## Info

Adding 273 to the Celsius scale gives the centigrade absolute scale, which like the Celsius scale is used in most scientific work, and in engineering practice other than in America. If 459 is added to the Fahrenheit scale, then the Rankine absolute scale, which is used in American engineering practice, is obtained. (The ratio of 273 to 459 + 32 = 273/491 = 5/9.) The reason for defining these absolute scales is given below.

Optional Section 5.2 Significance of General Gas Law

Helium gas is not the only substance that obeys the general gas law. Almost all normal gases (such as oxygen or nitrogen) at not too high pressures or not too low temperatures obey the same set of laws. Not too low temperatures means well above their boiling temperatures. (The boiling temperature for helium is -269° C and for nitrogen -196° C.) For all such gases there is a wide range of conditions for which it is found experimentally that when the product of their pressure P and volume V, that is, PV, is plotted against temperature, for example, as measured on the Celsius temperature scale, then a straight line is obtained. This is shown by the solid lines in Fig. 5.0.2. If these lines are extended toward the left of the figure (dashed lines in the figure) they all cross the horizontal temperature axis at —273° C (—459° F on the Fahrenheit scale). It does not matter whether the gas actually behaves this way at low temperatures (in fact, all real gases eventually deviate from the straight line relationship); but by definition the ' 'ideal'' or ' 'perfect'' gas follows the straight line behavior drawn in Fig. 5.0.2. The significant feature of the ideal gas behavior is that all the straight lines pass through the same point on the graph, with P = 0 and therefore PV = 0. At this point we remind ourselves that the choice of 0° C to be the freezing point of water was purely arbitrary. If we choose a new temperature scale, for which the freezing point of water is 273° and the boiling point of water is 373°, then the temperature at which PV = 0 will be 0°, and there will be no negative temperatures on this scale, which is Temperature (°C)

Figure 5.O.2. Use of general gas law to define "absolute" temperature scale.

Solid lines a, b, c, d represent product of experimental measurements of P and V. Dashed lines are extrapolations of the straight portions of the plots. The extrapolations all converge at -273° C. Adding 273 to all Celsius readings is equivalent to shifting the zero of the scale to the convergence temperature, thereby making it 0° absolute.

Temperature (°C)

Figure 5.O.2. Use of general gas law to define "absolute" temperature scale.

Solid lines a, b, c, d represent product of experimental measurements of P and V. Dashed lines are extrapolations of the straight portions of the plots. The extrapolations all converge at -273° C. Adding 273 to all Celsius readings is equivalent to shifting the zero of the scale to the convergence temperature, thereby making it 0° absolute.

commonly called an absolute temperature scale. It should be noted that if the temperature were less than 0° on the absolute scale, then either the pressure or the volume of the ideal gas would be negative. Both of these possibilities are physically impossible.

On this new absolute temperature scale, the straight lines representing the ideal gas all pass through the zero of the horizontal axis, and the equation describing them can be written

where C is a proportionality constant. Eventually it was recognized that the constant C depended in a simple way on the amount and kind of gas, so that C = nR and the general gas law can be written as

where n is the number of "moles" of gas, and R is called the general gas constant per mole. A mole of any pure substance is defined as an amount of the substance that has a mass in grams numerically equal to the sum of the atomic masses of the atoms making up one molecule of the substance. The quantity n is not required to be a whole number. A mole of any pure substance contains a specific number of molecules, called Avogadro's number, which is approximately equal to 6.02 X 1023.

The general gas law leads to three other laws, which were actually recognized before the general gas law, and which were used to infer the general gas law. These laws can be derived from the general gas law by keeping one of the quantities P, V, or T constant and unchanging.

If the temperature is held constant, the result is Boyle's Law, which simply states that if the temperature of a gas does not change, then the product of pressure and volume, PV, must remain constant. Thus if P is multiplied by a certain factor, then V must be divided by that same factor in order to keep the product constant. Similarly if P is divided by a certain factor, then V must be multiplied by that same factor. Figure 5.1 shows plots of Boyle's Law at four different temperatures.

On the other hand, if P is held constant, increasing the temperature causes an increase in the volume of the gas that is exactly proportional to the increase of temperature, if the absolute temperature scale is used. This is called Charles' Law or Gay-Lussac's Law, V/T = constant. Basically this law says that gases expand significantly when heated. It is this law that is responsible for the flight of hot air balloons, since ' 'expanded air'' is lighter than ordinary air.

The third law derived from the general gas law has no particular name, but it says that if the volume is held fixed, then the "absolute" or true pressure of gas is proportional to the absolute temperature, that is, PIT = constant. This is the law that explains why it is dangerous to throw used spray containers into a fire.

2.3 Temperature Differences and the Flow of Heat When a thermometer is used to measure temperature, the thermometer must come into thermal equilibrium with the system whose temperature is to be measured. However, it is, in principle, possible to measure the temperature of the system even before the thermometer reaches thermal equilibrium. If two systems at different temperatures are interacting thermally with each other, heat will flow from the higher temperature (hotter) system to the lower temperature (colder) system. Thus a nurse can tell that the patient has a fever by feeling the patient's forehead. If the patient has a fever, the forehead will feel hot; that is, heat will flow from the patient's forehead to the nurse's hand. If, on the other hand, the patient's temperature is well below normal, the forehead will feel cool; that is, heat will flow from the nurse's hand to the patient's forehead. This very fact that temperature differences can cause heat to flow is very significant in understanding the concept of entropy and how heat energy can be transformed into other forms of energy.

### C. The Natural Flow of Heat

Everyone knows that heat flows from hotter objects to colder objects. If an ice cube is dropped into a cup of hot coffee, the ice cube becomes warmer, melts into water, and the resulting water becomes warmer, whereas the coffee becomes cooler. The heat "flows" out of the hot coffee into the cold ice cube. As already discussed in Chapter 4, this was easily explained by the erroneous caloric theory, which also indicated that heat cannot by itself flow from colder to hotter objects, because that would be making the caloric fluid flow "uphill." Of course, the caloric theory is wrong, and so it is not possible to explain so simply why heat flows from higher temperature bodies to lower temperature bodies, and not vice versa. In terms of the distinction between heat and temperature made in Section B1 above, energy flows from high concentrations (high temperature) to low concentrations (low temperature); that is, energy "spreads out" by itself. This fact is the essential content of the second law of thermodynamics. Like Newton's laws of motion or the law of conservation of energy, this law is accepted as a basic postulate. There are several different ways of stating the second law, all of which can be shown to be equivalent to each other.

In terms of the natural flow of heat, the second law of thermodynamics is stated as follows: In an isolated system, there is no way to systematically reverse the flow of heat from higher to lower temperatures. In other words, heat cannot flow "uphill" overall. The word overall is very important. It is, of course, possible to force heat "uphill," using a heat pump, just as water can be forced uphill using a water pump. An ordinary refrigerator and an air conditioner are examples of heat pumps. However, if a refrigerator is working in one part of an isolated system to force heat "uphill," in some other part of the system more heat will flow "downhill" or further "downhill," so that overall the net effect in the system is of a "downhill" flow of heat. If one system is not isolated, a second system can interact with it in such a way as to make the heat flow ' 'uphill'' in the first system, but of course there will be at least a compensating "downhill" flow of heat in the second system.

Although the second law of thermodynamics as stated above seems obvious and not very important, it leads ultimately to the result that heat is a form of energy not completely convertible to other forms. Indeed the ideas involved in the second law of thermodynamics provide a deep insight into the nature of the physical universe. Some general applications of the second law and different ways of stating it are discussed further below.

D. Transformation of Heat Energy Into Other Forms of Energy

### 1. Heat and Motion

Long before heat was recognized to be a form of energy, it was known that heat and motion were somehow related to each other. The friction that is often associated with motion always results in the generation of heat. Steam engines, which require heat, are used to generate motion. By the middle of the nineteenth century it was recognized that a steam engine is an energy converter, that is, a system that can be used to convert or transform heat energy into some form of mechanical energy—work, potential energy, kinetic energy—or electrical energy or chemical energy, and so on.

Figure 5.3. Schematic representation of heat flows for a heat engine, (a) All the heat from the source passes through the engine, (b) Some of the heat from the source bypasses the engine.

Figure 5.3. Schematic representation of heat flows for a heat engine, (a) All the heat from the source passes through the engine, (b) Some of the heat from the source bypasses the engine.

Even before the true nature of heat as a form of energy was understood, as discussed in the preceding chapter, Carnot realized that the steam engine was only one example of a heat engine. Another familiar example of a heat engine is the modern gasoline engine used in automobiles. Carnot realized that the essential, common operating principle of all heat engines was that heat flowed through the engine. He pointed out that the fundamental result of all of Watt's improvements of steam engines was more effective heat flow through the engine.

In the spirit of Plato, in an effort to get at the "true reality," Carnot visualized an "ideal heat engine," whose operation would not be hampered by extraneous factors such as friction or unnecessary heat losses. He therefore considered that this ideal engine would be reversible (could move the fluid uphill) and would operate very slowly. The principle of the engine was that heat had to flow through the engine to make it operate, and it was necessary to ensure that all the available heat would flow through the engine. This is shown diagrammatically in Fig. 5.3. In Fig. 5.3a all the heat from the source flows through the engine. In Fig. 5.3b some of the heat from the source is diverted, bypassing the engine, going directly to the heat sink, and thus is "wasted."

### Optional Section 5.3 The Reversible Heat Engine

Carnot proved that the best possible heat engine, what is called the ideal heat engine, is also a reversible heat engine. In the context used here reversible means more than moving heat from lower temperatures to higher temperatures (this is what refrigerators and air conditioners do). If a heat engine runs in the "forward" direction, a certain amount of "heat" moves from a higher temperature reservoir to a lower temperature reservoir (see Fig. 5.4) and work is done by the engine. If the engine runs in a "backward" direction, work is done on the engine and heat is moved from a lower temperature to a higher temperature reservoir. If the engine is truly reversible, the exact same amount of work as it puts out in the forward direction can be used to drive the engine "backward," and the exact same amount of heat moves from the low-temperature reservoir to the high-temperature reservoir as had previously moved from high temperature to low temperature. Everything is then exactly the same as before the engine operated in the forward direction.

This means of course that there is zero friction in all of the moving parts of the engine and that there is no air resistance to the motion of any parts. Friction and any other resistance to motion results in irreversibility of the heat engine because (1) it reduces the work output of the engine when running in the forward direction, and (2) it increases the amount of work needed to run in the backward direction. Note the analogy to Galileo's suppression of friction effects in order to get at the underlying nature of motion.

Carnot also inferred from Watt's introduction of the condenser and use of continued expansion after the steam valve was closed (see the discussion in Section B1 of Chapter 4 of Watt's improvements of the Newcomen engine) another cause of irreversibility—heat flow between engine and heat reservoirs had to be accomplished at as small a temperature difference as possible. For example, a temperature difference between heat reservoirs and engine of 1° is much more desirable than a temperature difference of 50° because it "wastes" less of the heat flow's capability to drive the process (compare the two parts of Fig. 5.4). In the ideal case the temperature difference between engine and reservoir when heat transfer takes place is zero degrees. This is the same kind of idealization as saying that friction and air resistance to motion can be totally eliminated only at a speed of zero miles per hour. Indeed, the reversible engine must operate very slowly.

Carnot then specified a particular type of reversible heat engine, now called the Carnot engine, which would operate between two particular temperature reservoirs, that is, it would take in heat at a ' 'hot'' temperature Tn and discharge heat at a "cold" temperature Tc and would satisfy these requirements. The operation of this engine proceeded in a cycle of four steps, called the Carnot cycle, as described further below.

There are other possible reversible engine designs that can operate over the same temperature range as the Carnot engine (and that have a different set of steps in their cycle), but they have the exact same efficiency as the Carnot engine provided all the steps in the cycle are reversible. The key element is reversibility.

Carnot then made an analogy between the ideal engine and a well-designed waterwheel or water mill through which water flows, causing rotation of a shaft, which can therefore operate any attached machinery. In fact, because the caloric theory considered that the heat fluid flowed from high temperatures to low temperatures just as water flows from high levels to low levels, the analogy was very close. The heat engine could be considered to be a heat wheel, which could be designed just like a waterwheel. (Carnot's father had made many suggestions for improving the design of waterwheels, and Carnot was influenced by these.)

Waterwheels operate by virtue of having water fall from a high-level reservoir to a low-level reservoir, and similarly heat wheels operate by virtue of having heat flowing from a high temperature or hot "heat reservoir" to a lower temperature or relatively cooler ' 'heat reservoir." For most efficient operation, the water-wheel should be tall enough to take in water exactly at the level of the high reservoir (see Fig. 5.4). The water should stay in the "buckets" of the waterwheel and not spill out as the wheel turns (otherwise it would be wasted and its motive power lost), until the buckets are about to dip into the low-level reservoir, at which time they can discharge their water. The buckets must discharge all their water at the low-level reservoir, and not carry any water back up again or have any water fall into them until they get back up to the high-level reservoir.

Carnot reasoned further that the ideal or most efficient heat engine must be operated in exactly the same way as the most efficient waterwheel: Heat must be taken in only when the engine is at the temperature of the high-temperature reservoir. The engine must be thoroughly insulated while its temperature is changing from high to low, so that no heat will leak out (be diverted) and its motive force be lost. The heat must then be discharged at the temperature of the low-temperature reservoir, so that no heat will be carried back "uphill."

He concluded that the efficiency (i.e., the relative effectiveness of the engine in providing motion or work) of his ideal heat engine would depend only on the Figure 5.4. Carnot's water wheel analogy for a heat engine. HLR—high-level reservoir; LLR—low-level reservoir, (a) An inefficient design: 1—Caloric fluid being taken into the engine at a temperature far below that of the high-level reservoir. 2—Fluid being lost through "spillage" (leakage). 3—Fluid being discharged before buckets reach temperature of low-level reservoir. 4—Caloric fluid being carried back up to high-level reservoir. (b) An efficient design: 5—Fluid taken in only at temperature of high-level reservoir. 6—No loss of fluid as buckets move down to lower level. 7—Fluid discharged only when buckets reach temperature of low-level reservoir. 8—No fluid carried up as buckets move from low-level to high-level reservoir.

Figure 5.4. Carnot's water wheel analogy for a heat engine. HLR—high-level reservoir; LLR—low-level reservoir, (a) An inefficient design: 1—Caloric fluid being taken into the engine at a temperature far below that of the high-level reservoir. 2—Fluid being lost through "spillage" (leakage). 3—Fluid being discharged before buckets reach temperature of low-level reservoir. 4—Caloric fluid being carried back up to high-level reservoir. (b) An efficient design: 5—Fluid taken in only at temperature of high-level reservoir. 6—No loss of fluid as buckets move down to lower level. 7—Fluid discharged only when buckets reach temperature of low-level reservoir. 8—No fluid carried up as buckets move from low-level to high-level reservoir.

difference in temperature of the two reservoirs, and not on whether it was a steam engine using water as the working substance or some other kind of engine using some other working substance (such as helium gas or oil or diesel fuel or mercury vapor). He concluded, however, that for practical reasons the steam engine was probably the best one to build.

2. First Law of Thermodynamics: Conversion of Heat to Another Form of Energy

About 1850, the English physicist Lord Kelvin (as discussed in Chapter 4) and the German physicist Rudolf J. E. Clausius both recognized that Carnot's basic ideas were correct. However, Carnot's theory needed to be revised to take into account the first law of thermodynamics (i.e., the principle of conservation of energy), which means that heat is not an indestructible fluid, but simply a form of energy. Although heat must flow through the engine, the waterwheel analogy had to be discarded. The amount of heat flowing out of the engine into the low-temperature reservoir is less than the amount of heat flowing from the high-temperature reservoir, the difference being converted into work or mechanical energy. In other words, when a certain amount of heat energy flows into the engine from the high-temperature reservoir, part of it comes out as some other form of energy, and the rest of it flows as heat energy into the low-temperature reservoir. All the energy received must be discharged because after the engine goes through one complete cycle of operation, it must be in the same thermodynamic state as it was at the beginning of the cycle. Thus its internal energy parameter, U, must be the same as it was at the beginning. Therefore, according to the first law of thermodynamics, all the energy added to it (in the form of heat from the high-temperature reservoir) must be given up by it (as work or some other form of energy and as heat discharged to the low-temperature reservoir). This is shown schematically in the energy flow diagram of Fig. 5.5.

3. Alternative Forms of Second Law of Thermodynamics

The essentia] correctness of Carnot's ideas, according to Clausius, lay in the fact that at least part of the heat flowing into the engine must flow on through it and be discharged. This discharged heat cannot be converted to another form of Figure 5.5. Energy flow for a heat engine. Energy input as heat; energy output as heat and work or other useful forms of energy.

energy by the engine. In other words, it is not possible to convert heat entirely into another form of energy by means of an engine operating in a cycle. To do so would violate the second law of thermodynamics (see section C above), which states that heat cannot flow "uphill" in an isolated system. This is shown schematically in Fig. 5.6, which illustrates what would happen if an engine could convert heat entirely into mechanical energy, for example.

Suppose such an engine were being driven by the heat drawn from a heat reservoir at a temperature of 30° Celsius (86° Fahrenheit). The assumption is that the engine draws heat from the reservoir and converts it completely into mechanical energy. This mechanical energy could then be completely converted into heat once again, as Joule was able to show by his paddle wheel experiment. In particular, the mechanical energy could be used to stir a container full of water at 50° with no other change, because the engine working in a cycle returns to its original state. The "driving force" for the entire process was the heat in the 30° reservoir. Heat would have "flowed uphill" by itself, which is impossible according to the second law of thermodynamics.

As will be discussed in Section E2, the flaw in this process is in the assumption of total transformation of heat into mechanical energy. Some heat must move "downhill" untransformed to allow for the possibility that the portion of the heat that is transformed into mechanical energy could be retransformed into heat energy at a higher temperature than the reservoir from which it was originally derived. Overall there must be a net "downhill" movement of heat. Section I below provides a detailed discussion of how this "overall downhill movement" of heat is calculated. Although it is possible in general to show that total conversion of heat to the other forms of energy is not possible, as has just been indicated, further insight into this principle can be obtained from a study of engine output

Figure 5.6. Schematic of an impossible machine. Heat taken from a large body of water at 30° Celsius is completely converted into mechanical energy. The mechanical energy is then reconverted into heat by stirring a container full of water at 50° Celsius with the resulting overall effect that heat flowed "uphill."

output

Figure 5.6. Schematic of an impossible machine. Heat taken from a large body of water at 30° Celsius is completely converted into mechanical energy. The mechanical energy is then reconverted into heat by stirring a container full of water at 50° Celsius with the resulting overall effect that heat flowed "uphill."

cycles, as is given in Section E below and in the appendix at the end of the chapter.

Sometimes the second law of thermodynamics is stated specifically in terms of these limitations on the convertibility of heat to another form of energy. As just shown, such statements are equivalent to the original statement in terms of irreversibility of heat flow:

1. It is impossible to have a heat engine working in a cycle that will draw heat from a heat reservoir without having to discharge heat to a lower temperature reservoir.

2. It is impossible to have a heat engine working in a cycle that will result in complete (100%) conversion of heat into work.

Whenever energy is released in the form of heat at some temperature (e.g., by burning some coal or oil or by nuclear reaction), some of the heat must eventually be discharged to a lower temperature reservoir.

### 4. Perpetual Motion of the Second Kind

Any violation of the second law of thermodynamics is called "perpetual motion of the second kind." If it were possible to build a perpetual motion machine of the second kind, some beneficial results could be obtained.

For example, an oceangoing ship could take in water from the surface of the ocean, extract heat from the water, and use that heat to drive the ship's engines. The result would be that the water taken in would be turned into ice and the ship propelled forward. The ice would be cast overboard, the ship would draw in more ocean water, and the process would continue. The ship would not need any coal, fuel oil, or nuclear energy for its operation. The only immediate by-products of the operation of the ship would be the ice, plus various previously dissolved salts and minerals that would have crystallized out of the ocean water during the heat-extraction process. All of these could be returned to the ocean, with the only net result being that the ocean surface is very slightly, essentially insignificantly, colder.

An even more spectacular example can be conceived. Imagine electricity generating stations located on the banks of the various polluted rivers of the world: the Cuyahoga in Ohio, the Vistula in Poland, the Elbe in Germany and the Czech Republic, the Ganges in India, and so on. These stations would take in polluted river water and extract heat from the water for conversion into electrical energy, thereby lowering the temperature of the water to below its freezing point. The process of freezing the water results in a partial separation of the pollutants from the water. The now cleaner frozen water is allowed to melt and is returned to the river, and the separated pollutants can be either reclaimed for industrial or agricultural use or stored with proper precautions. The result of all this would be a significant improvement of the environment as a by-product of the generation of electrical energy.

Neither of these schemes will work, because although they would take heat from an existing heat reservoir, they do not discharge any heat to an even lower temperature heat reservoir to conform to the requirement that heat must flow "downhill." Unfortunately, the second law of thermodynamics complicates the solution to environmental and energy problems.

E, Efficiency of Heat Engines

### 1. Efficiency of Any Heat Engine

The efficiency of any engine is simply the useful energy or work put out by the engine divided by the energy put into the engine to make it operate. In a heat engine, coal or oil or gas might be burned to generate the energy input in the form of heat. This is called Q H. Suppose that in a particular case 50,000 kilocal-ories (a common unit of heat) were released by the combustion of fuel to be QH. Suppose that of this amount 10,000 kilocalories were converted into electricity. This is called W for work. (In the following discussion, the word work is used for any form of energy other than heat. As already pointed out, work is completely convertible to any other form of energy.) The fractional efficiency is W/Qu = 10,000/50,000 = 1/5 = 20 percent. Because only 10,000 kilocalories were converted into electricity, 40,000 kilocalories (the remainder) were discharged to a lower temperature reservoir. This is called Qc.

It can be shown that the fractional efficiency can also be calculated from 1 — <2c/<2h- In the example at hand, 1 - QCIQU = 1 - 40,000/50,000 = 1 - 0.8 = 0.2 = 20 percent. This relationship is true for any heat engine, whether it is ideal or not. The efficiency of the large Newcomen engines was only about V2 percent, and for Watt's improved engine was about 3 or 4 percent. The fractional efficiency of the heat engines in a modern electric generating station is of the order of 0.4 = 40 percent. Table 5.1 lists the efficiencies of several types of engines. The temperatures given in Table 5.1 are based on the Kelvin scale, which is discussed in Section F.

### 2. Carnot or Thermodynamic Efficiency

The second law of thermodynamics states that even the very best heat engine conceivable—the ideal heat engine—cannot be 100 percent efficient. It is natural to ask how efficient this best heat engine actually is. This is an important practical question, because the ideal heat engine sets a standard against which other practical engines can be measured. If a particular engine comes close to the ideal efficiency, then its builders will know that they cannot improve it much more; on the other hand, if the engine has only one third or less of the ideal efficiency, then perhaps it can be improved significantly. The ideal efficiency is sometimes called the Carnot or thermodynamic efficiency.