KAHIBARO
Table of Contents
Why materials expand when heated
When a material is heated, its particles gain more internal energy. In solids, atoms are not fixed perfectly still, they vibrate around equilibrium positions. As temperature rises, these vibrations become larger. Because the forces between atoms are not perfectly symmetric, the average separation between atoms usually increases. This is why most materials expand when heated.
In liquids and gases, heating also increases the average separation between particles, so volume usually increases. Thermal expansion is therefore a change in size caused by a change in temperature.
Thermal expansion means that a material's dimensions change when its temperature changes. In most cases, heating causes expansion and cooling causes contraction.
Linear expansion
For long, thin objects such as rods, wires, or rails, the most important change is often the change in length. This is called linear expansion.
If an object has original length $L_0$ and its temperature changes by $\Delta T$, then the change in length $\Delta L$ is approximately
$$
\Delta L = \alpha L_0 \Delta T
$$
Here, $\alpha$ is the coefficient of linear expansion. Its unit is typically $\mathrm{K^{-1}}$ or $\mathrm{^\circ C^{-1}}$.
The new length is
$$
L = L_0 + \Delta L = L_0(1 + \alpha \Delta T)
$$
This formula works well for small temperature ranges where $\alpha$ can be treated as constant.
For linear expansion,
$$
\Delta L = \alpha L_0 \Delta T
$$
and
$$
L = L_0(1 + \alpha \Delta T)
$$
Use $\Delta T = T_{\text{final}} - T_{\text{initial}}$.
Area and volume expansion
When a two dimensional object such as a sheet is heated, its area increases. When a three dimensional object is heated, its volume increases.
For area expansion,
$$
\Delta A = \beta A_0 \Delta T
$$
For volume expansion,
$$
\Delta V = \gamma V_0 \Delta T
$$
Here, $\beta$ is the coefficient of area expansion and $\gamma$ is the coefficient of volume expansion.
For isotropic solids, which expand the same way in all directions,
$$
\beta \approx 2\alpha
\qquad \text{and} \qquad
\gamma \approx 3\alpha
$$
So the new area and new volume are
$$
A = A_0(1 + \beta \Delta T)
$$
and
$$
V = V_0(1 + \gamma \Delta T)
$$
For isotropic solids,
$$
\beta \approx 2\alpha, \qquad \gamma \approx 3\alpha
$$
These are useful approximations for small expansions.
Coefficients of expansion
Different materials expand by different amounts for the same temperature change. Metals usually have noticeable expansion. Glass and concrete expand less. This matters in engineering because connected materials must be able to expand without breaking.
A larger coefficient means a larger change in size for the same initial size and temperature change.
| Material | Typical $\alpha \, (\mathrm{K^{-1}})$ |
|---|---|
| Steel | $1.2 \times 10^{-5}$ |
| Aluminum | $2.4 \times 10^{-5}$ |
| Copper | $1.7 \times 10^{-5}$ |
| Glass | $9 \times 10^{-6}$ |
| Concrete | $1.2 \times 10^{-5}$ |
These values are approximate and can vary with composition and temperature.
Expansion of liquids
Liquids do not have a fixed shape, so volume expansion is the main effect to consider. If a liquid is heated, its volume usually increases according to
$$
\Delta V = \gamma V_0 \Delta T
$$
Liquids often expand more than solids. This is why the level of a liquid in a thermometer rises when temperature increases. The liquid expands more than the glass container, so the height of the liquid column changes.
A special and important exception is water near $4^\circ \mathrm{C}$. Between $0^\circ \mathrm{C}$ and $4^\circ \mathrm{C}$, water contracts when heated, and above $4^\circ \mathrm{C}$ it expands normally. This unusual behavior is important in nature because lakes freeze first at the top.
Expansion of gases
Gases generally expand much more than solids and liquids when heated. If a gas is free to expand, its volume can increase significantly with temperature. The exact relation for gases is usually studied with gas laws, so here the key idea is simply that gases are highly sensitive to temperature changes and show strong thermal expansion.
Everyday examples
Thermal expansion appears in many familiar situations. Railway tracks are built with small gaps so they can expand in hot weather. Bridges often use expansion joints for the same reason. Electric power lines sag more in summer because the metal wires become longer when heated. Jar lids can sometimes be loosened by warming them, since the metal lid expands.
A bimetallic strip is another useful example. It is made of two different metals bonded together. When heated, one metal expands more than the other, so the strip bends. This effect is used in thermostats and temperature switches.
Thermal stress
If an object is free to expand, its size changes without much difficulty. But if it is prevented from expanding, internal forces appear. These forces can produce thermal stress. This is why heated glass can crack if expansion is uneven, and why structures must be designed to allow for temperature changes.
The important physical idea is that temperature change can create force when expansion or contraction is constrained.
A temperature change can produce mechanical stress if expansion or contraction is prevented.
Solved example
Consider a steel rod of length $L_0 = 2.0 \,\mathrm{m}$ heated from $20^\circ \mathrm{C}$ to $70^\circ \mathrm{C}$. Take $\alpha = 1.2 \times 10^{-5}\,\mathrm{K^{-1}}$.
The temperature change is
$$
\Delta T = 70 - 20 = 50^\circ \mathrm{C}
$$
The change in length is
$$
\Delta L = \alpha L_0 \Delta T
$$
$$
\Delta L = (1.2 \times 10^{-5})(2.0)(50)
$$
$$
\Delta L = 1.2 \times 10^{-3}\,\mathrm{m}
$$
So the rod expands by
$$
\Delta L = 0.0012\,\mathrm{m} = 1.2\,\mathrm{mm}
$$
The final length is
$$
L = 2.0 + 0.0012 = 2.0012\,\mathrm{m}
$$
Important ideas to remember
Thermal expansion is the change in size of a material due to temperature change. Solids can expand in length, area, and volume. Liquids and gases are usually treated mainly through volume expansion. The amount of expansion depends on the original size, the temperature change, and the material coefficient.
Key formulas:
$$
\Delta L = \alpha L_0 \Delta T
$$
$$
\Delta A = \beta A_0 \Delta T
$$
$$
\Delta V = \gamma V_0 \Delta T
$$
For isotropic solids:
$$
\beta \approx 2\alpha, \qquad \gamma \approx 3\alpha
$$
Understanding thermal expansion helps explain both simple daily observations and important engineering designs.