## A

Figure 4.4. Joule's churn. Apparatus used to determine the mechanical equivalent of heat.

ture of the water just by rotating the paddle. Joule even set up the experiment to measure just how much heat is produced by a certain amount of mechanical work. Instead of just turning the paddle by hand, he set it up so that a metal cylinder suspended by a rope over a pulley could be allowed to turn the paddle as it fell a measured distance. In Section A2 of this chapter, work is defined as the product of force multiplied by distance. For Joule's experiment, the force is just the weight of the metal cylinder and the distance is how far it is allowed to fall, so he could measure the mechanical work exactly. He could determine the amount of water in his experiment and, from the temperature rise and the specific heat of water, deduce the amount of heat that must have been generated. Thus Joule determined the mechanical work equivalent of heat. He did this for a number of different materials such as various types of oils, and he always found the same value for the mechanical equivalent of heat. Joule eventually determined a value that varies less than 1 percent from the modern accepted value. These experiments represented a landmark in the history of science.

There are two extremely important consequences of Joule's experiments. The first is that mechanical work can produce heat. Heat is not conserved. The caloric theory cannot be correct and in fact there is no caloric fluid that cannot be created or destroyed. We need another theory of heat. Heat must be a form of energy. Recall that mechanical work can produce kinetic energy or potential energy. Joule demonstrated that work can also produce heat. It is clear that heat must also be a form of energy.

The second important consequence of Joule's experiments is the clue about how to construct a new theory of heat. Because of the vanes in the cylinder of water, the rotation of the paddles can do nothing to the water except agitate it. This restriction was an intended aim by Joule. Thus it must be concluded that agitated water is warmer water. But what is being agitated in the water? The answer is the molecules of which the water is comprised. These molecules have the familiar chemical symbol H20, meaning they consist of two atoms of hydrogen and one of oxygen. It must be molecules that are being agitated, and apparently the more agitated they are, the warmer the substance. Temperature must be related to the energy of the microscopic motions of these molecules. Thus temperature can be understood only in terms of molecular motions, as described by the molecular model of matter.

### Optional Section 4.2 Heat Capacity and Latent Heat

Let us consider quantitatively the example of heating water in order to make it boil. Suppose we have a pot of water with 1 liter of water in it. As discussed above, the unit of heat, one calorie, is defined as the amount of heat necessary to raise the temperature of one gram of water by one degree Celsius. Suppose the water starts at about room temperature, 25° C (77° F). If the pot is placed on a burner that can produce 30,000 cal/min of heat and if about one-half of the heat produced is transferred to the water, then about 15,000 cal/min of heat goes into the water. The total amount of heat (in calories) required to heat the water up to its boiling point (100° C) is heat = specific heat X mass X change in temperature or symbolically,

and for this example,

The amount of time required to bring the water up to the boiling point is then

1 R 15,000 cal/min

Now the latent heat of water, L, has been determined to be 539 calories per gram. This is the amount of heat (per gram) required to convert the water into steam (often called the heat of vaporization). If we wish to completely boil the water away, using the same burner, the total heat required would be

and the time required to do this would be

So we see that actually it would take about seven times as long to boil the water away as to raise the temperature to the boiling point. The amount of energy (i.e., heat) required to overcome the mutual attractive forces between the molecules in the water is actually quite significant.

2. The Kinetic-Molecular Model of Matter and the Microscopic Explanation of Heat

As early as 400 B.C., Democritus, a Greek philosopher, suggested that the world was made up of a few basic building blocks, too small for the eye to distinguish, called atoms. It was originally believed that there existed only a few different kinds of atoms. Later, it was erroneously thought that there was a different kind of atom for every different material such as wood, rock, air, and so on. We now know that there are hundreds of thousands of different chemical substances, which are now called compounds. The smallest amount of a compound is called a molecule, not an atom. Molecules, however, are made up from atoms, of which there are only about 100 chemically different kinds. Each naturally occurring atom is associated with one of the elements from hydrogen to uranium. Molecules and their constituent atoms obey the laws of mechanics and also the laws of conservation of energy and momentum. The dominant force is electrical and not gravitational.

It should be noted that we now know that even the objects identified as atoms are not the fundamental building blocks of nature as originally supposed. Atoms are themselves built up from protons, neutrons, and electrons; furthermore, the protons and neutrons are built up of even smaller units called quarks. In order to understand heat, however, one need not be concerned with the structure of a molecule.

Consider what happens to the molecules of a substance, such as water, as the substance goes from a gas (steam) to a liquid (water) and finally to a solid (ice). When the substance is a gas, the molecules are essentially not bound to each other at all. They are each free to move in any direction whatsoever, and to spin and vibrate. For a gas at normal pressures (such as the pressure of the atmosphere around us), molecules will not proceed very far in any one direction before colliding with other molecules. They will bounce off each other, obeying Newton's laws of motion and the laws of conservation of momentum and kinetic energy. If we add one gas to another (such as by spraying perfume in one corner of a room), the molecules of the new gas will, after many collisions with the molecules of the original gas, diffuse out in all directions. Soon the perfume can be detected across the room, as the perfume molecules become completely mixed with the original air molecules.

As the gas is cooled, the molecules move more slowly, and usually do not collide with each other with as much speed as when the gas was warmer. When the gas is cooled enough, we know it will condense into the liquid state. In the liquid state, the molecules are still fairly free to move about, colliding with each other as they move. However, there exists a small force of attraction between all the molecules (which is electrical in origin), and this small force from all nearby molecules adds up, preventing an individual molecule from escaping all the others and leaving the liquid. Thus liquids will remain in a container, even without a lid, for a long time. We do know that most common liquids will slowly evaporate. This is explained by noting that occasionally a molecule will be struck by two or more other molecules in rapid succession and driven toward the surface of the liquid. This gives the molecule an unusually large velocity, which allows it to overcome the net force acting on it from all the other molecules, and it will (if there are no intervening collisions) escape from the liquid.

If the liquid is cooled enough, it will finally "freeze" into a solid form. In a solid, the molecules are moving so slowly that the small forces on each one from the nearby molecules hold it in one place. The molecule is no longer free to move around among all the other molecules. The only motion it can have is a vibrational motion about its fixed place in the solid.

The molecular description of the states of matter has been verified repeatedly by many careful experiments. No "problem phenomena" are known to exist that question the accuracy of the account of how a material passes from the gaseous to the solid state as the temperature is lowered. (Many of the details of these transitions are not yet fully understood, however.) Thus gases, liquids, and solids are all composed of particles (the molecule) that are moving and are subject to the laws of mechanics.

Many of the earlier discussions can now be understood in detail. Heat input causes the molecules to move faster; heat removal causes them to move more slowly. If the molecules are moving very rapidly, the material is "hot." In fact, the temperature of a gas (whose molecules interact only by elastic collisions) is simply a measure of the average kinetic energy of random translational motion for all the molecules (recall that kinetic energy is V2mv2). The average speed of an air molecule at normal room temperature is found to be greater than 1000 mph! Because there are about a million billion molecules in one cubic foot of air, gas molecules collide with each other frequently (each molecule is struck by other molecules about 100 billion times per second).

It is interesting to note that one of the first direct verifications of the molecular nature of matter was provided by Albert Einstein, who explained the observed motion of very small solid particles suspended in a liquid solution. This motion, called Brownian movement, is very irregular, with the particle moving a short distance in one direction and then quickly in another direction and quickly again in another, and so forth. Einstein explained (and showed quantitatively) that this motion was caused by bombardment of the small particle from the even smaller individual molecules of the liquid.3 The same kind of motion can be seen for the dust particles that make a "sunbeam" visible.

One can now also understand why a gas can exert a pressure such as it does in an automobile tire. The individual molecules are very light, but they are moving very fast and there are a large number of them hitting the wall of the tire every second. The pressure represents the average force exerted by molecules on the wall as they hit and rebound. Of course, the air molecules on the outside of the tire are also exerting a pressure on the other side of the walls of the tire at the same time that the molecules inside exert their pressure. There is a greater density of molecules on the inside, however, and thus the inside pressure is greater than the outside pressure by the amount necessary to hold up the vehicle. This extra pressure inside is accomplished by forcing (from an air pump) more molecules per unit volume inside the tire than there are in the air outside of the tire. Note that if the temperature of the gas (air) inside the tire were to be raised, the pressure would also increase, because the molecules would have more kinetic energy. Such an increase in pressure is known to occur when a tire becomes warmer, on a long

3The particles observed in Brownian motion can be seen with a low-powered microscope. Molecules, however, are so small that they cannot be seen in any significant detail with the most powerful microscopes presently available. Until Einstein's work, many scientists felt that there were no convincing proofs of the existence of molecules. Einstein published this work in the same year (1905) in which he published his first paper on relativity theory (see Chapter 6, Section D3).

drive on a hot day, for example. The flexing of the tire results in more agitation of the molecules, just as in Joule's experiment.

Specific heat can now be understood in detail. Recall that the specific heat of a material is the amount of heat required to raise the temperature of one gram one degree Celsius. This means increasing the average kinetic energy of the molecules. Depending on the actual geometric arrangements of the atoms in a molecule or solid, different modes of motion are possible (such modes may include internal vibrations). Thus the specific heats of various compounds will be different because the averaging will be different. Similarly, one can understand latent heats. In order to make a liquid boil, enough heat must be supplied to overcome the net force on an individual molecule produced by all the other molecules still in the liquid state, so we do not see a rise in temperature. Once enough heat has been absorbed to overcome the bonds between the molecules and all the liquid boils into a gas, then any additional heat will serve to increase the kinetic energy of the molecules and a rise in temperature will occur.

The kinetic-molecular model of matter reduces the problem of understanding heat and the states of matter to understanding systems of particles that have certain forces between them and that obey the laws of mechanics. The success of this model made it appear that Newton's laws of mechanics permit a unified description of the behavior of all objects, large and small, and in complete detail. Only the development of quantum and relativity theories in the twentieth century (which are discussed in Chapters 6 and 7) placed limitations on the universality of Newton's laws.