Thermal Expansion

As discussed previously almost all solids expand when heated. The equation governing the change in a single dimension (linear expansion) is given by

\[ \rm \mathbf{ \frac{\Delta L}{L} = $\alpha$ \Delta T} \]

where $\alpha$ is called the coefficient of thermal expansion. The units for this coefficient are inverse ^$\circ$C or ^$\circ$C^-1. Typical values for metals are 10^-6 to 10^-5. For liquids they are 10 times bigger.

Here is an interesting video demonstration that shows linear thermal expansion at work:

This, by the way is the second way we have encountered that one can put strain on a material. Here we are dealing with thermal strain, whereas we has previously encountered mechanical strain.

The thermal coefficients $\beta$ and $\gamma$ for a 2-dimensional or 3-dimensional expansion can be easily derived from the equation for linear expansion:

Area expansion:

A = L² = L₀²(1+$\alpha\Delta$T)² = A₀(1+2$\alpha\Delta$T+$\alpha$²$\Delta$T²) A₀(1+2$\alpha\Delta$T)
We neglect terms with higher powers of $\alpha$, because $\alpha$ << 1.
=> $\Delta$A := A-A₀ = A₀2$\alpha\Delta$T => $\Delta$A/A = 2$\alpha\Delta$>T
You could also set up the area expansion using a generic coefficient of area expansion,
\[ \rm \mathbf{ \frac{ \Delta A}{A} = $\beta$ \Delta T} \]
When you combine the last derived equation and this definition equation, then you get:

$\beta$ = 2$\alpha$

Volume expansion:

V = L³ = L₀³(1+$\alpha\delta$T)³ = V₀(1+3$\alpha\delta$T+...)
=> $\Delta$V := V-V₀ = V₀3$\alpha\Delta$T => $\Delta$V/V = 3$\alpha\Delta$T

You could also set up the volume expansion using a generic coefficient of volume expansion,
\[ \rm \mathbf{ \frac{ \Delta V}{V} = $\gamma$ \Delta T} \]
When you combine the last derived equation and this definition equation, then you get:

$\gamma$ = 3\alpha