The reason for all this is **thermal expansion**. The article will discuss the underlying physics behind thermal expansion, and investigate some **real applications** of thermal expansion in physics and engineering.

## 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!

## 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.

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.

Some materials do not expand as we would expect. Water for example does not behave like most other liquids. Above a temperature of \(4^\circ \mathrm C\), water expands with **increasing** temperature just like most other materials. However, in the range of \(0-4^\circ\mathrm C\), water actually expands with **decreasing*** *temperature. This is called the anomalous expansion of water.

This explains why lakes freeze from the top down rather than the bottom up. The density of water is at its maximum at \(4^\circ\mathrm C\) (as the volume of the water would expand if the temperature is increased or decreased from \(4^\circ\mathrm C\)) so when the water is cooled from this temperature, it becomes less dense than the water beneath it. Remember, less dense fluids rise, so the cooler water remains on the surface until it freezes. Essentially, the surface of the lake freezes while the water beneath is still liquid!

## 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.

## 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.

**Question 1:**

An aluminum metal bar of length \(0.5\,\mathrm{m}\) is heated from a temperature of \(20^\circ\mathrm C\) to \(100^\circ\mathrm{C}\). What is the final length of the bar? The coefficient of linear thermal expansion of aluminum is \(2.30\times 10^{-5}\,\mathrm{K}^{-1}\).

**Answer 1:**

We can use the equation for linear thermal expansion stated above to find the change in length.

\[\Delta L=L\alpha \Delta T\]

We know the original length and the coefficient of linear thermal expansion. All we need is the temperature change, which is the difference between the final and initial temperatures. This is equal to \(80^\circ\mathrm{C}\). As the value of one Kelvin is equal to the value of one degree Celsius, we do not need to change the units of the change in temperature or the coefficient.

Using all of the given values in the equation above gives a change in length of \(\Delta L=0.00184\,\mathrm m\).

The final length will be equal to the change in length added to the original length.

\[L_F=F+\Delta L=0.502\,\mathrm m\]

**Question 2: **

A cube of gold is initially at a temperature of \(20^\circ\mathrm C\). If it is heated to a temperature of \(2000^\circ \mathrm C\), what would the ratio of its final volume to its initial volume be? The *linear *coefficient of* *thermal expansion of gold is \(1.40\times 10^{-5}\,\mathrm{K}^{-1}\).

**Answer 2:**

The change in volume due to the thermal expansion of the gold is\(\nabla V\) and the initial volume is \(V\). Then the ratio \(R\) of the final volume to the initial volume can be given by the following formula:

\[R=\dfrac{V+\nabla V}{V}\]

In order to find the change in volume, we can use the equation for volume thermal expansion stated above.

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

We can substitute this formula above into the ratio equation and cancel the initial volume terms to find the ratio in terms of the change in temperature and the volume thermal expansion coefficient.

\[R=\dfrac{V+V\alpha_V\Delta T}{V}=1+\alpha_V\Delta T\]

However, we are not given the volume expansion coefficient in the question, only the linear coefficient. Remember that they are related by the following formula:

\[\alpha_V=3\alpha_l\]

Using the value for \(\alpha_l\) given in the question we find that \(\alpha_V=6.9\times 10^{-5}\,\mathrm{K}^{-1}\). We then only need the temperature difference, which is equal to \(1800^\circ\mathrm C\). We can substitute these values into our equation for the ratio of the final and initial volumes to get the answer of \(R=1.12\).

## Thermal Expansion - Key takeaways

- Thermal expansion is the process of how matter will change its length, area, volume, or density as a result of heating.
- Heating a material causes the atoms to gain kinetic energy and move away from each other.
- The ratio of the change in length to the initial length of a heated object is equal to its coefficient of linear expansion multiplied by the change in temperature.
- The coefficient of area expansion is two times the coefficient of linear expansion.
- The coefficient of volume expansion is three times the coefficient of linear expansion.
- A higher thermal expansion coefficient means that a material will expand more per unit length when it is heated.
- Metals usually have a high coefficient of linear expansion than other materials.
- Mercury thermometers work on the basis of thermal expansion.

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##### Frequently Asked Questions about Thermal Expansion

What is thermal expansion?

Thermal expansion is when a material expands due to being heated.

How to calculate thermal expansion?

The ratio of the change in length to the initial length of a heated object is equal to its coefficient of linear expansion multiplied by the change in temperature.

What are the different types of thermal expansion?

The different types of thermal expansion are linear, area and volume expansion.

What causes thermal expansion?

Thermal expansion is caused by a material being heated.

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