Chapter 6: Forms of the Second Law Chapter 6: Forms of the Second Law...56 6.1 Introduction...56 6.2 Perpetual Motion Machines...57 6.3 The Principle of Kelvin...58 6.4 Maximum Work...59 6.5 Heat Engines and the Carnot Efficiency...60 6.5.1 The Carnot efficiency...61 6.5.2 The Carnot cycle...62 6.5.3 The ultimate energy crisis...63 6.6 The Principle of Clausius...64 6.7 The Principle of Caratheodory...64 6.1 INTRODUCTION The presentation of the Second Law that was given in Chapter 5 is based on a postulate suggested by Buchdahl (Concepts of Classical Thermodynamics) and is adopted here because it is, in my opinion, the simplest and most closely tied to experiment. However, it is worthwhile to pause and consider some of the other approaches that have been taken to arrive at the Second Law. In the literature of thermodynamics the Second Law is usually motivated in one of three ways. The most direct approach is to simply postulate the existence of the entropy with a definition that includes enough of its properties to imply the others. This method is increasingly used in texts on classical thermodynamics, for example, Callen (Thermodynamics), Tisza (Thermodynamics), and Truesdell (Rational Thermodynamics). It is advantageous in that it dramatically shortens the theoretical development and lets one move immediately to applications. The postulate is usually justified on the grounds that the existence of entropy is well established and can be accepted without reviewing the work that led to it. The disadvantage, which I believe is more important, is that postulates that are simply asserted obscure the experimental foundations of the Second Law. Entropy is one of the least transparent of the scientific concepts one is expected to master, and there is, at least in my view, a profound advantage in recognizing that it rests on something more than hot air. The second approach that is widely used in modern texts is to infer the Second Law from Statistical Mechanics. This method is particularly common in the physics literature and is used, for example, by Landau and Lifshitz (Statistical Physics). We shall discuss it when we develop Statistical Thermodynamics. It has the advantage of economy, since one needs to learn statistical thermodynamics in any case, and has the additional advantage of phrasing the entropy in terms of a statistical "randomness" that many students find rela- page 56
tively easy to grasp. But there are also disadvantages. The most important to the engineer is the reliance on relatively simple atomic models that are deceptive in both their generality and their depth. The Second Law was, historically, extracted from experimental observations on macroscopic systems of the sort one treats in engineering. It predates the atomic theory, does not depend on it, and can be applied with confidence to experimental situations for which an atomic model can hardly be written down. In fact, the statistical foundations of the Second Law are still not set in a form that is completely satisfactory to many theoretical physicists (see, for example, Prigogine and Stengers, Order Out of Chaos, and note the periodic appearance of technical articles that offer new arguments to exorcise Maxwell's demon). The lingering ambiguities in the microscopic basis of the Second Law do not trouble most physicists; they accept the Second Law and are perfectly willing to believe that any shortcomings in the microscopic theory will eventually be sorted out. The third approach is to base the Second Law on some general postulate that is directly related to experiment. We have done that here. However, the postulate used in Chapter 5 is only one of many that might be taken as a point of departure. [A good recent historical survey is given in Bailyn, A Survey of Thermodynamics.] Three alternate formulations are of particular interest because of their historical importance and their continued use. The first is due to Carnot and Kelvin, and postulates the impossibility of constructing a "perpetual motion machine of the second kind": a device that converts heat directly into useful work. The Carnot-Kelvin principle leads immediately to the ultimate limit on the efficiency of heat engines that ensures an eventual energy crisis. The second, simpler postulate is due to Clausius, and asserts the impossibility of directly transferring heat from a colder to a hotter body. Its corollary governs the direction of heat flow in systems that are out of thermal equilibrium. The third, and most elegant mathematically, is due to Caratheodory, and simply asserts that there are states in the neighborhood of any given state that cannot be reached by quasi-static adiabatic processes. This formulation has the advantage that Second Law can be established without an explicit appeal to the First. Otherwise it resembles the one we have used. [That is not by accident. Caratheodory was largely responsible for the basic framework used in most modern developments of the laws of thermodynamics, including that adopted by Buchdahl and used here.] We shall consider the three classical postulates in turn. They are equivalent to the postulate we have used in that they lead to the same conclusions and can, in fact, be derived as consequences of it. In the course of the discussion we shall also derive several useful relations that govern the performance of engines. 6.2 PERPETUAL MOTION MACHINES The Second Law was born as a by-product of the effort to improve the efficiency of steam engines in the late eighteenth and early nineteenth centuries. This work culminated in the assertion, by Sadi Carnot, that there is a limiting value of the efficiency beyond which one cannot go. Carnot's principle frustrates the attempt to design a "perpetual motion machine" that produces unlimited quantities of useful energy with essentially no input. page 57
To place the problem in context it is important to distinguish two kinds of perpetual motion machines. The most valuable economically would be a perpetual motion machine of the first kind, a device that does an arbitrary amount of work without fuel. To accomplish this the device would have to be capable of a cyclic transition that returned it to its original state and had no consequence other than doing a positive net work on its environment. Such a device is forbidden by the First Law. For a complete cycle ÎE = Q + W = 0 6.1 If the environment were required to supply heat it would be a fuel; hence Q must be negative or zero. It follows that W must be positive or zero and the device cannot do net work on its surroundings. A perpetual motion machine of the first kind cannot be constructed. (Simply stated, there is no such thing as a free lunch.) The impossibility of constructing a perpetual motion machine of the first kind has only slightly dampened the enthusiasm of inventors who are determined to do so. The U.S. Patent Office annually reviews dozens of applications for patents on devices that claim to do precisely that. Since the First Law requires that a working machine use fuel the search for the perfect machine devolves into a search for a machine that uses the most available fuel in the most efficient possible way. The most available fuel is the internal energy of the universe, which is there in an essentially inexhaustible supply. The internal energy of at least the ambient part of the universe can, in theory, be extracted by cooling the machine so that heat is automatically transferred to it. If this heat can be converted into work with perfect efficiency the machine will remain cool, and can go on doing useful work indefinitely. A machine of this sort is called a perpetual motion machine of the second kind: a device that can extract an arbitrary amount of energy from its surroundings in the form of heat and convert it into useful work. The Second Law has the consequence that no such machine can be constructed. (Stated simply, there is not such thing as a cheap lunch.) The impossibility of constructing a perpetual motion machine of the second kind is most compactly stated in the Principle of Kelvin. 6.3 THE PRINCIPLE OF KELVIN Kelvin's Principle can be stated as follows: It is impossible to devise an engine which, working in a cycle, produces no effect other than the extraction of heat from a reservoir and the performance of an equal amount of useful work. (The wording used here follows Pippard (Classical Thermodynamics). Essentially identical wording appears in several other texts. I have not seen the original reference.) page 58
To derive Kelvin's Principle from the Second Law as it was given in Chapter 5 let the engine receive a quantity of heat, Q, from the reservoir and do an amount of work, W. By the Carnot inequality, equation 5.62, the net entropy change in the process is ÎS Q T 6.2 which is positive definite. The engine cannot return to its original state without transferring heat to some other system to decrease its entropy, and hence cannot operate in a cycle when fueled by a single reservoir. In fact, the hypothetical engine cannot do work at all. The net work done for each increment of heat received is, from the First Law, W = E - Q 6.3 If the only change in the engine is the receipt of heat its geometric coordinates are constant. Hence and W = 0. E = T S = Q 6.4 It follows that the direct conversion of heat into work is not possible; the internal energy of the universe cannot be used as a fuel. Again, this does not prevent inventors from trying. Perpetual motion machines of the second kind are proposed almost daily. 6.4 MAXIMUM WORK While it may not be possible to convert the internal energy of the universe into useful work, it certainly is possible to extract work from the spontaneous processes that occur in the universe, as when water flows downhill from one reservoir to another or heat flows from a high-temperature bath to a colder one. If one encloses the systems that spontaneously exchange energy in a single composite system that also includes the mechanism that produces work, the composite system is adiabatic. It is then possible to set an upper limit on the work that can be produced as the internal constraints that set the initial state of the composite system are relaxed and it evolves toward an unconstrained equilibrium. Let a composite system be composed of a number of subsystems that are separated by partitions that maintain them in states that are not in equilibrium with one another. For example, the subsystems may have different temperatures, pressures, or chemical potentials. Now let the partitions be removed, relaxed or changed so that the composite system evolves toward an unconstrained equilibrium state, and let there be a mechanism that is driven by the spontaneous change of state to produce a net useful work. For example, the systems may be reservoirs at different heights and the mechanism may be a water wheel in page 59
the concourse through which the water flows when the sluices that constrain the upper reservoir are opened. Since the system is adiabatic, the net work done by it in an incremental transition is - W = - E 6.5 where E is its internal energy change. Hence E should be as negative as possible. But for a given step toward equilibrium, as characterized by a given change in the geometric coordinates of the subsystems, E = E(S) - E 0 6.6 where S is the entropy of the final state. Since E S > 0 6.7 the maximum work is done when the entropy change is as small as possible, and the maximum possible work is done when the process is isentropic. From the perspective that entropy is time, the maximum possible work is done when the system changes in such a way that it remains in the present. As the system "ages" it loses the capacity for useful work. 6.5 HEAT ENGINES AND THE CARNOT EFFICIENCY While perfect engines do not exist, useful engines do. It is important to know how to design them so that they will perform as well as possible. The working mechanism of a useful engine should operate in a cycle so that it periodically returns to its initial state. Otherwise the engine would run down. The engine is expected to do a net useful work per cycle. Since energy is conserved, the only way it can do this is to receive heat and convert it into work. The results of Section 6.3 show that it cannot simply receive heat from a reservoir since the heat added monotonically increases its entropy. To return to its original state the engine must interact with some third system to lower its entropy to the original value. The net effect of the third system is to extract a quantity of heat from the engine. The whole device can be modeled as a composite system that includes a high-temperature reservoir to supply heat, a low-temperature reservoir to receive heat, and a cyclic engine that connects the two and does useful work as heat flows between them. The composite system is called a heat engine. While it may not be obvious that all useful engines can be treated as heat engines, this is, in fact, the case. Take, for example, the water wheel that operates between reservoirs of different heights. To keep the engine working, one must have some mechanism for replenishing the water in the upper reservoir. The natural process that accomplishes this page 60
is evaporation. Heat is absorbed by the water in the lower reservoir, converted into gas, cooled and recondensed in the form of rain into the upper reservoir. If one includes enough of the universe into the system it becomes adiabatic, and operates as a heat engine. 6.5.1 The Carnot efficiency A heat engine is an adiabatic system that evolves toward equilibrium as heat is transferred from high temperature to low. The work done by the engine per cycle of the working mechanism is - W = (Q 1 + Q 2 ) 6.8 where Q 1 is the heat supplied to the mechanism from the high-temperature reservoir at T 1 and Q 2 is the heat supplied from the low-temperature reservoir at T 2. The efficiency of the engine is defined as the net work done per unit of heat supplied by the high-temperature reservoir: = - W Q 1 6.9 The results of Section 6.4 show that the work done, and hence the efficiency, is maximal if the total entropy change is zero. In this case, ÎS = ÎS 1 + ÎS 2 Hence = - Q 1 T 1 - Q 2 T 2 = 0 6.10 Q 2 = - Q 1 T 2 T 6.11 1 The sign of Q 2 is opposite to Q 1 ; heat is rejected to the lower reservoir. Using equations 6.8 and 6.11 in 6.9 gives the maximum possible efficiency of a heat engine The work done per cycle is, then, = T 1 - T 2 T 1 6.12 - W = Q 1 T 1 - T 2 T 1 6.13 page 61
The limit 6.12 is called the Carnot efficiency after Sadi Carnot, who first obtained it. Any engine that is based on the extraction of heat or the combustion of fuel can be cast into the form of the heat engine described here by properly defining the system. All practical engines are of this type. It follows that the Carnot efficiency is the limiting efficiency for practical engines. 6.5.2 The Carnot cycle The Second Law places an upper limit on the efficiency of heat engines, but does not ensure that an engine that achieves this efficiency can be designed. It is important to recognize that there are engine designs that achieve Carnot efficiency, at least in the limit of zero friction. The classic example is an engine that operates through the Carnot cycle, which is diagrammed in Fig. 6.1. T 1 T 1 2 4 3 T 2 V Fig. 6.1: The Carnot cycle plotted on a T,V diagram of a one-component gas. Steps 1 and 3 are isothermal. Steps 2 and 4 are adiabatic. Let an engine consist of a working gas equipped with a piston that can do work on it and with two reservoirs at temperatures T 1 > T 2 that can be brought into contact with it. Let the cycle start with the gas at the temperature, T 1, and in contact with the high-temperature reservoir. Let the piston be withdrawn so that the volume of the gas increases isothermally. In the process the gas extracts a quantity of heat, Q 1, from the reservoir. Let the gas then be expanded adiabatically by isolating it from the reservoir. The gas does work on its environment by displacing the piston. Its temperature decreases during the expansion, which is continued until the temperature falls to T 2. The gas is then brought into diathermal contact with the low-temperature reservoir, and compressed isothermally. During the compression the gas transfers a quantity of heat Q 2 to the low-temperature reservoir. The gas is finally compressed adiabatically to restore its initial volume and temperature, completing the cycle. During the adiabatic compression the environment does work on the system. Assuming that friction can be neglected and that the heat transfers at T 1 and T 2 are done quasi-statically, the overall process is isentropic. The net work done is page 62
given by equation 6.13, the relative quantities of heat are fixed by equation 6.11, and the Carnot efficiency is achieved. The Carnot cycle is closely approximated in modern steam engines and gas turbines. Because of its efficiency, the gas turbine is the engine of choice for electric power stations, whether they use coal, petroleum or nuclear fuels. As one can see from equation 6.12, the efficiency of a Carnot engine increases dramatically as the temperature difference between the two reservoirs is increased. Since heat must ultimately be released into the ambient, the lowest temperature of the low-temperature reservoir is essentially fixed. The improvement in turbine power generators hence depends on increasing the operating temperature at the hot side. The limitation is the development of turbine materials that can accept extremely high temperatures without degradation or mechanical failure. Internal combustion gasoline engines do not operate in a Carnot cycle and do not achieve Carnot efficiency. 6.5.3 The ultimate energy crisis As we have seen, it is a consequence of the Second Law that waste heat is always generated during the production of power. The heat must be conducted away lest it accumulate and raise the temperature of the low-temperature reservoir. Taking the earth as a whole as a system, the only mechanism for releasing heat is its radiation into space. The surface temperature of the earth is determined by a balance between the heat absorbed from the sun, the heat produced by processes on the earth, both natural and artificial, and the heat radiated into space. Assuming that the heat load from solar and natural processes is constant, the heat produced in any artificial process that generates heat necessarily raises the heat load on the earth and increases its temperature. In fact, since all work is eventually returned to heat through friction and other irreversible processes, the whole heat generated by combustion of fuel, induced radioactive decay, or enhanced solar absorption through photovoltaics or solar collectors is eventually added to the heat load of the earth. There are, therefore, rather tight upper limits on how much power can be produced on the earth without raising its temperature to a level that would be intolerable for its inhabitants. There are so many unknowns in the equation, for example, atmospheric changes that alter the net absorption of solar radiation, that it is not easy to predict precisely where this upper limit lies. Realizing that the standard of living is closely tied to energy consumption (machines replace men and do more and more work) the upper limit on energy production places an upper limit on the aggregate material standard of living, which must be divided in some way over the population of the earth. In the long run the efficiency of energy production and distribution, which controls heat pollution, is a more important consideration than the projected cost of the fuel. page 63
6.6 THE PRINCIPLE OF CLAUSIUS The starting point for Clausius' treatment of the Second Law was the somewhat more intuitive postulate that heat must flow from a hotter body to a colder one. However, the Clausius Principle is also phrased in terms of the properties of heat engines: It is impossible to devise an engine which, working in a cycle, produces no effect other than the transfer of heat from a colder to a hotter body. (Again, the wording follows Pippard.) The proof of the Clausius Principle is straightforward. Let the engine be combined with the hot (T 1 ) and cold (T 2 < T 1 ) reservoirs in a composite system. Since there is no net interaction with the environment, the system can be taken as isolated. If the quantity of heat, Q, is transferred from the cold reservoir to the hot one the change in entropy is ÎS = Q 1 T - 1 1 T 6.13 2 Since ÎS 0, Q must be negative or zero, which establishes the Principle of Clausius and, incidentally, guarantees that heat flows down a temperature gradient in the absence of other effects. It is, of course, possible to transfer heat from a cold body to a hot one. The devices that do this are known as heat pumps. In accordance with the Second Law a heat pump transfers heat by doing mechanical work on the system. An example of an ideally efficient heat pump is a Carnot cycle that is run backwards. The net work that must be done to deposit the quantity of heat, Q 1, in the reservoir at T 1 can be easily computed from the relations given in Section 6.5. Good heat pumps are commercially available, and are widely used as efficient furnaces in homes in cold climates. 6.7 THE PRINCIPLE OF CARATHEODORY The Caratheodory development of the Second Law is the most popular among classical theorists, both because of its mathematical elegance and because it employs the minimum set of postulates; it obtains the Second Law without assuming the First.. We shall avoid most of the mathematics in this discussion, but one should be aware of the structure of the Caratheodory development. [There is a very good recent discussion in Bailyn, A Survey of Thermodynamics.] To set the Caratheodory development in the context of what we have done, recognize that the definition of the entropy as a state function S = S ({x}) 6.14 page 64
gives it a simple geometric interpretation. If we let the n constitutive variables, {x}, span an n-dimensional space then the states that have a given value of S fall on an (n-1) dimensional hypersurface in that space. The states on a hypersurface of constant entropy are connected by quasi-static adiabatic transitions. The paths that represent such transitions lie on surfaces of constant S and are solutions to differential equations of the form ds = 0 = k S x k dx k 6.15 Assuming that the partial derivatives of S are non-zero, equation 6.15 is solvable for any one of the dx k. It follows that only (n-1) of the dx k can be chosen independently; equation 6.15 has (n-1) "degrees of freedom", as it should to describe a path that lies on an (n-1) dimensional surface. Since the paths of all quasi-static adiabatic processes that begin from a given initial state lie on a particular surface, they cannot connect a state that lies on a particular constant entropy surface to any state that is off of it, however close the latter may be. It follows that, given a state Í on the surface S(Í) there are an arbitrary number of distinct states, Í', in the immediate neighborhood of Í that cannot be reached from it by any quasi-static adiabatic process. The relationship is illustrated in Fig. 6.2. E Í S(Í) q Fig. 6.2: Adiabatic accessibility. All states in the shaded neighborhood of Í are adiabatically inaccessible from it, except those that actually lie on S(Í). Caratheodory's development of the Second Law starts from the recognition that this reasoning can be done in reverse. Caratheodory began with the following postulate: In the immediate neighborhood of any equilibrium state of a system there are states that are inaccessible from it by any quasi-static adiabatic process. page 65
He then proved a mathematical theorem (Caratheodory's Theorem) that if the Caratheodory postulate holds then the paths of the quasi-static adiabatic transitions that initiate from a given state generate a unique surface of dimension (n-1). Since the states themselves are dense, these surfaces must be dense and non-intersecting. Hence the surfaces define a continuous state function that is called the entropy. Since these conclusions apply to any system the absolute temperature and metrical entropy can be defined as was done in Chapter 5, and the remainder of the consequences of the Second Law follow. Caratheodry's analysis is sufficient to define an entropy, but does not, by itself, establish the law of increase of the entropy or the entropy maximum principle. To do this, the Caratheodory postulate must be supplemented by a second postulate that sets the direction of change of the entropy in an irreversible, adiabatic transformation, i.e., that identifies the side of the entropy surface on which the inaccessible states lie. The postulate that is used is equivalent to the statement that work can be done on a system, but not extracted from it, which is the posulate we have made here. I have not followed the Caratheodory treatment because it is mathematically complex, and the complexity is unnecessary. The First Law exists, and is well established experimentally. There is no obvious reason why we should not use it, particularly since its use dramatically simplifies the formulation of the Second Law. page 66