Obtain the relation between the coefficient of volume expansion $(\alpha_V)$ and the coefficient of linear expansion $(\alpha_l)$.

(A) Consider a cube of side length $l$. When its temperature is increased by $\Delta T$,it expands equally in all dimensions.
From the volume formula,$V = l^3$.
The change in volume is given by $\Delta V = (l + \Delta l)^3 - l^3$.
Expanding this,we get $\Delta V = l^3 + 3l^2 \Delta l + 3l(\Delta l)^2 + (\Delta l)^3 - l^3$.
Since $\Delta l$ is very small,$(\Delta l)^2$ and $(\Delta l)^3$ are negligible. Thus,$\Delta V \approx 3l^2 \Delta l$ ... $(1)$.
From the definition of linear expansion,$\Delta l = \alpha_l l \Delta T$ ... $(2)$.
Substituting equation $(2)$ into equation $(1)$:
$\Delta V = 3l^2 (\alpha_l l \Delta T) = 3l^3 \alpha_l \Delta T$.
Since $V = l^3$,we have $\Delta V = 3 V \alpha_l \Delta T$ ... $(3)$.
Comparing equation $(3)$ with the general equation for volume expansion,$\Delta V = \alpha_V V \Delta T$,we get:
$\alpha_V = 3 \alpha_l$.