Most materials are subject to thermal expansion: a tendency to expand when heated, and to contract when cooled. For this reason, bridges are built with metal expansion joints, so that they can expand and contract without causing faults in the overall structure of the bridge. Other machines and structures likewise have built-in protection against the hazards of thermal expansion. But thermal expansion can also be advantageous, making possible the workings of thermometers and thermostats.
HOW IT WORKS
Molecular Translational Energy
In scientific terms, heat is internal energy that flows from a system of relatively high temperature to one at a relatively low temperature. The internal energy itself, identified as thermal energy, is what people commonly mean when they say "heat." A form of kinetic energy due to the movement of molecules, thermal energy is sometimes called molecular translational energy.
Temperature is defined as a measure of the average molecular translational energy in a system, and the greater the temperature change for most materials, as we shall see, the greater the amount of thermal expansion. Thus, all these aspects of "heat"—heat itself (in the scientific sense), as well as thermal energy, temperature, and thermal expansion—are ultimately affected by the motion of molecules in relation to one another.
MOLECULAR MOTION AND NEWTONIAN PHYSICS.
In general, the kinetic energy created by molecular motion can be understood within the framework of classical physics—that is, the paradigm associated with Sir Isaac Newton (1642-1727) and his laws of motion. Newton was the first to understand the physical force known as gravity, and he explained the behavior of objects within the context of gravitational force. Among the concepts essential to an understanding of Newtonian physics are the mass of an object, its rate of motion (whether in terms of velocity or acceleration), and the distance between objects. These, in turn, are all components central to an understanding of how molecules in relative motion generate thermal energy.
The greater the momentum of an object—that is, the product of its mass multiplied by its rate of velocity—the greater the impact it has on another object with which it collides. The greater, also, is its kinetic energy, which is equal to one-half its mass multiplied by the square of its velocity. The mass of a molecule, of course, is very small, yet if all the molecules within an object are in relative motion—many of them colliding and, thus, transferring kinetic energy—this is bound to lead to a relatively large amount of thermal energy on the part of the larger object.
MOLECULAR ATTRACTION AND PHASES OF MATTER.
Yet, precisely because molecular mass is so small, gravitational force alone cannot explain the attraction between molecules. That attraction instead must be understood in terms of a second type of force—electromagnetism—discovered by Scottish physicist James Clerk Maxwell (1831-1879). The details of electromagnetic force are not important here; it is necessary only to know that all molecules possess some component of electrical charge. Since like charges repel and opposite charges attract, there is constant electromagnetic interaction between molecules, and this produces differing degrees of attraction.
The greater the relative motion between molecules, generally speaking, the less their attraction toward one another. Indeed, these two aspects of a material—relative attraction and motion at the molecular level—determine whether that material can be classified as a solid, liquid, or gas. When molecules move slowly in relation to one another, they exert a strong attraction, and the material of which they are a part is usually classified as a solid. Molecules of liquid, on the other hand, move at moderate speeds, and therefore exert a moderate attraction. When molecules move at high speeds, they exert little or no attraction, and the material is known as a gas.
Predicting Thermal Expansion
COEFFICIENT OF LINEAR EXPANSION.
A coefficient is a number that serves as a measure for some characteristic or property. It may also be a factor against which other values are multiplied to provide a desired result. For any type of material, it is possible to calculate the degree to which that material will expand or contract when exposed to changes in temperature. This is known, in general terms, as its coefficient of expansion, though, in fact, there are two varieties of expansion coefficient.
The coefficient of linear expansion is a constant that governs the degree to which the length of a solid will change as a result of an alteration in temperature For any given substance, the coefficient of linear expansion is typically a number expressed in terms of 10−5/°C. In other words, the value of a particular solid's linear expansion coefficient is multiplied by 0.00001 per °C. (The °C in the denominator, shown in the equation below, simply "drops out" when the coefficient of linear expansion is multiplied by the change in temperature.)
For quartz, the coefficient of linear expansion is 0.05. By contrast, iron, with a coefficient of 1.2, is 24 times more likely to expand or contract as a result of changes in temperature. (Steel has the same value as iron.) The coefficient for aluminum is 2.4, twice that of iron or steel. This means that an equal temperature change will produce twice as much change in the length of a bar of aluminum as for a bar of iron. Lead is among the most expansive solid materials, with a coefficient equal to 3.0.
CALCULATING LINEAR EXPANSION.
The linear expansion of a given solid can be calculated according to the formula δL = aL OΔT. The Greek letter delta (d) means "a change in"; hence, the first figure represents change in length, while the last figure in the equation stands for change in temperature. The letter a is the coefficient of linear expansion, and L O is the original length.
Suppose a bar of lead 5 meters long experiences a temperature change of 10°C; what will its change in length be? To answer this, a (3.0 · 10−5/°C) must be multiplied by L O (5 m) and δT (10°C). The answer should be 150 & 10−5 m, or 1.5 mm. Note that this is simply a change in length related to a change in temperature: if the temperature is raised, the length will increase, and if the temperature is lowered by 10°C, the length will decrease by 1.5 mm.
Obviously, linear equations can only be applied to solids. Liquids and gases, classified together as fluids, conform to the shape of their container; hence, the "length" of any given fluid sample is the same as that of the solid that contains it. Fluids are, however, subject to volume expansion—that is, a change in volume as a result of a change in temperature.
To calculate change in volume, the formula is very much the same as for change in length; only a few particulars are different. In the formula δV = bV OδT, the last term, again, means change in temperature, while δV means change in volume and V O is the original volume. The letter b refers to the coefficient of volume expansion. The latter is expressed in terms of 10−4/°C, or 0.0001 per °C.
Glass has a very low coefficient of volume expansion, 0.2, and that of Pyrex glass is extremely low—only 0.09. For this reason, items made of Pyrex are ideally suited for cooking. Significantly higher is the coefficient of volume expansion for glycerin, an oily substance associated with soap, which expands proportionally to a factor of 5.1. Even higher is ethyl alcohol, with a volume expansion coefficient of 7.5.
Most liquids follow a fairly predictable pattern of gradual volume increase, as a response to an increase in temperature, and volume decrease, in response to a decrease in temperature. Indeed, the coefficient of volume expansion for a liquid generally tends to be higher than for a solid, and—with one notable exception discussed below—a liquid will contract when frozen.
The behavior of gasoline pumped on a hot day provides an example of liquid thermal expansion in response to an increase in temperature. When it comes from its underground tank at the gas station, the gasoline is relatively cool, but it will warm when sitting in the tank of an already warm car. If the car's tank is filled and the vehicle left to sit in the sun—in other words, if the car is not driven after the tank is filled—the gasoline might very well expand in volume faster than the fuel tank, overflowing onto the pavement.
Another example of thermal expansion on the part of a liquid can be found inside the car's radiator. If the radiator is "topped off" with coolant on a cold day, an increase in temperature could very well cause the coolant to expand until it overflows. In the past, this produced a problem for car owners, because car engines released the excess volume of coolant onto the ground, requiring periodic replacement of the fluid.
Later-model cars, however, have an overflow container to collect fluid released as a result of volume expansion. As the engine cools down again, the container returns the excess fluid to the radiator, thus, "recycling" it. This means that newer cars are much less prone to overheating as older cars. Combined with improvements in radiator fluid mixtures, which act as antifreeze in cold weather and coolant in hot, the "recycling" process has led to a significant decrease in breakdowns related to thermal expansion.
One good reason not to use pure water in one's radiator is that water has a far higher coefficient of volume expansion than a typical engine coolant. This can be particularly hazardous in cold weather, because frozen water in a radiator could expand enough to crack the engine block.
In general, water—whose volume expansion coefficient in the liquid state is 2.1, and 0.5 in the solid state—exhibits a number of interesting characteristics where thermal expansion is concerned. If water is reduced from its boiling point—212°F (100°C) to 39.2°F (4°C) it will steadily contract, like any other substance responding to a drop in temperature. Normally, however, a substance continues to become denser as it turns from liquid to solid; but this does not occur with water.
At 32.9°F, water reaches it maximum density, meaning that its volume, for a given unit of mass, is at a minimum. Below that temperature, it "should" (if it were like most types of matter) continue to decrease in volume per unit of mass, but, in fact, it steadily begins to expand. Thus, it is less dense, with a greater volume per unit of mass, when it reaches the freezing point. It is for this reason that when pipes freeze in winter, they often burst—explaining why a radiator filled with water could be a serious problem in very cold weather.
In addition, this unusual behavior with regard to thermal expansion and contraction explains why ice floats: solid water is less dense than the liquid water below it. As a result, frozen water stays at the top of a lake in winter; since ice is a poor conductor of heat, energy cannot escape from the water below it in sufficient amounts to freeze the rest of the lake water. Thus, the water below the ice stays liquid, preserving plant and animal life.
THE GAS LAWS.
As discussed, liquids expand by larger factors than solids do. Given the increasing amount of molecular kinetic energy for a liquid as compared to a solid, and for a gas as compared to a liquid, it should not be surprising, then, to learn that gases respond to changes in temperature with a volume change even greater than that of liquids. Of course, where a gas is concerned, "volume" is more difficult to measure, because a gas simply expands to fill its container. In order for the term to have any meaning, pressure and temperature must be specified as well.
A number of the gas laws describe the three parameters for gases: volume, temperature, and pressure. Boyle's law, for example, holds that in conditions of constant temperature, an inverse relationship exists between the volume and pressure of a gas: the greater the pressure, the less the volume, and vice versa. Even more relevant to the subject of thermal expansion is Charles's law.
Charles's law states that when pressure is kept constant, there is a direct relationship between volume and temperature. As a gas heats up, its volume increases, and when it cools down, its volume reduces accordingly. Thus, if an air mattress is filled in an air-conditioned room, and the mattress is then taken to the beach on a hot day, the air inside will expand. Depending on how much its volume increases, the expansion of the hot air could cause the mattress to "pop."
VOLUME GAS THERMOMETERS.
Whereas liquids and solids vary significantly with regard to their expansion coefficients, most gases follow more or less the same pattern of expansion in response to increases in temperature. The predictable behavior of gases in these situations led to the development of the constant gas thermometer, a highly reliable instrument against which other thermometers—including those containing mercury (see below)—are often gauged.
In a volume gas thermometer, an empty container is attached to a glass tube containing mercury. As gas is released into the empty container, this causes the column of mercury to move upward. The difference between the former position of the mercury and its position after the introduction of the gas shows the difference between normal atmospheric pressure and the pressure of the gas in the container. It is, then, possible to use the changes in volume on the part of the gas as a measure of temperature. The response of most gases, under conditions of low pressure, to changes in temperature is so uniform that volume gas thermometers are often used to calibrate other types of thermometers.
Many solids are made up of crystals, regular shapes composed of molecules joined to one another as though on springs. A spring that is pulled back, just before it is released, is an example of potential energy, or the energy that an object possesses by virtue of its position. For a crystalline solid at room temperature, potential energy and spacing between molecules are relatively low. But as temperature increases and the solid expands, the space between molecules increases—as does the potential energy in the solid.
In fact, the responses of solids to changes in temperature tend to be more dramatic, at least when they are seen in daily life, than are the behaviors of liquids or gases under conditions of thermal expansion. Of course, solids actually respond less to changes in temperature than fluids do; but since they are solids, people expect their contours to be immovable. Thus, when the volume of a solid changes as a result of an increase in thermal energy, the outcome is more noteworthy.
JAR LIDS AND POWER LINES.
An everyday example of thermal expansion can be seen in the kitchen. Almost everyone has had the experience of trying unsuccessfully to budge a tight metal lid on a glass container, and after running hot water over the lid, finding that it gives way and opens at last. The reason for this is that the high-temperature water causes the metal lid to expand. On the other hand, glass—as noted earlier—has a low coefficient of expansion. Otherwise, it would expand with the lid, which would defeat the purpose of running hot water over it. If glass jars had a high coefficient of expansion, they would deform when exposed to relatively low levels of heat.
Another example of thermal expansion in a solid is the sagging of electrical power lines on a hot day. This happens because heat causes them to expand, and, thus, there is a greater length of power line extending from pole to pole than under lower temperature conditions. It is highly unlikely, of course, that the heat of summer could be so great as to pose a danger of power lines breaking; on the other hand, heat can create a serious threat with regard to larger structures.
Most large bridges include expansion joints, which look rather like two metal combs facing one another, their teeth interlocking. When heat causes the bridge to expand during the sunlight hours of a hot day, the two sides of the expansion joint move toward one another; then, as the bridge cools down after dark, they begin gradually to retract. Thus the bridge has a built-in safety zone; otherwise, it would have no room for expansion or contraction in response to temperature changes. As for the use of the comb shape, this staggers the gap between the two sides of the expansion joint, thus minimizing the bump motorists experience as they drive over it.
Expansion joints of a different design can also be found in highways, and on "highways" of rail. Thermal expansion is a particularly serious problem where railroad tracks are concerned, since the tracks on which the trains run are made of steel. Steel, as noted earlier, expands by a factor of 12 parts in 1 million for every Celsius degree change in temperature, and while this may not seem like much, it can create a serious problem under conditions of high temperature.
Most tracks are built from pieces of steel supported by wooden ties, and laid with a gap between the ends. This gap provides a buffer for thermal expansion, but there is another matter to consider: the tracks are bolted to the wooden ties, and if the steel expands too much, it could pull out these bolts. Hence, instead of being placed in a hole the same size as the bolt, the bolts are fitted in slots, so that there is room for the track to slide in place slowly when the temperature rises.
Such an arrangement works agreeably for trains that run at ordinary speeds: their wheels merely make a noise as they pass over the gaps, which are rarely wider than 0.5 in (0.013 m). A high-speed train, however, cannot travel over irregular track; therefore, tracks for high-speed trains are laid under conditions of relatively high tension. Hydraulic equipment is used to pull sections of the track taut; then, once the track is secured in place along the cross ties, the tension is distributed down the length of the track.
Thermometers and Thermostats
MERCURY IN THERMOMETERS.
A thermometer gauges temperature by measuring a temperature-dependent property. A thermostat, by contrast, is a device for adjusting the temperature of a heating or cooling system. Both use the principle of thermal expansion in their operation. As noted in the example of the metal lid and glass jar above, glass expands little with changes in temperature; therefore, it makes an ideal container for the mercury in a thermometer. As for mercury, it is an ideal thermometric medium—that is, a material used to gauge temperature—for several reasons. Among these is a high boiling point, and a highly predictable, uniform response to changes in temperature.
In a typical mercury thermometer, mercury is placed in a long, narrow sealed tube called a capillary. Because it expands at a much faster rate than the glass capillary, mercury rises and falls with the temperature. A thermometer is calibrated by measuring the difference in height between mercury at the freezing point of water, and mercury at the boiling point of water. The interval between these two points is then divided into equal increments in accordance with one of the well-known temperature scales.
THE BIMETALLIC STRIP IN THERMOSTATS.
In a thermostat, the central component is a bimetallic strip, consisting of thin strips of two different metals placed back to back. One of these metals is of a kind that possesses a high coefficient of linear expansion, while the other metal has a low coefficient. A temperature increase will cause the side with a higher coefficient to expand more than the side that is less responsive to temperature changes. As a result, the bimetallic strip will bend to one side.
When the strip bends far enough, it will close an electrical circuit, and, thus, direct the air conditioner to go into action. By adjusting the thermostat, one varies the distance that the bimetallic strip must be bent in order to close the circuit. Once the air in the room reaches the desired temperature, the high-coefficient metal will begin to contract, and the bimetallic strip will straighten. This will cause an opening of the electrical circuit, disengaging the air conditioner.
In cold weather, when the temperature-control system is geared toward heating rather than cooling, the bimetallic strip acts in much the same way—only this time, the high-coefficient metal contracts with cold, engaging the heater. Another type of thermostat uses the expansion of a vapor rather than a solid. In this case, heating of the vapor causes it to expand, pushing on a set of brass bellows and closing the circuit, thus, engaging the air conditioner.
WHERE TO LEARN MORE
Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.
"Comparison of Materials: Coefficient of Thermal Expansion" (Web site). <http://www.handyharmancanada.com/TheBrazingBook/comparis.html> (April 21, 2001).
Encyclopedia of Thermodynamics (Web site). <http://therion.minpet.unibas.ch/minpet/groups/thermodict/> (April 12, 2001).
Fleisher, Paul. Matter and Energy: Principles of Matter and Thermodynamics. Minneapolis, MN: Lerner Publications, 2002.
NPL: National Physics Laboratory: Thermal Stuff: Begin ners' Guides (Web site). <http://www.npl.co.uk/npl/cbtm/thermal/stuff/guides.html> (April 18, 2001).
Royston, Angela. Hot and Cold. Chicago: Heinemann Library, 2001.
Suplee, Curt. Everyday Science Explained. Washington, D.C.: National Geographic Society, 1996.
"Thermal Expansion Measurement" (Web site). <http://www.measurementsgroup.com/guide/tn/tn513/513intro.html> (April 21, 2001).
"Thermal Expansion of Solids and Liquids" (Web site). <http://www.physics.mun.ca/~gquirion/P2053/html19b/> (April 21, 2001).
Walpole, Brenda. Temperature. Illustrated by Chris Fair-clough and Dennis Tinkler. Milwaukee, WI: Gareth Stevens Publishing, 1995.
A number that serves as a measure for some characteristic or property. A coefficient may also be a factor against which other values are multiplied to provide a desired result.
COEFFICIENT OF LINEAR EXPANSION:
A figure, constant for any particular type of solid, used in calculating the amount by which the length of that solid will change as a result of temperature change. For any given substance, the coefficient of linear expansion is typically a number expressed in terms of 10−5/°C.
COEFFICIENT OF VOLUME EXPANSION:
A figure, constant for any particular type of material, used in calculating the amount by which the volume of that material will change as a result of temperature change. For any given substance, the coefficient of volume expansion is typically a number expressed in terms of 10−4/°C.
Internal thermal energy that flows from one body of matter to another.
The energy that an object possesses by virtue of its motion.
MOLECULAR TRANSLATIONAL ENERGY:
The kinetic energy in a system produced by the movement of molecules in relation to one another.
The energy that an object possesses by virtue of its position.
In physics, the term "system" usually refers to any set of physical interactions, or any material body, isolated from the rest of the universe. Anything outside of the system, including all factors and forces irrelevant to a discussion of that system, is known as the environment.
A measure of the average kinetic energy—or molecular translational energy in a system. Differences in temperature determine the direction of internal energy flow between two systems when heat is being transferred.
Heat energy, a form of kinetic energy produced by the movement of atomic or molecular particles. The greater the movement of the separticles, the greater the thermal energy.
A property in all types of matter that display a tendency to expand when heated, and to contract when cooled.
"Thermal Expansion." Science of Everyday Things. . Encyclopedia.com. (December 17, 2017). http://www.encyclopedia.com/science/news-wires-white-papers-and-books/thermal-expansion
"Thermal Expansion." Science of Everyday Things. . Retrieved December 17, 2017 from Encyclopedia.com: http://www.encyclopedia.com/science/news-wires-white-papers-and-books/thermal-expansion
Thermal expansion is the change in size of an object as its temperature changes. Normally, as the temperature increases, the size of an object also increases. Conversely, most objects shrink as the temperature drops. On a hot summer day, electrical power lines sag between power poles. The sag occurs because the wires grow longer as the temperature increases. Long bridges often have interlocking metal fingers along the joints where sections of the bridge are joined to each other. The metal fingers allow the bridge sections to expand and contract with changes in the temperature.
A relatively small number of substances contract when they are heated and expand when they are cooled. Water is the most common example. As water is cooled from room temperature to its freezing point, it contracts, like most other substances. However, just four degrees Celsius above its freezing point, it begins to expand. At its freezing point a gram of ice takes up more space than does a gram of liquid water. This change explains the fact that ice floats on top of water.
Factors affecting thermal expansion
Imagine that a long, thin metal wire is heated. The wire expands. The amount by which it expands depends on three factors: its original length, the temperature change, and the thermal (heat) properties of the metal itself.
Some substances simply expand more easily than others. If you heat wires of aluminum, iron, and tungsten metals—all the wires being the same size and heated to the same temperature—each wire will expand by a different amount. The ease with which a substance expands is given by its coefficient of expansion. For comparison, the coefficients of expansion for aluminum, iron, and tungsten are 23 × 10−6, 12 × 10−6, and 5 × 10−6 per degree Celsius, respectively.
The values given in the previous sentence actually refer to the coefficients of linear expansion. They measure how much a substance expands in only one direction. But suppose the above experiment were done with blocks of aluminum, iron, and tungsten rather than wires. In that case, the expansion would occur in all three directions: length, width, and depth. The measure of expansion in all three directions is called the coefficient of volume expansion.
Length and temperature. Suppose this discussion is limited to a single kind of material, say an iron wire. The amount by which that wire expands when heated depends on only two factors: its original length and the temperature to which it is heated. An iron bar that is 16 feet (5 meters) long will expand more than a bar that is 3 feet (1 meter) long. And a 16-foot (5-meter) bar will expand more if heated by 68°F (20°C) than a 16-foot (5-meter) bar that is heated by 50°F (10°C).
Engineers and architects must always take into consideration the fact that objects usually expand when they are heated. As a result, they have to design buildings, bridges, power lines, and other structures to compensate for expansion and contraction.
Thermal expansion also is used in the construction of certain appliances and devices in homes and industry. One example is the bimetallic strip. A bimetallic strip consists of two pieces of metal welded to each other. The two metals are chosen to have different coefficients of expansion. When the bimetallic strip is heated, it bends one way or the other as one metal expands faster than the other.
One use of bimetallic strips is in thermostats used to control room temperatures. As a room warms up or cools down, the bimetallic strip in the thermostat bends one way or the other. If it bends far enough, it comes into contact with a button that turns a furnace on or off.
"Expansion, Thermal." UXL Encyclopedia of Science. . Encyclopedia.com. (December 17, 2017). http://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/expansion-thermal
"Expansion, Thermal." UXL Encyclopedia of Science. . Retrieved December 17, 2017 from Encyclopedia.com: http://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/expansion-thermal
The term thermal expansion refers to the increase in size of an object as that object is heated. With relatively few exceptions, all objects expand when they are heated and contract when they are cooled. Perhaps the most important exception to this rule is water. Water contracts as it cools from its boiling point to about 39.2°F (4°C). At that point, it begins to expand as it cools further to its freezing point. This unusual effect explains the fact that ice is less dense than water.
Different materials expand or contract at different rates. In general, gases expand more than liquids, and liquids expand more than solids.
When an object is heated or cooled, it expands or contracts in all dimensions. However, for practical reasons, scientists and engineers often focus on two different kinds of expansion, or expansivity: linear expansivity (expansion in one direction only) and volume expansivity (expansion in all three dimensions). The amount by which any given material
expands in either way is known as its coefficient of linear (or volume) expansivity.
The choice of these two variables is a practical one. Scientists and engineers often want to know the amount by which some pipe, bar, wire, or other long object will expand. For example, how much longer will a line of telephone wire be on a hot summer day when the temperature is 86°F (30°C) compared to a cold winter day when the temperature is 14°F (−10°C)? The fact is that the wire expands in all three directions, but it is only the linear direction (the length) that is of interest in a real-life situation.
Only solids have a coefficient of linear expansion. They differ from each other widely, with the coefficient of linear expansion of aluminum having a value nearly 50 times as great as that of fused quartz.
Volume expansivity also has its practical applications. Suppose that someone wants to know how much a balloon will expand as its temperature increases. The answer to that question depends on the volume expansivity of the gas used. The volume expansivity of gases ranges from a relatively low value for air to a relatively high value for carbon dioxide and sulfur dioxide.
A great many practical devices and systems depend on the thermal expansion of materials. An example is the bimetallic strip. A bimetallic strip consists of two metals of different thermal expansivities welded to each other. When the strip is heated, one metal expands more rapidly than the other. The strip bends in the direction of the metal with the lower thermal expansivity.
Perhaps the most common use of the bimetallic strip is in a thermostat. When a room becomes cold, the two metals in the strip contract, one more than the other. At some point, the strip bends enough to come into contact with a metal button that closes an electrical circuit, turning on the furnace. As the room warms up, the bimetallic strip begins to bend in the opposite direction. Eventually it pulls away from the contact button, the circuit is broken, and the furnace turns off.
The thermal expansion of objects in the real world often requires the attention of scientists and engineers. For example, the joints used to hold a bridge together have to be designed to provide space for expansion and contraction of the bridge deck. And railroad tracks are built so that they can slide toward and away from each other on hot and cold days, making sure that they do not bend out of shape because of overheating.
"Thermal Expansion." UXL Encyclopedia of Science. . Encyclopedia.com. (December 17, 2017). http://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/thermal-expansion
"Thermal Expansion." UXL Encyclopedia of Science. . Retrieved December 17, 2017 from Encyclopedia.com: http://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/thermal-expansion