Physics · Ch 7 — Thermal Properties of Matter

Areal Expansion

7.5.2

Areal Expansion

For a substance in the form of a flat plate of area $A$ , the fractional change in area $\Delta A/A$ for a small temperature change $\Delta T$ (Fig. 7.5) is directly proportional to $\Delta T$ :

\frac{\Delta A}{A} \propto \Delta T \quad \text{or} \quad \frac{\Delta A}{A} = \beta\,\Delta T \quad \text{--- (7.13)}

where $\beta$ is the coefficient of areal (superficial) expansion, again material-dependent. With $A_0$ the area at 0 °C:

A_T = A_0(1 + \beta T) \quad \text{--- (7.14)}

and, numerically, $\beta = A_T - A_0$ when $A_0 = 1\text{ m}^2$ and the temperature rises by 1 °C -- the increase in area per unit original area (at 0 °C) per one-degree rise. Between any two temperatures:

$A_2 = A_1[1 + \beta(T_2 - T_1)] \quad \text{--- (7.15)}$ …

Figure 7.5Fig. 7.5: Areal expansion of a plate

What this figure shows. A before-and-after schematic of a flat rectangular plate viewed from above. The 'before heating' plate is drawn with its original area A outlined. The 'after heating' plate is drawn slightly larger on all sides (a slightly bigger rectangle), with the thin extra border/strip around the original outline shaded or marked to represent the small increase in area, labelled delta-A. As with Fig 7.4, the caption notes delta …

Misc Ex.10Area of an aluminium plate after heating

Worked out. A thin aluminium plate has area 286 cm^2 at 20 °C (beta(aluminium) = 4.9x10^-5 /°C); substituting into A2 = A1[1+beta(T2-T1)] with T2 = 180 °C, the example computes A2 = 286[1+4.9x10^-5x160] = 288.24 cm^2. …

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