Thermal Expansion of Solids, Liquids and Gases

# Thermal Expansion

Substances in all three phases (solid, liquid, gas) change their volume with temperature. The phenomenon is called thermal expansion.

We have already encountered this phenomenon in the chapter, Gas laws and ideal gas equation. We learned the relationship between the gas volume and the temperature at constant pressure. The aforementioned relationship is called the Gay-Lussac law. At constant pressure, gas expands with temperature - its volume increases. The change in volume with temperature can also be accurately calculated for an ideal gas using the gas equation:

The change in volume with temperature is therefore given as:

We divide equation 2 by 1 and get:

 We note that:

The factor is called the coefficient of volume expansion of an ideal gas. For ideal gases, it is strictly determined by the reciprocal of the absolute temperature before the observed change.

Similar to gas, the equation also applies to most liquids. Due to the temperature, the molecules collide and repel each other. The same applies to most solids. Due to the thermal fluctuation, the molecules occupy more space, and the total volume increases.

There are also exceptions when the volume of a substance decreases with temperature. We say they have a negative temperature expansion coefficient.

Example

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## Linear expansion

Let's take a rod of length and increase its temperature by . The rod will extend by as shown in the figure below.

Linear expansion

The quotient of the extension and the original length is called the relative extension. The relative extension is proportional to the change in temperature :

or

The factor is called the linear expansion coefficient.

The final length of the rod after the thermal expansion is given as:

 We note from equation 3 above that: We factor out in the expression on the right-hand side of the equation:

The relative linear extension is proportional to the change in temperature and is given as:

where is the linear expansion coefficient. Consequently, the final length of the heated rod is given as:

The table of the linear expansion coefficients for some materials is shown below.

 Materials Linear expansion coefficient [] Iron 12.2 Aluminium 23.8 Glass 8.5

Example

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## Cubic expansion

Let's take a cuboid with dimensions of length a, b, and c. When the cuboid is heated, the dimensions are increased by , , and as shown in the figure below. This also increases the volume.

Cubic Expansion

From the figure above, we can obtain an approximate expression of how much the volume has increased. The change in volume is given according to the figure above, as:

 We note from equation 3 above that: Therefore We factor out in the expression on the right-hand side of the equation: We note that: We divide both sides of the equation by : We note that: where is called the volume expansion coefficient as stated earlier

In the calculation, it can be observed that we neglected the small cuboids along the edges. The total volume of these cuboids is given according to the figure above, as:

The final volume of the cuboid after thermal expansion is therefore given as:

 We note from equation 4 above that: We factor out on the right-hand side of the equation:

The relative volume expansion of a body is proportional to the change in temperature and is given as:

where is the volume expansion coefficient and is approximately given as:

The final volume of a body after heating is given by the formula:

The table of volume expansion coefficients of some substances is given in the table below:

 Substances Volume expansion coefficient Water at Water at Water at Alcohol Kerosene or gasoline Mercury

Example

The example is available to registered users free of charge.

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