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Temperature

volumetric expansion

Consider a rectangular solid object with volume V=L₁×L₂×L₃ at temperature T. After being raised to a temperature T+ΔT the volume of the object is given by

V + ΔV = (L₁+ΔL₁)×(L₂+ΔL₂)×(L₃+ΔL₃)

When one substitutes here the expressions for ΔL₁, ΔL₂ and ΔL₃, one finds for the new volume

V + ΔV = L₁×L₂×L₃×(1+αΔT)³ = V×(1+αΔT)³

This can be casted in the form

V + ΔV = V×{ 1+3αΔT+3(αΔT)²+(αΔT)³}

But, since αΔT<<1 one has

(αΔT)³<< (αΔT)²<< αΔT<<1

Hence, to a good approximation we may write

V + ΔV = V×{1+3αΔT}

From which we may conclude that

ΔV = 3α V ΔT

The coefficient for volumetric thermal expansion is thus to a good approximation equal to three times the linear thermal expansion coefficient.
For objects of other shapes, like a sphere the same conclusion holds.

pressure