Thermal Expansion

1. Introduction

When a body is heated, the amplitude of vibration of its molecules increases and the average separation between them grows, so the body expands. In solids this expansion appears as an increase in length, area and volume:

• increase in length is linear expansion;
• increase in area is superficial (areal) expansion;
• increase in volume is cubical (volume) expansion.

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2. Coefficient of Linear Expansion

Let a rod of length $L_1$ at temperature $\theta_1$ be heated to temperature $\theta_2$, so its length becomes $L_2$. Experiment shows the increase in length $\Delta L = L_2 - L_1$ is:

• directly proportional to the original length, $\Delta L \propto L_1$;
• directly proportional to the temperature change, $\Delta L \propto \Delta\theta$ where $\Delta\theta = \theta_2 - \theta_1$.

Combining:

$$\Delta L = \alpha L_1 \Delta\theta$$

where $\alpha$ is the coefficient of linear expansion (linear expansivity). Hence:

$$ \begin{aligned} L_2 &= L_1(1 + \alpha,\Delta\theta) \ \qquad \alpha &= \frac{\Delta L}{L_1,\Delta\theta} \end{aligned} $$

The coefficient of linear expansion is the increase in length per unit original length per unit rise in temperature. Its SI unit is $\text{K}^{-1}$ (or $^{\circ}\text{C}^{-1}$).

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3. Coefficient of Superficial Expansion

Let a plate of area $A_1$ at temperature $\theta_1$ be heated so its area becomes $A_2$. The increase in area $\Delta A = A_2 - A_1$ is proportional to $A_1$ and to $\Delta\theta$:

$$ \begin{aligned} \Delta A &= \beta A_1 \Delta\theta \quad\Rightarrow\quad A_2 \ &= A_1(1 + \beta,\Delta\theta) \ \qquad \beta &= \frac{\Delta A}{A_1,\Delta\theta} \end{aligned} $$

The coefficient of superficial expansion $\beta$ is the increase in area per unit original area per unit rise in temperature. SI unit: $\text{K}^{-1}$.

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