Physics · Ch 7 — Thermal Properties of Matter

Volume Expansion

7.5.3

Volume Expansion

For a substance in the form of a cube (or any three-dimensional solid) of volume $V$ , the fractional change in volume $\Delta V/V$ for a small temperature change $\Delta T$ (Fig. 7.6) is directly proportional to $\Delta T$ :

\frac{\Delta V}{V} \propto \Delta T \quad \text{or} \quad \frac{\Delta V}{V} = \gamma\,\Delta T \quad \text{--- (7.16)}

where $\gamma$ is the coefficient of cubical (volume) expansion. With $V_0$ the volume at 0 °C:

V_T = V_0(1 + \gamma T) \quad \text{--- (7.17)}

and between two temperatures $T_1, T_2$ with volumes $V_1, V_2$ :

V_2 = V_1[1 + \gamma(T_2 - T_1)] \quad \text{--- (7.18)}

Numerically, if $V_0 = 1\text{ m}^3$ and the rise is 1 °C, $\gamma = V_T - V_0$ : the increase in volume per unit original volume (at 0 °C) for a one-degree rise. Table 7.2 lists typical $\gamma$ values -- note liquids such as alcohol, gasoline and mercury have coefficients roughly an order of magnitude larger than solids like steel or aluminium; Invar, a special alloy, is deliberately engineered to have an extremely small $\gamma$ .

Since fluids (liquids and gases) have a definite volume but take the shape of their container, only their VOLUME expansion is meaningful. Because fluids are always held in some container, the container's own expansion must also be taken into account: if the fluid expands MORE than its container, the excess overflows from an open vessel, or, in a closed vessel, builds up extra pressure on the walls (this is exactly why a sealed can or balloon can burst on a hot day). …

Figure 7.6Fig. 7.6: Volume expansion of a cube

What this figure shows. A before-and-after schematic of a solid cube. The 'before heating' cube is drawn with its original volume V labelled. The 'after heating' cube is drawn as a slightly larger cube (all three dimensions marginally increased), with the thin outer shell/layer between the two cube outlines shaded or marked as the small increase in volume, labelled delta-V. As with the earlier expansion figures, delta-V is exagger …

Table 7.2Table 7.2: Coefficient of volume expansion (gamma) for some common materials (0-100 °C)

Material | gamma (K^-1)

Invar | 2x10^-6

Glass (ordinary) | 2.5x10^-5

Steel | (3.3-3.9)x10^-5

Iron | 3.55x10^-5

Gold | 4.2x10^-5

Brass | 5.7x10^-5

Aluminium | 6.9x10^-5

Mercury | 18.2x10^-5

Water | 20.7x10^-5

Paraffin | 58.8x10^-5 …

Figure 7.7Fig. 7.7: Coefficient of volume expansion of copper as a function of temperature

What this figure shows. A graph with the coefficient of volume expansion gamma on the vertical axis and temperature on the horizontal axis, specifically for copper, plotted over a wide temperature range starting from very low (near absolute zero) temperatures. The curve starts near zero (gamma is very small) at very low temperatures, then rises steeply through an intermediate range, and finally FLATTENS OUT / becomes nearly constant (a near-horizontal line) at high temperatures. The shape visually demonstrates the text's point that gamma is not strictly a constant but depends on temperature, and only approaches a genuinely constant va …

Misc Ex.11Liquid overflow from a heated glass beaker filled to the brim

Worked out. A liquid at 0 °C completely fills a 600 cm^3 glass beaker (gamma(liquid) = 1.75x10^-4/°C, gamma(glass) = 2.75x10^-5/°C); heating to 90 °C, the example separately computes the increase in the glass container's own volume (1.485 cm^3) and the increase in the liquid's volume (9.45 cm^3) using delta-V = gamma x V1 x (T2-T1) for each, then subtracts to find the liquid that overflows since the container did not expand as much as the liquid: 9.45 - 1.485 = …

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