Thermal Expansion

Calculating Linear Expansion With ΔL = L0αΔT

Most materials grow slightly longer when heated and shrink slightly shorter when cooled — the linear thermal expansion equation, ΔL = L0αΔT, finds exactly how much. Enter the original length, pick (or enter) a coefficient of expansion, and give the starting and ending temperature, and this calculator finds the change in length.

Every material expands and contracts at its own rate. This is why engineers add expansion joints to bridges and sidewalks, and why railroad tracks historically left small gaps between rails — a long steel rail can grow by several millimeters on a hot day, and something has to absorb that movement.

The Formula

$$\Delta L = L_0 \times \alpha \times \Delta T$$

Where $\Delta L$ is the change in length, $L_0$ is the original length, $\alpha$ (alpha) is the material’s coefficient of linear thermal expansion (per degree Celsius), and $\Delta T$ is the temperature change.

Worked Example

A 10-meter steel rail (coefficient of expansion 0.000012 per °C) heats up from -10°C to 30°C:

1. Temperature change: $30 - (-10) = 40$°C.
2. Change in length: $10 \times 0.000012 \times 40 = 0.0048$ m, or about 4.8 millimeters — the rail grows from 10 m to 10.0048 m.

Cooling the same rail back down to -10°C would shrink it by the same amount, back to its original 10 meters.

Key Factors to Consider

• Different materials expand at very different rates, which is why the coefficient value matters so much. Metals like aluminum and copper generally expand more per degree than steel, while materials like glass and concrete expand relatively little — choosing the correct coefficient for the specific material is the single biggest factor in getting an accurate result.
• This formula computes LINEAR (one-dimensional) expansion, which is different from volumetric (three-dimensional) expansion. For an object expanding in all three dimensions (like a liquid filling a container), the volumetric expansion coefficient is roughly three times the linear coefficient for the same material — using the wrong one gives a result off by that same rough factor.
• Real-world engineering deliberately plans around thermal expansion, rather than trying to prevent it. Expansion joints in bridges, sidewalks, and railroad tracks are specifically designed to absorb this predictable length change — trying to rigidly constrain a material against its natural thermal expansion can build up enormous internal stress and cause structural damage or failure.
• The coefficient of expansion itself can vary somewhat with temperature range and specific material composition/alloy. The published coefficient values used here are good general approximations for the temperature ranges most everyday and engineering applications operate in — for a very precise, safety-critical application, consulting a material-specific engineering reference for the coefficient at the exact temperature range involved is worthwhile.

Common Mistakes

• Using a volumetric coefficient in a linear formula, or vice versa. These two figures for the same material differ by roughly a factor of three — plugging a volumetric coefficient straight into this linear equation overstates the length change by about that same factor.
• Forgetting to convert temperatures to a consistent scale before subtracting. Mixing Fahrenheit and Celsius values when computing ΔT produces a temperature change that’s wrong by a large margin, since the two scales don’t share the same degree size.
• Assuming thermal expansion is negligible for short lengths or small temperature swings. Even a modest length can shift by a measurable amount over a wide seasonal temperature range — this is exactly why expansion joints exist even on short spans of bridge deck or sidewalk.

Useful to Know

• Curious about the energy required to actually cause that temperature change, not just its length effect? Heat Energy Calculator calculates the heat needed to raise or lower a material’s temperature.
• Working with a gas instead of a solid, where volume, pressure, and temperature are all linked? Ideal Gas Law Calculator covers that relationship directly.