THERMAL EXPANSION The use of an equation CHAPTER 14

THERMAL EXPANSION Most substances expand with increasing temperature and contract with decreasing temperature. This thermal expansion is usually quite small, but it can be an important effect.

Suppose the length of a solid rod is L o at some reference temperature T o. If the temperature is changed by an amount  T = T – T o then the length changes by an amount  L = L – L o Experiment shows that under usual circumstances the change in length is proportional to the temperature change, at least for a small temperature change.

We expect that the change in length should be proportional to the reference length L o. That is, if the change in length of a rod 2m long is 0.4mm, then the change in length of a 1m rod should be 0.2mm. The change in length also depends on the type of material. For example, copper and iron rods of equal length at one temperature have different lengths at other temperatures.

THE EQUATION These features can be put into equation form by introducing a coefficient that is characteristic of the material. The average coefficient of linear expansion is denoted by . The change in length  L for a temperature change  T is given by  L =  L o  T

THE EQUATION Although  depends on the temperature interval  T and the reference temperature T o, that dependence is usually negligible for moderate temperature changes. The coefficient  does not depend on the length L o. The dimension of  is reciprocal temperature, and the commonly used unit is reciprocal degrees Celsius ( o C -1 ).

Note that this unit is the same as the SI unit, reciprocal Kelvin (K -1 ), because we are using temperature changes. The next table lists the values of  for several common substances. Linear expansion Substance (solid) , o C -1 Aluminium2.4 Copper1.8 Steel1.1 Glass Concrete0.7 – 1.4

Our discussion of thermal expansion has been based on the change in length of a rod, but the equation applies to any linear dimension, such as the diameter of a cylinder or even the radius of a circular hole in a plate. You can think of thermal expansion as analogous to a photographic enlargement in which every linear feature of an isotropic substance changes proportionally. (An isotropic substance has the same properties in all directions.)

A bimetallic strip bends as its temperature is increased. The strip is a complete of two strips of different metals bonded together. Why does it bend?

Ball-and-ring thermal expansion demonstration. The ball barely fits through the ring when both are at room temperature. If the temperature of the ball alone is increased, it will not fit through the ring. If the temperatures of both the ball and the ring are increased, the ball again fits through the ring. This shows that when the ring expands, the size of the hole increases.

and for LIQUIDS? For liquids, as well as for solids, it is convenient to consider volume changes that correspond to temperature changes. If V o is the volume of a substance at a reference temperature T o, then the change in volume  V that accompanies a temperature change  T is given by  V =  V o  T where  is the average coefficient of volume expansion.

The following table shows some values of  for some liquids. Volume expansion Substance (liquid) , o C -1 Methanol113 Glycerin49 Mercury18 Turpentine90 Acetone132

Since the product of three linear dimensions gives a volume, it is not surprising that linear expansion and volume expansion are related. Experiments shows that  = 3  for an isotropic substance.

Characteristics of water Notable by its absence from the tables is liquid water. The positive values of  and  for the substances in that table indicate that they expand with increasing temperature. Water also expands (but not linearly) with a temperature increase in the temperature range from about 4 to 100 o C. However, between 0 o C and about 4 o C, water contracts with a temperature increase.

This behaviour is shown in the following figure in which the volume of 1 kg of water is plotted versus temperature. This variation of volume or of density with temperature is responsible for the stratification that sometimes occurs in large bodies of fresh water. The anomalous thermal expansion of water is ultimately due to the interaction of the unusually shaped water molecules.

Example 1 Expanding concrete. A concrete slab has length of 12m at -5 o C on a winter day. What change in length occurs from winter to summer, when the temperature is 35 o C? Solution. From 1 st table,  for concrete is around 1 × C -1. Using the first equation, we have  L =  L o  T = (1 × C -1 )(12 m)(40 o C) = 5 mm Adjacent slabs in highways and in sidewalks are often separated by pliable spacers to allow for this kind of expansion.

Exercise 1 1.A copper rod lengthens by 5mm when its temperature increases by 40 o C. What is the original length of the rod? 2.A metal rod has a length of 1m. It is heated through 200 o C. If the coefficient of linear expansion (  ) of the metal is / o C, find the expansion. 3.50m copper piping is heated through 70 o C. What is the expansion?

Example 2 Volume expansion of a sphere. An aluminium sphere has a radius R of mm at o C. What is its volume at 0.0 o C? Solution. The volume of sphere (4  R 3 /3) at 100 o C is V = mm 3. From the 1 st table  = 2.4 × o C -1 and  = 3  = 7.2 × o C -1 Applying 2 nd equation, we obtain  V = ( 7.2 × o C -1 )( mm 3 )( o C ) = mm 3 The volume at 0 o C is – 0.8 = mm 3.

An alternative approach is to evaluate the radius of the sphere (a linear dimension) at 0 o C and hence calculate the volume from V = 4  R 3 /3 with the new radius obtained. The answer should be the same in any way.

Exercise 2 1.What temperature change would cause the volume of mercury to change by 0.1 percent?