Thermal Expansion in Solids, Liquids, and Gases
Thermal expansion is the increase in the size of a substance when its temperature rises, because the particles vibrate or move more vigorously and take up more space; for a solid rod the change in length is ΔL = αL0ΔT, where α is the coefficient of linear expansion. In brief: gases expand the most, liquids less, solids the least, for the same temperature rise. Expansion must be allowed for in bridges, railway tracks and pipes. Water is unusual: it contracts when heated from 0 °C to 4 °C.
Gaps are left between sections of bridges and railway lines, and power lines sag more in summer. This guide explains expansion at the particle level, describes the behavior of solids, liquids and gases, gives the formulas, compares materials in a table, works through calculations and corrects common mistakes.
Why do substances expand?
The particles in a substance are bound by attractive forces. The potential energy curve is not symmetrical: the repulsion at short distances is stronger than the attraction at larger ones. When the temperature rises, particles vibrate with a larger amplitude, and the average distance between them increases. The particles themselves do not get larger. In gases, higher temperature makes particles move faster, and at constant pressure the gas takes up more volume.
How do solids expand?
For a small temperature change, the change in length is proportional to the original length and the temperature change:
ΔL = α L0 ΔT
The coefficient α (per kelvin, K−1) depends on the material. For area, ΔA ≈ 2αA0ΔT, and for volume, ΔV ≈ 3αV0ΔT. If the expansion is prevented, large forces arise: this is why railway tracks can buckle in hot weather. Engineers provide expansion joints in bridges, rollers under bridge ends, loops in pipelines, and gaps in pavement. A bimetallic strip consists of two metals with different α bonded together. When heated, it bends toward the metal with the smaller α, and is used in thermostats and old fire alarms.
How do liquids expand?
Liquids expand more than solids, with ΔV = βV0ΔT, where β is the coefficient of volume expansion. This is the principle of the liquid-in-glass thermometer: mercury or alcohol rises in a narrow tube. Water is an exception: between 0 °C and 4 °C, it contracts when heated, and it has the greatest density at 4 °C. As a result, ice floats, and the water at the bottom of a frozen lake stays at about 4 °C, which allows fish to survive the winter.
How do gases expand?
Gases expand much more than solids or liquids. At constant pressure, the volume of a fixed amount of gas is proportional to the absolute temperature (Charles’s law): V ∝ T (in kelvins). Approximately, 1/273 of the volume at 0 °C is added for each degree. A gas heated in a closed container instead increases in pressure. Hot air balloons rise because the heated air expands and becomes less dense.
Coefficients compared
| Material | State | Coefficient |
|---|---|---|
| Glass (borosilicate) | Solid | α ≈ 3 × 10−6 K−1 |
| Steel | Solid | α ≈ 12 × 10−6 K−1 |
| Aluminum | Solid | α ≈ 23 × 10−6 K−1 |
| Mercury | Liquid | β ≈ 1.8 × 10−4 K−1 |
| Ethanol | Liquid | β ≈ 1.1 × 10−3 K−1 |
| Ideal gas | Gas | β ≈ 1/T ≈ 3.4 × 10−3 K−1 at 293 K |
Worked example: expansion calculations
- Steel bridge. A steel girder 100 m long experiences a temperature rise from −10 °C to 30 °C (ΔT = 40 K). ΔL = αL0ΔT = 12 × 10−6 × 100 × 40 = 0.048 m = 4.8 cm.
- Comparing with aluminum. For the same length and temperature change: 23 × 10−6 × 100 × 40 = 0.092 m = 9.2 cm, almost twice as much.
- Mercury in a thermometer. 100 cm3 of mercury heated by 50 K expands by ΔV = 1.8 × 10−4 × 100 × 50 = 0.90 cm3.
- Gas. A balloon contains 1.00 m3 of air at 293 K (20 °C). At constant pressure, heated to 353 K (80 °C), V = 1.00 × 353 ÷ 293 = 1.20 m3, an increase of 20 percent for a rise of 60 K.
How was expansion used in devices?
John Harrison invented a bimetallic compensation for the effect of temperature on clocks in the 1700s, which allowed ships to find longitude with accurate chronometers. Invar, an alloy of iron and nickel with very low expansion (α about 1 × 10−6 K−1), was discovered by Charles Édouard Guillaume in 1896 and used in precision instruments, which won him the Nobel Prize in Physics in 1920.
Which mistakes are common?
Mistake 1: Thinking that atoms themselves get bigger
The average spacing between particles increases, but the particles do not grow.
Mistake 2: Forgetting that holes expand as well
A hole in a metal plate expands as if it was filled with the same metal.
Mistake 3: Using Celsius for gas expansion
Use kelvins for V ∝ T.
Mistake 4: Assuming that water always expands when heated
Between 0 °C and 4 °C it contracts.