Volume or Cubical Expansion
When a solid is heated and its volume increases, then the expansion is called volume expansion (or) cubical expansion.
Consider a rectangular parallel piped of slides l₁, l₂ and l₃.
Its initial volume (V₀) = l₁ x l₂ x l₃.
Differentiating both sides with respect to T, we get:
\(\frac{dV}{dT}={{l}_{2}}{{l}_{3}}\frac{d{{l}_{1}}}{dT}+{{l}_{1}}{{l}_{3}}\frac{d{{l}_{2}}}{dT}+{{l}_{1}}{{l}_{2}}\frac{d{{l}_{3}}}{dT}\).
dV/ dT = l₂l₃ (αl₁) + l₁l₃ (αl₂) + l₁l₂ (αl₃) = 3 αl₁l₂l₃
dV = 3 αV₀dT … (1)
If γ is constant over the temperature range of under consideration, then equation (1) can be interchanged.
\(\int\limits_{{{V}_{i}}}^{{{V}_{f}}}{dV}=3\alpha {{V}_{0}}\int\limits_{{{T}_{i}}}^{{{T}_{f}}}{dT}\).
ΔV = 3αV₀ΔT (or) γV₀ΔT
Where,
γ = Coefficient of superficial expansion.
Thrice the coefficient of linear expansion α.
Final area of the parallel piped, V = V₀ (1 + γΔT).
The average coefficient of superficial expansion, \(\gamma =\frac{\Delta V}{{{V}_{0}}\Delta T}\) and unit is same as α i.e., °C⁻¹ or K⁻¹.