Thermal expansion is the way matter changes size—growing longer, wider, or larger in volume—when heated. This change affects the material's size and density, but it usually does not include changes during phase transitions, like melting or freezing. In simple terms, when something is heated, it expands. Most materials shrink when cooled, but a few materials may expand slightly when cooled under certain temperature conditions. The unit used to measure thermal expansion in the International System of Units (SI) is the inverse kelvin (K).
Temperature is directly related to the average energy of motion in the molecules of a substance. Simply put, temperature measures how hot or cold something is, based on how fast its particles move. When energy increases, particles move faster, which weakens the forces holding them together and causes the material to expand. When a substance is heated, its molecules vibrate and move more, increasing the space between them.
The amount of expansion (called strain) divided by the temperature change is known as the material's coefficient of linear thermal expansion. This value usually changes as temperature increases. The coefficient of thermal expansion is not the same at all temperatures. Generally, it increases with higher temperatures because more heat weakens the forces between atoms, allowing them to move further apart.
Prediction
When an equation of state is available, it can be used to find out how much a material expands at different temperatures and pressures that are needed. It can also help determine other properties that depend on the state of the substance.
Contraction effects (negative expansion)
Some materials shrink when heated within certain temperature ranges. This is called negative thermal expansion, not "thermal contraction." For example, water's expansion rate decreases to zero when cooled to 3.983 °C (39.169 °F) and becomes negative below this temperature. This means water is most dense at this temperature, which helps bodies of water keep this temperature at the bottom during very cold weather.
Other materials also show negative thermal expansion. Pure silicon has a negative expansion rate between about 18 and 120 kelvins (−255 and −153 °C; −427 and −244 °F). ALLVAR Alloy 30, a type of titanium alloy, shows anisotropic negative thermal expansion over a wide temperature range.
Factors
Solid materials usually keep their shape when they expand due to heat, unlike gases or liquids. When bonds between atoms are stronger, materials expand less when heated. Stronger bonds also mean higher melting points, so materials that melt at higher temperatures usually expand less. Liquids generally expand a little more than solids when heated. Glasses expand slightly more than crystals when heated. At a certain temperature called the glass transition temperature, changes in an amorphous material cause sudden changes in how much it expands and how much heat it can hold. These changes help scientists find the temperature where a supercooled liquid becomes a glass. When materials absorb or desorb water, their size can change. Many organic materials, like plastics, change size more because of this than because of heat. Over time, common plastics in water can expand by several percent.
Effect on density
When a substance is heated, its particles spread out, increasing the space between them. This causes the substance's volume to change, while its mass changes very little. As a result, the substance's density changes. This change in density affects the buoyant forces that act on the substance. This process is important for the movement of fluids that are heated unevenly, such as air and water. It helps explain how wind and ocean currents form.
Coefficients
The coefficient of thermal expansion describes how the size of an object changes when the temperature changes. It measures the change in size as a fraction of the original size for each degree of temperature change, assuming pressure remains constant. Materials with lower coefficients expand or contract less when heated or cooled. There are three main types of coefficients: volumetric (for volume changes), area (for surface changes), and linear (for length changes). The choice of coefficient depends on the situation and which dimensions are important. For solids, changes in length or area might be the focus.
The volumetric thermal expansion coefficient is the most basic and is most important for liquids and gases. Most substances expand or contract in all directions when heated or cooled. Materials that expand equally in all directions are called isotropic. For isotropic materials, the area expansion coefficient is about twice the linear coefficient, and the volumetric coefficient is about three times the linear coefficient.
For gases, liquids, and solids, the volumetric thermal expansion coefficient is calculated using the formula:
α = α V = 1/V (∂V/∂T)p.
The subscript "p" means pressure is kept constant during the expansion, and the subscript "V" indicates this formula applies to volume changes, not length. For gases, keeping pressure constant is important because gas volume changes significantly with both temperature and pressure. This can be seen using the ideal gas law.
The table below summarizes thermal expansion coefficients for common materials. For isotropic materials, the relationship between the linear coefficient (α) and the volumetric coefficient (α V) is α V = 3α. For liquids, only the volumetric coefficient is usually listed, and the linear coefficient is calculated for comparison.
For many metals and compounds, the thermal expansion coefficient is inversely related to the melting point. For example, the linear coefficient for metals is roughly α ≈ 0.020 divided by the melting point (Tm). For halides and oxides, the formula is α ≈ 0.038 divided by Tm minus 7.0 × 10⁻⁶ K⁻¹.
The expansion coefficient (α) varies depending on the material and temperature. For example, the coefficient for hard solids can be as low as 10 K⁻¹, while for organic liquids, it can be as high as 10 K⁻¹. Some materials have large changes in α with temperature. For example, the volumetric coefficient of a semicrystalline polypropylene (PP) changes significantly with temperature and pressure, and the linear coefficient of certain steels (such as ferritic, martensitic, carbon, duplex, and austenitic steels) also changes with temperature. The highest linear coefficient in a solid has been recorded for a Ti-Nb alloy.
The formula α V ≈ 3α is commonly used for solids. Volumetric coefficients that do not follow this rule are highlighted in the table.
In solids
When calculating thermal expansion, it is important to determine if the object can expand freely or is restricted. If the object can expand freely, the change in size due to temperature can be calculated using the coefficient of thermal expansion.
If the object is restricted and cannot expand, internal stress will develop when the temperature changes. This stress can be calculated by considering the strain that would occur if the object could expand freely and the force needed to stop that strain, using the relationship between stress and strain described by Young's modulus. For solid materials, changes in external pressure usually do not significantly affect the size of the object, so pressure changes are often not considered.
Most engineering materials used in common applications have thermal expansion coefficients that do not change much over the temperature ranges they are designed for. Therefore, for calculations where very high accuracy is not needed, an average value of the coefficient can be used.
Linear expansion refers to changes in one dimension, such as length, rather than changes in volume. The change in length due to temperature is related to the linear thermal expansion coefficient (CLTE), which is the fractional change in length per degree of temperature change. Assuming pressure has little effect, the formula for CLTE is: α_L = (1/L) * (dL/dT), where L is a length measurement and dL/dT is the rate of change of length with temperature.
The change in length can be estimated using: ΔL/L = α_L * ΔT, where ΔL is the change in length and ΔT is the change in temperature. This formula works well if the expansion coefficient does not change much during the temperature change and the length change is small compared to the original length. If these conditions are not met, a more complex equation must be used.
For solid materials with significant length, such as rods or cables, the thermal expansion can be described by material strain, defined as: ε_thermal = (L_final – L_initial)/L_initial, where L_initial is the length before the temperature change and L_final is the length after the change.
For most solids, thermal expansion is proportional to the temperature change: ε_thermal ∝ ΔT. This means the strain can be estimated using: ε_thermal = α_L * ΔT, where ΔT = (T_final – T_initial) is the temperature difference, and α_L is the linear expansion coefficient. The units for α_L depend on the temperature scale used (e.g., per degree Fahrenheit, Celsius, etc.). In continuum mechanics, thermal expansion is treated as eigenstrain and eigenstress.
The area thermal expansion coefficient relates changes in area to temperature changes. It is the fractional change in area per degree of temperature change. Ignoring pressure, the formula is: α_A = (1/A) * (dA/dT), where A is the area of interest and dA/dT is the rate of change of area with temperature.
The change in area can be estimated using: ΔA/A = α_A * ΔT. This formula works well if the area expansion coefficient does not change much during the temperature change and the area change is small compared to the original area. If these conditions are not met, the equation must be integrated.
For solids, pressure effects can be ignored, and the volumetric (or cubical) thermal expansion coefficient is given by: α_V = (1/V) * (dV/dT), where V is the volume and dV/dT is the rate of change of volume with temperature.
For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature increases by 50 K, showing a 0.2% expansion. If the volume is 2 cubic meters, it would expand to 2.004 cubic meters under the same conditions, also a 0.2% increase. The volumetric expansion coefficient would be 0.2% for a 50 K temperature change, or 0.004% per K.
If the expansion coefficient is known, the change in volume can be calculated using: ΔV/V = α_V * ΔT, where ΔV/V is the fractional change in volume and ΔT is the temperature change. This formula assumes the expansion coefficient does not change significantly with temperature and the volume change is small compared to the original volume. If these conditions are not met, the equation must be integrated: ln((V + ΔV)/V) = ∫_{T_i}^{T_f} α_V(T) dT, and ΔV/V = exp(∫_{T_i}^{T_f} α_V(T) dT) – 1, where α_V(T) is the volumetric expansion coefficient as a function of temperature, and T_i and T_f are the initial and final temperatures.
For isotropic materials, the volumetric thermal expansion coefficient is three times the linear coefficient: α_V = 3α_L. This is because volume depends on three dimensions. For example, a cube of steel with side length L has an original volume V = L³. After a temperature increase, the new volume becomes V + ΔV = (L + ΔL)³ ≈ L³ + 3L²ΔL = V + 3V(ΔL/L). This shows that the volumetric expansion is three times the linear expansion for small changes.
In gases
Gases fill the entire container they are in, so the volumetric thermal expansion coefficient at constant pressure, α V, is the most important one to study.
For an ideal gas, a formula can be found by using the ideal gas law, pVₘ = RT, where p is pressure, Vₘ is molar volume (Vₘ = V/n, with n being the number of moles), T is absolute temperature, and R is the gas constant. By using math, this equation becomes p dVₘ + Vₘ dp = R dT.
When pressure is constant (isobaric), dp = 0, so the equation simplifies to p dVₘ = R dT. This leads to the isobaric thermal expansion coefficient: α V = 1/V (∂V/∂T)ₚ = 1/Vₘ (∂Vₘ/∂T)ₚ = R/(pVₘ) = 1/T. This shows that the expansion coefficient depends strongly on temperature. If temperature doubles, the expansion coefficient is halved.
Between 1787 and 1802, Jacques Charles (unpublished), John Dalton, and Joseph Louis Gay-Lussac discovered that ideal gases change volume linearly with temperature at constant pressure (Charles's law). They found that gases expand or contract by about 1/273 of their volume for each degree Celsius change between 0°C and 100°C. This suggested that gas volume would reach zero at around −273°C.
In October 1848, William Thomson, a 24-year-old professor at the University of Glasgow, published a paper titled "On an Absolute Thermometric Scale." In a footnote, he calculated that "infinite cold" (absolute zero) was equivalent to −273°C (he called this the "temperature of the air thermometers" of the time). This value was linked to the point where ideal gas volume becomes zero. By using the linear relationship between gas expansion and temperature, scientists calculated absolute zero as the negative reciprocal of 0.366/100°C—the average expansion rate of ideal gases between 0°C and 100°C. This gave a value very close to the currently accepted −273.15°C.
In liquids
The thermal expansion of liquids is usually greater than that of solids because the forces between molecules in liquids are weaker, and the molecules move more freely. Unlike solids, liquids do not have a fixed shape and instead take the shape of their container. Because of this, liquids do not have a fixed length or area, so their linear and areal expansions are only meaningful in specific contexts, such as measuring temperature changes or estimating rising sea levels due to climate change. Sometimes, the linear expansion coefficient (α L) is calculated using the measured volume expansion coefficient (α V).
In general, liquids expand when heated, except for cold water. Below 4 °C, cold water contracts, resulting in a negative thermal expansion coefficient. At higher temperatures, water behaves like other liquids, with a positive thermal expansion coefficient.
The expansion of liquids is often measured in a container. When a liquid expands, the container also expands. This means the observed increase in volume (as shown by the liquid level) is not the actual increase in the liquid’s volume. The expansion of the liquid compared to the container is called apparent expansion, while the true expansion of the liquid is called real or absolute expansion. The coefficient of apparent expansion is calculated by comparing the apparent increase in volume per degree of temperature change to the original volume. Absolute expansion can be measured using methods like ultrasonic techniques.
Historically, measuring the thermal expansion of liquids was difficult because direct measurements of liquid height changes due to heating only showed apparent expansion. This meant experiments measured both the liquid’s expansion and the container’s expansion at the same time. For example, when a flask with a narrow stem containing some liquid is placed in a heat bath, the liquid level in the stem first drops slightly before rising. This initial drop is not caused by the liquid contracting but by the flask expanding as it warms.
Soon after, the liquid warms and begins to expand. Since liquids typically expand more than solids for the same temperature change, the liquid’s expansion eventually exceeds the flask’s expansion, causing the liquid level to rise. For small, equal temperature increases, the real expansion of a liquid equals the sum of its apparent expansion and the container’s expansion. The absolute expansion of the liquid is the apparent expansion adjusted for the container’s expansion.
Examples and applications
When designing large structures, using tape or chain to measure distances for land surveys, creating molds for hot materials, or in other engineering tasks where temperature changes cause significant size changes, the expansion and contraction of materials must be considered.
Thermal expansion is used in mechanical systems to fit parts together. For example, a bushing can be placed over a shaft by making its inside diameter slightly smaller than the shaft’s diameter. The bushing is then heated until it expands enough to fit over the shaft. Once cooled, it contracts and holds the shaft tightly, creating a "shrink fit." A common industrial method for this is induction shrink fitting, which heats metal parts between 150 °C and 300 °C to cause expansion, allowing parts to be inserted or removed.
Some alloys have very small linear expansion coefficients, meaning they change size very little with temperature. One example is Invar 36, which expands about 0.6 × 10 K. These materials are useful in aerospace applications where large temperature changes occur.
Pullinger’s apparatus is used in laboratories to measure the linear expansion of a metal rod. The device includes a metal cylinder (called a steam jacket) with steam inlet and outlet tubes. Steam from a boiler is sent through the jacket to heat the rod. A thermometer is placed in the cylinder to monitor temperature. The rod is placed inside the jacket, with one end fixed and the other touching a screw. A micrometer or spherometer measures the rod’s position as it expands.
To find a metal’s linear thermal expansion coefficient, a pipe made of that metal is heated using steam. One end of the pipe is fixed, and the other rests on a rotating shaft connected to a pointer. A thermometer records the temperature, allowing scientists to calculate how much the pipe’s length changes with each degree of temperature.
Controlling thermal expansion is important for brittle materials like glass and ceramics. Uneven temperatures cause uneven expansion, leading to stress and possible breaking. Ceramics often need to work with other materials, so their expansion must match the application. Glazes must also match the thermal expansion of the ceramic body to avoid cracks or damage. Products like CorningWare and spark plugs rely on carefully controlled thermal expansion. The expansion of ceramic materials can be adjusted by firing them to create specific crystals or by using materials with desired expansion properties. Glazes are controlled through their chemical makeup and firing process. Adjusting thermal expansion often involves balancing it with other material properties.
Thermal expansion affects gasoline stored in above-ground tanks. In winter, gasoline in above-ground tanks may be more compressed than in underground tanks, while in summer, it may be less compressed.
Heat-induced expansion is important in many engineering fields. Examples include:
– Metal-framed windows need rubber spacers to handle expansion.
– Rubber tires must work well in different temperatures, whether from road surfaces, weather, or mechanical movement.
– Metal hot water pipes should not be used in long straight sections to avoid stress.
– Large structures like railways and bridges need expansion joints to prevent damage from temperature changes.
– A gridiron pendulum uses different metals to keep its length stable despite temperature changes.
– Power lines sag on hot days and tighten on cold days because metal expands when heated.
– Expansion joints in piping systems absorb thermal expansion.
– Precision engineering requires careful attention to thermal expansion. For example, a scanning electron microscope must account for tiny temperature changes, like 1 degree, which can shift a sample’s position.
– Liquid thermometers use mercury or alcohol in a tube to show temperature changes as the liquid expands or contracts.
– A bi-metal thermometer uses a strip made of two metals that bend due to their different expansion rates.