Question

Show that β ≈ 3α, by calculating the change in volume ΔV of a cube with sides of length L.

Short answer

The formula$\mathrm{\beta }=3\mathrm{\alpha }$ is proved below.

Step-by-step solution

Introduction

We calculate the change in length by the formula for linear expansion in solids and find new length and we further calculate change in volume and find new volume.

formula for linear expansion and volume expansion

Formula for length expansion of solids $∆\mathrm{L}=\mathrm{\alpha L}∆\mathrm{T}$

Formula for volume expansion of solids $∆\mathrm{V}=\mathrm{\beta V}∆\mathrm{T}$

Here,$\alpha ,\beta$are the coefficient of linear and volume expansion respectively, L, V is original length and volume respectively, and $∆\mathrm{T}$is the change in temperature.

New length = $\mathrm{L}+∆\mathrm{L}$

New volume = $\mathrm{V}+∆\mathrm{V}$

But for cube,

Volume = (Length)³

Equate new volume

$\begin{array}{rcl}{\left(L+∆L\right)}^{\mathbf{3}}& \mathbf{=}& \mathbf{V}\mathbf{+}\mathbf{∆}\mathbf{V}\\ {\left(L+\mathrm{\alpha L}∆T\right)}^{\mathbf{3}}& \mathbf{=}& \mathbf{V}\mathbf{+}\mathbf{\beta V}\mathbf{∆}\mathbf{T}\\ {\mathbf{L}}^{\mathbf{3}}{\left(1+\alpha ∆T\right)}^{\mathbf{3}}& \mathbf{=}& \mathbf{V}\left(1+\beta ∆T\right)\\ {\mathbf{L}}^{\mathbf{3}}\left(1+{\alpha }^3∆T^3+3\alpha ∆T+3{\alpha }^2∆T^2\right)& \mathbf{=}& \mathbf{V}\left(1+\beta ∆T\right)\end{array}$

Since α is very small, second and fourth terms in bracket on LHS are negligible. Also, wkt.V=L³. So the equation becomes

$\begin{array}{rcl}\left(1+3\alpha ∆T\right)& \mathbf{=}& \left(1+\beta ∆T\right)\\ \mathbf{3}\mathit{\alpha }\mathbf{∆}\mathbf{T}& \mathbf{=}& \mathbf{\beta }\mathbf{∆}\mathbf{T}\\ \mathbf{3}\mathbf{\alpha }& \mathbf{=}& \mathbf{\beta }\end{array}$

Therefore, it is proved that$\mathrm{\beta }=3\mathrm{\alpha }$