Physical principles


If you are interested and want to learn about the physical principles behind thermal expansion and density, then this section is for you. There is a short introduction to both quantities and some simple physical explanation.

If you want more information we encourage you to visit the Literature section of this technical area. The physics books all have some information on expansion and density.


All materials change size when the temperature is changed. The extent to which this occurs is controlled by the atomic arrangement of the material’s structure. The outcome is technologically very important. In solid materials it controls dimension changes in processing and the joining of materials. In liquids it can cause bursting in sealed vessels. In gases, it can cause large changes in pressure at constant volume, or large changes in volume at constant pressure.

Thermal expansion, for example, is a key factor when building a road over a bridge, since the length of the bridge varies with temperature (seasonal changes) causing the pavement to crack. Density determines how much something weighs with respect to its volume and this knowledge can be used to optimise various components, e.g. in the automobile or aerospace industry where weight costs in terms of performance, fuel efficiency and pollution.

Warning: underestimating thermal expansion can lead to embarrassing results, like these bent railway lines.

As can be seen from the photo above for solid train rails, all materials in all physical states are affected by thermal expansion, and in turn a change in density. It is not limited to only one industry sector but to all parts of our daily life, although the connection is often barely visible.

THERMAL EXPANSION - what happens when materials expand

A physical explanation for thermal expansion is commonly based on a simplified model of crystalline solids, where the atoms are held together in a regular array (the lattice) by forces of electrical origin. The forces between atoms are like those that would be exerted by a set of springs connecting the atoms. At any temperature the atoms of the solid are vibrating. When the temperature increases, the amplitude of these vibrations increases and thus the average distance between neighbouring atoms itself increases. This leads to an expansion of the whole solid body as the temperature is increased. For gases, this situation changes a little bit, as the atoms or molecules are no longer situated in a regular array or lattice but can be seen as freely moving within the gas. In this case, an increase in temperature causes an increase in kinetic energy of the particles (which equals an increase in velocity at which the atoms move within the gas) and thus, unless there is a corresponding increase in volume, the pressure in the gas increases.

Temperature increases

Temperature increases






The three figures above demonstrate how temperature leads to expansion: in the solid state (red spheres) the atoms are all regularly ordered in the lattice. As temperature increases the average distances between the atoms starts growing until the material melts and becomes liquid (blue spheres). The liquid state can be described as a state where an actual lattice no longer exists but there is still some sort of short-distance ordering between the atoms. As a result, the liquid material usually covers a larger volume than the corresponding solid. If temperature is further increased the material will become gaseous (yellow spheres) which means even the short-distance order is destroyed and all atoms can move much more freely and thus expand into a larger volume.

Linear and volumetric thermal expansivity - some equations

For solid materials, a change in size is usually defined in terms of a fractional change in dimensions when heated or cooled over a given temperature range, T1 to T2, and is called the mean linear expansion coefficient, defined as: 

where L is the dimension and δL is the size change.

Illustrating linear thermal expansion: a
bar of length L at
temperature T1 expands to length L + δL at temperature T2.

Alternatively, the temperature dependence of the fractional size change at a given temperature, T, is called the linear thermal expansivity:

In either case, because behaviour is usually non-linear, data MUST have appended temperature information, otherwise they are meaningless.

For solids and liquids, and sometimes for gases too, the volumetric expansion can be expressed in a similar manner, but with V substituted for L. In the cases of liquids and gases, the pressure needs to be defined and kept constant. The mean volumetric expansion coefficient is given by , and the mean volumetric expansivity is given by β(T). For a solid material, the fractional change in volume is given by:

Since in most cases the numerical value of α(T) for solid materials is small, typically in the range –1 x 10-6 °C to +250 x 10-6 °C, the higher order terms in the above expansion can be ignored:

Hence, the volumetric expansion coefficient is about three times the linear expansion coefficient.

Note: negative values for expansion coefficients are found in many anisotropic crystalline materials. Different crystallographic directions show different expansion coefficients as a result of distortions on heating or cooling. The net effect in a polycrystalline material can be negative, especially if there is additional cracking between grains. There are other anomalous effects leading to negative expansion coefficients. Water, for example, shows a negative volumetric expansion coefficient between the freezing point and 4 °C.

DENSITY - what determines density

Solids: The physical property ‘density’ of a solid material is determined by the equilibrium distance of the constituent atoms. Such an equilibrium condition (in a lattice type structure) is, as always, based on a competition between the forces of attraction and repulsion between neighbouring atoms. The mass of all the atoms ordered in such a regular (equilibrium) lattice can be calculated, leading to a value for the density if the volume is known.
(NOTE: this explanation holds for solids only.)

Wood, having a lower density than water,
floats - which is wonderful news for loggers, who can transport logs over long distances
by river.


Liquids and gases: In liquids and especially in gases, the term density can only be understood in a statistical way: since the atoms are not located in a relatively fixed lattice structure the mass of a liquid or a gas is defined as the mean number of atoms per unit time in a given volume. If the mean mass and volume are known then density can be calculated in the same straightforward manner as for solids.

The density equation

The mass (m) per unit volume (V) of a medium, whether solid, liquid or gas, is termed the density (ρ). This definition is:

The volume of most materials increases upon heating and, since the mass remains constant, the density must correspondingly decrease.