Suppose we have a sample of solid material that expands when the temperature rises. This is the usual case, but some solids expand more per degree Celsius than others. The extent to which the height, width, or depth of a solid (its linear dimension) changes per degree Celsius is known as the thermal coefficient of linear expansion.

For most materials, within a reasonable range of temperatures, the coefficient of linear expansion is constant. That means that if the temperature changes by 2°C, the linear dimension will change twice as much as it would if the temperature changed by 1°C. But there are limits to this. If you heat a metal up to a high enough temperature, it will become soft, and ultimately it will melt, burn, or vaporize. If you cool the mercury in an old-fashioned thermometer down enough, it will freeze. Then the simple length-versus-temperature rule no longer applies.

In general, if s is the difference in linear dimension (in meters) produced by a temperature change of T (in degrees Celsius) for an object whose initial linear dimension (in meters) is d, then the thermal coefficient of linear expansion, symbolized by the lowercase Greek letter alpha (a), is given by this equation:

When the linear size of a sample increases, consider s to be positive; when the linear size decreases, consider s to be negative. Rising temperatures produce positive values of T; falling temperatures produce negative values of T.

The coefficient of linear expansion is defined in meters per meter per degree Celsius. The meters cancel out in this expression, so the technical unit for the thermal coefficient of linear expansion is a little bit arcane: per degree Celsius, symbolized/°C.

Imagine a metal rod 10 m long at 20.00°C. Suppose this rod expands by 0.025 m when the temperature rises to 25.00°C. What is the thermal coefficient of linear expansion?

The rod increases in length by 0.025 m for a temperature increase of 5.00°C. Therefore, s = 0.025, d = 10, and T = 5.00. Plugging these numbers into the formula above, we get:

a = 0.025/(10 x 5.00) = 0.00050/°C = 5.0 x 10-4/°C

We are justified in going to only two significant figures here, because that the limit of the accuracy of the value we are given for s.

Suppose a = 2.50 x 10-4/°C for a certain substance. Imagine a cube of this substance whose volume V1 is 8.000 m3 at a temperature of 30.0° C. What will be the volume V2 of the cube if the temperature falls to 20.0°C?

It's important to note the word "linear" in the definition of a. This means that the length of each edge of the cube of this substance will change according to the thermal coefficient of linear expansion. The volume changes by a larger factor, because the change in the linear dimension must be cubed.

We can rearrange the above general formula for a so it solves for the change in linear dimension, s, as follows:

where T is the temperature change (in degrees Celsius) and d is the initial linear dimension (in meters). Because our object is a cube, the initial length, d, of each edge is 2.000 m (the cube root of 8.000, or 8.0001/3). Because the temperature falls by 10°C, T = -10.0. Therefore:

s = 2.50 x 10-4 x (-10.0) x 2.000 = -2.50 x 10-3 x 2.000 = -5.00 x 10-3 m = -0.00500 m

That means the length of each side of the cube at 20°C is equal to 2.000 - 0.00500 = 1.995 m. The volume of the cube at 20.0°C is therefore 1.9953 = 7.940149875 m3. Because our input data is given to only three significant figures, we must round this off to 7.94 m3.

When matter is heated or cooled, it often does things other than simply expanding or contracting, or exerting increased or decreased pressure. Sometimes it undergoes a change of state. This happens when solid ice melts into liquid water, or when water boils into vapor, for example.

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