Thermal Expansion

Thermal Expansion Definition

Thermal expansion is the process of how matter will change its length, area, volume, or density as a result of heating. Solids, liquids, gases, and plasma all undergo an increase in volume due to thermal expansion.

You may have noticed that you can loosen a tight, metal lid on a glass jar by running hot water over it. This is because the hot water transfers heat to the atoms in the metal and the glass, causing the atoms within the jar to gain kinetic energy and expand. However, this effect is greater in metal than with glass, so the lid expands more than the jar and therefore it becomes easier to twist the lid off!

Fig. 1. Running hot water over a jar lid can help loosen it.

Thermal Expansion Formula

When heat is applied to a material, it will expand in the directions that it is able to. Matter can be constrained in their expansion in a particular direction such as by a container or the surface it rests on. This means that we discuss the thermal expansion of materials in either one, two, or three dimensions depending on the context.

Linear Expansion

Consider a metal rod that is exposed to a change of temperature. For expansion in only one dimension then the thermal expansion is considered linear. The formula for linear thermal expansion is given below:

\[\Delta L=L\alpha_l \Delta T\]

where \(\Detla L\) is the change in length of the rod, \(L\) is the initial length, \(\alpha_l\) is the coefficient of linear expansion of the material and is measured in inverse Kelvin, \(\mathrm{K}^{-1}\), and \(\Delta T\) is the change in temperature of the material. The formula states that the change in length is equal to the product of the initial length of the material, the coefficient of linear expansion, and the change in temperature.

Area Expansion

If a solid can only expand in two dimensions and is held in place in the other direction then its area in these two dimensions will expand. The formula for area thermal expansion is given by:

\[\Delta A=\alpha_A A\Delta T\]

where \(\Delta A\) is the change in area of the object, \(\alpha_A\) is the coefficient of area expansion, and \(A\) is the initial area of the object. The other quantities are the same as before. The coefficient of area expansion is two times greater than the coefficient of linear expansion.

Volume Expansion

If a solid is not restricted in any spatial dimensions when its temperature increases, then the volume of that solid expands. Similarly, liquids and gases can also expand volumetrically but in real-world contexts are often partially restricted in their expansion by their containers.

The formula for volume thermal expansion is:

\[\Delta V=V\alpha_V\Delta T\]

where \(\Delta V\) is the change in volume, \(V\) is the initial volume and \(\alpha_V\) is the coefficient of volumetric expansion, which has the same units as the coefficient of linear expansion \(\alpha_l\). The coefficients of volume and linear expansion are related by the formula:

\[\alpha_A=2\alpha_l\]

\[\alpha_V=3\alpha_l\]

The coefficient of volumetric expansion is equal to three times the coefficient for linear expansion. You should see a pattern with the area coefficient as well. They are both just the coefficient of linear expansion multiplied by the number of dimensions in which expansion is happening.

Thermal Expansion coefficient

The coefficient of linear expansion depends on the material that is being heated. Different materials expand by different amounts depending on their thermal coefficients. A higher thermal expansion coefficient means that a material will expand more per unit length when it is heated. This can be seen by rearranging the thermal expansion formula.

\[\dfrac{\Delta L}{L}=\alpha\Delta T\]

In one dimension, the ratio of the change in length of an object compared to its original length is equal to the linear expansion coefficient multiplied by the temperature change. For a given temperature change, a larger coefficient will lead to a greater increase in length. Some approximate thermal expansion coefficients of various materials are shown in the table below.

Table 1. This table shows the linear expansion coefficient and the area expansion coefficient for several common materials.

The table above helps us understand our example of opening a tight, metal lid on a glass jar. The table shows that metals generally have a higher linear thermal expansion coefficient than glass and so will expand more when heated. Heating a material causes the atoms to gain kinetic energy and spread out, which depends on the strength of the bonds between the atoms. Materials have different bonding forces which will result in different thermal expansion coefficients. The values of these coefficients change slightly on the temperature of the material, but they can be taken to be constant in most cases as the change is normally insignificant.

Fig. 2. Lakes freeze from top to bottom due to the anomalous expansion of water, Wikimedia commons.

Thermal Expansion Examples

Thermal expansion occurs in nearly every material known to humankind. There are many applications of thermal expansion which can be used to our advantage.

Mercury Thermometers

You will have almost definitely used a mercury thermometer before. They are the most common thermometer used in school laboratories. They consist of a thin glass tube containing liquid mercury with temperature markings on the side. When the temperature of the thermometer changes, the mercury will expand and the liquid will rise up the thermometer. When it stops rising you can read off the new temperature.

Fig. 3. A mercury thermometer is used for measuring

temperature, publicdomainpictures.

Mercury is used because it has a high coefficient of thermal expansion of \(1.8\times 10^{-4}\,\mathrm{K}^{-1}\), which means that the expansion is visible to the eye when the liquid rises in the tube. Also, the coefficient of thermal expansion of mercury barely changes with temperature so the change in temperature recorded using the thermal expansion will be accurate.

Railway Tracks

Railway tracks are made with gaps between each rung due to thermal expansion. The temperature of railway tracks varies a lot so if there were no gaps, the material would expand when it becomes hotter and would eventually expand enough so that the tracks break apart. Thermal expansion of solids also causes cracks on roads and bridges.

Fig. 4. Railway tracks are made with gaps in order to prevent damage from thermal expansion, publicdomainpictures.

Thermal Expansion Calculations

The equations for the different types of thermal expansion can be used in practice problems. You must work out which equation is the right one to use and do not forget the factors relating to the coefficients for the linear, area, and volume expansion.