Module 6
PlanningGuide

Lesson 2.7

. Try This
. Concepts
. Equations
. Examples
. Exercises
. Answers
. Definitions

Lesson 2.5
Lesson 2.6
Lesson 2.7
Lesson 2.8
Lab 2.5
Lab 2.6
Lab 2.7
Lab 2.8
Project 6

Lesson 2.7 Expansion

Objectives
This lesson deals with thermal expansion of solids liquids and gases. After completing the lesson, you should be able to explain why materials expand when heated. You should be able explain how bimetallic strips are used in thermostats and temperature sensing devices. You should also be able to use the coefficient of linear expansion for a solid to calculate the increase in length when heated.

Overview
Solids, liquids and gases expand when heated. Large forces can be generated when solids, liquids and gases expand. Expansion joints need to be included in structures and large pieces of equipment to avoid damage at higher temperatures.

When the temperature of a substance increases, the particles that make up the material move faster. The rate at which these particles collide with adjacent particles increases and this causes the particles to move further apart. The exception is water below 4ºC. Water expands as it cools from 4ºC to 0ºC and this is due to the formation of microscopic ice crystals in the water below 4ºC. Ice crystals are structured in such a way that the density of ice is less than that of water.

MINI LAB

CHOICE OF ACTIVITIES

Bimetallic strip: Open a thermostat or temperature-indicating device that uses a bimetallic strip and show how the strip bends when heated.

Hold the ends of rods or pipes made from different metals in a flame (gas cooker) and compare the rates at which heat is transferred through the different materials. See how the thickness of the material affects the rate of heat transfer.

Expansion & Kinetic Theory
In liquids and gases, the particles (molecules) can move around but they also vibrate and rotate. In solids, the particles (atoms, molecules, ions) vibrate and rotate. When a material is heated, the increase in energy causes the particles that make up a substance move faster and further. The movement of each particle has an effect on the adjacent particles. Increased movement of particles tends to drive the particles further apart.

In liquids and solids, the forces of attraction between particles are weakened by the increased movement of the particles. The volume of the material increases. Gases expand to occupy the space in a container. The pressure in the container is a measure of the rate at which gas molecules collide with each other and with the surface of the container. When a gas is heated, the increased rate of movement of the particles results in an increase in pressure. If the volume of the container can change (e.g. in a balloon), the volume of the gas will increase as the temperature increases.

Water & Ice
Water is unique in that it contracts when heated from 0ºC to 4ºC. Water also expands when it changes phase from liquid to solid. That is why ice cubes float in water and pipes break when the water inside of them freezes.?This also explains why a lake freezes at the surface, and not from the bottom up.

The expansion of water as it cools from 4ºC to 0ºC is due to the formation of microscopic ice crystals in the water below 10ºC. Ice crystals are structured in such a way that the density of ice is less than that of water. In the absence of these microscopic ice crystals, water would contract normally as it cools to 0ºC but as it cools below 10ºC, more and more microscopic ice crystals with a density lower than that of water form. The overall density of the mixture continues to increase until it reaches 4ºC. After that the density starts to decrease.

Expansion Forces
Solids and liquids are essentially incompressible. If a liquid completely fills a rigid container and it is heated, the expansion forces will rupture the container. Hydrocarbons such as gasoline and liquid propane tend to expand more than water does. That is why propane containers need to be filled very carefully to ensure that the container is not completely filled and that space is available for the liquid to expand. Steel and concrete structures also need to be designed and constructed to allow for expansion and contraction with changes in temperature. Large forces can be created when solids expand and contract.

The Bimetallic Strip
The degree of expansion for a particular increase in temperature differs from material to material. Each material has its own coefficient of expansion. This is a measure of the amount of expansion per degree increase in temperature.

If the flat surfaces of two metal strips are joined together and the metals have different coefficients of expansion, a change in temperature will create large forces that will cause the shape of the combined strip to be distorted. The strip will bend in one direction when heated and bend in the opposite direction when cooled.

Many temperature sensing and controlling devices make use of bimetallic strips. Dial-type thermometers use a coiled bimetallic strip attached to a pointer to indicate the temperature of the bimetallic strip. Thermostats use bimetallic strips to control temperature by using the mechanical movement of the strip to open and close electrical contacts.

Coefficient of Linear Expansion
Different materials expand by different amounts for the same increase in temperature. The coefficient of linear expansion is a measure of the amount by which a solid will increase in length as a result of a 1º increase in temperature. The actual increase in length depends on the original length of the object and the coefficient is defined as follows:

The coefficient of linear thermal expansion of an object is the ratio of the change in length per ºC to its original length.

Because the value is small, the coefficient is usually reported in units of 10⁶ ºC^-1 or 10⁶ K^-1

a is usually used to denote the coefficient of linear thermal expansion. If l₀ is the initial length,

then l_t , the length at temperature t is given by: l_t = l₀(1 + a .t)

................ D L
or a = --------
.............. L₀ D T

Where L₀ is the original length, D L is the change in length and D T is the change in temperature.

TYPICAL COEFFICIENTS

Material ................Temperature Range ........Coefficient of Linear Expansion

Aluminum ......................20ºC ...........................22 x10^-6 ºC^-1.

....................................100ºC ...........................24 x10^-6 ºC^-1.

Steel .........................0 to 100ºC .......................11 x10^-6 ºC^-1.

Brass ........................0 to 100ºC .......................19 x10^-6 ºC^-1.

Copper ......................0 to 100ºC ........................17 x10^-6 ºC^-1.

Glass ........................0 to 100ºC ..........................8 x10^-6 ºC^-1.

Lead ..........................0 to 100ºC .........................31 x10^-6 ºC^-1.

Coefficient of Volumetric Expansion

If DV is the change in volume of a quantity of material due to change in temperature (DT),

DV = b V DT

where V is the initial volume of the material
b is the coefficient of volume expansion
DT is the temperature change in °C (or K)

TYPICAL COEFFICIENTS

Linear expansion:

a = (l_t – l₀) / (l₀ t) : Where: a = coefficient of linear expansion (ºC^-1)

l₀ = Initial length (m)

l_t = length at temperature t (m)

t = temperature of object (ºC)

. . . . . . . . D L
or a = ----------
. . . . . . L₀ D T

Where: L₀ is the original length,

D L is the change in length and

D T is the change in temperature.

Volumetric expansion:

DV = b V DT where V is the initial volume of the material (m³)
DV = change in volume (m³)
b is the coefficient of volume expansion (ºC^-1)
DT is the temperature change in °C (or K)

Example 2.6.1 Coefficient of Linear Expansion

The coefficient of linear expansion of steel is 11 x10^-6 ºC^-1. If a steel rail with an initial length of 20 meters at 20ºC is heated to 60ºC, by how much will its length increase?

Solution

. . . . . D L
a = ----------
.. . . L₀ D T

= 11 x10^-6 ºC^-1

L₀ = 20 m.

D T = (60 – 20) = 40ºC

D L = 11 x10^-6 ºC^-1 x 20m x 40ºC = 8.8 x10^-3 m

Example 2.6.1 Coefficient of Volume Expansion

The coefficient of volume expansion of paraffin is 590 x10^-6 ºC^-1. By how much will the volume of 3 cubic meters of paraffin increase if it is heated from 5ºC to 20ºC?

Solution

DV = b V DT where V is the initial volume of the material (m³)

DV = change in volume (m³)

b is the coefficient of volume expansion (ºC^-1)

DV = 590 x10^-6 ºC^-1 x 3 m³ x (20ºC – 5ºC) = 2.655 x10^-2 m³.

Questions

1. Most materials expand when their temperature is increased. What causes the expansion?
2. Do liquids usually expand more than solids if their temperatures are increased by the same extent?
3. Why do gases generally expand to a greater extent than liquids or solids?
4. Water is unusual in that it contracts when its temperature is increased from 0ºC to 4ºC. Why does water contract when it is heated in this temperature range?
5. What is a bimetallic strip?
6. Why do they bend in one direction when heated and bend in the other direction when cooled?
7. How is the coefficient of linear expansion of a solid defined?
8. If the coefficient of linear expansion of aluminum is 25 x10^-6 ºC^-1 and that of steel is 12 x10^-6 ºC^-1, will aluminum expand to a greater extent than steel when the temperatures of both substances are increased by the same amount?
9. The coefficient of linear expansion of aluminum is 25 x10^-6 ºC^-1. If an aluminum pipe with a length of 5 meters at 15ºC is heated to 150ºC, by how much will its length increase?
10. The coefficient of volume expansion of gasoline is 950 x10^-6 ºC^-1. By how much will the volume of 1 cubic meter of gasoline increase if it is heated from 10ºC to 25ºC?