.

$\begin{array}{l}\mathrm{\Delta }V\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{V}_{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{V}_{1}\phantom{\rule{0.147em}{0ex}}\mathit{and}\\ \\ \mathrm{\Delta }T\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}{t}_{2}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{t}_{1}\\ \\ \mathrm{\Delta }V\phantom{\rule{0.147em}{0ex}}\propto \phantom{\rule{0.147em}{0ex}}{V}_{1}\mathrm{\Delta }T\\ \\ \mathrm{\Delta }V\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{\gamma }{V}_{1}\mathrm{\Delta }T\end{array}$

Rearranging the above equation,

$\begin{array}{l}\mathrm{\gamma } =\phantom{\rule{0.147em}{0ex}}\frac{{V}_{2}-{V}_{1}}{{V}_{1}\left({t}_{2}-{t}_{1}\right)}\\ \mathit{Where},\\ {V}_{1}\mathit{and}\phantom{\rule{0.147em}{0ex}}{V}_{2}\phantom{\rule{0.147em}{0ex}}\mathit{are}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{volumes}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{material}\phantom{\rule{0.147em}{0ex}}\mathit{before}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{after}\phantom{\rule{0.147em}{0ex}}\mathit{heating}\phantom{\rule{0.147em}{0ex}}\mathit{respectively}.\\ {t}_{1}\mathit{and}\phantom{\rule{0.147em}{0ex}}{t}_{2}\phantom{\rule{0.147em}{0ex}}\mathit{are}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{temperatures}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{material}\phantom{\rule{0.147em}{0ex}}\mathit{before}\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathit{after}\phantom{\rule{0.147em}{0ex}}\mathit{heating}\phantom{\rule{0.147em}{0ex}}\mathit{respectively}.\\ \mathrm{\gamma }\phantom{\rule{0.147em}{0ex}}\mathit{is}\phantom{\rule{0.147em}{0ex}}\mathit{the}\phantom{\rule{0.147em}{0ex}}\mathit{volumetric}\phantom{\rule{0.147em}{0ex}}\mathit{coefficient}\phantom{\rule{0.147em}{0ex}}\mathit{of}\phantom{\rule{0.147em}{0ex}}\mathit{expansion}.\end{array}$

Relation between $\mathrm{\alpha },\phantom{\rule{0.147em}{0ex}}\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}\mathit{and}\phantom{\rule{0.147em}{0ex}}\mathrm{\gamma }$:

$6\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}4\mathrm{\beta }\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}2\mathrm{\gamma }$