Question

Define the coefficient of linear, areal and cubical expansion of solid.

Answer 1

Hint: Recall that heating a body results in increase in its dimensions. This change in dimensions is experimentally found to be proportional to the original dimension and the change in temperature. These coefficients are introduced as the proportionality constant in these relations. Now you could accordingly define each of these coefficients.

Complete answer:
We know that, on heating a body, an increase in dimensions happens for the body. This phenomenon in which the body expands on heating is called expansion of solid. In general, the solids undergo three types of expansions, namely, linear expansion or longitudinal expansion, superficial expansion or areal expansion and the cubical expansion or volumetric expansion.
Coefficient of linear expansion of solids:
If the body undergoes an increase in length due to heating, then the resultant expansion is linear or longitudinal. Let ${{l}_{0}}$ be the length of a rod at $0{}^\circ C$ and let $l$ be the length at $t{}^\circ C$, then, it is experimental proven that the change in length of the rod $\left( l-{{l}_{0}} \right)$ is directly proportional to its original length$\left( {{l}_{0}} \right)$ and change in temperature$\left( \Delta t \right)$, that is,
$\Rightarrow l-{{l}_{0}}\propto {{l}_{0}}\Delta t$
$\Rightarrow l-{{l}_{0}}=\alpha {{l}_{0}}\Delta t$
Where, ‘\[\alpha \]’ is the coefficient of linear expansion of a solid. It can be defined as the increase in length per unit original length at ${{0}^{{}^\circ }}C$per unit rise in temperature.
Coefficient of areal expansion of solids:
Similar to linear expansion, in case of areal expansion due to heating, the change in area is directly proportional to the original area and the change in temperature, that is,
$\Rightarrow A-{{A}_{0}}\propto {{A}_{0}}\Delta t$
$\Rightarrow A-{{A}_{0}}=\beta {{A}_{0}}\Delta t$
Where, ‘$\beta $’ is the coefficient of areal expansion. It is defined as the increase in area per unit original area at $0{}^\circ C$ per unit rise in temperature.
Coefficient of cubical expansion of solids:
Similarly in volume expansion due to heating, the change in volume is directly proportional to the original volume and also the change in temperature, that is,
$\Rightarrow V-{{V}_{0}}\propto {{V}_{0}}\Delta t$
$\Rightarrow V-{{V}_{0}}=\gamma {{V}_{0}}\Delta t$
Where, ‘$\gamma $’ is the coefficient of volume expansion due to heating. It can be defined as the increase in volume per unit original volume at $0{}^\circ C$ per unit rise in temperature.

Note:
These coefficients are actually found to be related to each other. The relation between $\alpha $ and $\beta $ is given by,
$\beta =2\alpha $
Also, the relation between $\alpha $ and $\gamma $ is given by,
$\gamma =3\alpha $
Hence,
$6\alpha =3\beta =2\gamma $