CHAPTER 6

THERMAL EXPANSION The use of an equation

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 Lo at some reference temperature To. • If the temperature is changed by an amount T = T – To then the length changes by an amount L = L – Lo • 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 Lo. • 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 = Lo T

THE EQUATION • Although  depends on the temperature interval T and the reference temperature To, that dependence is usually negligible for moderate temperature changes. The coefficient  does not depend on the length Lo. • The dimension of  is reciprocal temperature, and the commonly used unit is reciprocal degrees Celsius (oC-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) , 10-5 oC-1 Aluminium 2.4 Copper 1.8 Steel 1.1 Glass 0.1 - 1.3 Concrete 0.7 – 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 Vo is the volume of a substance at a reference temperature To, then the change in volume V that accompanies a temperature change T is given by V =  Vo T where  is the average coefficient of volume expansion.

• The following table shows some values of  for some liquids. Volume expansion Substance (liquid)

, 10-5 oC-1

Methanol

113

Glycerin

49

Mercury

18

Turpentine

90

Acetone

132

• 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 oC. • However, between 0oC and about 4oC, 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 -5oC on a winter day. What change in length occurs from winter to summer, when the temperature is 35oC? • Solution. From 1st table,  for concrete is around 1 × 10-5 C-1. Using the first equation, we have

L

= Lo T = (1 × 10-5 C-1)(12 m)(40 oC) = 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 40oC. What is the original length of the rod? 2. A metal rod has a length of 1m. It is heated through 200oC. If the coefficient of linear expansion () of the metal is 0.00002 /oC, find the expansion. 3. 50m copper piping is heated through 70oC. What is the expansion?

Example 2 • Volume expansion of a sphere. An aluminium sphere has a radius R of 3.000 mm at 100.0 oC. What is its volume at 0.0 oC? • Solution. The volume of sphere (4R3/3) at 100oC is V = 113.1 mm3. From the 1st table  = 2.4 × 10-5 oC-1 and  = 3 = 7.2 × 10-5 oC-1 Applying 2nd equation, we obtain V = (7.2 × 10-5 oC-1)(113.1 mm3)( -100.0 oC) = - 0.81 mm3 The volume at 0oC is 113.1 – 0.8 = 112.2 mm3.

• An alternative approach is to evaluate the radius of the sphere (a linear dimension) at 0oC and hence calculate the volume from V = 4R3/3 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?

Questions 1)

a) b)

c)

A steel rule is calibrated at 22oC against a standard so that the distance between numbered divisions is 10.00 mm. What is the distance between these divisions when the rule is at -5oC? If a nominal length of 1m is measured with the rule at this lower temperature, what percent error is made? What absolute error is made for a 100-m length?

Questions 2)

a) b) c)

d)

A copper plate at 0oC has thickness of 5.00mm and a circular hole of radius 75.0mm. Its temperature is raised to 220oC. Determine the values at this temperature of: the thickness of the plate, the radius of the hole, the circumference of the boundary of the circular hole, the area of the hole in the plate.

Questions 3)

a)

b)

A steel shaft has diameter 42.51mm at 28oC. It is to be fitted to a steel pulley with a circular hole of diameter 42.50mm at that temperature. By how much must the temperature of the shaft be reduced so that it can fit in the hole? Suppose the temperature of the entire structure is reduced to -5oC after the shaft has been fitted. Will the shaft come loose? Explain.

Questions 4) a)

b) c) d)

The density of aluminium is 2692 kg/m3 at 20oC. What is the mass of an aluminium sphere (V = 4R3/3) whose radius R at this temperature is 25.00mm? What is the mass of the aluminium at 100.0oC? What is the density of aluminium at 100oC? What are the answers to parts (b) and (c) if the aluminium is in the shape of a cube?

Questions 5)

A glass ( = 2.2  10-5 oC-1) bulb is completely filled with 176.2ml of mercury ( = 2.2  10-5 oC-1) at 0.0oC. The bulb is fitted, as illustrated in the following figure, with a glass tube of inside diameter 2.5mm at 0.0oC. How high does the mercury rise in the tube if the temperature of the system is raised to 50.0oC? The change in diameter of the glass tube may be neglected. Why?