Show that volume expansion coefficient is three times the linear expansion coefficient for a solid.

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Solution

Let the initial dimensions of the cuboid be $l_{1}, l_{2}$ and $l_{3}$ .
Initial volume $V = l_{1} l_{2} l_{3}$
Final dimensions are,
$l_{1}^{'} = l_{1} (1 + α Δ T)$ $l_{2}^{'} = l_{2} (1 + α Δ T)$ $l_{3}^{'} = l_{3} (1 + α Δ T)$
Final volume $V^{'} = l_{1}^{'} l_{2}^{'} l_{3}^{'}$
Or $V^{'} = l_{1} (1 + α Δ T) l_{2} (1 + α Δ T) l_{3} (1 + α Δ T)$
$V^{'} = V (1 + α Δ T)^{3}$
Since $α$ is very small, so $(1 + α Δ T)^{3} \approx 1 + 3 α Δ T$
$⟹ V^{'} = V (1 + 3 α Δ T) = V (1 + γ Δ T)$
$⟹ γ = 3 α$

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