Physics · Ch 7 — Thermal Properties of Matter

Linear Expansion

7.5.1

Linear Expansion

For a substance in the form of a long rod of length $l$ , a small temperature change $\Delta T$ produces a fractional change in length $\Delta l/l$ that is directly proportional to $\Delta T$ (Fig. 7.4):

\frac{\Delta l}{l} \propto \Delta T \quad \text{or} \quad \frac{\Delta l}{l} = \alpha\,\Delta T \quad \text{--- (7.10)}

where $\alpha$ , the coefficient of linear expansion, depends on the material. Rearranging with $l_0$ the length at $0\,°\text{C}$ and $l_T$ the length at temperature $T$ :

l_T = l_0(1 + \alpha T) \quad \text{--- (7.11)}

If $l_0 = 1$ and $T = 1\,°\text{C}$ , then numerically $\alpha = l_T - l_0$ : the coefficient of linear expansion is defined as the increase in length per unit original length (at 0 °C) for a one-degree rise in temperature. Its unit is per degree Celsius or per kelvin, and, since it varies only slightly with temperature, it is treated as constant over ordinary working ranges. Equation (7.11) is then more conveniently written between any two temperatures $T_1$ and $T_2$ with lengths $l_1$ and $l_2$ :

$l_2 = l_1[1 + \alpha(T_2 - T_1)] \quad \text{--- (7.12)}$ …

Figure 7.4Fig. 7.4: Linear expansion of a rod

What this figure shows. A simple before-and-after schematic of a straight rod lying horizontally. The 'before heating' rod is drawn with its original length l labelled along its length. The 'after heating' rod (drawn either below or overlapping the first, extended further to the right) shows the same rod slightly longer, with the small extra increase in length marked and labelled as delta-l (the change in length) at the far end. The caption explicitly notes that delta-l is exaggerated in the drawing for clarity, since in reality the fractional change in …

Table 7.1Table 7.1: Coefficient of linear expansion (alpha) for some common materials (0-100 °C)

Material | alpha (K^-1)

Carbon (diamond) | 0.1x10^-5

Glass | 0.85x10^-5

Iron | 1.2x10^-5

Steel | 1.3x10^-5

Gold | 1.4x10^-5

Copper | 1.7x10^-5

Silver | 1.9x10^-5

Aluminium | 2.5x10^-5

Sulphur | 6.1x10^-5

Mercury | 6.1x10^-5 …

Misc Ex.8Coefficient of linear expansion of a metal rod from measured lengths

Worked out. A metal rod is 4 cm long at 27 °C and 4.02 cm long at 387 °C; substituting into l2 = l1[1+alpha(T2-T1)] and solving for alpha gives alpha = (0.02x10^-2)/(4x10^-2 x 360) = 1.39x10^-5 /°C. …

Misc Ex.9Temperature needed for an iron rod to reach a given length

Worked out. An iron rod is 4.256 m long at 27 °C (alpha(iron) = 1.2x10^-5 K^-1); the example rearranges l2 = l1[1+alpha(T2-T1)] to solve for the final temperature T2 at which the rod's length becomes 4.268 m, finding T2 = 261.96 °C. …

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