A New Perspective on Thermodynamics

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Author: Bernard H. Lavenda

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A New Perspective on Thermodynamics

Bernard H. Lavenda

A New Perspective on Thermodynamics

123

Bernard H. Lavenda Università Camerino Via del Bastione, 2 62032 Camerino Italy [email protected]

ISBN 978-1-4419-1429-3 e-ISBN 978-1-4419-1430-9 DOI 10.1007/978-1-4419-1430-9 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009940574 c Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

To Ethan and Ronit

Preface

More than to any other single individual, thermodynamics owes its creation to Nicolas-Léonard-Sadi Carnot. Sadi, the son of the “great Carnot” Lazare, was heavily influenced by his father. Not only was Lazare Minister of War during Napoleon’s consulate, he was a respected mathematician and engineer in his own right. Mathematically, Lazare can lay claim to the definition of the cross ratio, a projective invariant of four points. Lazare was also interested in how machines operated, emphasizing the roles of work and “vis viva,” or living force, which was later to be associated with the kinetic energy. He arrived at a dynamical theory that machines in order to operate at maximum efficiency should avoid “any impact or sudden change.” This was the heritage he left to his son Sadi. The mechanics of Newton, in his Principia, was more than a century old. It dealt with the mechanics of conservative systems in which there was no room for processes involving heat and friction. Such processes would ruin the time reversibility of mechanical laws, which could no longer be derived by minimizing the difference between kinetic and potential energies. When Sadi wrote his only scientific work in 1824, there were no laws governing the mechanical effects of heat. In fact, caloric theory was still in vogue, which treated heat as an imponderable fluid that was conserved. Although this interpretation led naturally to the interpretation of latent and specific heats, it was at a loss to account for phenomena related to the transport of heat, like that which occurs in steam engines. Carnot’s great achievement was the recognition that motive power is due to the passage of heat from a hotter to a colder body. He used caloric theory in analogy with the drop in height of water used to turn a water-wheel. Water is conserved, just like caloric, and just as the water at the bottom has a smaller potential energy to do work, so, too, the caloric at the lower temperature can do less work than if it were at a higher temperature. The difference in height represented the difference in temperature and the fall in height represented fall of heat through this temperature difference that could produce work on any material system that was capable of undergoing dilation and contraction. ´ Sadi graduated from the Ecole Polytechnique and served in the Army Engineers doing boring jobs in the provinces. When he retired on half-salary in order to devote himself entirely to following his scientific pursuits, his father was in no

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position to further his son’s academic career under the restored monarch. Lazare was to die in exile in 1823, a year before the publication of his son’s monumental work. Sadi was not a member of the establishment, and, in his life time, did not receive the honors to which he rightly deserved. He was not a member of the Academie Francaise and did not frequent the circles of the illustrious giants of his time. The fate of his book was a telling tale. Not one of France’s illustrious scientists, such as Fourier, Laplace, Gay-Lussac, Legendre, and Ampére, even mentioned Carnot’s book or took up his ideas, although it was given a favorable review by Pierre-Simon Girard writing in the well-known journal, Revue Encyclopedique. Being ignored is a fate far worse than being criticized. The achievements of men such as Laplace, Poisson, and Fourier are easily quantified. We talk of the Laplace transform, the Poisson distribution, and the Fourier series. But, with recognition came the fossilization of these great minds, for in time they became set in their ways and were no longer receptive to new ideas that were put forward by the younger generation. Carnot was not alone in his solitude, for there was a young brilliant mathematician who was to die in a duel only days before Carnot. Though the career of Evariste Galois was short, his name has been attached to “Galois’s theory” of groups, not to mention his development of the connections between algebraic equations and transcendental functions. Galois had the same treatment that was bestowed on Carnot: Fourier lost one of the papers he submitted to the Academy, while Poisson had rejected another. It is as Planck said, progress is made not by the conversion of the old generation to the new ideas of the younger generation, but, only when that generation dies out. However, Carnot’s real achievements are less cut and dry and less quantifiable, for they started a chain reaction that is still going on. Out of Carnot’s analysis can be distilled very general principles to explain the working and evolution of any system that operates on a difference in temperature. Most scientists would agree that Newton laid in peace till the coming of Planck, but Cardwell (1971) points out, this is simply not true. The realization that heat could be transformed into work, and their difference represents the internal energy, which is a conserved quantity over a complete and reversible cycle, shook Newton from his resting place at least 50 years prior to the discovery of blackbody radiation and, subsequently, quantum theory. It was done by amateurs, or better outsiders, such as Carnot, Mayer, and Joule. Toward the end of his short life, Carnot rejected the caloric theory, even going so far as to calculate the mechanical equivalent of heat. Yet, like Gauss’s unpublished discovery of hyperbolic geometry, where more than two parallel lines pass through a point, the discovery of the mechanical equivalent of heat was attributed to a little-known physician from Heilbronn. In a lecture to the Royal Institution in 1862, John Tyndall credits Mayer with the discovery of the interconversion of heat and work, to the chagrin of Joule. Joule protested that he too should be credited independently with the discovery of energy conversion to which Tyndall agreed entirely. All could have ended happily were it not for a certain Peter Guthrie Tait, professor of

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Natural Philosophy at Edinburgh University, who with Kelvin, objected to Tyndall’s remarks. The controversy was newly alighted with Mayer’s claims utterly rejected and Tyndall’s scientific capacities called into question. Tait was fond of pointing the finger at Mayer claiming that his determination of the mechanical equivalent of heat from the isothermal expansion of a gas was critically flawed because all the heat does not have to be used in external work. According to Tait, some of the heat could be used in “loosening the bonds between the gas molecules.” Mayer did not have at his disposition the data on the Joule– Thomson porous plug experiments. But how many times are scientific discoveries made for the wrong reasons? Certainly Tait had a unique position in the history of science. Probably due to an inferiority complex next to people like Clerk-Maxwell and Kelvin, Tait thought his role would be in defending his fellow countrymen against what he saw as an attack by foreigners. We have Tait to thank for anglo-saxonizing the ideal gas laws of Mariotte and Gay-Lussac, whom he referred to as Boyle’s and Charles’s laws, respectively. And Tait accomplished all of this not by swaying public opinion, but, rather, by writing textbooks for physicists. Textbooks still undiscriminately refer to these laws by their English names. As Cardwell (1971) concludes: “even if Tait had never lived it would have been necessary to have invented him.” To uphold British scientists at the expense of their German and French colleagues, Tait published in 1868 his most influential Sketch of Thermodynamics in which he excuses his lack of impartiality and then goes on to vindicate the British claim to the dynamical theory of heat. Mayer’s determination of the mechanical equivalent of heat was ridiculed by Tait on the basis that as a gas expands isothermally there is no reason to believe that all the heat has been used for external work. Some of the heat, Tait tells us, could have gone into breaking the bond of the gas molecules that tie them together. Clausius is subordinated to Kelvin by replacing “maximum entropy” by “least dissipation of energy.” Tait even gets the second law inequality reversed, which had resounding consequences on Clerk-Maxwell’s little black book, Heat, first published in 1877, which was to go through ten editions, with the last corrections being made by Lord Rayleigh after Clerk-Maxwell’s death. Notwithstanding the enormous success of Clerk-Maxwell’s Heat, he fell pray to Tait’s confusion about entropy. This is revealed in his 1873 letter to Tait where only lately under the conduct of Professor Willard Gibbs that I have been led to recant an error from your [Sketch], namely that the entropy of Clausius’s is unavailable energy while that of Thomson’s is available energy. The entropy of Clausius is neither one nor the other.

Textbooks are passed down from one generation to the next, while public lectures and writings die with their audiences. Tait, in effect, institutionalized the scientific mafia, like no one before him. He replaced Clausius’s entropy, and its unending tendency to increase through thermal interactions, by Kelvin’s principle of the minimum dissipation of energy, whereby all the available energy of the universe must diminish without end. The universe rather than suffering a heat death would be frozen out in an energeticalness state. Once scientists become prisoners of a single doctrine, there is no longer any room for a Carnot. Creative minds of individuals are replaced by adhesion to party lines,

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and this marks the death kneel of science in general. It is therefore fitting to title this book A New Perspective on Thermodynamics in the fervent hope that it will give birth to new Carnots. If we are to take our cue by what happened in England at the turn of the twentieth century, we might take some comfort. Although Tait’s Sketch influenced a whole generation of English-speaking scientists, it brought about its own demise. For at the turn of the century the pendulum was swinging elsewhere: mainly on the continent, but, with a little help from J. Willard Gibbs in America. The time was ripe for the likes of Planck and Boltzmann. Planck, the conservative, a disciple of Clausius, was about to use thermodynamics to discover the quantum while Boltzmann, the radical, lowered the second law to a statement in probability and just missed the quantum by taking the continuous limit at the end of his calculations. Boltzmann’s psychological problems would not let him see that day, while Planck lived long past to see developments that certainly bewildered and confounded his conservative, and yet very creative, mind. Even against all odds, science seems to regenerate itself. It is a pity that it takes more than one’s lifetime. The present day is no exception; it has seen a proliferation of Taits, thanks to expanded educational systems and modern communication. It was not a mere coincidence that the giants of nineteenth century science were noblemen or men of good economic stature. The third Lord Rayleigh was a landed proprietor of independent means, who could afford to vacate the chair at Cambridge left empty on Maxwell’s death, just 5 years after accepting it. Others, like James Prescott Joule, could leisurely dabble in experimental science because his father owned a brewery in Manchester. These scientists did science not for promotion but because they were genuinely interested in what they were doing, having a child-like curiosity to understand the workings of Nature. Sadly, all this has changed in more modern times with governments providing the financial backing of science projects that are dictated by lobby groups. The economic barrier has been broken down and science has been opened up to the masses. This does not mean that the science being performed today is good science. Here we discuss in detail three lines of research, two of which have acquired a rather large following: the so-called “endoreversible” engine and its generalization to finite-time thermodynamics, and the possibility to obtain nonequilibrium work from equilibrium free energy differences, which has found resonance in the biological community. A third area deals with the geometrization of thermodynamics, claiming that Gibbs’s goal has been finally achieved. The first two are bound up with the misconceived notion that thermodynamic processes can be accomplished in finite time. The third has rediscovered thermodynamic fluctuation theory and has wrongly attributed to it a Euclidean metric geometry (Weinhold 1975). Politics and plagiarism have gone to such heights that it makes original discoveries in science almost go unnoticed. The reason is again to be found in how textbooks are written and to whom they are addressed. The splintering away of the scientific branches, and their compartmentalization, has aggravated the situation. A textbook is not an encyclopedia where one goes to find a specific formula, or handy definition. Rather, a textbook should develop a logical line of thought and show how

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out of misconceptions and confusion arose the universal truths that remain “for all times and for all cultures, including extraterrestrial and non-human ones,” to borrow a phrase from Planck (1900) in describing his “natural units.” Even Clerk-Maxwell’s celebrated little black book, The Theory of Heat, first published in 1877, which could be found on the benches in almost all laboratories in Europe, is tainted with Tait’s influence. We have already seen how Clerk-Maxwell had been misled by Tait on the notion of entropy. He was also confused on the relation between the absence of the exchange of heat and the conservation of entropy for he writes (Maxwell 1891) “when there is no communication of heat this quantity remains constant, but when heat enters or leaves the body the quantity increases or diminishes.” For the quantity Clerk-Maxwell is obviously referring to is entropy, and he is equating adiabatic and isentropic processes, which is not always true. Notwithstanding these minor blemishes, the aim of Clerk-Maxwell’s Theory of Heat was to teach, while Tait’s Sketch was out for vendetta, and in so doing, sparked interest in a subject that was not to be equaled by any other theory, and probably never will be. It is the intention of the present volume to draw from both these “textbooks” by uniting (Maxwell 1891) “the doctrines of heat (that) have been developed” with an attempt to give (Tait 1868) “a rough sketch of the history of a grand physical theory, especially one of so modern a date” without the difficulty of finding it “impossible to be strictly impartial.” So is it only of historical interest? I think not. For the claims of what thermodynamics can do have come back to haunt us on more than one occasion. Truesdell (1980), writing in his apology to the “spectators” in the Tragicomical History of Thermodynamics 1822–1854, claims that “Only now could a real history of thermodynamics be written, since only in the last twenty years have the expressed aims of the creators of thermodynamics have been achieved.” He continues “much of what I write now about the classical papers on thermodynamics I could not have written twenty years ago, because I did not then have the grasp of rational thermodynamics that today we may and do teach our beginning students.” So it appears that the Taits of this world are still out there, fervently working to sway the unsuspecting “spectator” that they, and they alone, have found the holy grail of thermodynamics. If there is something “rational” about Rational Thermodynamics it is certainly well hidden (Lavenda 1978), and interest in it has not stood the test of time. Even though almost 200 years have past since Carnot enunciated his principle that work can be obtained only by having heat “fall” between two temperatures, heat and work are often mangled beyond anything that Carnot would recognize. A partial list of modern day confusion includes the following: The absorption of heat from a heat bath at a temperature other than that of the

heat bath, and applying a condition that the cycle be reversible (e.g., the so-called “endoreversible” engine in Sect. 4.2.4). The distinction between equilibrium and nonequilibrium processes (e.g., thermoelectricity in Sect. 4.3). Equilibrium changes in the Helmholtz free energy are caused by nonequilibrium work done in a finite amount of time (e.g., Jarzynski’s inequality in Sect. 5.3.1).

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Camouflaging the condition of microscopic reversibility as a fluctuation-

dissipation relation (Crooks’s equality in Sect. 5.3.7). The generalization of extensive to nonextensive thermodynamics (Chap. 6). Violations in the second law for small, but finite times. Claims that “fulfillment of a goal envisioned by Gibbs, the laws of ther-

modynamics have been written down in the form of a Euclidean metric geometry: : :” (Weinhold 1975) (cf. Sect. 6.11). What have been proposed as new and exciting results are, in reality, the result of confusion. A clear example of this can be found in Fermi’s classic text on thermodynamics (Fermi 1956), which we will discuss in greater detail in Chap. 5. Fermi gives Kelvin’s postulate regarding the second law as follows: A transformation whose only final result is to transform heat extracted from a source which is at the same temperature throughout entirely into work is impossible.

Fermi tries to rebut this by considering the isothermal expansion of an ideal gas. Since the internal energy is a function solely of the temperature, here we have a process in which all the heat absorbed is converted entirely into work. But, as Fermi is quick to point out, this does not violate Kelvin’s postulate because the initial and final states are not the same. While technically this is true, it has nothing to do with Kelvin’s postulate for Kelvin was considering the work done in a closed cycle in which the system returns to the initial state at the end of the cycle. In particular, an efficiency cannot be determined for an isothermal expansion because it is not cyclic. Efficiency can only be determined by “paying the price” for the system to return to its initial state. In other words, proceeds can only be determined after the production costs have been taken into account. Fermi was, in fact, taking Kelvin’s postulate out of context. Following Carnot’s lead, work can only be obtained by “letting down” heat to a lower temperature. It is the cold reservoir that is foreign to the physicist where work is the integral of the force over the distance covered. That heat must be given up to the cold reservoir belongs solely to the realm of thermodynamics, as Carnot appreciated so well. For if we were to separate the cycle into individual steps, we could surely fabricate things so that there would be seeming violations of the second law. Early objections to the second law were voiced by Hirn. In both his examples there are compensating, and opposing, processes at work. In each of his examples it appears that two engines are at work, one proceeding in a forward cycle and the other in a reverse cycle. It is the former that drives the latter, since it is the transference of heat from a higher to a lower temperature that dominates. If there are violations to the second law they cannot appear on a macroscopic scale, as Clerk-Maxwell so imaginatively conceived them by creating a “demon” that is capable of sorting out the faster-moving molecules, which are hot, from their slower-moving counterparts, which are cold. It is this type of confusion, which persists to this day, that has led to many “remarkable” results in the thermodynamic literature. Trevignano Romano

Bernard H. Lavenda

Contents

Preface .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . vii List of Figures . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . xvii List of Tables. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . xix 1

The Predecessors of Carnot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1.1 The Dawn of the Science of Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1.2 From Isothermal to Adiabatic Propagation of Sound . . . . . . .. . . . . . . . . . . 1.3 Sources for Réflexions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 1.4 Carnot’s Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

1 1 6 8 9

2

Thermodynamics from Carnot to Clausius and Kelvin . . . . . . .. . . . . . . . . . . 17 2.1 Réflexions: An Ignored Masterpiece . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 17 2.2 Doctrine of Latent and Specific Heats . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 22 2.3 Interconvertibility of Heat and Work . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 25 2.3.1 Temperature Absolutely . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 29 2.3.2 Does a Gas Heat or Cool When Passed Through a Porous Plug? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 30 2.3.3 Kelvin’s Absolute Scale.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 31 2.3.4 Clausius’s Enunciation of Carnot’s Theorem and Its Corollary 34 2.4 Integrating Factors Galore.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 36 2.5 Carathéodory’s Attempt at the Second Law .. . . . . . . . . . . . . . . .. . . . . . . . . . . 41

3

Thermodynamics in a Carnot Equation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.1 Why Exterior Differentials? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2 Two-and-a-Half Commandments of Thermodynamics . . . . .. . . . . . . . . . . 3.2.1 Independent Variables .V; T / . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.2 Strange Carnot Cycles: An Interlude .. . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.3 Independent Variables .p; T / . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.2.4 Independent Variables .V; p/ . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.3 A Clash of Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 3.4 Carnot’s Real Efficiency Resurrected .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

47 47 48 49 51 55 57 63 66

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5

6

Contents

Equivalence of First and Second Laws . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.1 Laws of Thermodynamics: Local and Global Forms . . . . . . .. . . . . . . . . . . 4.2 Carnot’s Modified Axiom and His Criterion for Irreversibility . . . . . . . 4.2.1 Carnot Cycle.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.2.2 Otto Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.2.3 Brayton Engine .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.2.4 Endoreversible Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.2.5 Stefan–Boltzmann Law from the Carnot Cycle. . . . .. . . . . . . . . . . 4.2.6 Relativistic Carnot cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.2.7 Coefficient of Performance from the Complementary Efficiency of a Refrigerator . . . . . . .. . . . . . . . . . . 4.3 Beyond Steam Engines: Thermoelectricity .. . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.3.1 Thomsons’s Theory of the Electrical Specific Heat . . . . . . . . . . . 4.3.2 Tait’s Thermoelectric Diagrams . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.3.3 Clausius’s Thermoelectric Theory .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . 4.4 Irreversibility Viewed as Violations in the First and Second Laws.. . .

71 71 73 74 78 80 82 92 96 101 102 104 105 109 111

Work from Nonequilibrium Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.1 What is Work? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.2 Principle of Thermal Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.3 Principle of Maximum Work .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.4 Work and Free Energy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.4.1 The ‘Jarzynski Equality’ .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.4.2 Canonical with Respect to What? . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5.4.3 Microscopic Origins of the Carnot–Clapeyron Equation.. . . . . 5.4.4 From Adiabatic to Isothermal Relations and their Underlying Microscopic Counterparts .. . .. . . . . . . . . . . 5.4.5 The “Crooks Equality” .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

117 117 118 120 123 123 125 130

Nonextensive Thermodynamics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.1 Incomparable Thermodynamic Laws .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.2 Einbinder’s @-Theorem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.3 Unconventional Phase Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.4 Nonextensive Adiabatic Potentials. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.5 The Second Laws .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.6 An Equivalent Carnot Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.7 Confines of Thermodynamics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.8 Mathematical vs. Physical Inequalities . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.9 Comparable Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.10 Bounds on Mean Temperatures and Volumes . . . . . . . . . . . . . . .. . . . . . . . . . . 6.11 Geometry of Thermodynamics.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.12 Thermodynamics of Coding and Fractals . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.12.1 Optimal Coding Lengths .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 6.12.2 Pseudoadditive Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

145 146 153 155 159 162 166 168 171 172 174 176 181 183 186

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6.12.3 Multifractals to Strange Attractors . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 188 6.12.4 Exponential Entropies and the Correlation Dimension .. . . . . . . 191 Bibliography . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 195 Index . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 201

List of Figures

1.1

Carnot’s original cycle using infinitesimal temperature differences . . . . . 10

2.1

The original plate appearing in Réflexions that Carnot used to describe his cycle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . (Left) Position of the piston corresponding to the branches of the Carnot cycle. (Right) Closing of the Carnot cycle . . . . . .. . . . . . . . . . . Joule–Thomson experiment .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Contrast between adiabatic and isothermal curves . . . . . . . . . . . .. . . . . . . . . . . Clerk-Maxwell’s construction for determining the absolute temperature scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

2.2 2.3 2.4 2.5

18 19 30 32 32

3.1 3.2

Strange Carnot cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 53 A Carnot cycle in the region of negative latent heat . . . . . . . . . . .. . . . . . . . . . . 54

4.1 4.2 4.3 4.4 4.5 4.6 4.7

Carnot cycle . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Otto cycle . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Brayton cycle . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Endoreversible engine .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . The Carnot cycle according to Stefan and Boltzmann .. . . . . . . .. . . . . . . . . . . Radiation falling obliquely on a surface.. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . A Carnot cycle in the pV N -plane whose working substance is radiation . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Tait’s thermoelectric diagram .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . Thermoelectric analog of a Carnot cycle .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

4.8 4.9 6.1 6.2 6.3 6.4

Phase equilibrium diagram.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . A cyclic path such that the transition from A to B is irreversible while from B to A is reversible.. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . A cycle equivalent to Carnot’s cycle . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . The thermodynamic surface of water sent to Gibbs by Clerk-Maxwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .

74 79 81 82 93 93 94 105 106 158 165 167 177

xvii

List of Tables

4.1 4.2

Nonmaximal, Carnot, and observed efficiencies .. . . . . . . . . . . . . .. . . . . . . . . . . 76 Expression for the efficiencies of the Carnot, Curzon-Ahlborn, and isothermal engines . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 92

xix

Chapter 1

The Predecessors of Carnot

1.1 The Dawn of the Science of Heat To the ancients, fire was the lightest of the four known elements. The concept of fire as an element lasted down to the time of Lavoisier, which was finally abandoned in favor of caloric theory. Phenomena involving the transfer of heat were imagined as being the result of a fluid, called “caloric,” which permeated the gaps between atoms of a solid causing thermal expansion and whose loss through the surface could explain Newtonian cooling. The work of Count Rumford on the heat produced by the boring of cannons was interpreted as coming from a solid; when work is done on it, it behaves like squeezing a sponge full of water. At the close of the eighteenth century, technological advances in the creation of steam engines came predominantly from England. There was no consensus of what constituted heat, and the physics of the steam engine was virtually unknown. The dominating theory of the time was the caloric theory. Whereas in England caloric theory met its opposition by the semimechanical theories of Davy, Young, and Herapath, exerting their authority by invoking Newton, in France there was an almost complete acceptance of caloric theory, based on the authoritative personalities of the likes of Lavoisier, Poisson, and Fourier. According to Lavoisier, caloric is to be treated as an indestructible substance that is conserved in all thermal processes, while, at the same time, heat and work are taken to be equivalent. Carnot’s analogy with a drop in the level of water (chute d’eau) through a mill with the descent of heat (chute de calorique), in a reversible engine, embodies this notion of indestructibility and the necessity for its conservation. Yet, heat is known to be produced by friction, so it appears or disappears when one rubs, or stops rubbing, his hands. Notwithstanding how incongruous these two beliefs are, they became the origin of heat and entropy as we know them today. The quantity conserved in an isentropic process is now referred to as entropy, while the quantity conserved in an adiabatic process is the heat. Authors (Callendar 1911; La Mer 1953; Hirshfeld 1954; Kuhn 1955) have even gone so far as to read into Carnot’s work that every time “chaleur” is mentioned it is to be interpreted as “heat,” while when he talks about “calorique,” he is really intending “ entropy.”

B.H. Lavenda, A New Perspective on Thermodynamics, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1430-9 1,

1

2

1 The Predecessors of Carnot

These distinctions only emerged out of the later work of Clausius; at the beginning of the nineteenth century the two concepts were confused. The quantities of heat and caloric were used interchangeably. In the later writings of Clapeyron, who based his work on Carnot’s theory, the opposite usage was given to the quantity of heat, as opposed to the quantity of caloric. Since both worked with infinitesimal differences in temperature, there was more reason for confusing heat and entropy. In what was later to be referred to as the “doctrine of latent and specific heats” by Truesdell, the quantity of heat was taken as a two-component fluid. One of the two components was free and sensible to the touch; it would raise the temperature. The other was stored, or latent, heat that was imperceptible to the senses. This latent heat was assumed to be chemically bonded to the molecules in a body. Although the total quantity of heat remains the same when such a body undergoes adiabatic compression, some of the latent heat is converted into sensible heat as observed by a rise in its temperature. Poisson, following Laplace’s Méchanique Celeste, considered that a gas was completely defined by giving it a well-defined pressure, p, and volume, V . The change in heat, dQ, of such a body could then be expressed as1 dQ D

@Q @V

dV C

p

@Q @p

dp: V

The quantity of heat also depends on the temperature, but this could be accounted for by an equation of state so that

and

@Q @V

@Q @p

D

p

D

V

@Q @T

@Q @T

p

V

@T @V

@T @p

D Cp

p ; R

D Cv

V ; R

p

V

where R is the universal gas constant. Setting dQ D 0, for an adiabatic process, led to pV D const., where D Cp =Cv was referred to as the ratio of adiabatic to isothermal “elasticities.” was shown by the measurements of Clément and Désormes, on the one hand, and GayLussac and Welter, on the other, to be the ratio of the two heat capacities, and it had to be greater than unity. Clément and Désormes based their analysis on the doctrine of caloric-of-space whereby air that is allowed to enter an evacuated vessel brings with it its own heat, thereby causing a rise in temperature. Knowing the total volume of air, and knowing the volume of void, these workers could calculate the specific heat of the void in terms of the specific heat of air. Gay-Lussac and Welter showed further how the experiment could be used to measure the ratio of the two specific heats. 1

In those times no distinction was made between total and partial derivatives.

1.1 The Dawn of the Science of Heat

3

Another way of deriving the adiabatic relation pV D const., which utilizes the derivatives of the ideal gas equation of state, was given by Poisson (Mach 1986). A quantity of heat Q contained in a unit mass of gas is a function solely of the pressure p and the density , i.e., Q D f .p; /. It was known in Poisson’s time that the so-called permanent gases followed a number of empirical laws rather closely. The three empirical laws were as follows: 1. Mariotte’s law, which was anglo-saxonized into Boyle’s law by Tait, asserts that at constant temperature the product of the pressure and volume remains constant pV D const.

(1.1)

2. Gay-Lussac’s law, which again was anglo-saxionized into Charles’s law by the same Tait, states that for all permanent gases, the empirical temperature t is proportional to the fractional increase in volume at constant pressure, tD

1 V V0 ; ˛ V0

(1.2)

where ˛ is a constant, known as the coefficient of thermal expansion, which was the same constant for all permanent gases. Combining (1.1) and (1.2) gives pV D p0 V0 ˛.1=˛ C t/:

(1.3)

Whereas the pressure and temperature can neither depend on the mass, nor the total extension of a gas, the volume can and does depend upon the mass. The unit of mass is known as the gram-molecule, or simply “mol.” A mole is just molecular weight in grams. If there are n moles of substance, the specific molal volume is v D V =n. If we call R D p0 V0 ˛, then (1.3) can be written as pV D nR.1=˛ C t/, and R is referred to as the gas constant. We will normally refer to one mole of substance. 3. Avogadro’s number, which is the third of the three empirical laws, says that the molal volume, v, is nearly the same for all permanent gases under “normal” conditions of p D 1 atm, t D 0ı C, and v0 D 22; 414 cm3 . Therefore, the gas constant is also universal. One mole of any substance contains the same number of molecules NA , known as Avogadro’s number. It is roughly 6 1023 , and Boltzmann’s constant, which was actually discovered by Planck in his work on blackbody radiation (Lavenda 1991), is the gas constant per molecule, D R=NA . The gas constant is roughly R D 82 atm cm3 deg1 mol1 , or 8:3 107 erg deg 1 mol1 , and, therefore, the gas constant per molecule D 1:37 1016 erg deg1 mol1 . If p D const. in the combination of Mariotte’s and Gay-Lussac laws, (1.3), then ˛ @ D ; @t 1 C ˛t

4


while if the density D const., then @p ˛p D : @t 1 C ˛t So for the heat capacity at constant pressure, Poisson found Cp D

@Q ˛ @Q @ D ; @ @t @ 1 C ˛t

while for the heat capacity at constant density (or volume) he derived Cv D

@Q ˛p @Q @p D : @p @t @p 1 C ˛t

Dividing one by the other, and assuming that the ratio Cp =Cv D D const., Poisson obtained the partial differential equation p

@Q @Q C D 0: @p @

Solving this, say by the method of Lagrange, where the auxiliary equations are dp d dQ D D ; p 0 led to two independent solutions Q D a and p 1= = D b, where a and b are arbitrary, but positive, constants. The general solution to the partial differential equation is therefore Q D f .p 1= V /, where the volume V replaces 1=. Poisson’s relation, p 1= V D const., provides the relation between pressure and volume when there is no loss of heat. The Poisson relation has withheld the test of time, even though it was based on caloric theory, where a body behaves as a thermal sponge insofar as when it is compressed it would give up thermal substance, while, when it is dilated, it would absorb heat. Having gotten this far Poisson reasoned that the heat capacity should vary as p 1= V =T , or p 1 , when the ideal equation of state is used. According to Delaroche and Bérard (Cardwell 1971) Everyone knows that when air is compressed heat is disengaged. This phenomenon has long been explained by the change supposed to take place in its specific heat; but the explanation was founded upon mere supposition, without any direct proof. The experiments which we have carried out seem to us sufficient to remove all doubts upon this subject.

A given volume of air at p D 1006 mm was found by them to have a heat capacity of 1:24 times that of the same quantity of gas at p D 740 mm. This was in flagrant contradiction of the fact that the heat capacities of an ideal gas do not vary with temperature. Both Carnot and Clapeyron were led down the wrong path by this,

1.1 The Dawn of the Science of Heat

5

and Mendoza (Mendoza 1960) muses that no other bad experiment has had such a catastrophic effect on the development of thermodynamics, for these workers assumed that the heat capacity was given by the logarithm of the ratio of the volumes when the gas underwent compression or expansion. It would have been apparent that no such variation was possible if Delaroche and Bérard had conducted their experiments at pressures higher than atmospheric pressure. This could have been the cause of Carnot’s lack of use of Laplace’s theory of adiabatic sound propagation in which the ratio of heat capacities, , is definitely supposed constant. Other than this, the caloric theory proved an admirable basis for performing the calculations on the segments of the cycle of a steam engine: An important point, emphasized by Carnot, is that it is necessary to close the cycle. When Carnot talks about caloric it is necessary to divide the heat capacities by the absolute temperature. No such modification is needed in Clapeyron’s work since he introduces the unknown function C.T /, which Kelvin was later to identify as the absolute temperature on the basis of a suggestion given to him by Joule. Fourier used the following model of caloric to explain the processes associated with heat conduction. A body was placed in thermal equilibrium, at constant temperature, with a radiation gas that comprised particles (much later identified as photons). When the particles of radiation were attached to the molecules of the body, the heat was latent. Only when the particles of radiation were transferred between the molecules did it become sensible heat, and the density of such radiation particles determined the temperature of the body. Laplace accepted this model whole heartedly, and used it to calculate the coefficients of absorption and emission of radiation by the molecules. Why were there no alternatives to the caloric theory? In particular, why could not heat be considered the random kinetic energy, or what was referred to as the vis viva, or living force, of the molecules within a body. First, it would take more than a century to show the existence of real molecules, as opposed to some figment of the imagination that was useful in visualizing processes, but could hardly be considered as being real. Second, even though Daniel Bernoulli had given the correct interpretation of the pressure of a gas in terms of the collisions of the molecules against the walls of the enclosure as far back as 1738, French physicists were not prepared to accept such a kinetic model. Mechanics held sway, and through it everything could be explained by action at a distance that reduced the interaction of molecules with the sides of their containers as central forces. Even if they could work out the energetics of such a process there was something that was missing, and that something was entropy, and its conservation in a reversible process. The caloric theory mixed the two since energy and entropy conservation was one and the same thing, i.e., the conservation of caloric. Although Carnot did not accept the equivalence of heat and work at the time of writing Réflexions, in papers that were found posthumously, Carnot went even so far as to calculate the J factor connecting heat and work (cf. Sect. 2.1). But, there was nothing to guarantee that J was actually a constant for all temperatures. In an early paper of Kelvin, in which he analyzes the Carnot–Clapeyron equation for the latent heat of steam, he actually found that J did vary with temperature! If the caloric

6


theory needed the propagation of particles of radiant heat to be that of the velocity of light, then any other theory treating heat as the vis viva of the motion of molecules in a body required an ether in which the vibrations could be propagated!

1.2 From Isothermal to Adiabatic Propagation of Sound A major application of the calorimetric equation was made by Laplace in modifying the expression for the velocity of sound that was first proposed by Newton (1687). Newton argued that a sound wave applies a pressure, p D F=A, normal to a surface of area A, where F is the corresponding force. Dividing p by the density D V =m, where V is the volume of the container and m is the mass of the particles in the container, gives a quantity that has units of the square of a velocity, i.e., the product of acceleration and length. If this is identified as the velocity of sound, then s p : cs D

(1.4)

Newton calculated his velocity of sound (1.4) under isothermal conditions. By 1783 it had become clear to the Paris Academicians that experiments carried out at 7.5ı C gave a velocity of sound of 337:2 m/s, while Newton’s formula (1.4) predicted a velocity of only 283:4 m/s, or about a sixth too small. In order to reconcile theory with experiment, Lagrange assumed – without any motivation – that the pressure increases faster than the density, yet still requiring that the process be isothermal. It was Laplace who relinquished the assumption that the process be isothermal. At points where the wave coils up (i.e., condensations) there is a rise in temperature, which causes a corresponding increase in temperature, while at points where the waves are stretched (i.e., rarefactions), the temperature decreases and together with it, the pressure. Laplace argued that all differences in the pressure are greater by the ratio of the heat capacities, Cp =Cv , than Newtonphad supposed, and, consequently, the velocity of sound was increased by the factor .Cp =Cv /. How did Laplace arrive at this conclusion? Laplace began with the calorimetric equation of state, choosing V and p as independent variables, @Q @Q dV C dp @V p @p V @T @T D Cp dV C Cv dp: @V p @p V

dQ D

(1.5)

According to the caloric theory of heat, with every geometric increase in volume there was an uptake of heat, and with every decrease in volume there was a corresponding emission of heat. Thus, a mass under constant pressure that is undergoing

1.2 From Isothermal to Adiabatic Propagation of Sound

7

expansion as a result of increasing its temperature consumes more heat than the same mass whose volume remains invariant. The former was called the heat capacity at constant pressure, and it was measured by Delaroche and Bérard, while the later was called the heat capacity at constant volume. This quantity was difficult to measure, but these experimenters found an indirect way of measuring the heat capacity at constant volume. The very rapid sound vibrations presented an ideal way of carrying out the air pump experiment of Clément and Desormes, which, although had an entirely different aim, measured the ratio of the two heat capacities. In such an idealized experiment, the transmission of sound was so rapid that there could not be an equalization of heat through conduction, which is a much slower process. Laplace, therefore, considered the process to be adiabatic and set (1.5) equal to zero, thereby obtaining

@p @V

D Q

Cp Cv

D

@T @V

p

@p @T

V

@.T; p/ @.p; V / D @.V; p/ @.T; V /

@p @V

;

(1.6)

T

which relates the adiabatic compressibility to the isothermal compressibility, where D Cp =Cv . Laplace could state his result for the velocity of sound as v" s u # u @p @p t cs D V : DV @V Q @V T

(1.7)

In general the relation between these adiabatic and isothermal compressibilities is given by @T @p @p @p D C : (1.8) @V Q @V T @T V @V Q When (1.6) is inserted in (1.8) there results Cp Cv D Cv

@V @p

T

@p @T

V

@T @V

;

(1.9)

Q

which must be identical to Cp Cv D T

@V @T

p

@p @T

:

(1.10)

V

The well-known expression for the difference in heat capacities, (1.10), is neither a consequence of the first or second laws of thermodynamics, but rather to their difference, or Carnot’s formula. We shall derive this in the next chapter, but, for the time being, we accept it on faith.

8


In order that (1.9) equals (1.10), it is necessary that Cv

@V @p

T

@T @V

DT

Q

@V @T

: p

Introducing the relation between the ratio of the heat capacities and isothermal compressibilities, (1.6), into the left hand side of the above equation leads to Cp

@V @p

Q

@T @V

D Cp

Q

@T @p

DT

Q

@V @T

: p

Then the definition of the heat capacity at constant pressure Cp D T .@S=@T /p , and Maxwell’s relation, .@T =@p/S D .@V =@S /p , produce an identity. But, this was far from being known at the time. Laplace published his results in 1816. Using the experimental value, D 3=2, found by Delaroche and Bérard, he got the value 345 m/s for the velocity of sound, instead of the Newtonian value 283 m/s. The difference with the measured value was attributed by Laplace to experimental error. But, his big point was that the difference between the value he found and that of Newton could be used as a means of determining the ratio of the heat capacities, which he proceeded in doing. Laplace found the value D 1:4252. Remarkably, Laplace’s result (1.7) is independent of the caloric theory, or any microscopic picture of air and caloric attracting and repelling each other, or whether the ratio of the heat capacities is constant or not.

1.3 Sources for Réflexions Prior to Carnot, the fact was that steam engines (machine aJ vapeur), or more generally heat engines (machines aJ feu), which was Carnot’s primary interest, were not known to undergo cycles. It was known from the time of Watt, the inventor of the steam engine, that the expansion took place isothermally so that Mariotte’s law (1.1) could be used. However, 5 years before Réflexions appeared, Clément and Desormes broke with tradition in supposing that an adiabatic expansion occurs after the cutoff in the isothermal expansion. This would cause a drop in the temperature of the working substance. Fox (1986) conjectures that possibly these authors were looking for an application of their law whereby “a given weight of steam always contains the same amount of heat whatever pressure is created” (Carnot 1824). If isothermal conditions persisted after cut-off, it would not be known how much extra fuel would be required to maintain a constant temperature, whereas if adiabatic conditions were to apply, no extra fuel would be needed at all. The mathematical distinction between isothermal and adiabatic processes was not at all clear in the early 1800s. For example, Carnot cites the “theorem” that “when a volume of gas changes at constant temperature, the amounts of heat

1.4 Carnot’s Thermodynamics

9

absorbed or released by the gas will follow an arithmetical progression when increases or decreases in volume follow a geometrical progression” (Carnot 1824). What Carnot is referring to is the heat at constant temperature, and from this springs his error in expressing the heat capacity as the logarithm of the volumes: “The change in the specific heat of a gas resulting from a change in the volume depends solely on the ratio between the initial and the final volume” (Carnot 1824). Therefore, Carnot’s specific heats read more like “specific entropies” (Kuhn 1955). Poisson’s adiabatic relation, p 1= V D const., was not derived by Poisson until 1823, and there is no knowledge that neither Clément and Desormes, nor Carnot, were cognizant of it. Rather, Clément and Desormes argued that if the pressure remained constant during adiabatic expansion, and a state of saturation remained, for, according to caloric theory, there would be no possibility of riding the caloric in an adiabatic process, then the state of the system could be determined from the ideal equation of state, assuming it was applicable to saturated steam, and the pressure–temperature relation could be read off of the saturated-pressure tables. Between the time of the Clément–Desormes paper in 1819 and the appearance of Réflexions, Clément added an additional step of a condensation immediately following adiabatic expansion (Fox 1986). This, according to Lervig (1972) seems to be the “fingerprint” of Carnot, and it does not appear from Clément’s lectures at the Conservatoire that he was fully appreciative of the role that the condensation phase has. In his lectures of 1823, he calculates that the motive power lost is much less that gained in the isothermal and adiabatic expansions, while in his lectures of 1826 he ignores the condensation step altogether, thereby returning to his original treatment of 1819. Oddly enough, in his notes found posthumously, Carnot queries anew “why, in order to develop motive power by heat, a cold body is required, why motion cannot be produced by consuming the heat in a heated body?” In fact, the little that remains of Carnot’s writings points to the growing scepticism which he regarded caloric theory, even going so far as the write According to certain ideas that I have conceived on the theory of heat, the production of one unit of motive power necessitates the destruction of 2:70 units of heat.

Amazingly enough, this is only 11% lower (Hoyer 1974) than the currently adopted value.

1.4 Carnot’s Thermodynamics In his posthumous manuscripts published only in 1878, we have mentioned that Carnot finds the mechanical equivalent of heat of 0:370 kilogram-meters. This is because Carnot uses kilogram-calorie as the unit of heat and 1;000 kilogrammeters of the unit of work. Hence, the destruction of 2:7 units of heat gives 1=2:7 D 0:37 kilogram-meters of work. That is, the number of kilogram-meters divided by the number of kilogram-calories gives the mechanical equivalent of heat.

10

1 The Predecessors of Carnot p

Fig. 1.1 Carnot’s original cycle using infinitesimal temperature differences

A t = 1°C

Δp

B D t = 0°C

C

V

(

V 1−

1 116

)

Δv

(

V 1+

)

1 267

This is remarkably close to Mayer’s determination of the mechanical equivalent of heat in 1842, which we shall discuss shortly. It is all the more remarkable because Carnot determined it on the basis of caloric theory. Carnot always worked with infinitesimal temperature differences, which allowed the distinction of first- and second-order terms. It also produced confusion since dividing by a temperature difference of 1ı C was like not dividing by a temperature difference at all. Work always turned out to be a second-order term. Consider the Carnot cycle in Fig. 1.1.2 The isothermal segments are at t D 1ı C and t D 0ı C. Now, using the principle of conservation of heat we can take any closed path and demand that heat be conserved. The heat absorbed in the isothermal expansion A ! B must be equal to the sum B ! D and D ! A. But since D ! A is an adiabatic compression it follows that Z

Z

B

B

dQ D A

dQ:

(1.11)

D

2 Early in the history of thermodynamics, it was appreciated that temperature should have bounds of “absolute” hot and cold. Absolute cold was reached by extrapolating the data on real gases. In Carnot’s time, absolute cold was 267ı C.


11

For the heat necessary to heat 1 kg of air from 0ı to 1ı C, Carnot takes the experimental value of Delaroche and Bérard Z Cp D Q D

B

dQ D 0:267 kcal:

(1.12)

D

Carnot then determines the work, W , in the complete cycle A ! B ! C ! D ! A and divides it by the heat, Q, to obtain W=Q D 1:395 kg-m/kcal. With the heat given by (1.12), Carnot readily obtains 2:697 kcal equal to 1;000 kg-m. The question that Hoyer (1976) tries to provide an answer to is as follows: How can a correct result be obtained from an incorrect theory? His thesis is that, to the order that Carnot was considering his theory using infinitesimal transformations, the conservation of heat is correct to first order. Processes involving work enter only at second order. If Carnot knew that the internal energy was a state function then for the triangle A ! B ! D he would have written Z

Z

B

dQ D A

I

B

dQ C

p dV;

D

H but since the work pdV D 12 pV is a second-order term, it can be neglected in a first approximation yielding the conservation of heat. Then, on account that Carnot is working with infinitesimals, and a unit change in temperature, t D 1ı C, there results Z B Z B dQ D Cp dT D Cp t D Cp ; A

D

for the expansion under constant pressure. This gives back the incorrect result, (1.12), of Delaroche-Bérard. In his notes, Carnot attempts to give an analytical expression for his theory. He writes the work in the form W D F 0 .t/Qt: Carnot knows that the mechanical work, W , is proportional to the product of the difference in two temperatures of the heat reservoirs and the heat absorbed during the isothermal expansion A ! B. But, what he does not know is F 0 , which has become known as Carnot’s function. However, Carnot does know that it can only be a function of the temperature of the upper isotherm. Had Carnot known the first principle (conservation of internal energy) Z

Z

B

C

dQ D A

dQ C pV; D

(1.13)

12


and the second law (conservation of entropy) 1 t

Z

B A

1 dQ D t t

Z

C

dQ;

(1.14)

D

that is, the heat absorbed or rejected at the temperature at which this happens, which for a reversible processes must coincide with the temperatures of the heat reservoirs, he would have expanded the lower temperature in powers of t in (1.14) to obtain Z

B

A

Z C t C dQ D 1 C dQ: t D

Retaining only the first order, and comparing it with (1.13), would have resulted in W D pV D

1 tQ; t

the determination of his function, F 0 .t/ D 1=t. But, we are getting ahead of our story. Let us return to the origins of the first and second laws and see how they were deduced from Carnot’s cycle. To see how closely this comes to the modern day calculation of the mechanical equivalent of heat, we can, like Carnot, replace the adiabatic steps by isochoric ones since we are considering only small temperature differences. Then, the volume change, V , can be expressed as a Taylor series expansion in the difference in temperature, t, and since the latter is infinitesimal we can stop at first order to obtain V D A Vt; where A is the sum of the isobaric coefficient of expansion at 0ı C (Hoyer 1976), 1 ˇD V

@V @t

D 0:009; p

and the adiabatic coefficient of expansion at the same temperature (Hoyer 1976) 1 ˛D V

@V @t

D 0:00366: Q

The pressure under standard conditions is p D 10;335 kg/m2 (in comparison to Carnot’s value of 10;400), and the volume occupied by 1 kg of air under the same conditions V D 0:7734 m3 (in place of Carnot’s value of 0:77) gives the modern day value 1 kcal D 426:6 kg-m;


13

of the mechanical equivalent of heat when the work, pAVt, is equated to the heat absorbed in the isothermal expansion, A ! B. Since the heat absorbed in A ! B is equal to the heat absorbed in D ! C , Cp D

1 t

Z

B

dQ D 0:24 kcal=ı C;

D

the mechanical equivalent of heat is obtained by inserting the heat capacity times the temperature difference into the left-hand side of Z

B

dQ D pV .˛ C ˇ/t;

J A

and dividing through by t. In contrast, Mayer’s calculation of the mechanical equivalent of heat in terms of mechanical work is more direct and leads to a general expression for the proportionality factor, J , relating heat to mechanical work. Yet, Tait would certainly not have agreed. As the story goes (Cardwell 1971, p. 283), John Tyndall was addressing the Royal Institution in 1862 on the topic of the conservation of energy. Toward the end of his lecture he mentioned that the discovery of the mechanical equivalent of heat had been worked out by a “neglected little doctor, who was, even then, cultivating his vineyards in Heilbronn.” Joule felt slighted and brought his grievances up with Tyndall. Although Tyndall acquiesced to Joule’s demand for priority over the doctrine of energy conversion, the matter could have been laid to rest were it not for the intervention of our old friend Tait. Together with Kelvin, Tait wrote an article for a contemporary journal in which Mayer’s claim to the discovery of the mechanical equivalent of heat was flatly rejected, and Tyndall’s scientific capabilities were called into question. Mayor considers a gas expanding isothermally, in which the work, pV , is equal to the heat content. In Mayer’s own words: “the depression of a mercury column which compresses a gas is equal to the quantity of heat liberated by compression” (Hutchinson 1976). At constant pressure, the quantity of heat would be Cp T , but, Mayer considers .Cp Cv /T as the relevant quantity of heat. Truesdell simply glosses over this point, and a relevant discussion of why the difference in heat capacities should be used is difficult to come by in the literature. According to the old literature (Preston 1894), a perfect gas is one in which its molecules are outside the “sphere of attraction” of each other. Thus, when it expands the work will be entirely external since there will be no work consumed in separating its molecules. If there is a change in the volume of gas from V1 to V2 at constant pressure this external work will be p.V2 V1 /. Whereas the specific heat at constant volume, cv , is that quantity of heat required to raise the temperature of a unit of mass by 1ı C at constant volume, the specific heat at constant pressure, cp , is the quantity of heat required to raise the temperature of a unit mass by 1ı C at constant pressure. For cv measurements, the pressure increases at constant volume with no external work being done, while for cp the volume increases under constant pressure.

14


The amount of external work is measured by the product of the pressure and the volume change, and this must be equal to the difference in the specific heats, cp cv D adiabatic elasticity isothermal elasticity > 0; because cp exceeds cv “merely by the thermal equivalent of work done under constant pressure, while the temperature changes by 1ı C.” Thus, JQ D W; (1.15) which, in the present case, is J cp cv D p .V2 V1 / D R .T2 T1 / :

(1.16)

That is, when heat is converted into work, or vice-versa, J units of work are equivalent to one unit of heat. And because there is a unit change in temperature, independent of what scale is being used, J cp cv D R: This equation determines one specific heat when the other is known; if both are known it determines the mechanical equivalent of heat, J . And this is precisely what Mayer did: “If we put the ratio of the capacities of atmospheric air under the same pressure and volume equal to 1:421, : : : the descent of a weight from a height of about 365 m corresponds to the heating of an equal weight of water from 0ı to 1ı C.” It is astonishing that as late as the last decade of the nineteenth century, there was still confusion between heat capacities and heat. If the temperature difference had been included on the left-hand side of (1.16) there would not have been the necessity of considering a unit temperature difference that leaves the right-hand side equal to R. Mayer’s analysis is based on the fact that the amount of heat absorbed by a gas, as it expands at constant temperature, is exactly equal to the thermal equivalent of the mechanical work done by the gas as it expands. Reversing this process we conclude that the heat developed by compressing the gas at uniform temperature is the thermal equivalent of the work done to compress it. As Clerk-Maxwell remarked “This is by no means a self-evident proposition.” He cites a number of cases in which this is not true, and, therefore, concludes that “the dynamical equivalent of heat, which Mayer found on this proposition, at a time when its truth had not been experimentally proved, cannot be regarded as legitimate.” However, the same ideal gas was used by Clausius to determine the Carnot function in the Carnot–Claypeyron equation so that this criticism of Clerk-Maxwell’s can hardly be lodged against poor Mayer. Rather, Clerk-Maxwell attributes the correct determination of the mechanical equivalent of heat to Joule. For if p be the specific heat at constant pressure, which is the quantity of heat required to raise a unit of mass of a substance by one degree, expressed in “dynamical measure or foot-pounds,” then the dynamical equivalent of


15

heat is J D p =cp . According to Joule, it is equal to 772 foot-pounds at Manchester per pound of water. That is, the work done when a quantity of water falls 772 ft at Manchester raises that quantity of water by 1ı F. On the strength of the mechanical equivalent of heat, Joule claims that the heat required to raise a pound of water from 39ı to 40ı F is equivalent mechanically to 772 “foot-pounds” of work, i.e., “so many pounds raised one foot” (Tait 1868, p. 9). Clerk-Maxwell was quick to emphasize that nothing has been said, nor need be said, concerning the temperature of the body that has done the heating. It could be equally as well have been taken from “cold water at 500 F, (as) from a red-hot iron heater at 7000 F, or from the sun at a temperature far above any experimental determination, and yet the heating effect of the heat is the same whatever be the source from which it flows.” Using this argument, Clerk-Maxwell faults Carnot who believed that heat when extracted in a hotter body could produce more mechanical energy than when it is in a colder body. The mechanical equivalent is the same in both cases, “although when in the hot body it is more available for the purpose of driving an engine.” It is precisely the higher temperature that makes the hot body more attractive for purposes of driving engines because the heat supplied to the heat rejected is directly proportional to the ratios of their temperatures. The heat that is supplied or rejected must be at the same temperature as the reservoir in order that the cycle be reversible so that a greater difference in temperature results in more work being done.

Chapter 2

Thermodynamics from Carnot to Clausius and Kelvin

2.1 Réflexions: An Ignored Masterpiece In 1824 Sadi Carnot (1824) published a little read volume, entitled Réflexions sur la Puissance Motrice du Feu, in which he formulated an ideal cycle that would describe the operation of a steam engine. Carnot’s first claim is that the operation of the engine does not depend upon the nature of the working substance. To him it was evident that the reasoning would have been the same for all other gaseous substances, and even for all other bodies susceptible of changes in temperature through successive contractions and dilatations, which comprehends all natural substances, or at least all those which are adapted to realize the motive power of heat.

The gas is enclosed in a cylinder with a movable piston, and there are two heat reservoirs, A, and B, at temperatures tA > tB . The original plate appearing in Réflexions is shown in Fig. 2.1. In reference to the figure, Carnot enumerates the four steps in his cycle. Body A is the hot reservoir, and the initial volume of the body of air is abcd. When the system comes in contact with A the piston gradually rises to ef . The contact is such that the temperature remains constant throughout. A is then removed, and the piston continues to rise from ef to gh. The expansion produces a fall in temperature until it becomes equal to the body B. Compression then occurs until the piston arrives at cd . But, because the system is in contact with B, the temperature remains constant. B is then removed and compression brings the piston to i k with a rise in temperature until it is equal to body A. The pressure–volume graphical representation, which is still used today, can be found in Clapeyron’s paper that appeared in 1834, 2 years after Carnot’s death. The question that immediately arises is as follows: How did Carnot know how to close his cycle? We can confront Carnot’s piston cycle with the closed curve in the p; V -plane, whose area represents the useful work performed by the engine. Carnot begins by placing the gas in contact with the heat reservoir at t1 . The piston is now at position a in left side of Fig. 2.2, corresponding to state A0 on the right side of the figure.


17

18

2 Thermodynamics from Carnot to Clausius and Kelvin

g

b

e

f

c

d

i

k

a

A

e

f

e¢

f¢

c

d

c¢

d¢

a

b

a¢

b¢

b

B

e

c a

c

d

e

f

f

d b

a

b

Fig. 2.1 The original plate appearing in Réflexions that Carnot used to describe his cycle

The piston is raised and the volume is allowed to expand isothermally until it reaches position b, corresponding to state B in the p; V -plane. For a given ratio of temperatures, the only parameter that enters is the volume ratio, VB =VA0 . The gas is now allowed to expand adiabatically until the temperature falls to t2 . The piston is now at c and the state C in the corresponding diagram. The gas is then compressed isothermally at temperature t2 until it has returned to its initial position a, corresponding to state D. The volume VD is chosen so that it is equal to VA , but obviously the state has a lower pressure and temperature. The gas is then compressed adiabatically so that its temperature will rise. When it has reached the temperature t1 it will be in state A, which corresponds to e in the cylinder. An isothermal expansion takes the system from A back to A0 , which brings the piston back to a.

2.1 Réflexions: An Ignored Masterpiece

c P

19

A A’

b a

e

B D C V

Fig. 2.2 (Left) Position of the piston corresponding to the branches of the Carnot cycle. (Right) Closing of the Carnot cycle

No one except Klein [1976] has ever queried Carnot’s procedure for describing his cycle. Although Carnot still adhered to the caloric theory in Réflexions, the conservation of heat in no way entered his description of his cycle. The same is not true of Clapeyron who began the cycle in state A. The quantity of heat that enters in the isothermal expansion to state B is Q, as shown in Fig. 2.2. Clapeyron then follows Carnot’s procedure, except that the state D is now defined as that state in which the same amount of heat Q is given back to the cold reservoir as was absorbed by the hot reservoir. The final adiabatic compression takes the system back to the initial state A. It may seem odd that William Thomson, later to become Lord Kelvin, after reading Clapeyron’s paper in 1845 and finally managing to obtain a copy of Réflexions some 3 years later, should choose Clapeyron’s version of Carnot’s cycle over that of Carnot’s, insofar as he was well aware of the pitfalls of caloric theory. The point was addressed by his brother James in conjunction with a new effect in which an increase of pressure causes a decrease in the freezing point of water. Surprisingly, instead of destroying the caloric theory, the effect only reinforced the belief in it as heat consisting of some imponderable fluid. This, as Klein tells us, had an effect on how Carnot’s cycle was to be closed. According to James Thomson the state D should be chosen by the (unique) intersection of the isotherm at t2 and the adiabat that will take the system back to A when the gas has reached the temperature t1 . Thomson’s [1849] prescription was precisely this: Continue the motion [of the piston] till all the heat has been given out to the second lake at [t2 ], which was taken in during [the isothermal expansion] from the first lake at [t1 ].

This would have been little console to Carnot since he did not know the relation between adiabatic and isothermal curves. Notwithstanding its drawbacks, Carnot’s prescription for determining the state at which adiabatic compression should begin

20


was quite clear and easy to visualize: The state D should be such as to have an equal volume with the initial state A0 . An alternative procedure for determining this state was proffered by Clerk-Maxwell [1891] in his little black handbook of Theory of Heat. He simply started his cycle at B and the criterion that VC =VB should be equal to VD =VA , which, as we have mentioned, is the sole parameter in the theory. Carnot draws the analogy of motive power with the fall of water, which depends on height and quantity. So, too, the motive power of heat depends on the quantity of “caloric” employed, with the difference in temperature of the two reservoirs acting the role of the height of the fall. Although the evolution of Carnot’s thinking will lead to a refutation of caloric theory in the end, he was still a believer at the time he wrote his memoir: “The production of motive power is due: : : not to any real consumption of caloric, but to its transport from a warm body to a cold body.: : :” However, the notes that were found posthumously, written sometime after the publication of Réflexions, carried a very different tone, “Heat is nothing but motive power: : : whenever there is a destruction of heat, motive power is produced.” Carnot is also aware that his cycle had maximum efficiency. Suppose, says Carnot, that there is a more efficient engine that could render more work, Wmax , than the work W obtained from his engine with the same heat absorption, Q1 , between temperatures t1 and t2 . Since this engine renders a greater amount of work it could be used to run Carnot’s engine in a reverse cycle with a positive amount of work .Wmax W / left over. Repetition of the cycle would “not only [create] perpetual motion but of motive power being created in unlimited quantities without the consumption of caloric or any other agent.” Carnot, thus, establishes the supremacy of his engine, with maximal efficiency, for anything else would lead to perpetual motion. He even alludes to the possibility of other cycles in which during part of the time a working body is absorbing heat from the furnace, its temperature t1w is less than the temperature of the furnace t1 . Then heat will flow spontaneously “across the wall that ‘transmits caloric easily’ and so will become heat at the temperature [t1w ]” (Truesdell 1980). A “fall” in caloric will result in smaller amount of work being done because heat is being absorbed at a lower temperature. This results in a “true” loss of motive power. The same thing would happen at the refrigerator if the working substance would be at a temperature t2w higher than the refrigerator temperature t2 . According to Carnot, the heat would not have had the capability of performing the same amount of work had it remained in the body until it reached the temperature, t2 < t2w of the refrigerator. Although Carnot deduces this from caloric theory it stands even when the caloric theory falls. The caloric theory was at its best in the description of storage and transfer of heat as an inanimate object but failed to describe processes of frictional heating, as in the cannon-boring experiments of Rumford. The caloric theory predicted a definite “fall” of heat in order to do work. Kelvin, who followed in Carnot’s footsteps, tried to put the caloric theory to test by creating a hypothetical engine that would do work even though there was no “fall” in heat. He imagined that the working substance was water at 0ı C; water when

2.1 Réflexions: An Ignored Masterpiece

21

it freezes expands, and by alternating between freezing and melting, work could be done. As we have mentioned earlier, this was designed as a test to discredit caloric theory, yet, it came to its rescue for Thomson’s brother, James [1849], soon found a totally unexpected effect: An increase in pressure causes a lowering in the melting point of ice. So there was a small, but definite “fall” in heat that was responsible for the work (7:5 103 ı C for an increase in pressure of 1 atm). Carnot bases his analysis on two principles (Lervig 1972b): Heat is conserved. Work is less in non-Carnot cycles.

The caloric theory was definitely at odds with the possibility of the interconversion of heat and work. Early in 1850, Clausius [1850] sees a way out whereby Carnot’s analysis could be salvaged by retracting the caloric theory upon which it was supposedly founded. He withdraws the foundation upon which Carnot’s theory was based – without destroying the structure – and replaces it by the proposition that the work done during the cycle was equal to the difference between the heat absorbed at the furnace and the heat rejected to the refrigerator. That is, Carnot’s assertion about the fact that finite differences in temperature are needed to make engines operate remains true even if the caloric theory are abandoned. It was just as Joule [1850] claimed: Heat had to be consumed if work was to be done. But, in order to do work, heat had to pass from a hotter to a colder body, as Carnot asserted. From these two statements emerged the first and second laws of thermodynamics in which Clausius introduced two new functions, the internal energy and entropy. Clausius reasons that a gas which expands into empty space cannot be brought back to its original form without a decrease in entropy. Thus, Clausius [1868] concludes as follows: The energy of the universe is constant. The entropy of the universe tends to a maximum.

However, these grandiose statements seem hardly to apply to engines operating reversibly in cycles. The internal energy was needed in order to allow a partial conversion of heat into mechanical energy and vice-versa. Clausius calls transformations that proceed by themselves, or spontaneous, as “positive” transformations and those that required another compensating transformation as “negative.” Total conversion of heat into work would result in no change in the internal energy. If an irregularly heated body was adiabatically isolated from the rest of the world, all parts of the body would eventually arrive at a common final temperature. Because the heat is related to the entropy by the inverse temperature, the common temperature would necessarily be greater if the internal energy was maintained constant rather than the entropy. The inequality is a consequence of the fact that means are monotonically increasing functions of their order, and the final, uniform temperature must result in a mean of the initial temperatures. Consequently, if we require that no heat be given out to an auxiliary body at the common temperature reached, the process will result in maximum work being done by the system. Gibbs refers to such processes as isentropic;

22


entropy being conserved rather than heat, as Carnot was forced into accepting from the caloric theory. However, the distinction between adiabatic and isentropic processes is not always clear. In contrast, Kelvin couched his evolution principle, not in terms of the increase of entropy, but, rather, in the perpetual running down of the available energy of the universe. To some (Heaviside 1892, p. 488), “it is incredible that it can always have been going on, and dismal in its final result if uninterrupted. It is therefore the duty of every thermodynamician to look out for a way of escape.”

2.2 Doctrine of Latent and Specific Heats If a body undergoes a change in volume, dV , and a change in temperature, dT, then the heat that must be added for this to occur is dQ D Lv dV C mcv dT;

(2.1)

where Lv is the latent heat of expansion at constant temperature, and cv is the specific heat at constant volume, of body of mass m. Apart from the absence of the mechanical equivalent of heat, this calorimetric equation was familiar to early practitioners of thermodynamics. In a sense, (2.1) takes us back to caloric theory (Ivory 1827): the absolute heat which causes a given rise of temperature, or a given dilatation, is resolvable into two distinct parts; of which one is capable of producing the given rise in temperature, when the volume of the air remains constant; and the other enters into the air, and somehow unites with it while it is expanding: : : The first may be called the heat of temperature; and the second might very properly be named the heat of expansion; but I shall use the well known term, latent heat, understanding by it the heat that accumulates in a mass of air when its volume increases, and is again extricated from it when the volume decreases.

All this is in harmony with the corporal nature of heat in the caloric theory. The word “latent” was used in contrast to “sensible” heat because of the lack of increase in temperature when a gas absorbs heat upon expansion. Nowadays, latent heat is used in a much more restricted sense where it refers specifically to the latent heat of “fusion” or of “evaporation” (Cravalho 1981). If (2.1) were a perfect differential, it would require the specific heats at constant pressure and volume to depend upon the temperature [cf. (2.44) below]. This was extremely troublesome to Carnot because it contradicts his conclusion that “: : : both specific heats increase as the density of gas diminishes, but their difference does not change.” This could hardly be the case when it was known from Laplace’s formula for the speed of sound in air that their ratio was constant. Truesdell muses whether this is why Carnot failed to consider the case where the ratio of specific heats is constant (Truesdell 1980). If the heat capacities were temperature dependent, and if Carnot were to accept that their ratio were constant, then their difference could not also be constant.

2.2 Doctrine of Latent and Specific Heats

23

In this sense, Clausius’s transfer of the focus of attention from the heat function to the internal energy as a state function was welcome, since it was only the difference between the heat absorbed, dQ, and the amount of external work done, pdV , that would be required to be a perfect differential, dE D .Lv p/ dV C Cv dT;

(2.2)

where E is the internal energy, and the heat capacity at constant volume Cv D mcv . The integrability condition on (2.2), @Lv @Cv @p D ; @T @T @V

(2.3)

avoids the inevitable conclusion that the heat capacities be functions of the temperature, which for an ideal gas are certainly not. The second law is, in essence, a derivation of the Carnot–Clapeyron equation. Clapeyron, even more than Carnot, was a prisoner of the caloric theory, and his derivation of the equation named after him was based on the fact that the heat was a state function, which he assumes to be a function of p and V . Clapeyron [1834] writes as follows: @Q @Q dQ D dV C dp: (2.4) @V p @p V But, because the temperature is being held constant, Mariotte’s law requires (1.1), and so (2.4) becomes " # p @Q dQ dQ D dV: (2.5) @V p V @p V Then the ratio of the amount of “motive power” (work) .dp/V dV D R d T dV =V , where R is the gas constant, to the heat absorbed by an isothermal expansion, dT R dT D (2.6) @Q @Q C V @V p @p p

V

must, according to Carnot, be a product of d T and some universal function of temperature, C . In this way, Clapeyron arrives at the differential equation @Q @Q V p D RC; (2.7) @V p @p V where the right-hand side is some function of pV , again, because of Mariotte’s law. In other words, this ratio must be independent of the nature of the working substance and can only depend upon the temperature. Clapeyron finds the solution Q D R .B C ln p/ ; where B and C could, at most, depend on the temperature.

(2.8)

24


The method by which such a solution can be found is the method of characteristics. This method identifies special paths in the p; V -plane, along which the partial differential equation (2.7) reduces to a set of ordinary differential equations called auxiliary equations. The auxiliary equations corresponding to the partial differential equation (2.7) are dp dQ dV D D : (2.9) V p RC Two independent solutions are Q=C ln V D a and pV D b, where a and b are arbitrary constants, so the solution of (2.7) is ˆ.a; b/ D ˆ.Q=RC ln V; pV / D 0; or Q D RC ln V .pV /;

(2.10)

where ˆ and are arbitrary functions. It is clear from Mariotte’s law that the characteristics coincide with the isotherms in the p; V-plane. Inserting (2.10) back into the differential equation (2.7) results in 1 C

@Q @V

D p

p dp R D D ; V T dT

(2.11)

which follows from the ideal gas equation. Clapeyron was fully appreciative of the fundamental importance of the quantity C , referred to as the Carnot function. Notice that the total derivative appears on the right-hand side of (2.11). It generalizes Carnot’s treatment to give an expression for the variation of the steam pressure with temperature when it is in equilibrium with water. Why the ideal gas equation of state could be applied to such a phase equilibrium is that the volume of steam is far greater than the volume occupied by water so that to a first approximation steam could be considered as an ideal gas. Clausius [1850] was later to modify the Carnot–Clapeyron equation, (2.11), to read . @Q D d T =C: (2.12) R dT V @V T The left-hand side of (2.12) is the ratio of the maximum work that an ideal gas can do when it absorbs a quantity of heat by isothermal expansion. The specialization to an ideal gas meant that “during the expansion of a gas only so much heat becomes latent as is used in doing external work.” Clausius then retains Carnot’s principal assumption that there is always “a tendency to equalize temperature differences and therefore pass from hotter to colder bodies,” but rejects Carnot’s premise that “the production of motive power in a steam engine is due not to an actual consumption of caloric but to its passage from a hot body to a cold one.” Clausius uses this to equate the left-hand side of (2.12) with the infinitesimal temperature difference dT

2.3 Interconvertibility of Heat and Work

25

between the two bodies, and another function, 1=C , that “is a function of T only.” It is precisely the right-hand side of (2.12) which Clausius needed to arrive at Carnot’s principal assumption. The strong dichotomy in Clausius’s [1850] argument is apparent. On the lefthand side of (2.12) he talks of the latent heat in doing external work, i.e., an isothermal expansion. While, on the right-hand side he considers the maximum work that can be produced “by the transfer of a unit of heat from body A at the temperature t to body B at the temperature ,” i.e., the transfer of heat that occurs when a hot body is placed in contact with a cold body, at constant volume. This mode of heat transfer has nothing to do with the isothermal expansion “from A to the gas, and by the first compression: : : was given up to the body B.” There is no temperature difference between A and B, at least to first order, which is what Clausius considers. Apparently, Clausius has in mind the isothermal expansion and compression steps in Carnot’s cycle, but nowhere does he introduce the fact that they are occurring at different temperatures. Clausius now uses the interconversion of heat and work. In an isothermal expansion, the heat absorbed is equal to the work produced, dQ D pdV . From (2.12) he obtains an expression for the latent heat, which he then equates with the pressure to get RC @Q D p; (2.13) D @V T V which is what Truesdell [1980] attributes to Holtzmann [1848]. Finally, by appealing to Mariotte’s law, Clausius derives C D T .

2.3 Interconvertibility of Heat and Work After Clausius [1850] realized that the internal energy was required to render the heat function a total differential that it supplanted as the true state function, it took him another 4 years to realize that C 1 was the integrating factor for the quantity of heat absorbed, (2.1) (Clausius 1868). Even later, in Tait’s 1868 exposition in Sketch of Thermodynamics, the Carnot–Clapeyron equation, which is the result of the absolute temperature being the integrating denominator for the heat absorbed at the temperature at which it is absorbed, is derived instead by a geometrical construction. Multiplying both sides of (2.1) by C 1 the condition of integrability is @p d ln C Lv D ; dT @T

(2.14)

where the integrability condition for the internal energy, (2.3), has been employed. Truesdell [1980] implicates F 0 , equal to our d ln C =dT , to obtain the proportionality

26


between the work done and the heat absorbed. He writes the Carnot–Clapeyron equation as @p F 0 Lv D ; @T in our notation, and introduces it into the net work done over a cycle, ZZ W D A

F 0 Lv dV dT:

Assuming the existence of Carnot’s heat function, he adds zero to the integrand ZZ @Lv @Cv dV dT F 0 Lv C F @T @V A ZZ @ @ D FLv F Cv dV dT: @V A @T

W D

and uses Green’s theorem, I Z Z @M @N .M dx C N dy/ D dxdy; @x @y C A Z

to arrive at

Z

W D C

F .Lv dV C Cv dT / D

F dQ:

(2.15)

C

He then claims that (2.15) is to be rejected for any value other than F D constant. If this is true, then the Carnot–Clapeyron equation would vanish identically. It would also mean that the net work done over a cycle is zero since dQ is a function of state by assumption. Truesdell comes to the unwanted conclusion that if F were not a constant, it would be an “anti-integrating factor” for the quantity of heat absorbed, for otherwise W would vanish because dQ is a perfect differential. Truesdell (1980, p.115) then goes on to claim that if F is a function of T , “we see at a glance that this formula [for W] allows us to recover the equation in Carnot’s special axiom” W D ŒF .T1 / F .T2 / Q F .T1 / > F .T2 /

(2.16) if

T1 > T2 :

In fact, there is no need to distinguish between the Q at the two reservoirs since heat is conserved. How (2.16) can be extracted from a line integral over a closed circuit amounts to an act of faith. As Truesdell admits in a footnote on p. 102 “It is difficult to locate in his treatise any explicit statement of [(2.16)].” He then asserts, by negation, that if (2.16) were not true it would prove Carnot’s “main theorem,” Z VB R VB dV R D ln QD ; (2.17) d ln C =dT VA V d ln C =dT VA false.


27

Yet, what Truesdell refers to as Carnot’s [1824, p. 83] “main theorem” When a gas changes in volume without change of temperature, the quantities of heat absorbed or emitted by that gas are in arithmetic progression if the increments or decrements of volume are found to be in geometric progression

can actually be taken as an inadvertent admission on Carnot’s behalf of the interconvertibility of heat and work for an ideal gas under isothermal conditions. In differential form it says dQ D p dV , which is Clausius’s equation (17) in (Clausius 1850). Since p D RT =V , integration leads to setting Carnot’s C D RT . However, Carnot was not always clear between the distinction of heat and specific heat. Three pages later, Carnot (1824, p. 86) claims When the volume of a gas increases in a geometrical progression, its specific heat rises arithmetically.

Now, turning to Kelvin (1911), if we consider that temperatures of the furnace and refrigerator differ by an infinitesimal amount dT so that F .T C dT / D F .T / C F 0 .T /dT , then a small amount of work would be given by dW D F 0 .T /Q dT:

(2.18)

Kelvin then uses Joule’s prescription, dW D dQ;

(2.19)

apart from a multiplicative constant representing the mechanical equivalent of heat, to convert (2.18) into the differential equation dQ D QdT;

(2.20)

where he replaces F 0 by the symbol . But, he also replaces the significance of Q, which is no longer the heat absorbed from the furnace (Cropper 1986). Whereas, in (2.18), a small temperature difference produces a small quantity of work, dW , in (2.19), a small amount of heat, dQ, produces a small amount of work, dW . Surely, Kelvin was begging the question. Considering (2.20) as a differential equation and integrating it he got Q1 D ln Q2

Z

T1

.t/ dt:

(2.21)

T2

From (2.21) his reversibility condition, Q1 Q2 D ; T1 T2 [cf. (2.30) below] followed immediately by setting .t/ D 1=t. The Carnot efficiency, T2

C D 1 ; T1

28


[cf. (2.32) below] also resulted by setting the work equal to the heat consumed, W D Q1 Q2 . To borrow an expression of Lorentz, Kelvin’s mélange of finite and infinitesimal quantities might be called the “insensibilité de la thermodynamique.” However, Kelvin was reluctant to use what Helmholtz, Joule, and Clausius found for Carnot’s function, viz. the inverse temperature measured on the ideal gas absolute scale, because he could not find agreement with Carnot’s function calculated in its original formulation (Cropper 1987). Kelvin was later to claim that Carnot’s function, or for that matter, any continuous function of Carnot’s function, could be used for defining the absolute temperature scale. Although Kelvin claimed complete generality, his conclusions hold only when 1= is a linear function of the absolute temperature. Identifying C with the absolute temperature, as Kelvin later did, would have shown that it was the integrating denominator for the quantity of heat absorbed, leading to a new state function called entropy. In this way, modern thermodynamics would have had one founder, and not three, for Carnot was in need only of the last step. The function C in the Carnot–Clapeyron equation (2.14) is named after Carnot, and it was known to be a function of the temperature only. Carnot had proposed C as a measure of the work done per degree of temperature difference by a unit of heat at a given temperature. Clapeyron [1834] measured the Carnot function at five different temperatures, and Thomson [1852] extended the range of measurements by using Regnault’s data. Thomson was able to show that C diminished continuously with temperature. When Clausius [1854] brought C into his dynamical theory of heat, as we have just seen, a feud erupted over priority. Clausius [1856], complaining to the Editors of the Philosophical Magazine, had this to say about Thomson’s priority claim. His premise was as follows: : : : Prof. W. Thomson ascribed to Mr J. P. Joule the discovery of the theorem, that Carnot’s function, which Clapeyron expressed by C , and Thomson by the fraction 1= , “is nothing more than the absolute temperature multiplied by the equivalent of heat for the unit of work.”

Evidence to support his claim consisted of the following: Holtzmann established the same formula for the function C in a paper which appeared as early as 1845, and Helmholtz in his pamphlet in 1847, “On the Conservation of Force,” citing Holtzmann’s paper, calculated several values obtained by this formula, and compared them with those arrived at by Clapeyron in a different manner. But the views upon which Holtzmann founded his speculations do not agree with the mechanical theory of heat as at present received; so that after this had been recognized, the correctness of the formula found by him was, naturally rendered doubtful.On this account, in a paper communicated to the Berlin Academy in February 1850, “On the Moving Force of Heat,” in which I brought Carnot’s proposition in agreement with the mechanical theory of heat, I again endeavored to determine his function more accurately. Therein I arrived at the same formula as Holtzmann, and I believe that I then, for the first time, correctly explained the principles upon which this formula is based.

Clausius takes grave exception to Thomson’s statement: It was suggested to me by Mr Joule, in a letter dated December 9, 1848, that the true value of might be inversely as the temperature from zero.


29

and presents his conclusions: Against this I must beg to urge,—First, that, as far as I am aware, it is usual, in determining questions of priority in scientific matters, only to admit such statements as have been published. : : : Secondly, that since Thomson does not say that Mr Joule had proved the theorem, but only that he had offered it as an opinion, I do not see why this opinion should have the priority over that which Holtzmann had arrived at three years before.

So Clausius says that if you do not wish to offer me the recognition of being the discoverer of Carnot’s theorem, then at least give it to Holtzmann. But, Carnot’s theorem can only be understood within the mechanical theory of heat that Clausius has derived so, again, he is claiming priority.

2.3.1 Temperature Absolutely As we have mentioned in Sect. 1.1, the so-called permanent gases follow Mariotte’s law, (1.1), and Gay-Lussac’s law, (1.2), rather well. The zero of the temperature scale is chosen such that it corresponds to a volume V0 . Holding the pressure constant, the empirical temperature t is proportional to the increase in volume tD

1 V V0 ; ˛ V0

(2.22)

which is another way of writing Gay-Lussac’s law (1.2). A more rational way of choosing the zero point of the temperature scale is to shift the zero point of the empirical temperature scale by 1=˛ units, and write (Epstein 1937) T D t C 1=˛: (2.23) This is precisely what Clausius [1850] did. The zero of the empirical temperature is now T0 D 1=˛, and this makes “absolute” temperature proportional to the total volume of gas, viz. V =V0 D T =T0 . The equivalence between the measurements of absolute temperature and total volume implies that the volume, or some power of it, will have an analogous role to the absolute temperature in rendering the heat a perfect differential. Thomson and Joule [1853] proposed to determine C by the Joule–Thomson effect, and then define T by equating it to C , “a quantity which has an absolute value, the same for all substances for any given temperature, but which may vary with temperature in a manner that can only be determined by experiment”(Thomson 1851). Thomson relied too heavily on experiment, while Clausius not enough. In 1845 Joule carried out an experiment to test whether or not the internal energy was independent of the volume. If a gas gains or loses heat equivalent to the work done to compress it, or to the work it does when it expands, then there will be no internal work in the gas to change its volume. In other words, the specific heat will

30


be independent of the volume. According to Kelvin’s conjecture the internal work the gas does causes variations in its volume so that it is not identically zero, but, rather, should be very small. To test this conjecture, he modified Joule’s original experiment whereby two closed copper containers are coupled by a copper pipe containing a stop cock. With the stop cock closed one of the containers was filled with a gas under high pressure while the other container was evacuated. Both containers were placed in the same water calorimeter. Once the stop cock was opened, gas flowed into the empty container without doing work, but no change in temperature could be found. From this it was concluded that the dependence of the internal energy of the gas upon its volume is negligible; however, the accuracy of the experiment was reduced due to the enormous heat capacity of the calorimeter. In order to increase the accuracy of the experiment, it would be advantageous to remove the calorimeter entirely and measure the temperature of the expanded gas. However, once the stop cock is opened, the gas is in a turbulent state accompanied by variations in both temperature and pressure. To avoid such problems, Kelvin who carried out the experiments jointly with Joule over a 10-year period beginning in 1852 suggested that a porous plug should replace the stop cock so as to remove the cause of turbulence. As the gas passed through the porous plug, which damped out the kinetic energy of the gas through friction by making the velocity small enough so that its square would be negligible in comparison to other energy changes, a temperature difference could be registered by means of a thermoelectric couple. The tube was surrounded by insulating material to prevent heat exchanges with the environment.

2.3.2 Does a Gas Heat or Cool When Passed Through a Porous Plug? If we denote by primes, the quantities on the side of the porous plug, shown in Fig. 2.3, where the gas is expanding, the negative change in enthalpy is Z

p0

V dp Q:

H D p

piston maintains piston maintains Porous high pressure low pressure plug p v t

Fig. 2.3 Joule–Thomson experiment

p’ v’ t’

(2.24)


31

Performing an integration by parts in the first term on the right-hand side of (2.24) results in Z 0 H D pV p 0 V 0 C

V

p dV Q:

(2.25)

V

The difference .pV p 0 V 0 / is the total work done on the gas; the former is the work done by the pump to push the gas through the porous plug, while the latter is RV0 the work done by the gas upon expansion. The integral, V pdV is the work that is converted entirely into heat by friction as the gas passes through the porous plug. The quantity Q in (2.24) and (2.25) is the heat absorbed by the gas as it expands isothermally, and if this just balances the net work done on the gas there should be no change in the enthalpy. The change in enthalpy at constant pressure is Cp t. If the heat generated by external work is not sufficient to balance the heat absorbed by expansion, there will be a slight cooling of the gas, t > 0, as the gas passes through the porous plug. In the opposite case t < 0, there will be a slight heating of the gas. Now, according to the Carnot–Clapeyron equation, (2.35), the integral of the latent heat is Z V0 Z 0 Z V0 @ V @p @W dV D C : (2.26) Lv dV D C p dV WD C @t @t V @t V V Then rearranging (2.25) gives C D

W C pV p 0 V 0 C Cp t : @W=@t

(2.27)

If t D 0, and specializing to an ideal gas (1.3), where the coefficient of thermal expansion ˛ D 1=T0 , the inverse of the zero-point temperature, shows that the total work expended on the gas is zero, i.e., pV D p 0 V 0 . Since W D p0 V0 ˛.1=˛ C t/ ln V 0 =V ; and @W=@t D p0 V0 ˛ ln .V 0 =V /, it follows that C D 1=˛ Ct. Even if we drop these conditions, the value of C can be determined for each temperature on an arbitrary scale by determining t. In other words, in Kelvin’s new temperature scale the temperature “falling” from the furnace to the refrigerator through equal intervals of temperature would always produce the same motive power and would neither depend on the nature of the working substance nor the temperature at which the fall occurred (Carnot 1824, p. 41).

2.3.3 Kelvin’s Absolute Scale Clerk-Maxwell [1891] in his little black book, Theory of Heat, gives a graphical representation of Kelvin’s absolute scale that sheds some light on its meaning.

32


He considers isotherms and adiabats, which curve down to the right, as he shows in Fig. 2.4. He notes that when an adiabat crosses an isothermal one it does so at a greater angle with respect to the horizontal. This is to say that when a volume of gas is compressed, it will require a greater pressure when it is compressed adiabatically than when it is maintained at uniform temperature. Starting at point A on T , the gas expands by absorbing a quantity of heat Q till it reaches point B in Fig. 2.5. Then the gas is further expanded by absorbing another equal quantity of heat, Q, until it reaches state C . Consider now, p

Isothermal

Fig. 2.4 Contrast between adiabatic and isothermal curves

Adiabatic V

T T A

B

C

T‘ A‘

B‘

C‘

T‘‘ A‘‘

B‘‘

C‘‘

V Fig. 2.5 Clerk-Maxwell’s construction for determining the absolute temperature scale


33

says Clerk-Maxwell, the adiabats A A0 A00 , B B 0 B 00 , C C 0 C 00 through points A, B, and C , which give the relation between pressure and volume when the gas is allowed to expand without adding heat. Clerk-Maxwell draws the horizontal isotherms T 0 A0 B 0 C 0 and T 00 A00 B 00 C 00 corresponding to the lower temperatures T 0 and T 00 . In order that the isothermal be horizontal and easily distinguishable from the adiabats that slope downward toward the right, it is necessary to consider a phase equilibrium between liquid and gas. However, the nature of the discussion is totally independent of the working substance, as Kelvin fully appreciated, for Carnot’s function leads to a totally new method of defining temperature that does not depend upon the nature of the substance used to measure it. According to Carnot’s theorem, the work W that is achieved when a reversible engine works between temperatures T and T 0 by absorbing a quantity of heat Q depends only on the two temperatures. Because A B and B C both absorb the same quantity of heat Q, the areas A B B 0 A0 and B C C 0 B 0 must be the same since the areas represent the work W . The same is true of areas cut out by pairs of isothermal lines cutting the adiabatic curves. Thomson’s (1848) method of graduating the scale of temperature is to choose T , T 0 , and T 00 so that they cut out equal areas. In his own words The characteristic property of the scale which I now propose is, that all degrees have the same value; that is, that a unit of heat descending from a body A at the temperature T ı of this scale, to a body B at the temperature .T 1/ı , would give out the same mechanical effect, whatever by the number T . This may justly be termed an absolute scale, since its characteristic is quite independent of the physical properties of any specific substance.

In other words, the number of degrees between T and T 00 is proportional to the area A B B 00 A00 . However, two quantities still remain arbitrary: the size of the degree and the zero of temperature. Kelvin fixed this by making his absolute scale correspond with the Celsius scale between two standard temperatures, the freezing and boiling of water. To determine the absolute zero of temperature, let T 00 be the temperature at which all the heat is converted into work, W D Q. Surely no body can have a lower temperature. The area A B B 00 A00 represents the work W D Q .T T 00 / =C , where C is Carnot’s function. If the temperature is not T , but T 0 , the heat absorbed would be Q0 , and the work would be W D Q T 0 T 00 =C:

(2.28)

Based on the conservation of heat, Carnot supposes Q D Q0 . But, on the strength of the first law, Q0 D Q A B B 0 A0 , or Q0 D Q Q.T T 0 /=C . Therefore, concludes Clerk-Maxwell, the efficiency of the engine is Q .T 0 T 00 / =C W D 0 Q Q Q .T T 0 / =C T 0 T 00 : D C CT0 T

(2.29)

34


If the reservoir is at lowest temperature possible all the heat will be converted into work, W D Q0 . According to (2.29) this implies T 00 D T C . Label this the zero temperature, T 00 D 0, which implies T D C . That is, the temperature reckoned from absolute zero is precisely the Carnot function! Hence, Clerk-Maxwell was able to write Q instead of Q0 in (2.28) because C was still to be defined. And having found C D T , (2.28) can be written as W D Q0 T 0 T 00 =T 0 ; because of the condition of reversibility Q Q Q0 D D 0: C T T There is nothing preventing us from writing (2.28) as W D Q0 T 0 T 00 =C:

(2.30)

(2.31)

Writing the first law as Q0 D Q Q T 0 T 00 =C; and dividing the former by the latter gives W T 0 T 00 Q0 : D 0 Q Q .C T 0 C T 00 / If C D T 0 , as it should for a reversible process, maximum work, T0 W DQ 1 ; T p will result when T 0 D .T T 00 /. That is, if we leave the intermediate temperature 0 T indeterminate, and consider two bodies at temperatures T and T 00 to interact thermally, each containing two ideal engines, then the final temperature T 0 will be the geometric average of the highest T and lowest T 00 temperatures possible when the engines perform maximum work. We will come back to this in our discussion of endoreversible engines in Sect. 4.2.4.

2.3.4 Clausius’s Enunciation of Carnot’s Theorem and Its Corollary Kelvin credits Carnot with the remarkable theorem that the ratio of the work done to the heat absorbed in an isothermal transition, .@p=@T / dT dV ; Lv dV


35

is the same for all substances at the same temperature. The ratio of the work done to the heat absorbed at constant temperature is the Carnot efficiency RR RR .@p=@T / dT dV dpdV T1 T2 R R D D DW C ; Lv dV Lv dV T1

(2.32)

that cannot be matched by real engines. The proof of the first equality in (2.32) rests on Green’s theorem. Let C be a piecewise smooth simple closed curve in the V; T -plane, oriented clockwise, and let A be the area enclosed by C. Then by Green’s theorem it follows Z

Z

Lv dV C Cv dT ZZ ZCZ @Lv @Cv @p dV dT D dV dT; D @T @V @T A A

dQ D C

(2.33)

where the integrability condition on dE, (2.3), has been used in the last line. The extreme left-hand side of (2.33) is the difference in heat of the two reservoirs, while the extreme right-hand side of the same equation is the work output. This coincides with Clausius’s assumption that the work done is the difference in the heat absorbed at the furnace to that rejected at the cold reservoir, and (2.32) is a finite difference form of Clausius’s (2.12). It also corresponds to Carnot’s principle that the work should only depend upon the heat absorbed and the difference in temperatures of the two reservoirs, when the criterion for reversibility, (2.30), is introduced. Moreover, we note that (2.32) is the integrated form of the Carnot–Clapeyron equation (2.11). Using infinitesimal differences between the temperatures of the furnace and refrigerator, T1 D T2 C T , enables (2.32) to be written as ZZ

Z dp dV D

T dp dVT D dT T1

Z Lv dV;

(2.34)

which gives the Carnot–Clapeyron equation dp Lv D : dT T1

(2.35)

Written as a total derivative of the pressure, the Carnot–Clapeyron equation, (2.35), states that saturated vapor pressure of a liquid depends only on its temperature, and not on the volume of vapor. As the volume of vapor pressure varies, the liquid will either condense or evaporate, thereby leaving the vapor pressure constant, which is incompatible with Mariotte’s law, (1.1), because in that law .@p=@V /T ¤ 0. The Carnot–Clapeyron equation appears in Clapeyron’s 1834 paper in the form ı dp k D 1 C; dt

36


where k is the latent heat per unit volume of vapor, and ı and are the densities of vapor and liquid, respectively. Because vapor and liquid are a two-phase system, the pressure p must be independent of volume, a function of the temperature t alone. C , which is supposedly only a function of temperature, is the Carnot function. On page 41 of (Carnot 1824), Fox writes that Thomson showed that the reason why the degrees on his absolute scale became progressively smaller with respect to degrees measured on the gas scale, as the temperature decreased, was due to assuming Clapeyron’s C to be a constant. Apparently, Fox is talking about a different C . Even worse, the heat capacity at constant pressure is infinite. This can be understood by realizing that if the pressure is held constant and heat is absorbed, the temperature must remain fixed so that .@Q=@T /p is infinite (Landsberg 1961). It would, therefore, appear incorrect to evaluate the Carnot–Clapeyron equation (2.35) with the equation of state of an ideal gas. However, in the case of liquid– vapor equilibrium, the volume per particle of liquid is so much smaller than that of the vapor, which also has a much greater latent heat. This makes it permissible to neglect the liquid phase and treat the vapor phase as an ideal gas (Frenkel 1946).

2.4 Integrating Factors Galore In an analogous way that Carnot’s function C , introduced in Sect. 2.2, is an integrating denominator for the quantity of heat (2.1), suppose that F is integrating factor for same quantity, but which depends only on the volume. Then, the integrability condition is d ln F @p D Cv : (2.36) @T dV The unknown function has the same logarithmic form as in the Carnot–Clapeyron equation (2.14). Once one of these functions has been found, the other follows immediately from the adiabats. Consider the Carnot–Clapeyron equation (2.14). The net work over an entire cycle is ZZ W D A

@p dV dT D @T

ZZ A

d ln C Lv dV dT: dT

(2.37)

Carnot knows that the integrand in (2.37) is equal to mR=V . He obtains it by calculating the amount of heat required for a gas to expand while keeping its temperature constant, (2.17). This is relegated to a footnote in the 1824 edition of his memoir on page 76 of (Carnot 1824), where he goes on to explore—but not determine – his unknown function of the temperature C . Truesdell [1980, p. 110] elevates this footnote, where Q is determined, to Carnot’s “main theorem.” On differentiating Q with respect to the volume, Carnot finds Lv D mR=V d ln C =dT , which on rearranging gives the desired result.

2.4 Integrating Factors Galore

37

Regarding specific heats, Carnot’s expression

@Q @T

D Cv D V

R ln .Vb =Va / ; d ln C =dT

(2.38)

led to a great deal of confusion. Carnot claims [1824, p. 83] that the amount of heat absorbed or released at constant temperature “will follow an arithmetical progression when the increases or decreases in volume follow a geometrical progression” (2.17). Later on, on page 88, he claims the same thing for the heat capacity (2.38). In fact, Carnot had to contend with the recent findings of Dulong and Petit [1818], and on page 89 he expresses his reservations. It may be said that Clausius introduced the concept of internal energy to avoid the conclusion that the heat capacity, rather than being constant, was a logarithmic function of the volume. Clausius [1850] recognizes that the mutual attraction that acts in solid or liquids does not hold for gases, so gases do not have to overcome intermolecular attraction during expansion. Clausius’s conclusion is that a gas will take up as much heat at constant temperature to do external work of expansion @Q D p; (2.39) Lv D @V T which is actually Holtzmann’s hypothesis [1848] or, equivalently, dQ D mcv dT C p dV;

(2.40)

where cv can be a function of the temperature only. Clausius then mentions that “It is even probable that this magnitude cv , which represents the specific heat of the gas at constant volume, is a constant.” This was indeed prophetic since no one had measured cv directly (Truesdell 1980, p. 196). The determination of the heat capacities should have been at the forefront because, as Carnot emphasized, the key to the relation between heat and work lies with their determination. Then at constant volume, Clausius defines the heat capacity as

@Q @T

D mcv : V

Only if T is held constant in (2.40) does the first term on the right-hand side vanish, and one is left with Carnot’s [1824, p.83] law: “If a gas changes its volume without changing its temperature, the quantities of heat evolved or absorbed are in arithmetical progression, while the volumes are in geometrical progression.” But, then, nothing can be said about the heat capacity. Nevertheless, he finds that the difference between the heat capacities is constant, or at most a function of the temperature. Carnot does this by changing independent

38


variables in (2.1) from V; T to p; T . Considering p; T as independent variables in (2.1) transforms it into " dQ D Lv

@V @p

dp C

T

@V @T

#

dT

C Cv dT

p

D Lp dp C Cp dT;

(2.41)

where Lp is the latent heat with respect to pressure, Lp D T Lv V; and T D

1 V

@V @p

(2.42)

;

(2.43)

T

as the isothermal compressibility. For an ideal gas, Lp D .V =p/Lv , and cp is the specific heat at constant pressure, which is related to the specific heat at constant volume by cp cv D .R=p/Lv D .V =mT /Lv : The constancy of the difference in specific heats was established only later in 1845 by Holtzmann’s hypothesis (2.39) for an ideal gas. Carnot’s [1824, p.86] conclusion that The difference between the specific heat at constant pressure and the specific heat at constant volume is always the same, no matter what the density of the gas, provided its weight remains the same,

is in glaring contradiction to caloric theory, which claimed otherwise because the integrability condition for (2.41) is cp cv C T

@cp @ cp cv C p D 0: @T @p

(2.44)

Rather than following Carnot, we jump ahead in time and borrow from Holtzmann’s [1848] hypothesis, (2.39). Introducing this into (2.37) gives ZZ W D A

d ln C mRT dV dT: dT V

(2.45)

Carnot’s [1824, pp. 76–77] principle is as follows: The motive power of heat is independent of the working substances that are used to develop it. The quantity is determined exclusively by the temperatures of the bodies between which, at the end of the process, the passage of caloric has taken place.

In order to cancel the T in the numerator in the integrand of (2.45), a T in the denominator is needed. This would make the work proportional to the difference in temperatures of the furnace and refrigerator. And since the work must be positive,

2.4 Integrating Factors Galore

39

C must be proportional to the absolute temperature. Consequently, the net working over the cycle is Vb W D mR .T1 T2 / ln ; (2.46) Va where Va and Vb are the volumes of the working substance before and after the isothermal expansion. We can now use (2.46) to determine the function F in (2.36), since the work must obviously be the same no matter how it is calculated. According to Carnot’s axiom, the work Z T1 Z Vb mR @p F .Vb / D .T1 T2 / ln (2.47) W D dt dV @t s F .Va / T2 Va must be independent of the specific nature of the working substance. In order for this to be the case, and for (2.46) and (2.47) to coincide, we must put F .V / D V s , and identify 1=s as the number of half-degrees of freedom. For an ideal gas, s D .cp cv /=cv , but for gases whose internal energies are proportional to the volume, the pressure is a sole function of the temperature, and, consequently the specific heat at constant pressure, cp , cannot be defined. The role of the exponent in the volume term, by way of the logarithm, is to get rid of the last vestige of “the nature of the working substance that is used for developing motive power” (Carnot 1824, p. 94), i.e., cv D R=s, because “it really has no bearing on the amount of power [work] produced.” With this choice of the function F , (2.47) depends only on the temperature difference between the reservoir, and the ratio of the volumes that are connected by the isothermal expansion. Our corollary to Carnot’s theorem, (2.36), states that the ratio of the work done by an ideal gas, mRdV dT =V , to the quantity of heat, Cv dT , it receives, can only depend on the product of the change in volume dV and a quantity, F , that is a function only of volume, viz., dV mR dV dT D ; V Cv dT F

(2.48)

or

F : (2.49) V Since Cv D mR=s, we obtain F D V =s, and this does not tally with the expression we found previously, viz. F D V s . Therefore, we must discriminate between the differential form (2.48) and the integral form (2.47), that is, either dV =F or ln F . Clausius always worked with differentials. However, the adiabats are determined by V s T D constant, so that if T is an integrating denominator, V s will be an integrating factor for the quantity of heat absorbed or rejected. We must therefore choose the integral form, (2.47) (Helmholtz 1855). Truesdell [1980, pp. 118–119] would have us believe that the dimensionally invariant form of “Carnot’s special axiom” is as follows: Cv D mR

W D Q1 ln

T1 T2

(2.50)

40


because it is invariant under T ! AT “at least for all A in a small interval about 1.” However, (2.50) contradicts the fact that the work is the difference between the heat absorbed at the furnace and the heat rejected to the refrigerator, W D Q1 Q2 . Even what Truesdell’s refers to as Carnot’s “special axiom” (2.16) is in contrast with it. The latter has been arrived at by a faulty “extraction” from a derivation that uses the conservation of heat, as we have explained. Rather, from either (2.46) or (2.47) we can separate the right-hand sides into a product of the Carnot efficiency,

C D 1 T2 =T1 , and the heat absorbed Q1 D mRT1 ln .Vb =Va /. If we attempt a na¨ıve integration of the Carnot–Clapeyron equation ZZ

@p dV dT D @T

Z

Z

dT (2.51) T R we would get Truesdell’s proposal (2.50), where Q1 D Lv dV in that formula. However, what is overlooked is that the latent heat, Lv , is at least linear in the temperature, as it is for an ideal gas, according to Holtzmann’s hypothesis (2.39). In this section we have shown that ZZ ZZ d ln C d ln F Lv dV dT D Cv dV dT: W D (2.52) A dT A dV Lv dV

For an ideal gas (2.52) reduces to the identity: ZZ

1 mRT dV dT D T V

ZZ

s mR dV dT: V s

(2.53)

The integrating factor ln C cancels the temperature dependency of the latent heat and the volume dependence is determined by latent heat, whereas the integrating factor ln F determines the volume dependency in Carnot’s expression for the heat absorbed by an isothermal expansion. For gases with heat capacities that are power laws of the temperature, the latent heat with respect to volume is (Einbinder 1948; Lavenda 2005) Lv D " C p;

(2.54)

where " is the energy density. This generalization of latent heat is what Planck [1907] will later identify as the rest energy density. Now, the condition that (2.52) be satisfied results in the differential equation d ln " D ˇ d ln T;

(2.55)

where we have used the Grüneisen equation of state pV D sE.V; T /;

(2.56)

2.5 Carathéodory’s Attempt at the Second Law

41

to evaluate the latent heat, Lv , and set s D .ˇ 1/1 with ˇ > 1. The Grüneisen parameter s > 0 is the adiabatic exponent in the adiabatic relation V s T D const. The internal energy is now a power ˇ of the temperature. We will return to a discussion of power laws in Chap. 6.

2.5 Carathéodory’s Attempt at the Second Law We should spend a few words on Carathéodory’s formulation of the second law. In Carathéodory’s attempt at formulating the second law, the integrating factor for heat plays a primary role. Carathéodory couched the existence of the second law in the impossibility of some process taking place. The process considered was an adiabatic one, and the impossibility of such a process taking place meant that the entropy would always increase or always decrease. The criterion that he found was a certain integration constant had to be positive. Born having felt that the work of “engineers” in establishing the concepts of heat and work, and entropy and internal energy, was “a wonderful achievement : : : but far removed from physics,” urged his friend, Christian Carathéodory, a mathematician of Greek origin working in Germany, to remove the vulgar concept of steam engines and replace it with something more general that would not be tied down to mundane steam engines. In his own words Born [1949] writes as follows: But even as a student, I thought that they deviated too much from the ordinary methods of physics; I discussed the problem with my mathematical friend, Carathéodory, with the result that he analyzed it and produced a much more satisfactory solution. This was about forty years ago, but still all textbooks reproduce the “classical” method, and I am almost certain that the same holds for the great majority of lectures : : : This state of affairs seems to me one of unhealthy conservatism.

This is at least once where conservatism, whether it be healthy or unhealthy, has rightfully won out! The gist of Carathéodory’s argument is that there exists adiabatically inaccessible states no matter how near to a given state. What Carathéodory is saying is that there exists a family of constant entropy surfaces, which do not intersect so that states found between any two adjacent entropy surfaces cannot be reached by paths on either surface for they would no longer be adiabatic. His formulation deals with Pfaffian forms for which it is known that there exists an integrating factor in the case of two independent variables, but not in three or more. But, thermodynamically speaking, the properties of a pure substance are determined entirely by two state variables given that the number of particles is constant. And when it is not constant, it becomes a function of the temperature. So the generality that is made is really unnecessary but allows geometry to creep in and displace the physics of the problem. Regarding the increments in three independent variables as finite differences, they are the equations of planes through any given point. Now, if there exists a state function, these planes would all have to be tangent to the surface. Hence, if there

42


exists a state not lying on the surface it cannot be connected to a state that lies on the surface by an adiabatic path. Here, an adiabatic path means one of constant entropy. This may be the case when the integrating factor is a function of the temperature, but may not be when it is a function of the volume. Carathéodory (1909) considers a composite system with three independent variables, the volumes of the subsystems, and the common temperature. This would correspond to Carnot’s theorem stating that the ratio of the work to the heat absorbed isothermally is a universal function of the temperature. But, we have shown in Sect. 2.3.4 that there exists a corollary to that theorem to the effect that the ratio of the work to the quantity of heat absorbed isochorically can only be a universal function of the volume. For this situation the choice of variables are the two temperatures of the two subsystems and their common volume. In fact, Carathéodory replaces the two independent volumes by the two entropies of the subsystems. We can do the same by replacing the two independent temperatures by their entropies. For if dQ1 D d1 =1 is the increment in heat of the first subsystem and dQ2 D d2 =2 that of the second then the combined system would have dQ D dQ1 C dQ2 D d1 =1 C d2 =2 D d=:

(2.57)

If the three independent variables are the two empirical temperatures, t1 and t2 , and the total volume V occupied by the composite system, then, following Carathéodory they can be replaced by the two values of the potential, 1 and 2 , and V . But, according to (2.57), depends on 1 and 2 , but not upon V . Not even the quotients @ @ D D ; @1 1 @2 2 depend on V , i.e., @.=i /=@V D 0 for i D 1; 2. Thus, @ ln 2 @ ln @ ln 1 D D D f .V /; @V @V @V where f .V / must be some universal function of the volume of the combined system. Then, the doctrine of latent and specific heats gives dQ D Lv dV C Cv dT D d= D dL=V s : Specializing to an ideal gas, i.e., Holtzmann’s hypothesis, and rearranging give RV s1 T dV C V s Cv dT D

R d .T V s / D dL: s

which identifies L as the adiabatic potential, where we have used R D sCv . Instead Carathéodory finds dQ D Cv dT C RT dV =V D T dS;


or

43

dS D d ln T Cv V R :

Now, for an ideal gas, R D Cp Cv so that S , like L, is a function only of z D T V s , where the adiabatic exponent is s D .Cp Cv /=Cv , or 23 for an ideal gas. The difference between the adiabatic potentials L and S is that the former is nonextensive while the latter is; both are functions only of z D T V s . By choosing the empirical temperature as the common variable, Carathéodory is implicitly using its intensive property to state that 1 can only be a function of V1 and t. However, the most generalized integrating factor for a system with V and T as independent variables is T 1 g.T V s /, where g is an arbitrary function (Einbinder 1948). We could just as well choose g to be unity as we could to be equal to its argument. The essential point is that the total volume and common temperature are not independent of each other, but, rather, are related by the adiabatic condition T V s D const., for only in this way the composite system will be isolated from its environment and there will exist a family of surfaces of constant entropy. So Carathéodory’s formulation tells us nothing that we do not already know, and in much less physical terms. As an example, consider the adiabatic condition imposed on the doctrine of latent and specific heats in three independent variables dQ D p1 dV1 C Cv dT C p2 dV2 D 0; which we specialize to an ideal gas with 2s degrees of freedom T 1 T dV1 C dT C dV2 D 0: V1 s V2

(2.58)

We already know that this will not lead to any primitive, .V1 ; T; V2 / D const. What we do know is that a differential equation, T 1 dV1 C dT D 0; V1 s will lead to a primitive, .V1 ; T / WD V1s T D const. Then, there exists some quantity for which T @ @ D V1s D : D sV s1 T D ; @V1 V1 @T s This quantity is D sV1s . Since does not involve V2 , we define another quantity ƒ WD T =V2 D s=V2 , for which J D

@ƒ @ @ƒ @ D 0; @V1 @T @T @V1

(2.59)

44


is satisfied identically. That is, if both ƒ and are both functions of V1 and T , (2.59) is the condition that ƒ can be expressed as a function of alone (Forsyth 1956, p. 14). Rather, if (2.59) does not vanish identically we cannot reduce (2.58) to an equation in two variables for which we know there exists a primitive. Thus, the original relation (2.58) gives way to

T 1 T dV1 C dT C dV2 D d C ƒdV2 D 0; V1 s V2

which reduces to d C

s dV2 D 0: V2

Since only two variables are involved this gives the primitive D V2s D const.; and, consequently, D

@ @ ; ƒ D : @ @V2

This determines the quantity D V2s . Consequently,

T T 1 dV1 C dT C dV2 V1 s V2 D .d C ƒdV2 / DW d ;

(2.60)

or, in terms of the three original variables, this leads to the primitive L .V1 ; T; V2 / D .V1 V2 /s T D const.;

(2.61)

for which the condition of integrability is satisfied identically. Equation (2.61) defines a new adiabatic potential, L, which is nonextensive. It can be written as p L D VGs=2 T for the two-component volume, where VG D .V1 V2 / the geometric mean volume. For an n-component volume, it generalizes to L D VGs=n T , which diminishes the number of degrees of freedom. The nonextensive adiabatic potential is related to the entropy by Ln=s D S , and we will have occasion to discuss it in greater detail in Chap. 6. Now, where does Carathéodory’s axiom, that there are inaccessible points in the neighborhood of any given point by an adiabatic path, come in? Buchdahl (1949) gives the condition of inaccessibility that @=@T D 0

(2.62)

should be satisfied identically throughout the entire space of the coordinates , T , V2 . It is clear that (2.62) will be satisfied everywhere if D const. For he concludes:


45

“In that case [] and therefore [ƒ] are independent of [T ]: : : and Eq. [(2.58)] is obviously integrable.” But, this is not what (2.59) says. Rather, if (2.59) is satisfied identically, then there exists a relation between ƒ and that is satisfied “for all values whatever of [V1 ] and [T ]” (Forsyth 1956, p. 14). The condition that would satisfy (2.62) would be the adiabatic one V1s T D const. But, that would limit (2.62) to the adiabats, and it would not hold everywhere in space. We do not need the concept of entropy to define the adiabats, and from our discussion in Sect. 2.3.3 we fully recognize that there are states lying between the adiabatic curves that slope downward in the T; V -planes that are inaccessible to states lying on the curves by an adiabatic transition. That is, we do not need d D ƒ dV2 to tell us that through any point lying on the surface D const. there passes one curve (Born 1949, p. 145). Rather, the point is that there are potentials that are sole functions of adiabatic parameter z D V s T , the entropy being that adiabatic potential that is extensive. We have, moreover, shown that even if a greater number of “deformation variables,” to borrow the terminology of Carathéodory, is used the situation still remains the same. Furthermore, to show that the entropy does increase, Carathéodory replaces the third of the independent variables V1 , V2 , and t by S . Then he compares the initial state V10 , V20 , S 0 with the final state V1 , V2 , S to show that S > S 0 . He claims that the final state can be reached in two steps: 1. The volumes V10 , V20 are altered by a quasi-static-adiabatic process that keeps S 0 constant 2. The volumes are kept at their final values V1 and V2 , but the entropy is allowed to increase to S , notwithstanding the fact that dQ D 0. Here, there is a lot to criticize. By switching from t to S as the independent variable, Carathéodory is going from nonisothermal to nonadiabatic processes. While it is true that the heat can be a function of volume and temperature, it is difficult to see how it can be a function of volume and entropy, when the entropy has yet to be defined. Moreover, the total system is adiabatically isolated so that t r=s V D t r=s .V1 C V2 / D const. That is, t r=s V1 D const. and t r=s V2 D const., since V1 and V2 are considered independent. If the empirical temperature does not change neither will the subvolumes. Thus, S 0 cannot be kept constant unless the adiabatic conditions are violated. This brings us to his second step where the entropy supposedly increases even though the total system is adiabatically isolated. Carathéodory considers a frictional process, like rubbing or stirring, that would lead to an increase in entropy at constant values of the final subvolumes V1 and V2 . Such irreversible processes would generate heat leading to an increase in temperature. But, the temperature cannot increase if the sub-volumes are to remain constant since Q D const. Finally, Carathéodory has gone through all the trouble to show that an integrating factor exists for dQ, and

46


then considers dQ ¤ T dS , meaning that S must be defined in some other manner! Since adiabatic and isentropic surfaces no longer coincide what is the surface that must be considered in order to arrive at the given state not lying on that surface? We have shown that: Although the mathematical properties of differential expressions involving two

independent variables cannot be carried over to those involving more than two, there are no physical differences. Two different states, which are joined by an absorption of heat, cannot be reached from a common state by different adiabatic paths. The cycle, in which the heat absorbed is entirely converted into work, does not have to be outlawed by Kelvin’s principle (Zemansky 1966), but, rather, by the fact that adiabatic paths belonging to the same family do not intersect. Adiabatic surfaces exist irrespective of the definition of the entropy function. In fact, as we shall see in Chap. 6, the entropy is defined in terms of theses adiabatic invariants and is the only function of these variables that is extensive.

Chapter 3

Thermodynamics in a Carnot Equation

3.1 Why Exterior Differentials? Because of the nonconservative nature of thermodynamic fields, exterior differential forms seem like the natural formulation of thermodynamics. The so-called covectors of heat and work, H and F , are not gradients of any scalar functions. Information can be had by studying their “curls,” which are nonvanishing and measuring the deviations of these functions from “gradients,” which would lead to state functions that are path independent. These equations can be considered as being analogous to the first and second laws of circuitation in electrodynamics (Heaviside 1893). A great simplification is that the usual state space of thermodynamics is the plane in which the “cross product” or “wedge product,” ^, of covectors in the state plane can be interpreted as a vector pointing in the direction “perpendicular” to the plane (Hannay 2006). The resulting “covector,” which is not an arrow vector because directions and distances are not defined in thermodynamics, is “perpendicular” to the state plane, which can be treated, therefore, like a scalar quantity. Let us briefly review some elementary notions of differential forms. 0-forms are scalars, like f D f .x 1 ; x 2 /. 1-forms, or linear forms, like A1 dx 1 C A2 dx 2 , are also known as Pfaffians. And to complete the list we will have need of the 2-form, B12 dx 1 ^ dx 2 . A natural question is why the “wedge,” ^? (Flanders 1963). Naturally, we would like to integrateRthe which are the familiar line, R 1- and12-forms, R 1 2 2 A dx C A dx , and surface, B dx ^ dx , integrals, respectively. These 1 2 12 C A integrals are “oriented” integrals in the sense that the curve C in the line integral has a sense of direction and that the surface A has an “outward” normal to it. Reversing the normal changes the sign of the integral. In addition, we want a rule for a change of variables, which is given by the Jacobian transformation dx 1 ^ dx 2 D

@.x 1 ; x 2 / 1 dy ^ dy 2 : @.y 1 ; y 2 /

Interchanging rows in the Jacobian changes the sign of the Jacobian, and, consequently, we must have dx 2 ^ dx 1 D dx 1 ^ dx 2 , if the transformation of variables B.H. Lavenda, A New Perspective on Thermodynamics, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1430-9 3,

47

48

3 Thermodynamics in a Carnot Equation

is to remain invariant. Hence, the “wedge,” or “exterior product” of a linear differential form is a skew-symmetric product. In particular, dx ^ dx D 0, which says that the Jacobian vanishes when its rows are equal. 1 2 1-form, ! D A11 dx C2 A2 dx , we can always associate a 2-form, d! D To each 1 2 @A2 =@x @A1 =@x dx ^ dx . The line integral of the former around a closed curve C is equal to the integral of the region it encloses, A, viz., Z C

A1 dx 1 C A2 dx 2 D

Z Z A

@A2 @A1 2 @x 1 @x

dx 1 ^ dx 2 :

This is Green’s theorem, and we have already used it in calculating the work in the Carnot cycle [cf. (2.33)]. The “curl” of the covector fields, r ^ F , and r ^ H, are scalar functions and all take place in the plane, R2 . Its generalization, Stokes’s theorem, differs from Green’s theorem in three essential ways. First, Stokes’s theorem lives in R3 ; second, it involves a surface integral, instead of a double over a plane region, and, third, the curl of a covector in R3 is a vector, while its curl in R2 is a number. Since our state space is two-dimensional, Green’s theorem suffices, and we will now proceed to derive the laws of thermodynamics for the covector fields.

3.2 Two-and-a-Half Commandments of Thermodynamics The generalized displacement, dr, will be .dV; dT/, .dp; dT /, or .dV; dp/, if the independent variables are chosen to be .V; T /, .p; T /, or .V; p/, respectively. The work covector, F in dW D F dr will thus be given by .p; 0/, .V; 0/, or .p; 0/, respectively. Likewise, the heat covector H in dQ D H dr will be given by the independent partial derivatives, .Lv ; Cv /, and .Lp ; Cp /, for .V; p/ and .p; T / as independent variables. The gradient r D .@i ; @j / for the independent variables, .i; j /. In this notation, the first and second laws plus Carnot’s axiom in its differential form are r ^ H r ^ F D 0; 1 r ^ H C r ^ H D 0; r ^ H D r ^ F ;

(3.1) (3.2) (3.3)

where is the appropriate integrating factor for the heat covector, H. Carnot’s grand differential principle, (3.3), stands half-way between the first, (3.1), and second, (3.2), laws, being equal to their difference. Yet, we will see that it gives entire structure of equilibrium thermodynamics. Carnot’s grand integral principle can be expressed as Z

ZZ

W D

p dV D C

A

dp ^ dV D F .T1 ; T2 /Q1;2 ;

(3.4)

3.2 Two-and-a-Half Commandments of Thermodynamics

49

where F is a function of the two temperatures, and Q1;2 is either the heat absorbed, or rejected, in a given closed cycle C, that has area A.

3.2.1 Independent Variables .V; T / The characteristic potential is the internal energy, E. Differentiating dE D T dS pdV

(3.5)

with respect to the V at constant T , and using the Grüneisen equation of state, (2.56), gives the partial differential equation T

@p @T

V

V s

@p @V

D

T

1Cs s

p:

(3.6)

Employing the method of characteristics, we get the Lagrange, or auxiliary, equations dT s dV s dp D D : (3.7) T V 1Cs p Integrating the total differential equations, (3.7), we obtain the adiabats TV s D const.;

pV 1Cs D const.;

and p s=.1Cs/=T D const.;

(3.8)

of which any two determines the third. If we take D 1=T as the integrating factor, for the quantity of heat communicated to the system (2.1), then Carnot’s principle in differential form (3.3) is .Lv ; Cv / ^ .@v ; @T / T D T .@v ; @T / ^ .p; 0/;

(3.9)

where the heat field vector H has components .Lv ; Cv /. Equation (3.9) is none other than the Carnot–Clapeyron equation,

@p @T

D V

Lv : T

(3.10)

If we take, instead, D V s as the integrating factor, Carnot’s principle is .Lv ; Cv / ^ rV s D V s r ^ F or

@p @T

D V

sCv : V

(3.11)

50


We now compare (3.10) and (3.11) with Carnot’s grand integral principle, (3.4), in the cases of ideal and generalized gases. Using (3.10) we construct Carnot’s grand integral principle, (3.4) as Z Z

Z Z

W D

dp dV D A

A

dp dT dV D dT

Z

V2

Z

V1

T1 T2

Lv dT dV: T

(3.12)

For an ideal gas, the latent heat of expansion is given by Holtzmann’s hypothesis, (2.39) and the work turns out simply W D R ln

V2 V1

.T1 T2 / D RT1 ln

V2 V1

T1 T2 T1

D Q1 C ;

(3.13)

where Q1 D RT1 ln .V1 =V2 / is the heat absorbed in the isothermal expansion, and

c D 1 T2 =T1 is the Carnot efficiency. Things become trickier when we treat a generalized gas of which a photon gas is an archetypal example. From the first principle we know W D

4 ˚ 4

T1 .V2 V1 / T24 .V3 V4 / ; 3

where is the radiation constant. The fact that the pressure is one-third of the energy density for a photon gas brings in the factor 43 . Using the adiabatic constraints

3

T2 T1

D

V2 V3 D ; V3 V4

(3.14)

the work becomes W D

4 3

T .V2 V1 / .T1 T2 / D Q1 C : 3 1

If we use Carnot’s grand principle in its integral form, we get Z Z W D A

" 4 # 1 4 Lv T2 dT dV D T1 .V2 V1 / 1 ; T 3 T1

where the latent heat of expansion is Lv D 4p from (2.54), i.e.,

@E @V

D Lv p D 3p: T

The problem is that this has an efficiency,

D1

T2 T1

4 ;

(3.15)


51

that is much greater than Carnot’s. According to Carnot’s line of argument, this machine could be used to drive Carnot’s engine in reverse: : : : if it were possible by any method whatever to make the caloric produce a quantity of motive power greater than we have made it produce by our first series of operations, it would suffice to divert a portion of this power in order by the method just indicated to make the caloric of the body B return to the body A from the refrigerator to the furnace, to restore initial conditions, and thus to be ready to commence again an operation precisely similar to the former, and so on: this would be not only a perpetual motion, but an unlimited creation of motive power without consumption either of caloric or any other agent whatever.

If we separate the heat absorbed or rejected from the integral of the Carnot function, we are led to consider the heat rejected because the work cannot be greater than the total heat absorbed through isothermal expansion. Thus, if we write the work as Z

Z

V4

T1

dT T V3 T2 4 4 T1 D T2 .V3 V4 / ln 3 T2 4 3 T2 .V3 V4 / .T1 T2 / 3 4 D T13 .V2 V1 / .T1 T2 / D Q1 C ; 3

W D

Lv dV

(3.16)

we would find that its upper bound coincides with (3.15) based on the inequality ln x x 1 for x > 0. This clearly shows that nothing can beat the Carnot efficiency.

3.2.2 Strange Carnot Cycles: An Interlude With .T; p/ as the pair of independent variables, the doctrine of latent and specific heats, (2.1), is transformed into (2.41). It was always assumed by the forefathers of thermodynamics that Lv > 0, making Lp D Lv = .@p=@V /T D Lv V T < 0:

(3.17)

This will be true in the “normal region.” Now, from the Carnot–Clapeyron equation and the equality Lv D T

@p @T

D T V

@V @T

it follows that ˛T D Lv T ;

p

@p @V

; T

52


where ˛ is the coefficient of thermal expansion ˛D

1 V

@V @T

:

(3.18)

p

Now, the partial derivative in (3.18) can be written as

@V @T

p

D .@p=@T /V

ı

.@p=@V /T :

In the normal case, where .@p=@T /V > 0 and .@p=@V /T < 0, an increase in the temperature will cause an increase in volume along an isobar. However, an early discovery, in the seventeenth century, was that water, at atmospheric pressure, has a maximum at 4ı C. This means that V will decrease with T along the isobar of 1 atm to a minimum at 4ı C, and then the volume will increase as the temperature continues to drop, making (3.18) negative. This has been termed “the anomalous behavior of water.” Thus, we cannot attach any definite sign to .@p=@T /V .1 A point where

@p @T

D V

˛ D 0; T

(3.19)

has been termed a “piezotropic point” by Truesdell (1979), which he borrowed from meterology, where the equation of state p D p.V / often occurs. In the second line of (2.41) the heat capacity at constant pressure was found to be related to the heat capacity at constant volume by Cp Cv D T

@V @T

p

@p @T

D V T ˛ 2 =T > 0:

(3.20)

V

For an ideal gas, the right-hand side of (3.20) is just R, the universal gas constant. This is a well-known result that has been derived in a rather unconventional manner. Consequently, the doctrine of latent and specific heats can be expressed as dQ D Lp dp C Cp dT D ˛V T dp C Cp dT:

(3.21)

In the anomalous region where ˛ vanishes (3.21) shows that isotherms will coincide with adiabats so that no adiabat can cross the 4ı C isotherm. In fact, this is Sommerfeld’s (1956) solution to the following problem: Imagine a Carnot cycle with water as its working fluid operating between 2ı C and 6ı C, so that at 6ı C there is an isothermal expansion and isothermal compression at 2ı C. It is

1

That the isothermal compressibility be positive is a necessary condition for thermodynamic stability.


53

seen that heat is added during both processes, if the pressure is low enough, and so heat is converted completely into work in violation of the second law. How is it possible to resolve this contradiction?

Sommerfeld answers the contradiction by saying simply that the putative adiabats do not exist, which connect the 2ı C and 6ı C isotherms. In the T; V -plane, the adiabats are found from @V Cv D ; @T Q Lv which is obtained by setting dQ D 0 in the doctrine of latent and specific heats, (2.1). The slope of the adiabats is negative in the T; V -plane for Lv > 0, which is what one would normally expect: As the volume increases in an adiabatic expansion the temperature drops. However, in the region where Lv < 0, the slope is positive. At the minimum, the adiabatic is parallel to the V -axis. And through this line is the curve along which Lv D 0. This implies that .@p=@T /V D 0, which has been called a “neutral” curve by Truesdell and Bharatha (1979). It is also a piezotropic curve since the pressure is independent of the temperature. The adiabats, if they existed, would have to have the form of curves connecting the two isotherms at temperatures T> higher and T< lower than the absolute temperature, where water has maximum density, shown in Fig. 3.1. The neutral curve C separates the normal regions where Lv > 0 from abnormal region where Lv < 0. Heat would be rejected to the cold reservoir by an isothermal expansion, rather than an isothermal compression as in the case of the Carnot cycle. Although it is true that precise measurements do show that the temperature at which water has maximum density does vary with the pressure [at 100 atm is roughly 2ı C (Thomson 1962)], so that an adiabat will have a nonzero tangent in the T; p-plane making it distinct from an isotherm, the theoretical possibility of a “strange” Carnot cycle refuting the second law should be answered without any recourse to experiment. The paradox is resolved merely by recognizing that

T

Lv > 0

Q>0 T>

Q 0 C

Fig. 3.1 Strange Carnot cycles

V

54


in the anomalous region, Lv < 0, heat is rejected during isothermal expansion in contrast to its normal behavior where compression is needed in order to reject heat to the colder reservoir. Thus, a strange Carnot cycle will consist of two isothermal branches where the working fluid undergoes expansion, coupled to two adiabats (Thomson 1962). But, we have to make explicit allowance that the adiabats connecting the two isothermal expansions must necessarily cross the neutral curve separating normal from abnormal behavior, as shown in Fig. 3.1. Strange Carnot cycles have been around since 1928 (Trevor 1928) and have occasionally attracted attention (Thomson 1962). If both isotherms laid to the right of the neutral curve in the T; V-plane, then heat would be rejected as the system undergoes an isothermal expansion on the 6ı C isotherm, while it would be absorbed as the system undergoes an isothermal contraction at the 2ı C isotherm. But, upon crossing the neutral curve the heat that was being rejected at the higher temperature would suddenly be absorbed, and the heat that was being absorbed by compression along the isotherm at the lower temperature would suddenly be rejected. A Carnot cycle cannot be constructed whose cycle passes through a neutral curve. But, what about a Carnot cycle that lies wholly to the right of the neutral curve C , as depicted in Fig. 3.2? Heat would be rejected at the higher temperature by isothermal expansion and absorbed at the lower temperature through isothermal compression. However, this would violate Carnot’s principle that “work is derived from heat by lowering its temperature.” In other words, what we are proposing is that work could be obtained by raising the temperature of heat so that it would be absorbed at the low temperature and rejected at the high temperature. Moreover, the pressure would have to be greater at the lower temperature than at the higher temperature in order for positive work to be done. Therefore, it is not true that positive work can be derived from a body undergoing an ordinary Carnot cycle “in part where Lv < 0 and @p=@T < 0” (Truesdell 1979, Remark 2, p. 51).

T

Lv > 0

Lv < 0 Q Q=0

Q=0 C T< Q> 0

Fig. 3.2 A Carnot cycle in the region of negative latent heat

V


55

3.2.3 Independent Variables .p; T / The characteristic potential is the enthalpy, which as a function of p and T satisfies the differential equation

@H @p

V D

T

@Q @p

D Lp :

(3.22)

T

This is because the doctrine of latent and specific heats as a function of .p; T / can be written as dQ D Lv dV C Cv dT " # @V @V dp C Cv C Lv dT D Lv @p T @T p D Lp dp C Cp dT:

(3.23)

Equation (3.22) follows from equating the coefficients in the first terms of the second and third rows, while @V Cp Cv D Lv @T p by equating the coefficients of the second terms. The latter is none other than (3.20) above. With the displacement dr D .dp; dT /, there is the work W D F dr D .V; 0/ .dp; dT /. With the heat field vector given by H D .Lp ; Cp /, Carnot’s grand principle in differential form is

Lp ; Cp ^ .@p ; @T /T D T .@p ; @T / ^ .V; 0/: @V D ˛V T: Lp D T @T p

Latent heats of expansion are related by (3.17), and since the isothermal compressibility T > 0, it follows that Lv > 0, and Lp < 0. We thus obtain the Carnot–Clapeyron equation ˛V D

@V @T

D p

1 C

@Q @p

D

T

@S @p

;

(3.24)

T

with Carnot’s function C standing in for the absolute temperature. Clapeyron regarded his equation as “the most general consequence we can get from this [Carnot’s] axiom: It is absurd to suppose that force or heat can be created from nothing at no cost.” Although Clapeyron assumes that his C is a constant, all the data that he could muster indicated otherwise.

56


Moreover, replacing Q by T S leads to a Maxwell relation

@V @T

D

p

@S @p

:

(3.25)

T

Introducing (3.25) into (3.22) gives the partial differential equation .1 C s/p

@V @p

C sT

T

@V @T

D V;

(3.26)

p

which has the same characteristic curves, (3.8), as the partial differential equation for the pressure, (3.6). The work derived from Carnot’s grand principle is Z W D

p1

Z

p2

T1 T2

T1 T2 Lp p1 dT dp D RT1 ln D Q1 C : T p2 T1

(3.27)

The same expression can be obtained by using the integrating factor D p s=.sC1/. In contrast, Carnot’s principle in differential form is

Lp ; Cp ^ rp s=.1Cs/ D p s=.1Cs/ r ^ .V; 0/;

giving the Carnot–Clapeyron equation ˛V D

@V @T

D p

s Cp : 1Cs p

(3.28)

In integral form it gives the work Z W D

T1 T2

Cp dT

s 1Cs

Z

p1 p2

dp p1 D R .T1 T2 / ln ; p p2

which is the same as that in (3.27). Neither (3.24) nor (3.28) can be used to calculate the work in a Carnot cycle that uses a photon gas because T D 1 and Cp D 1, which are related by Cp T D : S Cv Both Carnot’s differential principle, (3.11), and Clapeyron’s companion equation, (3.28), are expressed in terms of heat capacities. From the former we get @p @T @p s D D ; V @T V @Q V @Q V


or T

@S @p

D V

57

V ; s

(3.29)

whereas the latter, which is equivalent to s 1 D 1Csp gives

@V @T

T

@S @V

p

@T @Q

D p

D

p

@V @Q

; p

1Cs p: s

(3.30)

Neither (3.29) nor (3.30) is a Maxwell relation, but both indicate that we should look at the V; p-plane.

3.2.4 Independent Variables .V; p/ With the generalized displacement as dr D .dV; dp/, the increment in the work will be given by dW D F dr D .p; 0/ .dV; dp/ D p dV:

(3.31)

Transforming the doctrine of latent and specific heat (2.41) to the independent variables .V; p/ gives "

# @T @T dQ D Lp dp C Cp dp C dV @p v @V p Cp Cp T dV: ˛V T dp C D Lv ˛V

(3.32)

Each time we transform from the independent variables .V; T / to .p; T /, and finally to .V; p/ the heat covector, H, becomes more complicated, and more conditions on the thermodynamic stability of the system need to be imposed. The heat covector is now Cp Cp T HD ˛V T ; ; ˛V Lv which when introduced into Carnot’s local principle, H ^ rT D T r.p; 0/;

(3.33)

58


shows that it is satisfied identically. The more complicated the expressions for the covectors become, the simpler become their corresponding Carnot–Clapeyron equations. Forgetting the specific form of the heat flux vector, Carnot’s local principle (3.33) can be written in Jacobian form as @.Q; T / D T: @.V; p/

(3.34)

Introducing (3.29) and (3.30) into (3.34) lead to the partial differential equation for the temperature, T D T .V; p/: .1 C s/p

@T @p

V

V

@T @V

D sT;

(3.35)

p

which has the same characteristic curves as (3.6) and (3.26). But, in this case, where the pair of independent variables are conjugate to one another, there is no characteristic potential. With the aid of the Carnot (3.10) and Clapeyron (3.24) relations we can write the Jacobian (3.34) as

@Q @V

@Q @p

D T

V

@p @V

C

T

@Q @V

: p

This is the general relation for the change of variable, which is held constant in a differential, viz., @z @w @z @z D C : @x y @w x @x y @x w Photons can be infinitely compressed, ˛ D 1, and this appears as flat isotherms in the p; V -plane. Carnot’s grand principle reduces to .@Q=@V /p .@T =@p/V D T; or, equivalently,

@Q @V

DT

p

@p @T

D 4p;

(3.36)

V

where we used the equation of state p D 13 T 4 of a photon gas. Integrating (3.36) in the T; V -plane gives ZZ A

dp dT dV D dT

Z

V2 V1

which is blatantly wrong!

Z

T1 T2

1 4p dT dV D T14 T24 .V2 V1 / ; T 3

(3.37)


59

If we separate the latent heat of expansion with respect to the volume from Carnot’s function, we cannot have W=Q1 ! 1

as T1 ! 1:

That is, if we calculate the work from W D Q1 ln

T1 D4 T2

Z

Z

V2

T1

p dV V1

T2

dT T

T1 4 D T14 .V2 V1 / ln 3 T2 4 3 T1 .V2 V1 / .T1 T2 / D Q1 C ; 3

(3.38)

we find that it is greater than that given by Carnot’s principle. Since such a process can be refuted by Carnot’s argument of a perpetual machine, we must determine the work from the heat rejected to the cold reservoir W D Q2 ln

T1 D 4 T2

Z

Z

V4

T1

p dV V3

T2

dT T

4 4 T1

T2 .V3 V4 / ln 3 T2 4 3 T2 .V3 V4 / .T1 T2 / 3 4 D T13 .V2 V1 / .T1 T2 / D Q1 C : 3 D

(3.39)

This does give the maximum work according to Carnot’s grand principle, where the heat absorbed in the isothermal expansion at the boiler is Q1 D

4 4

T .V2 V1 / : 3 1

Since ln.T1 =T2 / is bounded above and below by T1 T2 T1 T1 T2 ln ; T1 T2 T2 with the equalities holding if the temperatures are equal, we may, as a first approximation to ln.T1 =T2 /, use the mean of the upper and lower bounds, i.e., ln

T1 1 T1 T2 T1 T2 T 2 T22 C : D 1 T2 2 T1 T2 2T1 T2

We then obtain the work as W D TA .S2 S1 / C ;

(3.40)

60


where TA D 12 .T1 C T2 / is the arithmetic mean of the reservoir temperatures, and Si D 43 Ti3 Vi . This means that the temperature of the heat reservoir has been lowered to the arithmetic mean temperature. More work can be performed by increasing this temperature, and this is the significance of the inequality ln x x 1, unless x D 1 where all the temperatures are equal, and the equality applies. This can be confirmed, and its significance expounded upon, by the work performed by an ideal gas, viz., T1 V3 T12 T22 RT2 ln T2 V4 2T1 T2 V2 D RTA ln C ; V1

W D Q2 ln

where we have used the adiabatic condition, V3 =V4 D V2 =V1 . The temperature at which heat is absorbed in the isothermal expansion has been lowered from T1 to the arithmetic mean TA . The inequality ln x x 1 raises it to the temperature of the highest reservoir. Alternatively, we can keep the temperature of the hot reservoir at its maximum value and consider the efficiency " 2 # TA 1 T2

C C ; D

D 1 2 T1 T1

(3.41)

where the inequality is guaranteed by the fact that .x 1/2 0. That is, the efficiency has been decreased by a factor that is the ratio of the mean temperature if the two reservoirs were to thermally equilibrate without performing any work and the temperature of the hot reservoir. We then have the work W D Q1 D Q1 Q2 : Solving for Q1 in terms of Q2 gives the condition for reversibility as Q1 Q2 D 2 ; 2 T1 TM2

(3.42)

where TM2 is the mean temperature of order 2, viz., T M2 D

1=2 1 2 T1 C T22 : 2

Condition (3.42) appears to go against the grain of the second law of thermodynamics, which singles out the absolute temperature itself as the integrating denominator for the heat. Clearly, we cannot use the adiabats derived from the


61

partial differential equations for p and V since written into them is the relation dQ D T dS . If we replace T by any positive power of T , say T r , the characteristic equations for the adiabats are r

dV dE dT D s D ; T V E

(3.43)

in contrast to (3.7). Thus, the adiabats are T r V s D const., and E D V s .T ˛r V ˛s / ; for any ˛ > 0, where is a positive constant of proportionality, e.g., a radiation constant. This is due to the fact that we demand that the energy tends to zero uniformly with the temperature. The pressure is p D sE=V D s T ˛r V ˛s.sC1/ :

(3.44)

If we impose that the internal energy be first-order homogeneous, ˛s D s C 1, then we have the case of a photon gas since the isothermal compressibility is infinite. A finite, positive value of the isothermal compressibility, T , requires ˛s < s C 1. Using the first law and the fact that is an integrating denominator for the heat lead to the Carnot–Clapeyron equation 1Cs @p p; (3.45) D 0 @T V s where the prime means differentiating with respect to the argument, in this case the temperature. Introducing (3.44) into (3.45) reduces it to ˛s 0 D r : (3.46) 1Cs T Now, the first factor on the right-hand side can, at most, be unity when homogeneity is imposed. The exponent r is not so restricted, so that in the case where the internal energy is a first-order homogeneous function, the integrating denominator is D

const. ; Tr

(3.47)

which was not unexpected. In the case r D 1, the ratio ˛s=.s C 1/ < 1, on account of the thermodynamic stability condition on the isothermal compressibility, the integrating denominator admits only powers of T less than unity. In the case of a photon gas, the work accomplished between two temperatures, T1 > T2 , is W D Q1 Q2 i h .˛1/s .˛1/s .˛1/s .˛1/s T2˛r V3 : D .1 C s/ T1˛r V2 V1 V4

62


Evaluating the second term in brackets along the adiabats, T1r V1s D T2r V4s and T1r V2s D T2r V3s , gives r T2 Q1 : W D Q1 Q2 D 1 T1

(3.48)

For r > 1 this would contradict Carnot’s conclusion that his efficiency is the greatest possible. The only possibility is that T2 is the average temperature TM2 , which is necessarily larger than T2 for T1 ¤ T2 . And for r D 2, (3.48) coincides with (3.42). Now, it is well known (Hardy 1952) that the ratio of the smallest, T2 , to largest, T1 , value is always less than the ratio of means in which the numerator, r, is of a smaller order than the denominator, s, i.e., T2 T Mr T1 ; T1 T Ms T2

r < s:

However, the powers involved do not let us deduce anything other than the fact that the lowest temperature has been raised above T2 because of the monotonic increasing property of the order of the means, TM2 > TM0 . To find Lp given Lv , we use (3.36) and the integrability condition, r ^ Lv ; Lp D 1; of the first law dE D .Lv p/ dV C Lp dp: For upon introducing (3.36) into the first law the resulting integrability condition is @Lp @Lv 1D : @p @V

(3.49)

According to Holtzmann’s hypothesis, (2.39), an ideal gas absorbs heat equal to the amount of work it does in an isothermal expansion, or Lv D p. Hence, Lp is independent of the volume. However, for a photon gas Lv D 4p, and integrating the integrability condition (3.49) we find Lp D 3V , where we have set the arbitrary function of the pressure that results from integration equal to zero because Lp D 0 for vanishing volume. If we use D 1=T as the integrating factor, Carnot’s grand principle in differential form gives (3.36). Rather, if we employ D V s , then Carnot’s principle gives 3s D 1. This determines the adiabatic exponent for blackbody radiation.

3.3 A Clash of Inequalities

63

3.3 A Clash of Inequalities Had Carnot lived until 1850 he would have been 54. Even before his untimely death in 1832, his notes clearly show that he was well aware of the interconversion of heat into work. As we have seen in Sect. 2.3, Carnot even went so far as to calculate the mechanical equivalent of heat, which has fallen into oblivion because there is no longer a distinction between mechanical and thermal units. This and the second law could have been all of Carnot’s own doing and not that of Clausius. But, we have no reason to believe that Carnot would have gone further than this. Beginning in 1854, Clausius began moving into totally unexplored territory by investigating processes – irreversible ones – in which the maximum possible work was not achieved. The idea behind Clausius’s inequality is that irreversible processes should decrease the amount of work that the engine can perform. This was also recognized by Carnot, but it was not quantified by him. Apart from a sign, Clausius calls I dQ DN (3.50) T the “uncompensated” transformation occurring during a process, which is always negative unless it reaches its limiting value, 0, when the process is reversible. Consider a temperature T2 , which is the temperature at which the system rejects a quantity of heat Q2 to the reservoir. Since this temperature is constant, the quantity Q2 =T2 can be taken out of the integral in (3.50), which can then be written as: Z Q1 dQ Q2 D N; (3.51) T T2 0 where the negative sign takes into account that heat is being given up to the reservoir. Solving for Q2 , Z Q1 dQ T2 N; Q 2 D T2 T 0 and introducing it into W D Q1 Q2 , Clausius finds Z W D Q 1 T2

Q1 0

dQ C T2 N; T

(3.52)

which is the amount of work that the system can do. Now, assume that the whole process is reversible, says Clausius. Then Z Wrev D Q1 T2

0

Q1

dQ T

(3.53)

differs from (3.52) by the term T2 N . Since T2 > 0 and N < 0, the greatest amount of work that can be accomplished is when the process is reversible, and “every process which causes changes in a cyclical process that are not reversible diminishes the amount of work.”

64


Considering one boiler and one condenser, the inequality T2 Q1 ; W 1 T1 together with the first law over a cycle, W D Q1 Q2 , implies Q 2 T2

Q1 : T1

That is, more heat is given up to a reservoir at the temperature T2 . If adiabatic equilibration occurs at the temperature T2? then Q2 D Q2? D T2?

Q1 Q1 T2 D Q20 ; T1 T1

(3.54)

where Q2? is the heat given up at T2? T2 , and is necessarily greater than the heat Q20 that is given up to the condenser at temperature T2 . The inequality in (3.54) shows that in Clausius’s uncompensated transformations (3.50), the heat rejected to the condenser and the temperature of that reservoir may not coincide with the temperature of the system at which heat is rejected. More heat may be given off to the condenser than when the temperatures of the system and condenser are equal. The temperature appearing in (3.50) is always that of the reservoir. Enter Tait. Tait (1868) defines the “practical” value of the heat as T1 T2 dQ; T1

(3.55)

where T1 and T2 are the temperatures of the boiler and condenser, respectively, and dQ is the quantity of heat absorbed. Splitting the two terms, and integrating over an entire cycle, Tait gets the work performed as I W D Q 1 Q 2 T2

dQ : T

(3.56)

Now, if the cycle is reversible, the practical value of the heat is Q1 Q2 , which is just the work performed by the engine in a cycle. This is guaranteed by the first law, and it necessitates that I dQ D 0; (3.57) T which Clausius would fully agree with. But, what Tait implies here is that if the process is not reversible, the first law would be compromised. Tait continues But in general this integral has a finite positive value, because in non-reversible cycles the practical value of the heat is always less than [Q1 Q2 ]. Hence, the amount of heat H is lost needlessly, i.e., otherwise than to the refrigerator, or in producing work, is [T2 dQ=T ].

3.3 A Clash of Inequalities

65

This is Thomson’s (1852a) expression for the amount of heat dissipated during a cycle. It is, of course, an immediate consequence of his important formula for the work of a perfect engine.

Tait refers to Sect. 175 of his book. Let’s see what it says. Let Q and Q C dQ be the quantities of heat given off and taken in by a working substance at temperature T . According to the first law, the work done is dQ. In Tait’s first law the increment in the internal energy is missing, and his expression of the second law stems from Carnot’s equation, dW D

Lv dp dT dV D dV dT: dT T

(3.58)

Both expressions give the increment in the work, and so must be equal dQ D Q1 dT =T;

(3.59)

R where Q1 D Lv dV is the latent heat of expansion at the temperature of the hot reservoir. Rather than specifying what heat this is, Tait leaves it generically as Q and integrates (3.59) to obtain the heat as a linear function of the temperature. But, according to Carnot’s equation (3.58), this must be a finite heat if (3.59) is to make any sense. Consequently, integration leads to the unacceptable equation (3.56). The integration of infinitesimals to obtain finite quantities is subtle, if not deceptive! The temperature T appearing in the integrand of (3.56) is, according to Tait’s own words, “the temperature of the hot body,” which has the heat dQ. There is no sense in splitting the two terms in his “practical” value of the heat, (3.55), and integrating over a cycle. This did not pass the attention of Clausius, for in a letter to Thomson published in the Philosophical Magazine, Tait [179] writes that “I must introduce the subject by a reference to the comments made by Prof. Clausius upon a somewhat slipshod passage (178) of my little work on Thermodynamics. That passage refers to the integral Z dQ ; T to which I believe Rankine first called attention, but which is essentially connected to your doctrine.” The doctrine Tait is referring to is Thomson’s minimum dissipation of energy. Tait then admits to his error: I cannot altogether complain of Prof. Clausius’s comments, because I cannot account for my having called the above integral (in a way in which I have employed it) a positive quantity, except by supposing that in the revision of the first proof of my book I had thoughtlessly changed the word “negative” to “positive.” This might easily happen from my having used a novel term, “practical value,” in a somewhat ambiguous manner, at one place confounding it with “realized value.”

66


Tait then goes on to ask for Thomson’s intervention in the feud with Clausius by observing that there is a graver matter involved than any such mere slips of the pen; for Prof. Clausius asserts that the method I employ (and which I certainly obtained from your paper of 1852) is inapplicable to any but reversible cycles.

Thomson [1879a] rejoins by saying that “It [the doctrine] is certainly not confined to reversible cycles; but, on the contrary, it gives an explicit expression for the amount of energy dissipated, or as I put it, ‘absolutely and irrecoverably wasted,’ in operations of an irreversible character.” Thomson tried to supplant “entropy” by a word expressing the “availability” for work of heat “in a given magazine.” The word would have to express that quantity, which when wasted resulted in “dissipation.” In other words, he was looking for the antonym of entropy, undoubtedly as not to give Clausius the upper hand. Thomson [1879b] proffered the word “motivity,” but, like his formulation, it did not catch on, and in the end Clausius, with his “mechanical interpretation of heat,” prevailed.

3.4 Carnot’s Real Efficiency Resurrected Today, the Carnot efficiency is universally accepted as C D 1 T2 =T1 , and Carnot’s grand principle in integral form is as follows: W D Q1 C ;

(3.60)

where Q1 is the heat absorbed at the boiler, which is at temperature T1 . But, this is not the only form of the efficiency found by Carnot. In Carnot’s own words: If we use u to represent the motive power produced by the fall of a unit of quantity of heat from the temperature t to 00 , it follows (since. . . u must depend solely on t ) that u can be expressed by a function F .t /, and hence u D F .t /.

He then calculates the change in du as F 0 .t/dt, and when multiplied by e, “the amount of heat used in order to keep the temperature of the gas constant during the expansion” finds the motive power instead of that for a unit quantity of heat. He has already found the corresponding motive power to be ır D N ln v dt, where N D P =267, and P is the expansive force of air with volume 1 at a temperature 0ı . Carnot equates the two motive powers and obtains the heat as e D .N=F 0 .t// ln v. The prefactor depends only on t, and Carnot assumes “that the specific heat remains constant at all temperatures.” This means that N=F 0 .t/ D C t C C1 , where C is the constant appearing in the prefactor of the heat capacity at constant volume, and C1 is an integration constant. Carnot thus concludes F 0 .t/ D

N ; C t C C1

3.4 Carnot’s Real Efficiency Resurrected

and integrating finds

67

t ; F .t/ D A ln 1 C B

(3.61)

where A and B are two constants, the latter chosen so that F .t/ vanishes when t D 0ı . According to Truesdell, Carnot’s result (3.61) is “so objectionable as to have caused Carnot’s theory to be rejected, had they been perceived.” Since e is the quantity of heat absorbed at the boiler, it would be sufficient to raise t to get an indefinite amount of work from the engine. It is also considered common knowledge that the second law is insensitive to the heat rejected at the condenser, and it is only the first law that brings it in. Yet, all these statements are not foregone conclusions. We remarked above that there is a problem of interpreting Carnot’s global principle in that it is not always given by (3.3) on integrating the increment in work dW D .@p=@T /dV dT . As a preliminary remark, we observe that Q2 Q1 D T1 T2

(3.62)

is an adiabatic condition. The temperature in the denominators of (3.62) reduces the numerators to adiabatic conditions. For an ideal gas it is .V2 =V1 / D .V3 =V4 /, while for a photon gas the condition is T13 .V2 V1 / D T23 .V3 V4 / . To avoid unlimited work in the heat death, we calculate the work in terms of the heat rejected to the condenser and show that maximum work is given by (3.3). Into the expression for the work Z T1 Z V3 dT T1 D Q2 ln ; Lv dV (3.63) W D T T 2 V4 T2 we insert the adiabatic condition (3.62), and get

where the efficiency,

W D Q1 ;

(3.64)

D x ln .1=x/ 1 x;

(3.65)

is a concave function of the temperature ratio x D T2 =T1 . According to inequality (3.65), (3.64) is the nonmaximal work that the engine is capable of doing over a cycle. We can generalize (3.63) by considering two sets of n reservoirs, where heat is absorbed from QiC and rejected at Qi . Introducing (3.62) into (3.63), and summing give W D

n X

Qi ln

i D1

n X i D1 n X i D1

Qi

ln

QiC Qi n X i D1

QiC

n X i D1

QiC

X n

! Qi

i D1

Qi D Wmax ;

68


where the inequality in the second line is Jensen’s for a concave function. Jensen’s inequality for a convex function states that if all the numbers are positive then (Hardy 1952, p. 97 thm 117) x ln

x y xCy C y ln .x C y/ ln ; a b aCb

(3.66)

where the equality applies if and only if x=a D y=b, and .x ln x/00 > 0, i.e., the function is convex. For a concave function the inequality is reversed. We can also say something about irreversibility by considering the heat given up to the condenser when the heat absorbed by the boiler is the same in the two cases. Consider n cycles of a reversible engine that are required to extract a quantity of heat equal to n0 cycles of an irreversible engine, viz., n X

0

QiC

D

i D1

n X

QiC0 ;

(3.67)

i D1

which occurs at temperature T1 . The heat given up to condenser will, however, be different in the two cases. In the irreversible case, the work will be 0

0

W D

n X

0

QiC0

i D1

n X

Qi0

(3.68)

Qi :

(3.69)

i D1

while in the reversible case, W D

n X

QiC

i D1

n X i D1

The net work done is the difference between (3.68) and (3.69), 0 W D @

0

n X

Qi0

i D1

n X

1 Qi A :

(3.70)

i D1

To simplify matters, assume, without lossP of generality, that the same quantity of heat is extracted in each of the cycles so that niD1 QiC D nQ1 , and the same being P 0 true for the heat rejected, ni D1 Qi0 D n0 Q20 . Condition (3.67) thus becomes n0 Q10 D nQ1 ; and when this is inserted into (3.70) we find

Q20 Q2 W D 0 Q1 Q1

nQ1 :

3.4 Carnot’s Real Efficiency Resurrected

69

If this were positive, we could get more work in an irreversible process than a reversible one. Since this is impossible, we must have Q2 Q20 ; 0 Q1 Q1 which means that the irreversible cycle gives up more heat to the condenser at temperature T2 than the reversible one. Clausius’s fundamental inequality can also be derived from the complementary efficiency T2

D .1 x/ ln.1 x/ x D (3.71) T1 of the Carnot cycle by employing Kelvin’s postulate. The maximum work Wmax D

T2 Q1 T1

is equal to the heat absorbed Q2 from the condenser at temperature T2 . If we consider a set of n cycles, then Wmax D Q2 D

n X

Q2;i D T2

i D1

n X Qi i D1

Ti

;

(3.72)

whose net effect is to supply a quantity of heat Q2 at a uniform temperature T2 . However, this is in contradiction with Kelvin’s postulate that it is impossible to convert a quantity of heat at a uniform temperature all into work.

The only way out is to consider Q2 < 0, and, consequently, the work Wmax is negative. Since T2 > 0, we conclude from (3.72) that n X Qi i D1

Ti

0;

(3.73)

which is Clausius’s inequality. If the cycle is reversible, we can traverse the cycle in the opposite direction. This has the effect of changing all the signs of Qi , with the consequence that n X Qi i D1

Ti

0:

(3.74)

But, if Clausius’ inequality (3.73) is to hold at the same time as inequality (3.74) then it must be that the equality holds. Thus, for all reversible processes (3.57) holds in which every infinitesimal change of heat is accompanied by a change in the temperature. This is necessary in order for the cycle to be reversible. However, if the

70


cycle is irreversible the temperature of the engine that absorbs the heat need not be at the same temperature as the boiler. However, in this case there can be no criterion for reversibility, which is obvious from the fact that heat will flow spontaneously from hot to cold. This fact was not appreciated by the inventors of the endoreversible engine, and we will come back to this point in much greater detail in Sect. 4.2.4.

Chapter 4

Equivalence of First and Second Laws

4.1 Laws of Thermodynamics: Local and Global Forms Building on Carnot’s foundation for the determination of motive power in steam engines, Clausius introduced two functions of state and built around them the first and second law of thermodynamics. Perhaps due to his desire to formulate a “mechanical theory of heat,” he constructed the first law as a generalization of the conservation of energy in mechanics, resulting in the definition of the internal energy, and allowed for violations in the definition of the entropy as a function of state to be the indicators of the irreversibility of what Clausius termed “uncompensated transformations.” In this chapter, we will show that Clausius’s choice is, to a large extent, a matter of taste: The entropy function can be preserved, and violations in the internal energy can determine the “ordering of states” when less than maximum work is performed by the engine. We can say that Carnot knew the second law, but was ignorant of the first law. This is why he had to introduce his general and special axioms in order to determine the work done. In contrast, Clausius knew the first law, but allowed for violations in the second law to accommodate the possibility of wasted heat given off to the cold reservoir. Whereas both formulations attribute an increase in heat rejected to the cold reservoir as the origin of irreversibility, or the wasted portion of work that the engine was capable of doing, they do so in different ways. By reformulating Carnot’s general and special axioms, it will be possible to obtain an analytical expression for this wasted heat, whereas Clausius can only derive an inequality for his “uncompensated” transformations. We begin by formulating the local the first and second laws in terms of exterior calculus that we introduced in Sect. 3.1. The “natural” state plane is the temperature, volume, T; V -plane, as Carnot was the first to realize. The state vector, dr, has therefore the components .dT; dV /. Fundamental to all of thermodynamics is the calorimetric equation of state which expresses the heat, Q1 , a 1-form, in terms of the heat covector, H, with components .Cv ; Lv /, where Cv and Lv are the heat capacity at constant volume and the latent heat, respectively. Consequently, the calorimetric


71

72

4 Equivalence of First and Second Laws

equation of state, or what we have referred to as the doctrine of latent and specific heats, can be written as Q1 D H dr D Cv dT C Lv dV:

(4.1)

Recall the first law that introduces the notion of a work covector, F , with components .0; p/, where p is the pressure. The work done by the system, W 1 , also a 1-form, is given by (4.2) W 1 D F dr D p dV: With the aid of the heat and work covectors, the first and second laws can be formulated as r ^H Dr ^F (4.3) and r ^ H=T D 0 or r ^ V s H D 0; or r ^ H D T H ^ rT 1 ; r ^ H D V s H ^ rV s :

(4.4)

In fact, the second law (4.4) just specifies the curl of the heat vector, and any term which when multiplied by T to form an adiabatic invariant will do just as well in any other state plane containing T as a coordinate axis. Eliminating the curl of the heat covector between (4.3) and (4.4) yields the Clausius–Clapeyron equation r ^ F D T H ^ rT 1

or

r ^ F D V s F ^ V s ;

(4.5)

which Clausius needed, but not Carnot. Clausius formulates the first law (4.3) so that Carnot’s axiom would be deducible from it, and it would not appear as “independent principles in the theory of heat.” In component form, (4.3) is @Lv @Cv @p D ; (4.6) @T @T @V which when multiplied by the element of area, dV dT , and integrated over the limits of temperature and volume of the engine express the work in terms of the circulation in the heat current ZZ I @p W D dV dT D Q1 : (4.7) A @T C Recall that the path, C is transversed in the clockwise direction enclosing the area A according to Green’s theorem (2.33). The second laws (4.4) are given in component form as r^HD

Lv sCv D : T V

(4.8)

4.2 Carnot’s Modified Axiom and His Criterion for Irreversibility

73

It would appear that the second equality, unlike the first, depends on the specific nature of the working substance. This is contrary to Carnot’s axiom that the work done by the engine should be independent of the working substance. However, since sCv D R, the universal gas constant, the second inequality does not, indeed, violate Carnot’s axiom. Integrating (4.8) over a cycle gives I

Q1 D

ZZ

C

A

Lv dV dT D T

ZZ A

sCv dV dT: V

(4.9)

Whereas Clausius formulates the first law so as to deduce Carnot’s axioms, we will modify Carnot’s axioms to define a nonmaximal work function, which is bounded above by Carnot’s expression. This saves the second law and allows the energy balance equation to account for irreversibility.

4.2 Carnot’s Modified Axiom and His Criterion for Irreversibility Carnot [1824] states his general axiom in the following way: The motive power of heat is independent of the working substances that are used to develop it. The quantity is determined exclusively by the temperatures of the bodies between which, at the end of the process, the passage of caloric has taken place.

Carnot’s special axiom says The amount of work done by the engine is a product of the heat and a function of the two temperatures only.

Carnot does not specify whether the heat is that which is absorbed at the hot reservoir, Qh , or that rejected to the cold reservoir, Qc , since, according to caloric theory, the two are one and the same. Truesdell (1980) is correct in saying that Carnot’s general and special axioms are equivalent for engines working between infinitesimal temperature differences, but they fail to be equivalent when the temperature differences are finite. We will use this lack of equivalency for finite temperature differences to our advantage so as to transform the global expressions for the second laws into inequalities that comply with Carnot’s special axiom. In modified form, Carnot’s axiom can be stated as: The amount of motive power (work) is a product of the heat given off to the cold reservoir and either: (i) a function of the extreme temperatures which cannot depend upon their absolute values, or (ii) a function of the extreme values in the volume, which, again, cannot depend on their absolute values but only on their ratios.

The former condition is necessary in order that the work does not increase without bound when the high temperature, or maximum volume, is allowed to increase without limit. The latter conditions require the variables to appear only in a ratio.

74


Consequently, Carnot’s modified special axiom can be formulated as: W D F .x=y/Qc :

(4.10)

The function F .x=y/ obeys the functional equation F .x=y/ D F .x/ F .y/;

(4.11)

where if x D y, F .1/ D 0, so that a fall in temperature, or increase in volume, is necessary in order to produce motive power. If x D 0 (or y D 0), F .0/ D 0 would be the only solution. The solution to the functional equation is F .x/ D ln x for all values of x ¤ 0. The logarithm saves the universality of Carnot’s modified axiom, for, otherwise, the adiabatic exponent, s, of the volume would appear independently of the heat capacity, Cv . Since both depend upon the specific nature of the working substance, they would violate Carnot’s axiom. In other words, the heat released to the cold reservoir does not depend upon the details of the cycle, or the working substance, once the extreme temperatures, or volumes, are fixed. If the temperature is chosen, the natural cycle is Carnot’s, while if the volume is chosen, the Otto cycle appears as the natural one.

4.2.1 Carnot Cycle The well-known Carnot cycle, shown in Fig. 4.1 consists of four branches: 1. 1 ! 2: An isothermal absorption of heat Qh at the high temperature, Th . 2. 2 ! 3: An adiabatic expansion which reduces the temperature from Th to Tc . p

Th 1 Q>0

Tc Q=0

2 4

Q=0 3 Q 0;

(4.13)

for any positive m. Inequality (4.13) is very easily proven by a general method that determines the extremum of a function (Hardy 1952), which, in this case is .x/ D ln x x C 1. This function has a maximum at x D 1, where it vanishes. Consequently, .x/ 0 for all values of x > 0. It, therefore, follows that the nonmaximal work done by the engine is W D Qc ln

Th : Tc

(4.14)

Using the reversible amount of heat given up to the cold reservoir, Qc D Tc Qh =Th ;

(4.15)

76


(4.14) is seen to have the upper bound W D Qh

Tc Th Tc ln 1 Qh ; Th Tc Th

(4.16)

as a consequence of the inequality (4.13). This unambiguously shows that Carnot “knew the second law, without knowing the first” – contrary to what Truesdell (1980) claims. The last term in (4.16) defines the Carnot efficiency

C D 1

Tc ; Th

(4.17)

so that the middle term defines a nonmaximal efficiency

D

Tc Th : ln Th Tc

(4.18)

Table 4.1 compares Carnot, (4.17), the nonmaximal, (4.18), and the observed, efficiencies of three power plants. The nonmaximal efficiency comes much closer to the observed efficiency than the Carnot efficiency. As a first approximation to the efficiency (4.18), we may take the arithmetic average of its upper and lower bounds, Th Th Tc Th Tc ln ; T1 Tc Tc namely, ln

(4.19)

T 2 Tc2 Th h ; Tc 2Th Tc

(4.20)

which becomes better the closer Tc =Th is to 1. The work that the engine does is approximately W approx Qh ; (4.21) where

approx D

T2 1 1 c2 : 2 Th

(4.22)

Table 4.1 Nonmaximal, Carnot, and observed efficiencies Power source T2 (K) TG (K) T1 (K) TA (ı C) (%) (Carnot) (%) (Obs) (%) W Thurrock (Spalding 1966) CANDU (Griffiths 1974) Larderello (Chierice 1964)

298.15 499.9

838.15 568.15

36.8

64

36

298.15 413.38 573.15 162.5

34

48

30

353.15 429.82 523.15 165

26.5

32

16


77

The smaller the ratio, x WD Tc =Th , the greater the efficiency (4.18), which is a concave function of x, and, hence, is an object that is to be maximized. We may use the property of concavity to establish The greatest amount of work that can be done is the difference between the heat absorbed at the hot reservoirs and the heat rejected at the cold reservoirs, independent of whether the cycle is closed or not.

Denoting the heats absorbed at the hot reservoirs, Qh i , and the heats rejected at the cold reservoirs, Qc i , the work done over a cycle is W D

n X

Qc i ln

i

n X

Qc i ln

Qh i Qc i n X

i

Qh i

i

n X

Qh i

i

n X

n .X

Qc i

i

Qc i D Wmax :

(4.23)

i

The inequality in the second line of (4.23) is Jensen’s for a concave function, while the inequality in the third line follows from inequality (4.13). According to Carnot, the internal energy will exist as a function of state only when the maximum amount of work is performed during the cycle. That is, the change in the internal energy will vanish, and it is for this reason we say that Carnot was ignorant of the first law. In analogy with the first law – but without requiring the exactness of dE – we may construct the energy balance equation W D

n X i

Qh i

n X

Qc i E;

(4.24)

i

where E 0

(4.25)

for all irreversible transitions. Consequently, if Clausius salvaged Carnot’s theory by merely rejecting the conservation of heat, he could not have considered a general conservation of energy independent of the limitations set upon the engine. If state b is accessible from state a then Eb Ea > 0;

(4.26)

asserting that the only way a state can be reached is if work is expended in getting there. All other mechanical processes which exchange work and energy do not enter into Carnot’s formulation of engine efficiency. Moreover, any adiabatic process that is not coupled to a pair of isothermal, isochoric, or isobaric ones would violate

78


inequality (4.26). In other words, there can be no work done by any thermodynamic system that does not operate between two different temperatures, two different volumes, or two different pressures in a cycle. If the heat absorbed at the hot reservoir were used to calculate the work, there would be no correspondence between W and .Qh Qc /, and the need for giving off heat to the cold reservoir. It was caloric theory that required the cold reservoir for heat conservation. When it was realized that heat could be converted into work by Joule, Mayer, and Holtzmann, it only diminished the quantity of heat given off to the cold reservoir – but did not obliterate its importance! The accessibility of any state from a given one is limited by E Wmax :

(4.27)

From the balance of energy for nonmaximal work, W D Qh x ln x D .1 x/Qh E;

(4.28)

it follows that 0 E D 1 x C x ln x Qh .1 x/2 Qh D C Wmax :

(4.29)

The first inequality follows from 1 x x ln x;

(4.30)

while the second inequality follows from (4.13). Inequality (4.29) establishes a limit of accessibility of a state from a given one in terms of the maximum work performed by the engine.

4.2.2 Otto Cycle The Otto cycle, shown in Fig. 4.2, consists of four strokes: 1. 2. 3. 4.

1 ! 2: The gas in the cylinder is compressed adiabatically from Vmax to Vmin . 2 ! 3: Combustion modelled as heat absorption at constant volume, Vmin . 3 ! 4: The hot gas expands adiabatically from Vmin to Vmax . 4 ! 1: The pressure valve opens and temperature drops, modelled as heat rejection at constant volume, Vmax .

The second law is given by the second equality in (4.8) integrated over the limits in the T; V -plane Z T3 Z Vmax I sCv 1 dV dT: (4.31) Q D V C T2 Vmin

4.2 Carnot’s Modified Axiom and His Criterion for Irreversibility Fig. 4.2 Otto cycle

79

S

3

4

2

1

Vmin

Vmax

V

Now, the heat capacity is either a constant, as it is for an ideal gas, or a function of the adiabatic parameter, z D T V s , as it is for a degenerate gas, so that (4.31) can be converted into the inequality I

1

Z

Q D C

Z

T3 T2

Cv dT

Vmax

Vmin

dV s V

Z

T4

T1

Cv dT ln

V

max

s

Vmin

:

(4.32)

The first term on the right-hand side of (4.32) is the heat absorbed at constant volume, Qh , which models the combustion process. In order that work be done, this must be greater than the heat given off at constant volume, Qc . This is expressed by the inequality in (4.32). Consider the gas to be ideal, Cv D const. The adiabatic conditions s s T3 D Vmax T4 Vmin

s s Vmin T2 D Vmax T1

and

(4.33)

give the ratio of the temperatures as T3 =T2 D T4 =T1 , and hence, with the aid of inequality (4.13), we get the work done by the piston, according to Carnot’s modified special axiom, over a cycle as W D Cv .T4 T1 / ln

V

max

s

Vmin .T4 T1 / s s V Cv V max min s Vmin D O Qh D Wmax ;

(4.34)

where Qh D Cv .T3 T2 /, and the efficiency of the Otto cycle is

O D

V s Wmax T1 min D 1 D1 : Qh Vmax T2

(4.35)

The ratio, Vmax =Vmin , is known as the compression ratio (Cravalho 1981) for the engine.

80


This should be compared with the usual method of calculating the maximum work (Cravalho 1981). The heat absorbed as the engine goes from stage 2 to 3 is Qh D Cv .T3 T2 /. The heat rejected on going from 4 to 1 is Qc D Cv .T4 T1 /. Hence, the maximum amount of work that the Otto engine can perform is Wmax D Qh Qc T1 D C v T3 T2 1 T2 V s min D C v T3 T2 1 ; Vmax

(4.36)

which is precisely (4.34). The Otto engine is comparable to Carnot’s cycle. Using the adiabatic relations between the temperatures, and sCv D R, the work given in (4.34) can be made to read V s min W D

C Qh ; (4.37) Vmax where the Carnot efficiency is given by

C D 1

T2 T1 D1 ; T3 T4

(4.38)

and the heat absorbed in the isothermal expansion at the temperature of the hot reservoir, T3 , is V max : (4.39) Qh D RT3 ln Vmin

4.2.3 Brayton Engine The Brayton engine consists of two adiabats coupled by two isobaric curves, as shown in Fig. 4.3. The four strokes are: 1. 2. 3. 4.

1 ! 2: Adiabatic compression of fuel and air 2 ! 3: Heating by fuel combustion at constant pressure 3 ! 4: Expansion of the gas 4 ! 1: Rejection of the gas to the atmosphere

The 1-form of the heat will be given in terms of temperature and pressure as independent variables, (4.40) Q1 D Cp dT C Lp dp;

4.2 Carnot’s Modified Axiom and His Criterion for Irreversibility Fig. 4.3 Brayton cycle

81

S 4

3

1

2

p1

p2

p

where Cp is the heat capacity at constant pressure and Lp < 0 is the latent heat with respect to the pressure. The integrating denominator for (4.40) is p s=.sC1/ , and this gives a heat current Z

Q0 D

Z

C

pmax pmin

Z

T3

T2

s Cp dpdT: sC1 p

(4.41)

Again, we see that this is a universal expression since sCp =.s C 1/ D R. We can convert (4.41) into an inequality by replacing the heat absorbed in step 2 ! 3 by the heat rejected in step 4 ! 1 Z

Z

0

Q C

T4 T1

Cp dT ln

p

max

s=.sC1/ :

pmin

(4.42)

Specializing to an ideal gas with constant heat capacity, the work done over a Brayton cycle is p s=.sC1/ max W D Cp .T4 T1 / ln : (4.43) pmin Using the fundamental inequality (4.13) and the adiabatic conditions p

max

s=.sC1/

pmin

D

T1 T4 D T2 T3

(4.44)

gives the maximum work Wmax D B Qh , where the efficiency,

B D 1

p

max

pmin

s=.sC1/ ;

(4.45)

is given in terms of the pressure ratio (Cravalho 1981), and the heat produced by fuel combustion is Qh D Cp .T3 T2 /: (4.46)

82


Here, again, the roles have been inverted compared to Carnot’s cycle: The heat produced, (4.46), is given in terms of the ideal gas rather than the logarithm of the ratio of volumes, while the efficiency, (4.45), is expressed in terms of the pressure ratio, rather than the ratio of volumes under isothermal expansion.

4.2.4 Endoreversible Engine With the scope of determining real efficiencies that are realized in actual power plants, Curzon and Ahlborn (1975) published a highly influential paper that was eventually to become a new branch of thermodynamics called “finite-time thermodynamics.” What they did was to replace the isothermal expansion and compression steps in Carnot’s cycle by heat fluxes that are created by differences in temperatures between the working fluid and the temperatures of the boiler and condenser. The internal engine operated in a completely reversible fashion, and all irreversibilities were relegated to the coupling of the engine to the reservoirs. Such an engine has been called “endoreversible” (Rubin 1979). The Curzon – Ahlborn engine in Fig. 4.4 distinguishes between the temperatures of the reservoirs and the temperatures of the working fluid at which heat is absorbed and rejected. The temperature of the boiler is T1 and a quantity of heat Q1 is absorbed at the fluid temperature, T1w < T1 , where the subscript w stands for “warm.” However, we already know that according to Carnot’s argument, the heat engine will absorb a quantity of heat that is inferior to Q1 . Likewise, the temperature of the working fluid, when it comes in contact with the condenser, is T2w , which is higher than the temperature T2 of the condenser.

T T1

hot reservoir

Tw1 endoreversible engine

T2w T2

cold reservoir s

Fig. 4.4 Endoreversible engine


83

The cycle, shown in Fig. 4.4, consists of four steps: 1. The working fluid at the high temperature T1 is cooled to a temperature, T1w , at a constant minimum volume, Vmin . 2. By an adiabatic expansion, it is further cooled to the lowest temperature T2 where it reaches a maximum volume, Vmax . 3. The working fluid is heated isochorically to an intermediate temperature, T2w . 4. It is finally compressed adiabatically to reach the initial, high temperature, T1 . The simplest assumption for the kinetics of the heat transfer is that the heat transfer rate QP 1 is proportional to the temperature difference .T1 T1w / between the hot reservoir and the working fluid, viz., QP 1 D ˛.T1 T1w /;

(4.47)

where ˛ is the product of the area of the vessel and the thermal conductance (the inverse of the thermal resistance) per unit area of surface perpendicular to the direction of heat transfer. ˛ has units of kilowatts per degree Kelvin. Because steam engines involve the bulk flow of matter under steady flow conditions, we prefer, to make it more compatible with what follows, to use the capacity rate mc P v (Cravalho 1981a) as the coefficient of proportionality. This is to say that if the fluid can be modelled as an ideal gas, the change in energy due to heat transfer rate is a linear function of the temperature difference in the heat exchanger. The units of the capacity rate are the same as ˛. Curzon and Ahlborn associate the duration of the isothermal expansion, t1 , with the time of the flow, and multiply both sides of (4.47) by this time interval to obtain the total heat transfer Q1 D ˛t1 .T1 T1w / D m1 cv .T1 T1w /;

(4.48)

where m1 is the amount of mass transported in the time interval t1 . In the same way Curzon and Ahlborn assume a temperature difference between the working fluid and the cold reservoir to create an outward heat flux Q2 D ˇt2 .T2 T2w / D m2 cv .T2 T2w /:

(4.49)

Again, the heat rejected is assumed to be proportional to the difference in temperature between the working fluid and the temperature of the lower reservoir. We have set ˇt2 equal to the heat capacity, m2 cv , where m2 is the product of the mass flow rate, m P 2 , and its duration t2 . We know only their product, not how small or large the flow rate is or how large or small the time interval t2 is. Only their product is relevant, and m2 cv is determined by the ratio of the heat transferred, Q2 , to the difference in temperature .T2 T2w /. An adiabatic compression brings the working fluid back to its original state.

84


The crucial step in the Curzon–Ahlborn analysis is their condition for reversibility Q1 Q2 D : (4.50) T1w T2w Maximizing the power, P D

Q1 Q2 ; t1 C t2

(4.51)

with respect to the unknown intermediate temperatures, yields an efficiency, s

D1

T2 ; T1

(4.52)

which is less than the Carnot efficiency, (2.32). Their condition for maximum power (Curzon 1975, Eq. 9) is the first equality in T2w T2 D T1 T1w

s

˛T2 ˇT1

˛t1 T2w ˛t1 D D ˇt2 T1w ˇt2

s

T2 : T1

(4.53)

The second equality uses Curzon and Ahlborn’s condition for reversibility, (4.50), and the last equality expresses the ratio of the intermediate temperatures in terms of the reservoir temperatures. By equating the second and fourth terms in (4.53), Curzon and Ahlborn obtain the condition P 1 D .t1 =t2 /2 : ˇ=˛ D m P 2 =m

(4.54)

Furthermore, it has been claimed that the Curzon–Ahlborn engine has a net entropy gain (De Vos 1992) S D . C /

Q1 > 0: T2

(4.55)

So how can there be a criterion of reversibility (4.50) at the same time there is an increase in the entropy? It is therefore not unreasonable to ask what is the meaning of (4.50). From (4.54) there is a good reason to believe that something is amiss. We would have expected a relation such as P 1 t1 D m P 2 t2 D m2 D m; m1 D m

(4.56)

expressing the conservation of mass instead of (4.54). If the latter condition is used to evaluate (4.50), we obtain p (4.57) 2Ti w D Ti C .T1 T2 /; .i D 1; 2/


85

so that the sum,

p 1 .T1 C T2 / C .T1 T2 /; 2 is the sum of the arithmetic and geometric average temperatures. Introducing these values for the intermediate temperatures in (4.48) and (4.49) give p 1 p p (4.58a) T1 T1 T2 ; Q1 D mcv 2 p p p 1 Q2 D mcv T2 T1 T2 ; (4.58b) 2 T1w C T2w D

under the homogeneity condition (4.56). Their difference yields the maximum work output W D Q1 Q2 p 2 p 1 D mcv T1 T2 2 p 1

D mcv .T1 C T2 / .T1 T2 / ; 2

(4.59)

found by Curzon and Ahlborn from maximizing the power (4.51) with respect to the intermediate temperatures T1w and T2w whose optimal values are given in (4.57). The adiabatic conditions that relate the various temperatures to the extremum volumes are s s D T2w Vmax T1 Vmin

and

s s T1w Vmin D T2 Vmax :

(4.60)

Since the work cannot be superior to the heat transfer I

Q1 C

Z

T2w T2

Z

Vmax Vmin

V s sCv : dV dT D Cv T2w T2 ln max s V Vmin

(4.61)

Calling the right-hand side of (4.61) the work, its maximum value is T2 T2 D Cv T1 T1w G C T2 ; Wmax .T1w / D Q1 1 T1w T1w

(4.62)

which we take as a function of the unknown intermediate temperature, T1w . This temperature can be determined by requiring that the work, (4.62), be maximum (Leff 1987). The condition is p T1w D TG D .T1 T2 /; (4.63) which determines the optimal temperature of the engine as the geometric mean temperature. We know from the principle of maximum work that this is the lowest final temperature possible when the elements of an unevenly heated body are connected to

86


perfect reversible engines which do maximum work (Thomson 1853a). The expression for the maximum work D 2Cv .TA TG / D max Q1 max Wmax

is guaranteed by the arithmetic–geometric mean inequality, where s T2

max D 1 T1

(4.64)

(4.65)

is the Curzon–Alhborn efficiency, and the optimum heat flux into the system is Q1 max D Cv .T1 TG /:

(4.66)

We now inquire as to whether any work can be had from the heat rejected, and, in so doing, gain insights into the limitations of the second law. The work we want to consider is 2 T2w T2w T2 T2w W .T2w / D Q2 1 D Cv T2w T2 ; (4.67) C T1 T1 T1 and maximize it with respect to the intermediary temperature, just as we did with (4.62). The necessary condition for an extremum is: T2w D

1 .T1 C T2 / D TA ; 2

(4.68)

which results when perfect engines at different temperatures come into thermal contact and fail to perform any work. In order to perform work, the temperature must be lowered, which is possible by putting anew the system in thermal contact with the coldest of the surroundings objects at T2 . Introducing (4.68) into (4.67) gives the maximum work Cv 2 .T TG2 / D Q ; (4.69) W D T1 A where the efficiency and heat absorbed are

D 1 and respectively.

TA T1

Q D Cv .TA T2 /;

(4.70)

(4.71)


87

However, maximum work can always be expressed as a difference in heats, so (4.69) must be equivalent to: W D Qh Qc ; resulting in the reversibility condition: Qc Qh D TA TH

(4.72)

So if the temperature of the hot reservoir is Th D TA , the temperature of the cold reservoir will be the harmonic mean temperature, Tc D TH D TG2 =TA . Thus, in order to get work from a system that has already achieved a common temperature, TA , without any work being performed, it has to placed anew in thermal contact with the coldest of its surroundings. This echoes Kelvin’s original statement of the second law: It is impossible by means of an inanimate material agency to derive a mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.

To this we may add: The mechanical effect of a perfect engine is always inferior to that which would result from releasing heat to the coldest of its surroundings.

For, otherwise, we would have TH D T2 , and this would imply a vanishing temperature difference. Hence, we must always have TH > T2 if work is to be done by a fall in temperature. The efficiency and heat uptake are: TH TA

(4.73)

Q D C v TA ;

(4.74)

D 1 and

respectively. The final temperature TH , is a fraction, TG =TA , of the geometric mean temperature, TG . We can now determine the difference between the maximum and minimum work that the engine can perform. The greater the difference is the greater its capacity to do work. Letting x D T2 =T1 we can write Carnot’s expression for the work as W D Q1 x ln x D .1 x/Q1 E; (4.75) where the first term on the right-hand side is the Carnot expression for the absolute maximum work. In other words, the ratio of the change in energy to the heat absorbed, (4.76) E=Q1 D Œ1 x x ln.1=x/ D . C / ; equals the difference between the Carnot, or maximum, efficiency and nonmaximal efficiency (3.65) that we obtained by calculating the work by using the heat rejected to the cold reservoir.

88


The Flaw in the Endoreversible Engine There is a fatal flaw in the preceding analysis and it lies with (4.50): It is not the condition that the process be reversible. According to Thomson (1852b), a process is perfectly reversible when . . . the absolute values of two temperatures are to one another in the proportion of the heat taken in to the heat rejected in a perfect thermodynamic engine working with a source and refrigerator at the higher and lower of the temperatures respectively.

In symbols, Thomson claims that Q2 Q1 D T1 T2

(4.77)

is the true condition for the cycle to be reversible, which is not the Curzon–Ahlborn criterion of reversibility (4.50). We have already discussed how Thomson arrived at (4.77). Instead of (4.50), we should assume that an engine absorbs a quantity of heat Q1 at temperature T1 and emits the quantity of heat .TM =T1 /Q1 at any temperature TM < T1 , viz., TM Q1 : (4.78) Q2 D T1 Because there is a complete symmetry between the upper and lower reservoirs, we can equally well write TM Q2 ; (4.79) Q1 D T2 but, now, with the proviso TM > T2 . Equating the ratio of heats absorbed and rejected in (4.78) and (4.79) give the correct final temperature, TM D TG D p .T1 T2 /, and introducing the Curzon–Ahlborn criterion for reversibility (4.50) give T1w T1 TG D D : T2w TG T2

(4.80)

This fixes the ratio of the intermediary temperatures. The differences between their values and those of the two reservoirs is determined by introducing (4.57) into (4.48) and (4.49). We then obtain 1 mcv .T1 TG /; 2 1 Q2 D mcv .T2w T2 / D mcv .TG T2 /: 2

Q1 D mcv .T1 T1w / D

(4.81) (4.82)

The difference between the two quantities of heat, (4.81) and (4.82), is the maximum work (4.59). The work is always positive definite unless the two temperatures are equal; the inequality is a consequence of the arithmetic–geometric mean inequality. Several corrections to Tait’s Sketch (1868, pp. 101–102), which he attributes to Thomson’s (1853) paper, should be noted. Tait considers an element of mass dm of


89

an “irregularly heated body,” whose temperature is t and has specific heat c. Denoting T as the temperature to which “the whole body can be brought by means of perfect engines, so that all the heat lost is converted into work.” In the first equation on p. 101, Tait gives the adiabatic equilibration as Z 0D

Z

t

c dt

dm T

T : t

However, there should be no capital T in the integrand. As Tait correctly observes, this equation gives us a means of determining the final temperature T and anticipates the Cashwell–Everett paper by a hundred years! Tait then goes on to determine the work as Z Z t Z Z t t T dt D J dm c c dt; W D J dm t T T which he claims follows from the preceding equation. However, the integrand should just be c and not c.t T /=t. Here, he is confusing the work done J.Q Q0 / with JQ.t t0 /=t, where Q0 is the heat given out at the lower temperature t0 . An Isothermal Endoreversible Engine The heat conduction mechanism that Curzon–Ahlborn used for absorbing heat at the furnace and rejecting a smaller quantity at the refrigerator can be replaced by other mechanisms of heat absorption and rejection. Since the Curzon–Ahlborn engine absorbs heat by creating a difference in temperature between the boiler and the working fluid and rejects heat by a temperature difference between the working fluid and the condenser, it would, in Carnot’s opinion, absorb less heat than if the working fluid were at the same temperatures of the reservoirs of which they are in contact. We can replace this mechanism by a moving boundary which expands when in contact with the boiler and compresses when in contact with the condenser, at constant temperature, just as in the original Carnot cycle. However, both the internal energy and entropy are extensive quantities and are linear in the volume, and hence the first and second laws are incomparable (Lavenda 2005). In other words, no inequality results from means of the same order. The metrical entropy is additive, being a first-order homogeneous function, while all that is required of the “empirical” entropy is that it has the same value for all states that are accessible to it by quasistatic adiabatic transitions (Buchdahl 1966). What is required is another adiabatic potential that is not a first-order homogeneous function in order to allow us to compare means of different orders, and thereby establish the maximum property. Unlike the empirical entropy, the change in the metrical entropy, in a quasistatic transition of a composite system, is the sum of the changes of the entropies of the subsystems making up the composite system. For the ideal gas, the metrical entropy, S , is related to the empirical entropy z logarithmically S.z/ D mR ln z1=s ;

(4.83)

90


where R is the gas constant. The second law, mR d z mRT dz D D dQ; sz sV s

T dS.z/ D T S 0 .z/ dz D

(4.84)

shows that V s is also an integrating factor (Einbinder 1948; Lavenda 2005) for the heat [cf. p. 42], n

X mR dz D V s dQ D V s .Lvi dV C Cvi dT /: s

(4.85)

i

We make two simplifications: equal masses and assume the gas to be ideal, in which case Lvi D p, for each i , which is Holtzmann’s (1848) conjecture, the latter ensuring that the internal energy will be a function of the temperature alone. Then, adiabatic equilibration for an isothermal expansion, n Z VM n X X s (4.86) vs1 dv D T nVM Vis D 0; z D sT Vi

i

i

identifies the mean of order s, V Ms D

n 1 X

n

Vis

1=s ;

(4.87)

i

as the final volume of the composite system. The maximum heat absorbed in this isothermal expansion is Q D mRT

n Z X i

VMs Vi

where VG D

V dv Ms > 0; D nmRT ln v VG n Y

(4.88)

1=n Vi

(4.89)

i D1

is the geometric mean volume, G D M0 . Inequality (4.88) follows again from the fact that means are monotonically increasing functions of their order and s > 0. If n cells, initially at finite volumes V1 ; V2 ; : : : ; Vn were brought into mechanical contact, and left alone for an indefinite amount of time, the final, common, volume would be given by the geometric mean (4.89). Comparison of Endoreversible Engines In this paragraph, we compare the isochoric engine of Curzon–Ahlborn and the isothermal engine. The efficiency of the isothermal engine is

D1

VG V Ms

s :

(4.90)


91

The maximum work output W D Q1 D

s V s VM n s mRT1 1 sG ln s V Ms VGs

(4.91)

is the product of (4.90) and (4.88). In order to compare the two engines, we set n D 2 and use the adiabatic constraints 1 D V1s T1 D const. and 2 D V2s T2 D const. These adiabats convert the mean of order s for the volume, (4.87), into the inverse of the harmonic mean, H D M1 , for the temperature s VM D s

1 1 s 1 1 1 TA D .V1 C V2s / D C D 2; 2 2 T1 T2 TH TG

(4.92)

and the geometric mean of the volume, (4.89), into the inverse of the geometric mean of the temperature, VGs D

q

1 1 V1s V2s D p D : TG .T1 T2 /

(4.93)

Because we will be concerned only with their ratio, we have dispensed with the arbitrary constants in (4.92) and (4.93). Equation (4.93) highlights the very important point that the same adiabatic conditions that hold for the volume and temperature individually also hold for their means. The efficiency (4.90) can be expressed in terms of the ratio of the two mean temperatures as TG ; (4.94)

D1 TA which is always positive thanks to the geometric–arithmetic inequality. The maximum heat uptake, T 2 A ; (4.95) Q1 D mRTM1 ln s TG can be approximated for small differences between the arithmetic, TA , and geometric, TG , means by .TA TG / 2 Q1 ' mRTM1 ; (4.96) s TG where TM1 D T1 denotes the highest attainable temperature. Therefore, the approximate expression for the maximum work is W '

TM .TA TG /2 2 mR 1 : s TA TG

(4.97)

92


Table 4.2 Expression for the efficiencies of the Carnot, Curzon-Ahlborn, and isothermal engines

engine

T1 T2 Tp 1 Curzon–Ahlborn 1 .T1 T2 /=T1 Carnot

Isothermal

1

p .T1 T2 /= 12 .T1 C T2 /

The last factor has the form of Karl Pearson’s 2 D .TA TG /2 =TG , which is a measure of the deviation from the expected value, TG . In other words, 2 is expressed in terms of the “observed” frequencies, TA , and the “expected” frequencies, TG (Cramér 1946). For comparison purposes, we express the efficiency (4.52), heat uptake, (4.58a), and the work output, (4.59) of the Curzon–Ahlborn engine in terms of mean values,

D1

TG ; T M1

Q D mcv TM1

TA TG ; T1 TG

(4.98)

(4.99)

and W D mcv .TA TG /;

(4.100)

respectively. The efficiencies are shown in Table 4.2. For the West Thurrock coal fired steam plant (Spalding 1966), which has an efficiency of 36%, the Carnot efficiency is 64% and the Curzon–Ahlborn and isothermal efficiencies are 40% and 12%, respectively. This result is not surprising because the latter uses the average temperature of the two reservoirs as the higher temperature, as can be seen from (4.94), and the Curzon–Ahlborn engine uses the temperature of the hottest reservoir, which is apparent from (4.98). The former is clearly a very inefficient engine. The efficiencies are listed in Table 4.2. For an ideal gas, mcv D R=s, the heat uptakes of the two engines are roughly the same for the same power plant, and the work output of the isothermal engine is only 40% that of the Curzon–Ahlborn engine.

4.2.5 Stefan–Boltzmann Law from the Carnot Cycle In 1884 Boltzmann used a Carnot cycle to establish theoretically the empirical law found by Stefan for the dependency of the energy of thermal radiation upon the absolute temperature. Boltzmann envisioned a cylinder and piston containing thermal radiation. The piston moves without friction and both the walls of the cylinder as well as the piston are impervious to the flow of heat. There is an outlet at the side opposite to the piston, O, in Fig. 4.5, where heat can enter and leave.


93

C B

D A

A O

O

R1 T1

R2 T2

Fig. 4.5 The Carnot cycle according to Stefan and Boltzmann Fig. 4.6 Radiation falling obliquely on a surface

θ

A θ

B

O

Initially the piston is in contact with an isothermal reservoir at temperature T1 . If the outlet is opened, radiation will flow into the cylinder until there is the same energy density "1 in the cylinder as in the isothermal enclosure. To determine the pressure due to isotropic radiation, we consider radiation falling perpendicularly on a surface. If " is the energy density of the incoming waves, they will carry with them "=c units of momentum per unit volume. Thus, to each unit of area in unit time there will be c" units of energy per unit time and a momentum " per unit area per unit time. The pressure felt by the absorbing or reflecting surface is, therefore, ", since both pressure and energy density have the same dimensions. Now, if the radiation falls obliquely, at an angle with respect to the normal, the energy that crosses a unit area normal to the rays (AB in Fig. 4.6) has an area increased by an amount 1= cos than if they were falling normally on the surface (OB in Fig. 4.6). Moreover, the momentum suffers a decrease by an amount cos than if the rays were normal to the surface. Consequently, the radiation pressure is p D " cos2 : This radiation pressure must be averaged over an element of solid angle about the origin, O, 2 sin d , and, then to determine its mean value, this must be divided by half of the whole solid angle about O, which is 2. In this way, we find the average radiation pressure as Z pN D " 0

=2

cos2 sin d D

1 ": 3

(4.101)

94


However, Boltzmann did not arrive at (4.101) in this way. Rather, he reasoned that the energy density should be divided equally among the walls of the container, each wall receiving a pressure of "=3. To see why his reasoning was fallacious, he needed just to consider an ideal gas in which p D 23 "; that is, twice as large as the radiation pressure he was using. Surely, Boltzmann knew the result he was after! We may now consider the following cycle: 1. The piston starts at the initial position A in Fig. 4.7 with an initial volume V1 and pressure pN1 . The piston moves upward slowly until it reaches B, so as to heed Lazare Carnot’s warning that in order to determine maximum performance there should be no sudden changes or impacts, i.e., the process should be quasistatic. In order to maintain the radiation energy density constant, energy must be added through the opening, O, which places it in contact with reservoir R1 at temperature T1 . Just how much energy enters is governed by the following two considerations: Work is being done on the piston by pulling it up. If the temperature remains

at T1 , "1 , and likewise pN1 , will remain constant. Hence, the work will be W1 D

1 pN1 .V2 V1 / : 3

(4.102)

The interior volume of the cylinder has increased by an amount .V2 V1 /,

requiring an additional energy of "1 .V2 V1 /. Hence, the heat content that is added by radiation is 4 H1 D "1 .V2 V1 /: (4.103) 3 This is represented by the horizontal line AB in Fig. 4.7. Note that the isotherms are horizontal lines instead of being curves sloping downward as in the case of a perfect gas. However, this is still a perfect gas, but a perfect gas of a different kind, i.e., a radiation gas, and not a material gas that conserves particle number. The heat content added during the isothermal expansion is H1 , the enthalpy which must come from the external reservoir R1 in order to keep the working fluid at temperature T1 . p

T1

A

p1

B

dp p2

D

V1

C

T2

V2

V

Fig. 4.7 A Carnot cycle in the pV N -plane whose working substance is radiation


95

2. Upon arrival at B, the outlet O is covered by a perfectly reflecting tap thereby isolating the interior of the cylinder from the outside world. A further expansion to C is carried out. Without energy from the outside this work consumes energy from within: The energy density decreases from "1 to "2 , which is accompanied by a drop in temperature to T2 . Obviously, the pressure also decreases by a proportional amount. This adiabatic expansion is shown by the straight line BC in Fig. 4.7. If the expansion is a small one, we may replace the temperature difference, .T1 T2 /, by its infinitesimal, dT . This being the case, the change in the energy density ."1 "2 / may be replaced by d", and since (4.101) holds, we get dpN D

1 d"; 3

(4.104)

which is the change in the radiation pressure due to adiabatic expansion. 3. The working substance is now placed in contact with the isothermal reservoir R2 , which is at temperature T2 . The tap on the outlet is removed, in the third diagram in Fig. 4.5, and the piston is pushed downward until it reaches state D in Fig. 4.7. The compression serves to increase the density of radiation within the cylinder and since it must remain in equilibrium with the isothermal reservoir R2 , radiation must pass from the cylinder to the reservoir. But, because the compression is done so slowly the energy density is kept at "2 , for any change in the energy density would necessitate a corresponding change in the temperature. The heat content which leaves the engine is H2 , analogous to that which it absorbed at the higher temperature T1 . 4. Having reached D, which is determined by the ratio of the volumes – the only free parameter available – the radiation undergoes further compression which brings it back to its initial state A. The net effect has been to perform a closed cycle ABCD. If the pressure change N is small, the total work is approximately the area of the rectangle .V2 V1 /dp. Although this is not exactly true, because we are neglecting the two small triangles at the ends, it will be true asymptotically, viz., in the limit as dpN ! 0. The work accomplished by our engine will be dW D .V2 V1 / dpN D

1 .V2 V1 / d": 3

(4.105)

The efficiency of the Carnot engine is defined as usual

C D

dW T1 T2 dT : D D H1 T1 T

Inserting the value for the work, (4.105), and the expression for the heat content, (4.103), lead to 1 d" 3 D dT : 4 T1 "1 3

96


The subscripts are now superfluous and can be dropped since it must hold for all values of the energy density and temperature. The resulting ordinary differential equation d" dT D4 " T can be immediately integrated to give the well-known T 4 or, Stefan’s law " D aT 4 ; where a is a constant of integration. The emissive power of a blackbody, E, consisting of thermal radiation, is related to the energy density, " by " D 4E=c. Since the latter varies as T 4 so does the former, and we can write E D T 4 ; with another constant of proportionality, , known as the Stefan–Boltzmann constant, and fixes the constant of integration a D 4 =c.

4.2.6 Relativistic Carnot cycle In his first volume of Die Relativitätstheorie, von Laue (1919) proposes a Carnot cycle between a stationary system and one in relative motion to it. Because the proper temperature will be lower than the stationary temperature, relative motion will cause a temperature difference together with the possibility of performing work. It is rather odd that 15 years later, Tolman (1934) in his Relativity, Thermodynamics, and Cosmology was to reproduce von Laue’s cycle without even the slightest mention of who the discoverer really was! Due to the invariance of the pressure in special relativity, the cycle had to be contemplated as occurring under isobaric conditions. The relativistic Carnot cycle will occur in four steps: 1. 2. 3. 4.

A ! B Isobaric and isothermal expansion with heat uptake Q1 at T1 B ! C Reversible adiabatic acceleration to the velocity u C ! D Reversible and isothermal compression with heat rejection Q2 at T2 D ! A Reversible adiabatic deceleration which restores the system the original state

von Laue (as well as Tolman) worked with mechanical quantities thereby making the analysis much more complicated than necessary. It suffices to consider only internal changes with thermodynamic quantities. The heat uptake in the first step is Q1 D HB HA D .mB mA / c 2 ;

(4.106)


97

where Hi D Ei C pVi is the thermodynamic enthalpy of state i , and we have used the equivalence of heat and mass (Lavenda 2002) in establishing the second equality. According to Planck (1907), not only the energy density contributes to the inertial mass, but, in addition, there are stresses that are acting on the surface, and if these stresses are normal to the surface, the mass density will be %D

"Cp ; c2

with a corresponding momentum given by ." C p/u : c2 Likewise, the heat released to the condenser is Q2 D HC HD D .mC mD / c 2 ;

(4.107)

which is reckoned positive. From the fact that the enthalpy decreases in a state of motion, mC D 1 mB

and

mD D 1 mA ;

(4.108)

p where D 1= .1 u2 =c 2 / is the Lorentz factor, we may write the heat rejected, (4.107), in terms of the heat uptake as Q2 D 1 .mB mA / c 2 D 1 Q1 :

(4.109)

The work then will be the difference of (4.106) and (4.109), viz., W D Q1 Q2 D .mA mB / c 2 1 1 D Q1 1 1 D Q2 . 1/ :

(4.110)

From Carnot’s principle, we may deduce from the first equality in the second line of (4.110) that (4.111) T2 D 1 T1 : This says that a system in relatively uniform motion will have a temperature lower than that of the system in a state at rest. A comparison of (4.109) and (4.111), gives the condition of reversibility as: Q1 Q2 D : T1 T2 From (4.111) it follows that the Carnot efficiency is

C D 1 1 ;

(4.112)

98


which approaches unity as u ! c, since T2 ! 0. And from the second line of (4.110) we may deduce that the motion is hyperbolic, with uniform acceleration in the second and fourth steps, as we shall now show. In a uniformly accelerating frame in one dimension, the definition of hyperbolic (additive) time is

t C x=c =0 D ln t x=c

1 1Cˇ D ln K D ln ; 2 1ˇ

(4.113)

where the scale is set by 0 D c=g, and g is the acceleration constant of gravity at the surface of the earth. The second and last equalities testify to the fact that the motion is not uniform, i.e., u D x=t, but, rather uD

2x=t : 1 C .x=ct/2

(4.114)

This is the relative speed of two systems with equal and opposite velocities. Although known as the Einstein composition law for velocities, (4.114) was discovered by Poincaré. Taking the derivative of both sides of (4.113) with respect to t results in 1 dˇ g d D c dt 1 ˇ 2 dt

(4.115)

in a state of uniform acceleration. A moving clock will appear to go more slowly than one at rest (time dilatation), so that the proper time will be related to the coordinate time t by d D 1 : (4.116) dt Introducing (4.116) into (4.115) results in g=c D

d ˇ 1 dˇ D p : 2 3=2 dt dt .1 ˇ / 1 ˇ2

(4.117)

In a state of uniform acceleration, the Lorentz transform x 0 D x cosh ˛ ct sinh ˛;

(4.118a)

t D t cosh ˛ .x=c/ sinh ˛

(4.118b)

0

rotates the coordinate and time through an “imaginary” angle, ˛. Introducing the “rapidity” (Robb 1911), according to its definition ˇ D tanh ˛, which relates the


99

1 Euclidean measure of the relative velocity, p pˇ, to its hyperbolic measure ˛, we find 2 2 cosh ˛ D 1= .1 ˇ /, and sinh ˛ D ˇ= .1 ˇ /. We can thus write the Lorentz transform, (4.118a) and (4.118b), in the familiar form

x 0 D x cosh.g=2c/ ct sinh.g=2c/; t 0 D t cosh.g=2c/ .x=c/ sinh.g=2c/:

(4.119) (4.120)

From the second equality in (4.113) we have t C x=c D K.t x=c/; where

KD

1Cˇ 1ˇ

(4.121)

1=2 (4.122)

is the longitudinal Doppler shift, while from the first and third equalities, it follows that K D e g=c . Consequently, x=ct D tanh.g=2c/ D

cosh.g=c/ 1 : sinh.g=c/

(4.123)

But, this implies that the first of the transform equations, (4.119), vanishes identically, x 0 D 0, and the second relation, (4.120), reduces to t 0 D t sech.g=2c/ D t

p

.1 .x=ct/2 /:

(4.124)

Equation (4.124) can be considered as an expression of time dilatation in a frame of uniform acceleration. To see what these imply, we consider a “momentary rest frame.” This is an inertial frame whose velocity is equal to that of the particle. Viewed in this frame, the mass of the particle is its rest mass, but the particle is undergoing an acceleration g. Thus, its position, x 0 , as well as its velocity, xP 0 , is zero in the momentary rest frame, but, its acceleration is xR 0 D g. In contrast to the stationary frame, the acceleration is not constant, but, rather, decreases in time as: aD

g 1 C .gt=c/2

:

Introducing (4.123) into (4.114), and realizing that it is the double angle formula for the hyperbolic tangent, we get the rapidity, ˇ D tanh.g=c/;

1

As ˇ ! 1, ˛ ! 1, so there is no limit on the hyperbolic measure of the velocity.

(4.125)

100


showing that the velocity, u D dx=dt, still obeys the Poincaré addition law (4.114) of velocities even though we are not dealing with uniform motion. Returning to the equation of motion of uniform acceleration in time, (4.117), we integrate it, and using (4.125) we get: ˇ ; gt=c D sinh.g=c/ D p .1 ˇ 2 /

(4.126)

for u D 0 at t D 0. The fact that the hyperbolic time is the only additive time seems to have been overlooked in (Møller 1952). Multiplying (4.126) by (4.123) and using the second equality in the latter yield xg D c 2 .cosh.g=c/ 1/ :

(4.127)

Finally, multiplying both sides of (4.127) by the change in mass yields the work (4.110), viz., (4.128) W D mgx D Q2 .cosh.g=c/ 1/ ; where we have identified the heat (4.109) as the heat given up to the cold reservoir. From relation (4.126) it follows that p 1 D .1 C .gt=c/2 /: cosh.g=c/ D p .1 ˇ 2 /

(4.129)

Thus, (4.128) allows the heat absorbed at the furnace to be written as: p Q1 D mc .c 2 C .gt/2 /:

(4.130)

This provides a direct relation between heat and gravity. In the limit as g ! 1, where F D mg is the weight. The temper(4.130) becomes Q1 D F ct, p ature of the furnace is T1 D T2 .1 C .gt=c/2 /, which becomes infinite in the limit as g ! 1, meaning that work without limit can be extracted from such an engine! As approaches infinity, the work (4.110) also approaches infinity because T2 approaches zero since the velocity u approaches the speed of light. Taking the time derivative of (4.110), we find the power as i h p WP D QP 1 1 .1 ˇ 2 / C G a;

(4.131)

G D mu

(4.132)

where


101

is the momentum and a is the acceleration. The term G a is the power necessary to keep the system in a state of uniform acceleration. In the limit of infinite time, the rate of working is equal to the rate of heating, WP D QP 1 . Introducing the rate of change of the heat rejected at the cold reservoir, (4.109), into (4.131), and with the aid of (4.111), we come out with the important relation QP 1 QP 2 Ga D : T1 T2 T2

(4.133)

In words, (4.133) states that the algebraic sum of the rates of heating divided by the temperatures at which they are at is equal to the power necessary to keep the system in a state of uniformly accelerated motion, divided by the temperature of the condenser. For an inertial system, the acceleration vanishes, and (4.133) gives the kinetic P P D 0. G a is the minimum analog of Clausius’ criterion for reversibility, Q=T power consumption; for irreversible processes, T2 > 1 T1 , and the equality in (4.133) is transformed into an inequality, viz., QP 1 QP 2 Ga > : T1 T2 T2

(4.134)

Inequality (4.134) says that a greater entropy production will result when additional irreversible processes are present other than the minimum power consumption, G a=T2 , which is necessary to keep the system in a state of uniform acceleration.

4.2.7 Coefficient of Performance from the Complementary Efficiency of a Refrigerator The purpose of a refrigerator is to extract as much heat as possible, Qc , from the cold reservoir. This is known as the “refrigeration” effect: a positive heat transfer from the cold reservoir. In order to be able to compare reversible and irreversible cycles, Qc Qcrev , both cycles must experience the same negative net work transfer, and both must experience heat transfer from the same two reservoirs. Since the heat absorbed in an irreversible cycle is less than in a reversible cycle, it is necessary that the heat transfer to the hot reservoir be smaller in the algebraic sense for a reversible cycle than for an irreversible cycle. A complementary efficiency to the expression in the nonmaximal work (4.14) may be defined as W D Qh

T T T h c h ln ; Th Th Tc

(4.135)

102


where Qh > 0 is the heat transferred to the high temperature reservoir. Imposing the condition of reversibility (4.15), and using inequality (4.13), the upper bound to (4.135) is found to be W D

Qh C Qc ln

!.W / D

Th Th Tc

Tc Qh D Qc ; Th

(4.136)

where W D Qh C Qc

(4.137)

is the negative net work transfer, and ! WD

Tc Th Tc

(4.138)

is the coefficient of performance (COP). The COP is defined as the refrigeration effect or the positive heat transfer from the cold heat reservoir, Qc , to the hot reservoir, Qh . Inequality (4.136) shows that the refrigeration effect is smaller for an irreversible cycle than for a reversible cycle. In order to make a comparison between the two cycles, the same negative net work transfer must be made between the same two heat reservoirs. This requires the heat transfer to the hot reservoir to be smaller in the algebraic sense for the reversible cycle than for the irreversible cycle, since it absorbs more heat at the cold reservoir.

4.3 Beyond Steam Engines: Thermoelectricity After having mastered the thermodynamics of steam engines, both Thomson and Clausius turned their attention to other ways of producing work. Even in a perfectly homogeneous conductor there are intrinsic electric forces if it is not at a uniform temperature. And even when the temperature is the same throughout, there are intrinsic electric forces when the conductor is not entirely homogeneous. Strain which produces temperature differences also alters the intrinsic electric forces, and in crystalline materials the thermoelectric quantities will vary in different directions. When current passes from one material to another, or from cold to hot in a single material, there will be in addition to frictional heating, reversible thermal effects that produce heating or cooling depending on the direction of the current. As early as 1821, Seebeck discovered that in a closed circuit comprised of two metals when their junctions are maintained at different temperatures an electric current will flow around the circuit. If the metals are copper and iron and one of the junctions is heated to a temperature not exceeding 600ıC then an electric current is created across the hot junction in the direction from copper to iron. Such

4.3 Beyond Steam Engines: Thermoelectricity

103

thermoelectric circuits can be made to do mechanical work, and it was seen by both Thomson and Clausius as an alternative to a steam engine for generating power. But, the source of such work was not as obvious as in the heating of steam. In 1834 Peltier gave further insight into the origin of such an energy source. He discovered that when a current flows across a junction formed from two different metals, there is an absorption, or release, of heat. If a current flows in one direction across the junction and heat absorption occurs, it will inevitably generate heat when the current is reversed. Moreover, if the current flows in the same direction as the current at the hot junction, heat will be absorbed, while, if it flows in the same direction as the current in the cold junction, heat will be given off. The heat liberated or absorbed was found to be directly proportional to the quantity of electricity which passes through the junction. The amount of heat that is absorbed, or released, when a unit charge crosses the junction is referred to as the Peltier effect at the temperature of the junction. We now can fashion this thermal electrical effect as a Carnot engine. For suppose we place one iron–copper junction in a hot compartment and the other in a cold compartment. A current will be produced that flows from the copper to the iron in the hot compartment, while from the iron to the copper in the cold compartment. On the basis of Peltier’s discovery, this thermoelectric current will absorb heat in the hot compartment and release heat in the cold compartment. The thermoelectric current therefore plays the role of the working substance in the Carnot engine, so that the thermoelectric couple can be used in place of the ordinary steam engine. Both Thomson and Clausius came to this realization, but it was Thomson who gave a greater contribution to the formulation of the theory. Experiments carried out on thermoelectric currents can give ulterior confirmation of the view that one is dealing entirely with thermal energy. The current passing through the circuit absorbs heat at the hot junction and releases heat at the cold junction. There was no indication that other forms of energy were involved, such as chemical energy from changes in the nature of the junctions. But, could the thermoelectric circuit be treated as a reversible thermal engine? In order for the circuit to be reversible, the same thermal processes should occur in reverse order when the circuit is reversed. It was known from the work of Fourier that heat conduction occurs along metals when there exist a difference of temperature, and from the work of Joule there is a heating effect proportional to the square of the current which is impervious to the direction of the current. If these effects could be considered as second order, we may treat the circuit as a reversible engine to first order. In order for such to be the case, Thomson argued that the Peltier effect cannot be the only reversible thermal process in the circuit. Assume, for the moment, that the Peltier effect is the only reversible process that occurs in the circuit. Let Qc stand for the mechanical quantity of heat that is released at the cold junction which is kept at temperature Tc . Suppose that Qh is the Peltier heat absorbed at the hot junction which is maintained at temperature Th . Since the circuit is reversible, we require Qh Qc D ; Tc Th

104


but, this must be D

Work done when a unit charge traverses the circuit : Th Tc

Now, the work done when a unit of electricity travels around a closed circuit is the electromotive force E, so that the latter is given by ED

Qc .Th Tc / : Tc

(4.139)

If this were true, it would mean that the hotter the hot junction, the greater the electromotive force should be. However, Cumming showed that there were circuits in which the electromotive force decreased with increasing temperature of the hot junction until a point was reached where the electromotive force is reversed causing the current to flow in the opposite direction! Thomson was forced to conclude that there were other reversible thermal effects involved related to the flow of current along an unequally heated conductor. Through a painstaking series of experiments, Thomson was able to establish the reality of such reversible phenomena. Thomson discovered that when electricity flows along a wire whose temperature varies from point to point, heat is released when the current at that point flows in the direction of heat flow. That is, when the current is flowing from hot to cold, there will be a liberation of heat, while, if the current is flowing in the direction opposite to the heat flux, there will be an absorption of heat. This is true for copper, but, not for iron where the reverse holds. Consequently, when a current flows along an unequally heated copper wire, there is a tendency to diminish the differences in temperature, while when it flows along an iron wire it tends to accentuate those differences in temperature. This is called the Thomson effect.

4.3.1 Thomsons’s Theory of the Electrical Specific Heat Thomson found that he could conveniently express his effect by introducing what he termed “the specific heat of the electricity in the metal.” If p1 and p2 are any two points along the metallic wire at temperatures T1 and T2 and if this difference is assumed small, then the specific heat ce is defined by ce .T1 T2 / D Q;

(4.140)

where Q is the heat generated or absorbed in the segment p1 p2 when a unit of electricity passes from p1 to p2 . Another fact of experience is the following. If E1 is the electromotive force developed in a circuit when the cold junction is at temperature T0 and the hot junction at T1 and E2 is the electromotive force around the same circuit when the cold junction


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is at T1 and the hot junction at T2 , then the total electromotive force E1 C E2 is that of a circuit whose cold junction is at T0 and hot junction at T2 . That is to say, the sum of the electromotive forces selects the extremal temperatures of the separate electromotive forces. As a consequence, the total electromotive force around a circuit whose temperatures lie between T0 and T1 is Z ED

T1

….t/dt; T0

where ….t/ is referred to as the thermoelectric power at temperature t. Still another experimental fact concerns different circuits comprised of different metals all operating between the same extremal temperatures. If EMi Mj is the electromotive force for a circuit comprising of metals Mi and Mj and EMj Mk is the electromotive force for a circuit formed of metals Mj and Mk , then E Mi Mk D E Mi Mj E Mj Mk ; independent of the specific nature of metal Mj , provided only that all circuits are operating between the same extremal temperatures.

4.3.2 Tait’s Thermoelectric Diagrams We have Tait (1884, p. 196) to thank for the following graphical representation of thermoelectric power. Tait plots the thermoelectric power, ….T /, of a metal and some standard metal, such as lead, vs. the temperature. Characteristic plots are those shown in Fig. 4.8. For a small difference in temperature, … is assumed positive when the current flows from lead to the metal across the hot junction. From what has been said above, if the curves 1 and 2 are thermoelectric lines for metals M1 and M2 , then at a temperature Tw the circuit made up of these two metals will have a thermoelectric power given by EF in the figure. If the cold junction is Π E

D

1

A F

C

Tw

Th

2

B

0

Tc

Fig. 4.8 Tait’s thermoelectric diagram

T

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at Tc and the hot junction at Th , then the electromotive force for the whole circuit formed from metals M1 and M2 will be given by the area ABCD. Now consider the same two metals at temperatures T1 and T2 , where the temperature difference .T1 T2 / is very small. Then the work done by a unit charge as it moves around the circuit, or the electromotive force of the circuit, is given by the area ABCD. Now let us decompose this area into the following thermal factors. First, there is the Peltier effect at the junctions. Let area A1 stand for the mechanical equivalent of heat that is absorbed at the hot junction when a unit of electricity crosses from metal M2 to M1 . Analogously, the area A2 will represent the mechanical equivalent of heat released at the cold junction. Second, there is the Thomson effect due to the unequally heated metals. The mechanical equivalent of heat absorbed when a unit of electricity flows through M2 from hot to cold junctions is represented by the area B1 . Finally, the mechanical equivalent of heat released when a unit of electricity flows through M1 from hot to cold junctions is B2 . According to the first law, the work is equal to the differences in heat absorbed and rejected: (4.141) ABCD D A1 A2 C B1 B2 ; as shown in Fig. 4.9. According to the second law, if Q is the heat absorbed in any reversible engine at temperature T , where the algebraic sign of Q accounts for whether heat is absorbed or released, then XQ D 0: T If the temperatures at the two junctions differ little from one another, we may suppose that the temperature at which heat absorption takes place in the Thomson effect

D

a M1 b

A

c d

B

T2

Fig. 4.9 Thermoelectric analog of a Carnot cycle

M2

C

T1


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is the arithmetic mean of the two temperatures TA D 12 .T1 C T2 /. Thus, the second law can be stated as: A1 A2 B1 B2 C D 0: (4.142) T1 T2 TA Consequently, combining the first and second laws, (4.141) and (4.142), results in 1 ABCD D 2

A1 A2 .T1 T2 / : C T1 T2

Now, when T1 is near to T2 , the area ABCD ' CD .T1 T2 / ; or A1 ' CD T1 , which implies that A1 is given by the area CDac. The indices are superfluous, and we see that the Peltier heat at any temperature is Peltier force D … T D (thermoelectric power) (absolute temperature): According to the definition of the specific heat of electricity, the difference in the Thomson heats is B1 B2 D .ce1 ce2 / T1 T2 : But, according to the first law (4.141) and A1 D CDac, A2 D BAbd , we find B1 B2 D bADa dBC b D .tan 1 tan 2 / T1 .T1 T2 /; where 1 and 2 are angles with the tangents at A and B to the thermoelectric lines for M1 and M2 make with the abscissa along which the temperature is measured. Thus, it follows that ce1 ce2 D .tan 1 tan 2 / T1 :

(4.143)

When the temperature interval .T1 T2 / is not infinitesimal, the areas cCDa and BAbd will still represent the Peltier heats at the junctions, and the area dBC c will still give the absorbed heat when a unit of electricity flows along metal M2 from the point where the temperature is T2 to the point where it is T1 . In the experiments carried out by Tait (1884, p. 197), he found that the specific heat of electricity was directly proportional to the absolute temperature: ce D tan T:

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It was also known that the specific heat of electricity for lead was vanishingly small. Assume that ce is identically zero when it refers to lead. Thus, the thermoelectric power … for any metal with respect to lead is … D tan .T T0 /; where T0 is the so-called “neutral point” of the metal and lead. That is, T0 is the absolute temperature at which the lines of the metal and lead meet. If for any two metals, 1 and 2 are the angles that their lines make with respect to the lead line and T1 and T2 are their neutral temperatures, then their thermoelectric powers with respect to lead are: …1 D tan 1 .T T1 /; …2 D tan 2 .T T2 /: The thermoelectric power of the circuit that is made up of these two metals will be … D .tan 1 tan 2 /.T T0 /;

(4.144)

where T0 is their neutral temperature, T0 D

T1 tan 1 T2 tan 2 ; tan 1 tan 2

which is a sort of weighted average. Then, if Th and Tc are the temperatures of the hot and cold junctions, respectively, the electromotive force of the circuit is Z ED

Th

Tc

1 … dT D .tan 1 tan 2 / .Th C Tc / T0 .Th Tc /: 2

(4.145)

From this expression, Tait (1884, p. 415) concludes “that the expression for the electromotive force has no other variable factors than the two; the first [Th Tc ] of which was known to Seebeck, while the second [tan 1 tan 2 ] was discovered experimentally by Thomson.” Moreover, Tait notes that nothing “more than the algebraic difference between the specific heats of electricity in the two metals” can be measured since the same constant factor, .tan 1 tan 2 /, appears both in the expression for the electromotive force, (4.145), and in the Peltier effect. Furthermore, (4.145) shows that the electromotive force vanishes when the arithmetic mean temperature of the junctions is equal to the neutral temperature. Finally, if one of the junction temperatures is maintained constant and the other junction temperature is varied, the electromotive force will have a maximum or minimum value when the other junction temperature is at the neutral temperature.


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Although Tait followed Thomson in his exposition, we have chosen to follow (J.J.) Thomson (1921) in our treatment. Most modern treatments relegate the Peltier and Seebeck effects to the linear domain of irreversible thermodynamics (De Groot 1966). (J.J.) Thomson (1888) made even an earlier (somewhat unsuccessful) attempt to derive the properties of thermoelectricity from a classical mechanical approach involving a Lagrangian which distinguished between “controllable,” as opposed to “uncontrollable,” coordinates, where the former were related to work while latter obviously represented quantities related to heat. The fact of the matter is that the founders of thermodynamics were able to encorporate both the Seebeck and Peltier effects within the realm of classical thermodynamics through the employment of the Thomson effect.

4.3.3 Clausius’s Thermoelectric Theory Again a clash of ideas occurred between Clausius and Kelvin, the former claiming that the specific heat of electricity, (4.140), as far as he was aware, gave no explanation as to the cause of thermoelectric currents. According to Clausius, the electromotive forces that occur at junctions of different metals are of thermal origin. They are functions of temperature and vary in such a way that fall within the domain of the second law, i.e., reversible heating effects. If the junctions are at the same temperature, the electromotive forces balance one another, and no current flows. Rather, if the temperatures of the junctions are different, a current is setup, with the second law demanding that the contact force vary with temperature. The complete electromotive force is E D … .T1 T2 / ;

(4.146)

where … is a constant depending on the nature of the metals and T1 and T2 are the junction temperatures. Essentially (4.146) is Thomson’s (4.139). Deviations from normal thermoelectric behavior could occur at high temperatures which, according to Clausius, altered molecular composition. Clausius cites that the thermoelectric properties of strained and unstrained wires of the same material can be completely different. But, the supposed changes in structure were considered to be reversible, for, if, heating the material and then cooling it back to the original temperature the same structure is recovered. In place of the sum of terminal Peltier forces, … .T1 T2 /, Clausius considers the differences .…1 T1 …2 T2 /, where …1 and …2 are the values of … at the junction temperatures T1 and T2 . In metals such as copper, the forces are small due to small variations in …, so that the electromotive force between a junction at temperature T will be due to the product P of T and the small variation d… in …, viz., T d…. The sum over all junctions, Ti d…i , will represent the total electromotive force. The difference of potential, P D …T , or the Peltier force, varies as the temperature of the copper varies. Suppose we divide the copper wire into four pieces

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with Peltier forces P1 ; P2 ; P3 , and P4 . The terminal forces in the copper are joint to the lead, whose junctions have the potential differences, P and P4 . The intermediate forces are P1 P2 , P2 P3 , and P3 P4 . Their sum is simply P1 P4 , which cancels the junction force with lead, P4 P1 , and thereby leaves no resultant electromotive force. This is the characteristic property of a force which is derived from a potential: the integral around a closed circuit vanishes. In fact, if we consider the increment in the Peltier force, d .…T / D … dT C T d…; we get two terms, the first being the resultant increase in the electromotive force and the second is the integral force in the copper wire. If we consider five separate and distinct temperatures in the wire, T0 ; T1 ; T2 ; T3 , and T4 , we can write the sum of the first term as T0 …1 C .…1 …2 / T1 C .…2 …3 / T2 C .…3 …4 / T3 C …4 T4 : The first and last terms are thePPeltier forces at the junctions with lead, while the middle terms have the form T d…, which are the increments in the intrinsic forces. Now, the terms can be arranged to read .T1 T0 / …1 C .T2 T1 / …2 C .T3 T2 / …3 C .T4 T3 / …4 ; P which has the form …dT , or the increment in the electromotive force. It shows that the increment in the complete electromotive force is equal to the complete integral force within the copper wire, which can only be the case if the constitutive relations between … and T is linear and symmetric. Moreover, if we consider the force in the copper per unit length, T .d…=dx/, in the direction of decreasing … and consider similar variations in the temperature per unit length, then on the basis of Fourier’s law of heat conduction we will arrive at the linear laws of irreversible thermodynamics relating nonconjugate causes (forces) to effects (flows). Although these laws impose a symmetry in the transport coefficient matrix, the symmetry is other than Onsagerian, because their origins are not to be found in the principle of microscopic reversibility (Lavenda 1978). According to this principle, the forward and backward transition rates are equal at equilibrium. Although such relations apply to chemical reactions, they can hardly be expected to hold for a variety of thermoelectric effects which can be defined in terms of three transport coefficients, or in terms of the thermal and electrical conductivities, and the thermoelectric power.

4.4 Irreversibility Viewed as Violations in the First and Second Laws

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4.4 Irreversibility Viewed as Violations in the First and Second Laws The closest anyone has come to relating Carnot’s theory to (nonconventional) thermodynamics is Lervig (1972). He associates the “loss of work” in Carnot’s theory with the increase in entropy in Clausius’s theory. While it is true that irreversible processes are “described by the second law, they are in Carnot’s theory described by the first law.” However, in Carnot’s theory this requires the existence of the entropy function to show that all accessible states from a given state requires the increase in internal energy. Accessibility means that work must be done to cause a change in the state of a system and this cannot be done under adiabatic conditions. Thomson argues that work can only be done by “letting down” heat to a lower temperature. Carnot contends that the maximum amount of work is the difference in the heat absorbed to that rejected Wmax D Qh Qc :

(4.147)

He has no reason to doubt otherwise. Implicit in Carnot’s statement (4.147) is the second law. So Carnot accepts the second law, but is ignorant of the first law. In general, the work will be less than that given by (4.147) due to dissipation. But, there must still be a balance of energy, W D Qh Qc E;

(4.148)

where E > 0. That is to say, there cannot be any other sources of work other than what the engine does, for, otherwise, the heat given up to the cold reservoir would not be a limiting factor on the efficiency of the engine. According to Carnot, E < 0 would result in a perpetual motion of the second kind. Combining (4.147) and (4.148) results in W D Wmax E:

(4.149)

Therefore, in the presence of irreversibility Carnot negates the existence of a function of state called the internal energy. Now, Clausius accepts the first law W D Qh Qc ;

(4.150)

in all circumstances, whether they be reversible or not. However, he notes that, in general, (4.151) W Wmax D C Qh ;

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so that he is negating the existence of an entropy function in the presence of irreversibility. That is to say, inserting (4.150) into (4.151) results in Qh Qc D S 0; Th Tc

(4.152)

which is his famous inequality that bears his name. All that (4.152) says is that more heat is ceded to the cold reservoir than (4.15), in the event that the process was reversible. Now, introducing (4.150) into (4.152) results in W D Qh Qc D Wmax Tc S:

(4.153)

Thus, Clausius negates the existence of a function of state, called by him the entropy, in the presence of irreversible processes. Comparing Carnot’s negation of the first law, (4.149), with Clausius’s negation of the second law, (4.153), gives E D Tc S:

(4.154)

This represents the heat rejected to the cold reservoir in excess of the smallest possible quantity, (4.15), and it depends only on the temperature of the cold reservoir, Tc . Enter Thomson (1852a). We shall quote freely from Tait (1868). If dQ is the heat available for doing work, its practical value will be dW D

T Tc dQ; T

(4.155)

where Tc is the lowest possible temperature. Tait then goes on to say that in any cyclical process, if Qh is the heat absorbed and Qc is the heat released, then the practical value is Z dQ : (4.156) W D Q h Q c Tc T If the cycle is reversible, the work will be the difference .Qh Qc /, on the strength of the first law. This demands that the integral term in (4.156) vanish. In general, however, this integral has a finite positive value because “in nonreversible cycles the practical value of the heat is always less than Qh Qc :” Hence, the amount of heat lost needlessly, i.e., other than to the refrigerator or in producing work, is Z dQ 0: (4.157) Tc T This is Thomson’s expression for the amount of heat dissipated during the cycle.


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Thomson’s inequality is in blatant contradiction with Clausius’s (4.152). After having succeeded to confuse a great number of people, including Clerk-Maxwell himself, Tait (1884) retracts (4.157), but, without making any sense. According to Tait, the work will be simply the excess of the heat taken from some of the bodies over that given to others. This must always, except when perfect engines are employed, be less than the realizable value [(4.156)]. Hence, we see that the expression [(4.157)] is necessarily negative; except when perfect engines only are used, in which case alone its value is zero. This is Thomson’s expression for the heat dissipated during the cycle of operations.

As we have quoted from Clerk-Maxwell in the Preface, he was eventually set right by Gibbs, who was influenced by Clausius, and the German school, and not by Kelvin. Corrections to his little black book can be found in later editions; in particular, the tenth edition edited by Lord Rayleigh in 1891. Tait’s reasoning contradicts the fact that there is more excess heat available to perform work! The point Thomson and Tait missed was this: Their “practical” value, (4.155), is the maximum work that can be realized, since the cycle is reversible. No matter how many engines that are coupled to the system, it still remains the maximum work. Since they equate the work with the difference in the net heat, Thomson and Tait are adhering to the first law, viz. there exists a function of state, the internal energy. The last term in (4.156) can only be the difference between the maximum work, that is performed in a reversible cycle, and the actual work that is done. Hence, Thomson and Tait do, in fact, obtain Clausius’s inequality (4.152). Therefore, there was no reason why Tait’s book should have lost the credibility it did for having reversed Clausius’s inequality. Even in more contemporary, authoritative books on the history of thermodynamics (Cardwell 1971, pp. 266–267) one finds Clausius’s inequality reversed! Moreover, the heat Qc Tc Qh =Th must be given up to the cold reservoir, since there is no other means of getting rid of it. There is nothing in Carnot’s formulation that other dissipative processes exist which detract heat away from the ability to perform work. What about work that is performed in adiabatic processes? Again, there is no provision in Carnot’s conception of an engine to be able to get work from nothing. This is the reason that the specific nature of the adiabatic expansion and compression is not required. All that is important is that in order to be able to perform work, heat must be “let down” from a higher to a lower temperature. In Carnot’s sense, Carathéodory’s formulation of the second law is completely superfluous since work must be done in order that a change in the state of the system to occur. As we already know, as early as 1853, Thomson (1853) derives the amount of work that could be gotten when perfect thermodynamic engines are introduced into an irregularly heated body that reduce each of its parts to a single common temperature. If mi is the mass of the i th element of an irregularly heated body, with specific heat at constant volume, cv .Ti /, at temperature, Ti , then the perfect engines will operate in such a way that all the heat lost is transformed into work with the body

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reaching a common temperature TM . The condition that no heat should be given out to any other body at temperature TM is: n X

Z mi

TM Ti

i

cvi .t/ dt D 0: t

(4.158)

Thomson uses (4.158) to determine TM . Then he employs W D

n X

Z mi

i

Ti TM

cvi .t/dt

(4.159)

to determine the maximum work done. Observe that (4.158) is the condition that thermal equilibrium is achieved under adiabatic conditions, and the work (4.159) is the negative change in the internal energy so that (4.158) and (4.159) are expressions of the second and first laws when no heat transfer takes place. Denote by Tc the temperature of the coldest element. In this event all the heat will be converted into work, and there will be no heat that can be given off to a condenser above that in which adiabatic equilibration occurs. If the body resembles an ideal gas and the masses have a common specific heat, cv , then the maximum work is Wmax D cv D cv

n X i D1 n X

Z mi

Tf

( mi

i D1

Ti

Tc dt 1 t

n X i D1

m i Ti

n .X

) m i Tf

C Tc S;

(4.160)

i D1

where the change in entropy S D cv

n X i D1

n . Y 1= PniD1 mi mi mi ln Tf Ti :

(4.161)

i D1

1= PniD1 mi Q mi n T , more heat will be given off to the condenser than If Tf > i D1 i if it were at the geometric mean temperature, thereby decreasing the work. The minimum possible value of Tc is the geometric mean temperature, for which the entropy change (4.161) vanishes. This final common temperature is achieved by letting the hot reservoir at Th come into thermal contact with the cold reservoir at temperature Tc . In the Carnot cycle this is avoided by allowing the working fluid to undergo adiabatic expansion and compression which lowers and increases its temperature, respectively. Carnot knew fully well that the hot and cold reservoirs could not be allowed to come into thermal contact for, otherwise, no work could be achieved. This is none other than the principle of maximum work.


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The minimum change in energy is given by (4.154), which in this case is E D cv

n X

mi Tc ln

i D1

T c ; TG

(4.162)

1= PniD1 mi Q mi n where TG D T is the geometric mean temperature. Due to its i D1 i convex nature, (4.162) can be considered to contain information of the lowest mean temperature with respect to the geometric mean, TG . It is positive semidefinite, i.e., E 0 with equality if and only if Tc D TG . Since the cells are initially adiabatically isolated, we can transform (4.160) into the maximum work done by an “endoreversible” engine. The hottest cell at T1 absorbs an amount of heat Q1 . Since we have the hierarchy, T1 >

n X

m i Ti

n .X

i D1

m i Tc

i D1

n Y

m Ti i

1= PniD1 mi

> T2 ;

(4.163)

i D1

the maximum work is Wmax D cv

n X

( mi

) n . Y 1= PniD1 mi mi T1 Tc C Tc ln Tc Ti :

i D1

(4.164)

i D1

If adiabatic equilibration takes place, the last term vanishes, and for a system of two equal masses, m1 D m2 D m, (4.164) reduces to

Wmax D Q1 1

s

T2 ; T1

(4.165)

p where Q1 D 2mcv T1 and Tc D .T1 T2 /. Expression (4.165) gives precisely the efficiency of the endoreversible engine, which is what Thomson found for an unequally heated body which was allowed to come to a uniform final temperature and equipped with perfect engines. Thomson’s analysis precedes the endoreversible engine by a century and a quarter.

Chapter 5

Work from Nonequilibrium Systems

5.1 What is Work? It is common belief that in an adiabatic process, the first law asserts that a system can do work “at the expense of its internal energy” (Truesdell 1979). However, this is what Carnot’s general axiom precisely negates. If everything is at the same temperature no work can be done. Work is not associated with any single step in the Carnot cycle, but with the cycle itself. Hence, both adiabatic and isothermal processes are necessary to produce work. Fermi (1956) uses Kelvin’s statement A transformation whose only final result is to transform heat into work that has been extracted from a source at the same temperature throughout is impossible.

of the second law to show that work can be attributed to a single step in the Carnot cycle, as if it was a purely mechanical concept! Fermi’s interpretation is exemplary of the confusion in the interpretation of the first law as a purely mechanical law. He emphasizes the word only and cites as an example the isothermal expansion of a gas that absorbs heat at a single temperature thereby performing work. According to Fermi, the complete transformation of heat into work is not in violation of Kelvin’s statement because the gas occupies a larger volume at the end of the expansion. If this were the case we could carry out this process indefinitely thereby converting the heat absorbed entirely, and completely, into work. Rather, it is the net work that the first law is concerned with, and in order to calculate it, the system must be brought back to its initial state. The net change in the internal energy is zero but not the work which is equal to the difference between the heat absorbed at the boiler and the heat rejected at the condenser. The presence – and necessity – of the condenser is the last vestige of caloric theory. The possibility of doing work must be paid back by returning to the original state. This is precisely the role of the condenser. Only the net gain is pertinent. To Carnot (1824), the existence of a temperature difference is fundamental: “Wherever there is a difference of temperature, motive power can be produced.” Therefore, in order to produce motive power it is not enough to absorb heat, but, cold is also needed, for, otherwise, we would get unlimited work. The more heat returned to the cold reservoir the less work can be obtained. Without a difference B.H. Lavenda, A New Perspective on Thermodynamics, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-1-4419-1430-9 5,

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118

5 Work from Nonequilibrium Systems

in the heat absorbed and rejected, no work can be produced according to Carnot, whereas in the mechanical interpretation the decrease in the system’s internal energy can provide a source of work. Carnot outlaws such a process from occurring by producing an ordering of states which are accessible to the system, in much the same way that the second law uses the increase in entropy. The crucial point is that the work done by an engine cannot be considered in any one step; the net work is calculated after the engine has been returned to its initial state. The nonmechanical point of the energy balance equation is the partial return of the heat to the cold reservoir in completing the cycle. This is completely foreign to mechanics. It is quite remarkable that eventhough Carnot based his formulation on caloric theory, his formulation stood without its scaffolding. This is, in many ways, reminiscent of Clerk-Maxwell’s electromagnetic theory which used a deformable ether for the propagation of electromagnetic waves. Thomson (1881), later commenting on in the republication of his 1848 paper, admitted that his original paper was wholly founded on Carnot’s uncorrected theory, according to which the quantity of heat taken in the hot part of the engine: : :was supposed to be equal to that abstracted from the cold part: : :do not in any way affect the absolute scale of thermometry which forms the subject of the present article.

It is rather sad though that with Carnot’s growing disillusionment with caloric theory at the time of publication of Réflexions in 1824 that he would have gone so far as to dissuade potential readers from purchasing a copy which sold for only 3 francs. He did not have to try hard, for hardly anyone bought the book and hardly a bookseller heard of it (Mendoza 1960).

5.2 Principle of Thermal Resistance The second law correlates the flow of heat with the tendency of systems to evolve from a more constrained to a less constrained state. This has been formalized into a principle of “thermal resistance” by Oliver Heaviside (1892). Basically, Carnot found that work can be obtained from heat by decreasing its temperature. Heat is taken in at a higher temperature and pressure and given off at a lower temperature and pressure. The working substance thus does more work in expanding than compressing itself back to its original state so that when heat is lowered in temperature, work will be done. Two additional steps are required: one to lower the temperature and the other to raise it again so as to achieve the initial configuration. Clausius’s and Kelvin’s enunciations of the second law are, in fact, not so selfevident as that of Carnot’s, although they involve Carnot’s principle. Clausius claims that heat will not pass from a cold body to a hot body without compensation, but does not tell us what the compensation is. Kelvin says that we cannot get work out of a body by lowering its temperature below that of the surroundings. However, this is also included in Carnot’s statement that work can only be obtained by lowering the temperature of heat.

5.2 Principle of Thermal Resistance

119

Heaviside made the interesting observation that what ever be the nature of the change in the configuration of an object, its motion is always accompanied by (thermal) resistance opposing such a change. Such a change is always thermally resisted whether it be in heating or cooling of the object. Heaviside gives the following examples: Heating usually results in elongation and cooling in contraction. This is usually

the case, but not always. Take for example a rubber band where the elastic modulus (or stiffness) decreases with temperature. Above 4ı C water expands on heating, or heating opposes compression. However, below 4ı C, the opposite occurs, as we have seen in Sect. 3.2.2. The effect of compression is now to cool water. Water expands during evaporation so the pressure raises the boiling point of water. Water also expands in freezing so that pressure lowers the freezing point of water. As mentioned in Sect. 1.4, this effect was found by James Thomson (1849), which seemed to reinforce the caloric theory of heat as heat as some material substance. Twisting a wire can increase or decrease torsional rigidity. If heat decreases torsional rigidity it is a cooling effect, otherwise, it is a heating effect. In short, thermal resistance resists motion. If the motion is sudden, more work will have to be done than if it occurred in small installments. Hence, reversible processes are more efficient than nonreversible ones. Heaviside gives mathematical form to his principle in the following way. Suppose, says Heaviside, that an elastic body can be deformed while keeping its temperature constant at t. Work, W , has been done on the body, and heat Q has been taken from it. The principle of thermal resistance will then assert that .dW=dt/=Q is always positive, whatever be the signs of dW=dt and Q. Following in the footsteps of Carnot, Heaviside then discusses what would happen if his principle could be negated. After having strained the elastic body, it is brought back to its original form passing through the same series of intermediate deformations, but in reverse order, and at a slightly higher temperature, say t C dt. According to Heaviside (1892, p. 486) This requires there to be two other operations (2nd and 4th ), viz., to raise the temperature by dt in the second configuration, and to lower it by dt when it has got back to the first. Now the body does work W C .d W =dt /dt at the higher temperature. Hence, in a complete cycle, the body does work .d W =dt /dt . What else happens is that an amount of heat Q is lowered in temperature by an amount dt . Now, without any experience to guide us, Q and d W =dt may be algebraically positive or negative, and not of the same sign necessarily. But if they could be of opposite signs, work would be obtained through a substance by raising the temperature of heat. If then we take it as axiomatic that it is impossible by conveying heat from a cold to a hot body to obtain a mechanical effect, then we prove that the law of thermal resistance is universally true, at least for bodies in mass, and inanimate.

There is nothing to criticize in Heaviside’s conclusion, it is only that its mathematical formulation is found wanting. The 2nd and 4th operations, he refers to, are the adiabatic contraction and expansion that are necessary to heat and cool the body so as to bring it back to its original form. Thus, the body will be brought back to the

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same temperature t and not a slightly higher one, t C dt. Furthermore, if the body does work .dW=dt/dt over the entire cycle then this must be equal to the net heat acquired, viz., namely the difference between what it absorbs, at the higher temperature, and what leaves the body, at the lower temperature. There is no meaning to the algebraic signs of .dW=dt/ and Q since the work is defined only by completing the entire cycle, and not any part of it. The work, and even less its temperature derivative, cannot be equated to any single quantity of heat Q, since it is defined as the difference between the heat absorbed and the heat rejected. In other words, dW=dt and Q cannot be separately defined and their algebraic signs compared. There is no W for each individual step in the Carnot cycle, although Q can, and does, change sign. Thus, Heaviside’s “principle of thermal resistance, viz., Q dW D ; dt t where t is temperature according to the scale of equal dilatations of an imaginary perfect gas under constant pressure, whose energy, at constant temperature, is independent of its volume,” is not a statement of the second law. That “work is got by the lowering of the temperature of heat” is perfectly true, only that the work is determined when the cycle is completed, and the working substance is brought back to its initial state and temperature. Said slight differently, we cannot match W , or dW=dt, to a given Q, since the former has meaning only for the cycle. Heaviside’s real intent is to replace Clausius’s and Kelvin’s statements of the second law. According to Heaviside, Clausius’s statement that heat will not pass from a cold to a hot body by itself, or without compensation, is “true enough, by definition of cold and hot, if the cold and hot bodies be in contact. Otherwise, not evidently true, though a law of Nature.” Kelvin’s statement that work cannot be gotten from a body by cooling it down to the lowest temperature of the surroundings is “not self-evidently true” to Heaviside. He would like to replace it by the assertion that heat cannot be converted into work without the lowering of the temperature of heat. Here, we must agree with him since it has a closer resonance to Carnot’s original idea that work can be gotten by letting water fall in height. So work is not as primitive a concept as heat. We cannot pull out a piston, allowing heat to be absorbed at a constant temperature, and call it work if Heaviside’s statement of the second law is taken. It is only when heat is lowered in temperature that we can speak about work. And, in order to do so, we must complete a cycle so as to return the body to the state it existed in originally.

5.3 Principle of Maximum Work Where do maximum power considerations enter? Consider, for example, the formulation of the endoreversible engine. The multiplication of the rates of heat transfer, QP 1 in (4.47), and QP 2 , by the finite times, t1 and t2 , respectively, seems innocuous

5.3 Principle of Maximum Work

121

enough, but, it is the subsequent maximization of their difference that requires an indeterminate amount of time. The heat capacities, mcvi , in (4.48) and (4.49) are given in terms of the heats transferred to their respective differences in temperature. All concept of time has disappeared. Even if we were willing to hold onto the interpretation of ˛t1 and ˇt2 as products of thermal conductances and time intervals, their dimensions are joule per degree Kelvin, and not watt per degree Kelvin. So by starting from the rate of heat transfer and converting it into the heat transferred, Curzon and Ahlborn give the illusion that the times t1 and t2 are independent parameters. It would therefore appear to be reasonable to maximize power output in a finite time, and this procedure has resulted in a new branch of thermodynamics known as finite-time thermodynamics (Sieniutycz 1990). Their principle of maximum work can be paraphrased as follows. Maximize the work, (5.1) W D a.T1 T1w / b.T2w T2 / D ax by; under the condition that

by ax D ; T1 x T2 C y

(5.2)

which is (4.50), where a D ˛t1 and b D ˇt2 . The condition for an extremum in y is a p .T1 T2 / T2 D T2w T2 ; (5.3) aCb p which is (4.82) for a D b, and the geometric mean, .T1 T2 /, is the final mean temperature TM . That (5.3) is a maximum follows from the fact that y0 D

p .@2 W=@y 2 /yDy0 D 2b.a C b/=a .T1 T2 / < 0: So what Curzon and Ahlborn rediscovered was that the final mean temperature at which adiabatic equilibration occurs yields the maximum work. As we have previously discussed, this principle can be traced as far back as Thomson (1853), and stated as (Tait 1868, p. 101): p Let [ .T1 T2 /] be the temperature to which the whole body can be brought by means of perfect engines, so that all the heat lost is converted into work.

The time it takes to reach the geometric mean temperature is indeterminate so no maximum power output can be attributed in a finite time interval. This requirement is implicit in a little known paper by Cashwell and Everett (1967). A mean of order r is defined as: X Mr D

n X

pi xir

1=r ;

(5.4)

i D1

for an n-tuple of numbers x1 ; x2 ; : : : ; xn with corresponding probabilities p1 ; p2 ; : : : ; pn . Following the footsteps of Thomson (1853), without actually knowing

122


it, Cashwell and Everett draw an analogy to a thermodynamic system comprising a set of n isolated cells, or heat reservoirs, with initial temperatures Ti that are not all equal. The xi are the specific heats of the different cells, ci .T /, and the probabilities pi are the corresponding mass fractions, mi . This is what Thomson (1853) considered to be an irregularly heated body. Cashwell and Everett now take Clausius’s enunciation of the first and second laws to be: I. Energy is conserved II. Entropy increases

The conservation of energy implies that when these isolated cells are placed in thermal contact, a single uniform mean temperature TM will be reached in an indeterminate amount of time, such that there is no overall change in the internal energy E D

n X

Z mi

TM Ti

i D1

cvi .t/ dt D 0:

(5.5)

cvi .t/ dt > 0: t

(5.6)

It then follows that the change in entropy S D

n X i D1

Z mi

TM Ti

The inequality is a consequence of the fact that means are monotonically increasing functions of their order. The role of the integrating denominator is to lower the order of the mean, and the inequality follows from the property that means are increasing functions of their order (Hardy 1952), T M˛ > T Mˇ

˛ > ˇ;

(5.7)

unless all the Ti are equal. In order to determine the maximum work output, all parts of the body that were initially irregularly heated must come to a common, final temperature, which is the lowest possible temperature. In this case there can be no heat exchange with another body at the same or lower temperature. In this way we are assured that all the lost heat is converted into work (Thomson 1852b). This requires adiabatic equilibration at a lower temperature than the temperature that would have been reached if equilibration had come about through energy conservation (5.5). Under adiabatic equilibration, the decrease in the internal energy determines the maximum work. This interpretation is akin to Carnot’s assumption of heat conservation and the universal maximum efficiency of his engine. Only here, it is the conservation of entropy and not heat, that is being used so there is a distinction between adiabatic and isentropic processes. In short, we have no knowledge of how long it will take to reach the final, uniform, temperature to which the system equilibrates. The mean temperature is always greater than the final temperature of the coldest reservoir or coldest initial part

5.4 Work and Free Energy

123

of the system before it was allowed to equilibrate thermally. It is this difference that leads to an illusory increase in entropy (4.55). Equation (4.78) is a statement of reversibility according to the second law. The optimality of the Carnot efficiency is a consequence of the fact that the temperature of adiabatic equilibration, the geometric mean temperature, is greater than the temperature of the coldest reservoir.

5.4 Work and Free Energy Let us consider further an unequally heated body which can do work while coming to thermal equilibrium. We already know that the final equilibrium state that will be reached depends on the quantity of work that the system does. The transition to the final equilibrium in which maximum work is done on an external body corresponds to a reverse transition in which minimum work is done by the body on the system. If the volume and temperature of the system are maintained constant, then the minimum work done on the system is equal to its change in the Helmholtz free energy, F . In general, then, the work (Lavenda 1991) is W F:

(5.8)

Rather, if the temperature and pressure are maintained constant, then the lower limit to the work that can be done by the system is equal to the change in the Gibbs free energy. An equation of state involving constant values of T and V (or p) would mean that no process would occur at all. On the left-hand side of (5.8) we have a path-dependent term, while, on the righthand side there is a difference in a function of state, which, by definition, is path independent. Therefore, it would seem surprising that the inequality could be converted into an equality!

5.4.1 The ‘Jarzynski Equality’ It has been claimed that equilibrium information, like the change in the free energy, could be obtained by a series of nonequilibrium finite-time measurements which yield an average work function (Jarzynski 1997): [The] average over an ensemble of measurements of W , where each measurement is made after first allowing the system to equilibrate at temperature ˇ 1 , and where the parameters are fixed at [the initial state] A.

The proposed equality is eˇF D eˇ Wm ;

(5.9)

124


where the work function Z Wm D

ts 0

@Em P dt D @

Z

1 0

@Em d; @

(5.10)

and D t=ts is a (normalized) “protocol” parameter which affects the energy levels En of the system. F was identified as the difference in the Helmholtz free energies, .FB FA /, of two nonidentical, canonical distributions, A and B, with the same inverse temperature ˇ. The protocol parameter, , represents a generic external parameter that can be varied in time. That is, at a certain moment, the canonically distributed system, eˇ FA D

X

eˇ En ;

(5.11)

n

is isolated, and work is done on it up until a later time, where, again, it is canonically distributed, X eˇ Em : (5.12) eˇ FB D m

Dividing (5.12) by (5.11) results in eˇF WD eˇ.FB FA / D

XX m

eˇ.Em En / D eˇ W ;

(5.13)

n

or, equivalently, F D FB FA D ˇ 1 ln exp.ˇW /;

(5.14)

where (Jarzynski 1997) the overbar denotes an average over an ensemble of measurements of W , where each measurement is made after first allowing the system and reservoir to equilibrate at temperature T with parameters fixed at A.

For a thermally isolated system, “it is natural to identify the work performed on it with the net change in its internal energy” (Jarzynski 2006). Notwithstanding the claims made, there is absolutely no informational content in (5.13). Rather, if the overbar is the average D

eˇ W

E 0

D eˇ F0

X

eˇ.En CWn / D eˇ.FF0 / ;

(5.15)

n

where the subscript indicates an average over the unperturbed energy levels, or, equivalently, F D F F0 D ˇ 1 ln hexp.ˇW /i0 ;

(5.16)


125

then the situation is different. The free energy due to a perturbation Wm is eˇ F D

X

eˇ.Em CWm / ;

(5.17)

m

which can be cast into an average over the unperturbed energy levels eˇF D

X

E D ˇ Wm ˇ Wm p.0/ e D e ; m

(5.18)

0

m

by introducing the unperturbed probabilities .0/ D eˇ.F0 Em / : pm

(5.19)

5.4.2 Canonical with Respect to What? Jarzynski (1997) cites Zwanzig (1954) as saying that “the right-hand side of [eqn (5.14)] may be expanded as a sum of cumulants, or Thiele semi-invariants in powers of the inverse temperature.” However, it is not the variation in the inverse temperature which is under discussion, but the variation in the protocol parameter. Rather, it should be assumed that the system is canonically distributed with respect to the protocol parameter eln Z.0 ;ˇ / D

X

e0 ˇm :

(5.20)

m

The physical significance of the generating function eln Z.0 ;ˇ / D

X

e.0 /ˇm

(5.21)

m

is that the protocol parameter, 0 , which can be thought of as the unperturbed, equilibrium distance between the ends of a linear chain molecule, undergoes a stretching, due to the tension , so as to become elongated at a distance . Defining the unperturbed probabilities as .0/ D e0 ˇm =eln Z.0 ;ˇ / ; pm

(5.22)

the generating function (5.21) can be written as E D Z.0 ; ˇ/=Z.0 ; ˇ/ D eˇF D eˇ : 0

(5.23)

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It is the second equality which Jarzynski (1997) would have us believe that he proved. There is no mention of an equilibrium change in the free energy caused by a nonequilibrium work done in a finite-amount of time. This is what he read into (5.23) with absolutely no justification. The right-hand side of (5.23) can be expressed as an expansion in powers of the product ˇ in terms of the Thiele semi-invariants Mn ./ (Cramér 1946, p. 185), F D

1 X .ˇ/n Mn ./: nŠ nD1

(5.24)

Then, in order to obtain explicit expressions for the Thiele semi-invariants we expand the generating function (5.21) as Z.0 ; ˇ/ D

X

e0 ˇm

m

D Z.0 ; ˇ/

X .ˇm /n

X m

D Z.0 ; ˇ/

X m

n .0/ pm

nŠ X .ˇm /n

(5.25)

nŠ n .ˇm /3 .ˇm /2 .0/ C C ::: ; pm 1 C ˇm C 2Š 3Š

on account of the normalization of the unperturbed probabilities, X m

.0/ pm D

X

eˇ.FEm / D 1:

(5.26)

m

Taking the logarithm of both sides of the third line in (5.25) and expanding the logarithm in powers of ˇ give ln

o Z. 0 ; ˇ/ .ˇ/2 n˝ 2 ˛ D ˇ hi0 C 0 hi20 C Z.0 ; ˇ/ 2

(5.27)

Comparing terms of the same powers in ˇ in (5.24) with those in (5.27), we find explicit expressions for the Thiele semi-invariants. Only the first three Thiele invariants M1 ./ D hi0 ˝ ˛ M2 ./ D 2 2 0 ˝ ˛ M3 ./ D 3 3 0 ˝ ˛ M4 ./ D 4 4 0 3M22 ./; are related to the central moments, where D hi0 .


127

From (5.23) and (5.24) we obtain, in the high temperature, or small protocol, limit: ˇ2 ˝ 2 ˛ 0 C O ˇ 2 3 2 ˛ ˇ˝ W 2 0 C O ˇ 2 3 : D hW i0 2

F D hi0

(5.28)

The “fluctuation–dissipation” relation (5.28), given in (Jarzynski 1997), is attributed to Hermans (1991). But, Jarzynski fails to appreciate that F is not (5.14), and can never become it no matter how many terms we retain in (5.28). The condition that it is valid, as well as the general expansion in terms of the Thiele invariants (5.25), is that the work is smaller than the difference in the unperturbed energy levels. Even more important is that both (5.25) and (5.28) are independent of the P Retaining higher order terms in (5.28) does not “make it valid switching rate, . arbitrarily far from equilibrium,” contrary to what has been claimed in the literature (Liphardt 2002). These authors have even found that the change in free energy is independent of the switching rate. Hence, Jarzynski’s (1997) conclusion about extracting “equilibrium information (F ) from the ensemble of nonequilibrium (finite-time) measurements” is fallacious.

Thermodynamic Perturbation Theory In this paragraph we confirm the results obtained in the preceding section using thermodynamic perturbation theory. Identifying the work as the difference of energies of unperturbed states in two different canonical ensembles at the same temperature T is conceptually wrong. A simple quantum mechanical calculation shows that to second order, the mth perturbed energy level is (Landau 1959) .0/ C Wmm C Em D Em

X0

jWmn j2

n

.0/ Em En.0/

;

(5.29)

where the Ei.0/ are the unperturbed energy levels, assumed to be nondegenerate, and the prime indicates that the term n D m be omitted. The condition for the applicability of thermodynamic perturbation theory is (Landau 1958) .0/ En.0/ j; jWmn j jEm

which says that the perturbation coupling any two states must be much less than the absolute difference in the unperturbed energies of the states. Hence, the work can never be equal to the difference in energy of the unperturbed states.

128


Introducing (5.29) into the canonical distribution (5.12), taking logarithms of both sides, and expanding the logarithm give F D F0 C hWmm i

0 .0/ .0/ ˛ pn pm 1 ˝ 1X X 2 C : (5.30) jWmn j2 .0/ ˇ Wmm .0/ 2 m n 2 Em En

Since the diagonal term hWmm i is the mean value of the work in the mth quantum .0/ , state, it certainly does not equal the difference in the energies levels, En.0/ Em as Jarzynski (1997) would have us believe with the statement that “the work done on an isolated system is equal to the change in its energy.” Moreover, it follows that the change in the free energy is the smallest amount of work that can be done on the system F hWmm i ;

(5.31)

since the last two terms in (5.30) are negative semidefinite. That is, as a consequence of the monotonicity ofthe canonical distribution, the sign of difference of .0/ .0/ the unperturbed probabilities, pn pm , is the same as the sign of their energy .0/ difference, Em En.0/ . Let us now consider the “nonequilibrium” average of the exponential of (5.9). If Wm is small, thermodynamic perturbation theory (Landau 1959, 32) can be used to calculate the Helmholtz free energy given in the canonical distribution (5.17) of the body, viz., eˇF '

X m

1 eˇEm 1 ˇWm C ˇ 2 Wm2 ; 2

(5.32)

where the quadratic term needs to be retained in order to calculate first- and second-order corrections to the non-perturbed free energy, F0 . Following the same procedure as before, we take logarithms of both sides of (5.32) and expand the logarithm in series to the same order. We then obtain (5.28), which is none other than Eqn (32.3) of Landau and Lifshitz (1959)–long before it was referred to as a “fluctuation–dissipation” relation. ˛ Although it follows from the positive semidefi˝ niteness of the dispersion, W 2 0 0, that hW i0 F;

(5.33)

it does not follow from Jensen’s inequality for a convex function.

Adiabatic vs. Isothermal Switching Rates Two limiting cases have been known for a very long time: the infinitely fast and infinitely slow switching times (Kirkwood 1935; Zwanzig 1954).


129

The differential of the normalization condition, (5.26), of the canonical ensemble is d .ˇF / D ˇ

X

0 pm

m

Dˇ

@Em @

X @Em 0 pm Em d C dˇ @ m 0

d C hEi0 dˇ;

(5.34)

where the subscript zero indicates that the averaging is performed with respect to the unperturbed probabilities (5.26). Equation (5.34) can also be written as d hEi0 T dS D

@E @

d:

(5.35)

0

Integration of (5.35) from D 0 to D 1 can be performed in two cases only: In the isothermal case, where Z F WD F1 F0 D

1

0

@E @

0

d D hW i0 ;

(5.36)

and, in the adiabatic case, where Z E D 0

1

@E @

0

d D hW i0 ;

(5.37)

since, by definition, the thermodynamic energy is E D hE./i0 (Landau 1959, p. 38). In an isothermal process, the average work is equal to the change in the free energy, (5.36), while in an adiabatic process, the average work is equal to the change in the thermodynamic energy, (5.37). According to Jarzynski, the isothermal case corresponds to an infinite switching time, so that the time it takes for the protocol parameter D t=ts to evolve between 0 and 1 is infinite, while the adiabatic case corresponds to a zero switching time so that the change in is instantaneous. That is, ts is the time that the system remains in contact with the heat reservoir as the value varies from 0 to 1. On the other hand, an infinitely small value of ts would mean that the system is effectively decoupled from the heat reservoir. This distinction brings to mind the Newton versus Laplace calculations of the velocity of sound which we discussed in Sect 1.2. Recall that Newton used the isothermal approximation where the rarefactions and compression of the sound waves occurred so slowly that the system had the time to redistribute the heat evenly throughout, while Laplace assumed that they occurred so fast that the adiabatic approximation had to apply. However, this is in contradiction with the usual way that protocol parameters and their time derivatives are ordinarily defined (Landau 1959, p. 38). For an adiabatic process, the time derivative of the entropy, SP will depend on the rate P D 1=ts . For

130


P the entropy production can be expanded in powers of the rate. small values of , P Since S must tend to zero when P does, the entropy production must start off in quadratic terms in the rate. A linear term would mean that the entropy production P and this would violate the second law inequality SP 0. would change sign with , Consequently, the entropy production is quadratic in the rates to lowest order, SP D ˛ P 2 , where ˛ is some positive constant. In this way, the force, dS P D ˛ ; d

(5.38)

P the corresponding rate. The opposite limit, where P ! 1, or tends to zero with , ts ! 0, would necessarily correspond to the isothermal case for which the body has ample time to exchange heat with its surroundings thereby maintaining its temperature constant. It has been recognized since the time of Clerk-Maxwell (1891, p. 134) that “by means of diagrams of the isothermal and adiabatic lines the thermal properties of a substance can be completely defined: : :” There is nothing to justify the statement “for finite ts , our ensemble of trajectories lags behind the equilibrium distribution in phase space as H changes with time” (Jarzynski 1997). For thermodynamic stability, average tension must be an increasing function of protocol parameter, @ (5.39) hi0 > 0: @ This makes the average work a concave function of so we can apply Jensen’s inequality for a convex function to its negative so as to obtain: hexp.ˇW /i0 > exp .ˇ hW i0 / ;

(5.40)

which becomes an equality if and only if each realization gives hW i0 , i.e., in the absence of fluctuations. Applying the Jarzynski equality (5.9) to the left-hand side of (5.40) and (5.36) on the right-hand side, we immediately come out with the contraction that F F . Something is terribly amiss!

5.4.3 Microscopic Origins of the Carnot–Clapeyron Equation The doctrine of latent and specific heats in terms of the protocol parameter is dQ D L d C C dT:

(5.41)

The latent heat is L D

X n

@ hEi0 @E @pn0 D En ; @ @ @ 0

(5.42)


131

where the second equality follows from an integration by parts. The heat capacity at constant stretching is defined, as usual, by C D

@ hEi0 : @T

(5.43)

Thus, (5.41) misses to be a total derivative by the term

@E @

D 0

X n

pn0

@En D hi0 ; @

(5.44)

which is the definition of the mean tension hi in the canonical ensemble. This definition implies that probability distribution is insensitive to variations in the protocol parameter, as will be made clear below. For reference, the mean tension is defined in the microcanonical ensemble as @E @E D ; (5.45) hi0 WD @ 0 @ S which equates the mean of the derivative with the corresponding thermodynamic quantity evaluated at constant entropy (Landau 1959, p. 39). From these facts we draw the following two conclusions: 1. The differential increment in the heat absorbed, (5.41), dQ D

@ hEi0 @ hEi0 hi0 d C dT @ @T

(5.46)

becomes the total differential of the expected energy when the work term, hi0 d, is added to both sides. 2. Eqn (5.42) shows that the microscopic origins of the latent heat are the result of thermal fluctuations that prevent the average of a derivative from being equal to the derivative of the average. Working in the canonical ensemble and dividing d hEi0 D T dS C hi0 d by d we obtain

@ hEi0 @

The Maxwell relation,

D T

@ X 0 p En D T @ n n

@ hi0 @T

@S D @

@S @

T

C hi0 :

(5.47)

; T

(5.48)

132


for the enthalpy, together with the Grüneisen equation of state, s hEi0 D hi0 ;

(5.49)

where the parameter s has the same meaning as in (2.56), convert (5.47) into the partial differential equation T

@ hi0 @ hi0 D @T s @

1Cs s

hi0

(5.50)

for the average tension. We look for characteristic paths of (5.50) using the method of characteristics. The auxiliary equations which determine these paths are dT s d s d hi0 : D D T 1 C s hi0 The two independent solutions are s T D a; hi0

1Cs

D b;

(5.51a) (5.51b)

where a and b are two arbitrary, but positive, constants. These show that both the temperature and average tension decrease with the protocol parameter. The modulus of elasticity, a .1Cs/=s 1 1 @ D ; @ hi0 b.1 C s/ T analogous to adiabatic compressibility, .1=V / .@V =@p/S , shows that the elastic modulus decreases with increasing temperature. That the stiffness should increase is due to the fact that the mean tension increases with temperature, instead of the usual behavior where it decreases with increasing temperature. Integrals (5.51a) and (5.51b) represent the intersection of two surfaces, which is a curve; in fact, a family of curves is obtained by varying the arbitrary constants. One of these curves will pass through a point P , which is determined by a and b that result when the coordinates of P are substituted into (5.51a) and (5.51b). Suppose that P 0 is another point adjacent to P , then the direction cosines of PP 0 are proportional to dT 0 ; d0 ; d hi00 or to the values T; =s; .1 C s/ hi0 =s at P . Now, the condition that the curve PP 0 is perpendicular to the curve of (5.51a) and (5.51b) is that the inner product T dT

1Cs d C hi0 d hi0 D 0: s s

The solution of the differential equation (5.50) must include all curves which are perpendicular to the system (5.51a) and (5.51b) (Forsyth 1956).


133

An equation similar to (5.49) was written down by Lord Rayleigh in his theory of the pressure of vibrations. Rayleigh (1902) was concerned with the analogy of the newly (remember we are in 1902!) discovered pressure of vibrations and those of aerial vibrations. Rayleigh found that the mean additional force, analogous to our hi0 , of vibrations tends to push the piston that encloses them and is given by hEi0 . Here, represents the length of the vibrating string and hEi0 is the constant total energy or twice the mean kinetic energy of vibrations. He then went on to consider the case that would apply equally as well to a vibrating body or to a flexible vibrating rod. Now, represents the displacement from equilibrium, and the kinetic and potential energies were assumed to be homogeneous functions of orders n and m, respectively, in the displacement. Rayleigh then deduced the Grüneisen equation of state (5.49) with the parameter s D .m n/=2. For a flexible rod, m D 3 and n D 1, so that hi0 D 2 hEi0 =, while for a line of disconnected pendulums, m D n, and hi0 D 0. Finally, Rayleigh queried whether part of the energy of vibrations is “unavailable in the absence of appliances for distinguishing phases.” The phases are distinguished by the position of the ring which does not conserve the number of vibrations. Introducing the definitions of the mean energy, @ ln Z D hEi0 ; @ˇ and mean tension, 1 ˇ

@ ln Z @

ˇ

D hi0 ;

(5.52)

into the Grüneisen equation of state (5.49), convert it into the partial differential equation @ ln Z @ ln Z Cˇ D 0: (5.53) s @ @ˇ The definition of the force, (5.52), leaves invariant the probability distribution, pn0 , under variations in the protocol parameter since @E @ ln Z @ X ˇEn ln Z D 0: e D ˇ @ n @ 0 @

(5.54)

This is often referred to as the principle of adiabatic invariance (Lavenda 1991, pp. 187–188). Employing the method of characteristics, the adiabats are determined from the solution to the auxiliary equations s

d dˇ ln Z D D : ˇ 0

134


The two independent solutions are s =ˇ D a and ln Z D b, where a and b are arbitrary constants. Therefore, the general solution to (5.53) is ˆ.ln Z; s =ˇ/ D 0, where ˆ is an arbitrary function. Because ln Z must be convex in ˇ, the simplest solution is to set Z equal to a positive power m of the ratio .s =ˇ/ or what is equivalent ˇ : ln Z.; ˇ/ D m ln s This expression for the generating function is entirely consistent. Introducing the definition of the average energy, @ ln Z=@ˇ D m=ˇ D hEi0 , into the expression for the mean tension, @ ln Z=@ D sm= D ˇ hi0 , gives back the Grüneisen equation of state, (5.49). Now, the adiabatic Euler relation d hEi0 D hi0 d can be written as

d ln hEi0 D s; d ln

(5.55)

when the Grüneisen relation (5.49) is introduced into it. With the aid of the “adiabatic” (now intended in the mechanical sense of “slow variation”) relations hEi0 =! D const. and D const., where ! is the angular velocity and , the wave number, (5.55), can be expressed as ug d ln hEi0 d ln ! D D s; D up d ln d ln

(5.56)

where ug and up are the group and phase velocities, respectively. Since s varies between 13 , for an ultrarelativistic gas, and 23 for an ideal gas, ug < up . Their ratio is seen to be exactly the inverse of half the degrees of freedom of the system. Consequently, (5.49) can be written as ug hEi0 D hi0 : up

(5.57)

This relation is due to Lord Rayleigh (1902), and it comes as close as one can get to a dynamic statement about the system. The negative sign in (5.57) is due, as Rayleigh pointed out, to the fact that the mean tension must be applied in the direction which tends to diminish . Also, as Rayleigh observed, the mean tension, (5.57), is a measure of the tendency of the group of waves to disperse. We shall return to more general relations of this type in Sect. 5.4.4. Dividing both sides of (5.46) by T , the condition for integrability is T

@ hi0 @T

@E D L D @

0

@ hEi0 ; @

(5.58)


135

while multiplying both sides of it by s gives the integrability condition as sC @ hi0 D : @T

(5.59)

On the basis of the Maxwell relation (5.48) and (5.38), we conclude that the rate must be a increasing function of time 1 P D ˛T

@ hEi0 @E ; @ @ 0

and not a constant, as Jarzynski would have us believe. This shows the microscopic origin of the Carnot–Clapeyron equation: In the presence of fluctuations the derivative of the average will not be equal to the average of the derivative. Excluding neutral curves, which, as we have seen in Sect 3.2.2, appear in a certain range of temperatures where water displays anomalous behavior, the positive definiteness of the latent heat with respect to the protocol establishes a kind of Le Châtelier principle for first-order derivatives instead of the usual second-order ones, provided there is no distinction between the energy of the microcanonical ensemble and the average energy of the canonical ensemble. For if there were, there would be two forms of thermodynamics and not one as we know it. Using the canonical form of the probability distribution, (5.19), we can evaluate the latent heat with respect to the protocol parameter, (5.42), as L D

X m

Em

0 X @pm 0 Dˇ E m pm @ m

@Em @F @ @

:

(5.60)

Equating this expression with that found in (5.58) enables us to write the latent heat as ˛ ˝ ˇ .@=@/E 2 0 L D : 2 1 C ˇ hEi0 This shows that the latent heat is proportional to the negative of the average of the derivative of the second moment with respect to the protocol parameter. In contrast, the heat capacity at constant protocol is directly proportional to the second central moment, ˝ ˛ C D ˇ 2 E 2 0 : The microcanonical (adiabatic) ensemble pertains to an infinite switching time, or P ! 0, while the canonical (isothermal) ensemble applies to an instantaneous switching time, or P ! 1. Since there is no other ensemble intermediate between these two ensembles, Jarzynski’s inequality has absolutely no meaning. Even if it did, the canonical ensemble cannot be used for finite switching times since the process would no longer be isothermal! Even worse, Jarzynski mixes the two ensembles by imposing the adiabatic case on the isothermal one. Adiabats and isothermal never coincide at finite temperatures, as we have seen in Fig. 2.4 of 2.3.3.

136


5.4.4 From Adiabatic to Isothermal Relations and their Underlying Microscopic Counterparts Ehrenfest (Klein 1969) saw in the adiabatic principle the “royal road” to quantization because adiabatic invariant quantities were prime candidates for quantization. Ehrenfest’s theorem asserts that for any periodic system, the time integral of the kinetic energy when average over a period of the motion is invariant under any adiabatic change in a parameter upon which it depends. By “adiabatic” it is intended that the parameter will change sufficiently slowly over a period of the motion. Obviously, Ehrenfest borrowed the term from thermodynamics in the hope that these adiabatic invariant quantities would provide a broad enough foundation so that thermodynamics would withstand the onslaught of quantum theory, which was still in its infancy. We have seen on a number of occasions how the Carnot–Clapeyron equation T

@ @T

D L D

@E @

(5.61)

T

can be converted into a partial differential equation for the energy T

@E @T

s

@E @

DE

(5.62)

T

by the Grüneisen equation of state (5.49), and then be solved by the method of characteristics to give (Einbinder 1948) E D s .T s / ;

(5.63)

where is an arbitrary function. If we impose the adiabatic condition, Es D const:, we get the “cold” equation of state C (5.64) E D s; where C is a positive constant. The equation of state is devoid of the temperature and displays a repulsive nature among its constituents since .@E=@/ < 0. When does such behavior apply? For very low temperatures, the function in (5.63) will start out as a power law, say (5.65) E D C s.m1/ T m : Such a gas will be attractive, .@E=@/T D s.m 1/E= 0, where the equality sign implies that the gas is ideal for m D 1. Since there is no gas whose energy can increase more slowly than T , the ideal gas will separate repulsive (adiabatic), (5.64), from attractive, (5.65), equations of state. Do such relations exist on the microscopic level?


137

If we interpret the length as a wavelength, then the dispersion equation D c, for a nondispersive medium, converts (5.63) into the adiabatic condition E= D const., provided T = D const. The former is Planck’s quantization rule, while the latter is Wien’s displacement law. Ehrenfest (1923) conjectured that the search for other adiabatic invariants would lead to more general forms of quantization. Now, at high temperature in an isothermal regime, we may take as exponential. Then (5.63) becomes (5.66) E .T / D C ˛ e˛ =T ; which will readily be recognized as Wien’s formula, valid in the high-frequency limit of blackbody radiation (Lavenda 1991, Ch. 2), where C and ˛ are curve fitting parameters. Expression (5.66) is the average energy of an oscillator at temperature T that Planck obtained from Wien’s law. It carries with it a certain entropy S between frequencies and C d. It can be found from (5.66) by solving for the inverse temperature, and using the definition of the second law

E 1 @S 1 D ln D ; ˛ C ˛ T @E

(5.67)

in the frequency interval from to C d, where represents set parameters which control the relation between the system and its environment. Integrating (5.67) leads to E E ln 1 : S D ˛ C ˛ This expression for the entropy, via Gauss’s error law, leads immediately to the Poisson distribution, where the ratio E = is an adiabatic invariant proportional to the number of particles (Lavenda 1991, pp. 82–83). Alternatively, we can read off from (5.66) that the probability distribution is exponential (5.68) p 0 D C e˛ =T ; and use the Shannon–Gibbs entropy of the canonical distribution,1 X ˛ ˝ S1 D ln p 0 0 D p 0 ln p 0 D ln C C hEi0 =T;

(5.69)

to determine the normalization constant C D exp.F=T /. This clearly shows that the canonical distribution (5.68) is not valid in the P ! 0 limit, if we agree that this limit corresponds to the adiabatic limit. In this example, the adiabatic limit is characterized by Planck’s quantization condition and the Wien displacement law that pertain to the unique frequency for which the spectral energy distribution peaks.

1

The index on the entropy will become clear when we compare the Shannon–Gibbs entropy to an entire family of other entropies with the variation of a characteristic exponent [cf. (6.97) below].

138


This is a one-dimensional example where s D 1; in three dimensions the Grüneisen parameter, s, would only be a third as large. Photons may, therefore, be characterized by an energy E D

hc

T hc

;

which defines a thermal wavelength for photons as T D hc=T . The thermal energy will be a fraction of the rest energy mc 2 (Lavenda 1995), T D

C T

3s

mc 2 ;

where C D h=mc is the Compton wavelength and T is the sought after thermal wavelength. In the nonrelativistic limit, the Grüneisen parameter s D 23 , and the thermal wavelength turns out to be h ; (5.70) T D p .mT / whereas in the ultrarelativistic limit, s D 13 , and the thermal wavelength is T D

hc : T

(5.71)

Hence, from (5.63) the average energy will be E D

h2 m2

mT 2 h2

;

(5.72)

which would apply to slow moving electrons and not to the pressure of radiation. Yet, in prerelativistic times, it was precisely the adiabatic form of (5.72) that was applied to the pressure of radiation (Poynting 1910). A hundred years had passed since Thomas Young “killed” corpuscular theory and replaced it by the now universally accepted theory that light consists of waves. Three-quarters of a century later, Clerk-Maxwell showed that the electromagnetic phenomena consisted of electric and magnetic disturbances consisting “of tubes of electric force and tubes of magnetic force, which are rushing sideways against each other and against any surface which the beam strikes” (Poynting 1910, p. 21). That these waves must carry momentum with them when they strike any surface, just as if they consisted of small particles, was shown by the Italian physicist Bartoli in 1875. The action of these tubes of electric and magnetic disturbances was imagined to behave like springs and therefore contain energy. But, the action was entirely transversal, so that no mass could be associated with such waves. When these springs are compressed more energy is required, and it was believed that the energy density was proportional to the square of the wavelength. Experimental support for


139

this belief was supplied by Christian Doppler in 1842 who found that the wavelengths of moving objects were altered by their motion, shortened if the object is moving toward the observer, lengthened if moving away. Since the energy density was proportional to the square of the velocity, what they did get right was that the velocity, or more correctly the momentum, was inversely proportional to the wavelength, and not the expression for the energy density, which was also proportional to the inverse of the wavelength. Hence, de Broglie’s relation was known at least a quarter of a century before he discovered it! When waves strike a surface, they transfer momentum, and hence exert a pressure against the object. If the object is a perfect reflector, “the reflected waves press back just as much as the incident waves and so the pressure is doubled,” implying that s D 2 in the Grüneisen equation of state (2.56). However, the adiabatic condition, T V 2=3 D const. or T 2 D const., applies to the entropy of a degenerate electron gas and not to an ultrarelativistic gas, where T D const. would apply. In fact, the adiabatic condition, T 2 D const., defines the nonrelativistic thermal wavelength, as we have seen in (5.70)! Introducing the relation between wavelength and frequency into (5.72), we may write it as 2 I 2 ; (5.73) E D I T where I D m2C is a “microscopic” moment of inertia. The adiabatic relation, E2 D const., can now be written as E= 2 D const., in marked contrast with its ultrarelativistic analog. Bear in mind that we are considering an isolated system, “since the energy of such a body is by definition constant and does not fluctuate” (Landau 1959, p. 78). The situation is completely altered when we consider the isothermal case where becomes exponential. For then (5.72) becomes E D I 2 eF

0 =T I 2 =2T

;

(5.74)

where F 0 is the free energy in a system of coordinates rotating with the body. As Landau and Lifshitz (1959, p. 101) conclude, the statistical properties of a rotating system are affected only to the extent of an additional potential energy, 12 I 2 , due to the presence of the centrifugal force. We may now ask how the system, which is canonically distributed, responds when we alter the protocol parameter, . A change in will certainly alter the energy levels, E , which will, in turn, alter hEi0 . If the temperature is also varied during this change and the system remains canonically distributed throughout the change, we will have from (5.69) ˛ ˝ d ln p 0 0 D

@ ln Z @ ln Z C hEi0 dˇ C ˇ d hEi0 C d; @ˇ @ ˇ

(5.75)

140


where it is more convenient to work with inverse temperature ˇ, rather than the temperature itself, and we have written ln Z D ln C . The coefficient of dˇ vanishes on account of the definition of the average energy, and we are left with ˝ ˛ d ln p 0 0 D ˇ

@E d hEi0 d : @ 0

Hence, the work that is performed when the protocol parameter is varied is:

@E dW D @ and

d; 0

˝ ˛ T d ln p 0 0 D d hEi0 C dW

(5.76)

becomes the first law, if we interpret the term on the left as the heat added to the system that allows changes in temperature from T to T C dT and in the protocol parameter from to C d. In the case of equal a priori probabilities p 0 D 1 for all , where is referred to as the statistical weight, or the structure function, (5.76) (Lavenda 1991, Ch. 4) reduces to @ ln Z d: (5.77) d ln D ˇ dE C @ ˇ If (5.77) were to vanish so as to give (5.45), it would mean that is independent of the protocol parameter. This has been termed the principle of adiabatic invariance by Ehrenfest because “must always depend on those quantities which remain invariant under adiabatic influencing.” Consequently, Jarzynski’s equality (5.13) cannot handle the situation in which the protocol parameter is changed adiabatically precisely because the system does not remain canonically distributed.

5.4.5 The “Crooks Equality” Presumably motivated by expression (5.13), “that is valid in the far-fromequilibrium regime [which] is the recently discovered relationship between the differences in free energies of two equilibrium ensembles, F , and the amount of work, W , expended in switching between ensembles in a finite amount of time,” Crooks (1999) proffers a “new” equality PF .C!/ D eC! : PR .!/

(5.78)


141

Continuing to quote from Crooks, Here ! is the entropy production of the driven system measured over some time interval, PF .C!/ is the probability distribution of this entropy production, and PR .!/ is the probability distribution of the entropy production when the system is driven in a time-reversed manner.

Two comments are in order: First, the entropy production over some finite time interval is an entropy difference, and second, there is no such thing as a probability distribution for an entropy difference, or a work. Both are functions of the fluctuating variables, and are not fluctuating variables in their own right. In other words, work is not a stochastic process, but may be a function of such a process. Crooks then offers the following concrete example: : : : consider a classical gas confined in a cylinder by a movable piston. The walls of this cylinder are diathermal so that the gas is in thermal contact with the surroundings, which therefore act as the constant temperature heat bath. The gas is initially in equilibrium with a fixed piston. The piston is then moved inwards at a uniform rate, compressing the gas to some new, smaller volume. In the corresponding time-reversed process, the gas starts in equilibrium at the final volume of the forward process, and is then expanded back to the original volume at the same rate that it was compressed by the forward process. The microscopic dynamics of the system will differ for each repetition of this process, as will the entropy production, the heat transfer, and the work performed by the system. The probability distribution of the entropy production is measured over the ensemble of repetitions.

Presumably, heat will be released as the gas is compressed at constant temperature and absorbed when it expands at constant temperature. It is remarkable that in all the statements of the second law, there is no mention of any finite rates at which the process is carried out. Repeating Carnot’s words: if work is to be done by heat, some heat must pass from a hot body to a cold one,

so that no cycle can be completed at a single temperature and no work can be done. It also contradicts Kelvin’s pronouncement that It is impossible, by means of an inanimate material agency, to derive mechanical effect from any portion of matter by cooling it below the temperature of the surrounding objects.

The reference to an “inanimate material agency” has left many a reader perplexed, thinking that such a qualification would save the principle, insofar as some animate agency would ever be discovered that contradicts his principle. And not to mention Clausius, who uses Carnot’s “unmodified investigation of the relation of the mechanical effect produced and the thermal circumstances from which it originates, in the case of an expansive engine working within an infinitesimally small range of temperatures.” All would therefore agree that work, as we know it thermodynamically, cannot be defined by any engine working at a single temperature. Crooks then realizes that his (5.78) and Jarzynski’s (5.13) are “actually closely related,” since he

!

Z

C1

iD 1

PF .C!/e

!

Z

C1

d! D 1

PR .!/d! D 1:

(5.79)

142


Again, since ! is not the realization of a random variable in the interval between ! and ! C d!, it has no probability density PF .C!/. For if it were a true probability density function, then it would also be true ˝

˛ eC! D

Z

C1 1

PR .!/eC! d! D

Z

C1 1

PF .C!/d! D 1:

He then tells that his equality (5.78) “can be applied to systems that start in equilibrium, and that the entropy production ! for such systems is ˇF C ˇW .” First, it is not the entropy production, !, but, rather, the entropy difference; second, inserting the first law W D E Q into such an expression would lead to ! C ˇQ D ˇ .E F /, which is totally foreign to thermodynamics – even under isothermal conditions; and third, F D 0 is not the criterion for a process to be time symmetric! Moreover, inserting this expression into (5.79) and “noting that the free energy difference is a state function, and can be moved outside the average,” Crooks would find E D (5.80) eˇW D eCˇF ; 0

and not Jarzynski’s (5.13). Elementary concepts, both in probability theory and thermodynamics, are being distorted beyond recognition in order to arrive at a predetermined goal. Finally, Crooks’s “fluctuation–dissipation” theorem is none other than the condition of microscopic reversibility at equilibrium. It can be found in almost any book on statistical physics. It applies to Markov processes, which are stochastic processes in which the present determines the future, independent of the entire history of the process. It relates the transition probability density p.x; jy/ for a transition from the state y to another (nonequilibrium) state x in time to the reverse transition probability density p.y; jx/ according to (Lavenda 1985) p.x/ p.x; jy/ D D eŒS.x/S.y/ ; p.y; jx/ p.y/

(5.81)

where p.x/ is the invariant probability density that is given in terms of Boltzmann’s principle as ln p.x/ D S.x/ C const., in energy units where Boltzmann’s constant is unity. The fluctuation–dissipation theorem guarantees that the joint probabilities p.x; jy/p.y/ and p.y; jx/p.x/ are equal at equilibrium. These remarks also apply to other “nonequilibrium” fluctuation–dissipation theorems (Evans 1994; Gallavotti 1995). In a follow-up paper, Crooks [2000] obtains results that are “directly applicable to systems driven far from equilibrium.” He considers a system that is perturbed from its equilibrium state “for a finite amount of time” and then allowed to relax back to equilibrium. This process is supposedly accompanied by an absorption of heat Q and an increase in entropy of ˇQ equal to the entropy lost by the heat bath, ˇQ. The “change in the entropy of the system will be ˇE ˇF . Therefore,


143

the total change in entropy of the universe resulting from the perturbation of the system is ˇQ C ˇE ˇF D ˇW ˇF D ˇWd ; (5.82) the dissipated work.” Since the system together with the heat bath are isolated and the entropy decrease of the heat bath is equal to that gained by the system at the same temperature, we can only conclude that Wd D 0. Moreover, who tells us that the action of the perturbation for a finite time, or the relaxation of the system from its nonequilibrium state back to equilibrium, will occur with the reversible absorption of an amount of heat Q at temperature 1=ˇ? Surely, Crooks is begging the question for he is subtracting the change in the free energy from the expression of the first law E D Q CW applied to the system and not the combined system C heat bath. In other words, he needs the difference between the entropy of the system ˇ.E F / and the heat Q in order to call it the “dissipated” work. But, where is the dissipation if the system absorbs the quantity of heat Q at the temperature of the bath? Crooks has the exponent in the fluctuation–dissipation relation (5.81) replaced by the heat, which “is a functional of the path, and odd under time reversal.” We can even stretch our imagination to consider “heat” as the work communicated through “unconstrainable” coordinates (Thomson 1888), but to consider heat as a functional of the path of a fluctuation and odd under time reversal really requires something more than imagination.

Chapter 6

Nonextensive Thermodynamics

Carnot [1824] fully appreciated that when the volume of a gas changes at constant temperature the amount of heat absorbed or released by a gas will follow an arithmetical progression when increases or decreases in volume follow a geometrical progression.

Replacing the word “heat” by “entropy” results in the extensivity of the latter. Although this applies to the common case of an ideal gas, we are not bound to it, and in this chapter we consider more generalized mean relations. This has the effect of destroying extensivity of the thermodynamic potentials, except in some special cases, and in so doing, we will obtain nonextensive ones. Specifically, we supplement the early approach to thermodynamics discussed in Chap. 2 with more general considerations involving general means of temperature and volume, other than arithmetic and geometric mean values of classical thermodynamics (Lavenda 2005). These are the most common since they lead to logarithmic forms for the fundamental relation of entropy in which the entropy is a function of all the extensive independent variables. The Grüneisen equation of state (2.56), which makes the elimination of the chemical potential between the energy and total number of particles superfluous, will identify the free energies as the potentials from which all others can be derived (Einbinder 1948). Traditional thermodynamics provides the relation between probability and entropy, through Gauss’s law of error, which identifies the entropy as the potential of the distribution and gives the probability distributions belonging to the exponential family of distributions when the average and most probable value of the variate coincide (Lavenda 1991). The normal, Poisson, binomial, and negative binomial all belong to the exponential family. This family has some remarkably simple and general properties, undoubtedly the most important of which is that the data is summarized by a single statistic which contains all the relevant information supplied by the sample (Kullback 1959, p. 18). Already with Stefan’s law of blackbody radiation the logarithmic form for the fundamental relation for the entropy is lost, together with the connection with probability via Gauss’s error law. Planck’s search for the form of the entropy and the associated distribution of blackbody radiation can be summed up by saying that he was looking for a new form of the entropy that would have a logarithmic form,


145

146

6 Nonextensive Thermodynamics

which he found in every frequency interval. Although we remain within the macroscopic world of thermodynamics, we realize that there are also repercussions for fluctuation theory, and come close to relating the mean entropies to well-known entropies of information theory. The new mean entropies that we will be dealing with satisfy all the topological requirements for a distance (metric), including the triangle inequality, and play important roles in such seemingly disparate areas as coding theory and multifractals. The property that means are monotonically increasing functions of their orders will show that the entropy tends to increase when a system evolves from a moreto a less-constrained equilibrium states, while other adiabatic, nonextensive potentials manifest the opposite tendency. These properties are related to the concavity and convexity of these potentials, and we will provide a characterization of these potentials. Undoubtedly the most celebrated of all these relations is the arithmetic– geometric mean inequality, and we can turn the tables and offer a thermodynamic proof of the arithmetic–geometric mean inequality. This is well-known and has been rediscovered many times in the literature (Sommerfeld 1956; Landsberg 1980; Sidhu 1980). The arithmetic–geometric mean inequality is just one of an infinite number of inequalities based on the properties of means to which this chapter is dedicated.

6.1 Incomparable Thermodynamic Laws Choosing T and V as independent variables, the integrability condition for the entropy, dS D .1=T /dE C .p=T /dV 1 @E 1 @E C p dV C dT; D T @V T T @T V

is T

@p @T

D

V

@E @V

C p:

(6.1)

T

If the product pV measures the absolute temperature scale, i.e., pV / T , then it follows that @E D 0: (6.2) @V T This is to say that in order for Mariotte’s law to hold, it is both necessary and sufficient that the internal energy be a linear function of the absolute temperature alone.

6.1 Incomparable Thermodynamic Laws

147

However, if the pressure were to vary as some power ˛ of the absolute temperature [cf. (6.13)], it would follow from (6.1) that

@E @V

D T

T p

@p @T

1 p D .˛ 1/p:

(6.3)

V

Thus, the ideal gas condition (6.2), implying that pV measures the temperature, is a limiting case corresponding to Holtzmann’s hypothesis. For ˛ > 1 the gas manifests an attractive nature. The region where ˛ < 1 is ruled out because the energy would increase more slowly than the temperature itself, and since the temperature is the average of the kinetic energy of the molecules this would be impossible. Any gas of noninteracting particles will obey an equation of state having the form of the Grüneisen equation (2.56). Introducing (2.56) into the integrability condition (6.1) for the entropy converts it into a differential equation EDT

@E @T

V

V s

@E @V

:

(6.4)

T

One way of solving (6.4), which we have used time and time again, is the method of characteristics which selects out characteristic paths in the T; V -plane, where the partial differential equation reduces to an ordinary differential equation. The auxiliary set of differential equations dV dE dT D s D ; T V E

(6.5)

gives the characteristics, which are none other than the adiabats: T V s D c, and either E=T D a or EV s D b, with a, b, and c as arbitrary, but positive, constants. We thus see that the Grüneisen parameter s is, indeed, an adiabatic exponent. The general solution to (6.5) is 1 .a; c/ D 1 .E=T; T V s / D 0, viz., EDT

1

.T V s / ;

(6.6)

or 2 .b; c/ D 2 .EV s ; T V s / D 0 which says that E D V s

2 .T V

s

/;

(6.7)

where i and i are arbitrary functions. The coefficients of E, namely 1=T and V s , will soon be shown to be the integrating factors for the quantity of heat added to the system. The arbitrary functions i are solutions to the adiabatic equations T

@ i @T

V

V s

@ i @V

D 0: T

(6.8)

148


whose auxiliary equations are dT sdV d i D D : T V 0 Again, there are two independent solutions: i D a and T V s D b. The general solution to (6.8) is simply i D i .z/ in which T and V appear only through the product z D T V s . The variable z has a very special significance, as we will now show. Introducing the Grüneisen equation of state (2.56) into the first law, dQ D dE C p dV , gives dQ D dE C s

E dV: V

Multiplying both sides by V s results in V s dQ D d .EV s / DW dL:

(6.9)

since the right-hand side of (6.9) is a total derivative, so, too, must be the left-hand side. Now, according to (6.7), EV s is a sole function of T V s or z. The only way this can occur is if dQ is a product of T and a total derivative, which we know to be dS . Consequently, (6.10) z dS D d .EV s / D d 2 .z/: Integrating gives

L.z/ D EV s D

2 .z/:

(6.11)

A plethora of adiabatic potentials can be derived of which the entropy is one member, albeit, a very important one. The adiabatic potential (6.11) is both nonextensive and convex, and will be seen to have properties that are diametrically opposite to those of the entropy. But, what can we say about the arbitrary function 2 .z/? What we do know is that (6.12) T V s ; EV s ; and pV 1Cs all have the same values for all states which are accessible from each other by quasistatic, adiabatic transitions. They are referred to as the Poisson adiabatics, which we discussed in Sect. 1.3. Landau and Lifshitz (1959, p. 150) claim that, in general, the exponents in (6.12) “are here unrelated to the ratio of specific heats, since the relations Cp =CV D 53 and Cp Cv D R are not valid.” But, things work the same as if (6.12) were the Poisson adiabats for an ideal gas; only the numerical factors change [cf. discussion following (6.90)]. Any one of (6.12), or any function of them, would be a valid candidate for an “empirical entropy” (Buchdahl 1966, p. 70), whose definition we just gave. The empirical entropy is not extensive, and this is the reason for defining a “metrical” (Buchdahl 1966, p. 73) entropy as that entropy of a compound system which is the sum of the metrical entropies of the constituent systems. One way of getting


149

a metrical entropy from an empirical entropy is by taking the logarithm of the latter (Buchdahl 1966, p. 84). But, this is like pulling rabbits out of a hat because we know the logarithmic form that the entropy of an ideal gas must have. This recalls Carnot’s [1824, p. 86] proposition, mentioned in the introduction, that “When the volume of a gas increases in a geometrical progression, its specific heat rises arithmetically,” where we should read entropy for specific heat. The thermodynamic stability condition that the pressure cannot increase with volume will place an upper bound on the adiabatic exponent s D R=Cv . In view of the power law in (6.7), 2 can be a power law, say of exponent ˛. We then obtain

N0 p D sA V

1Cs Cs

C ˛ .˛1/s1 T V ; ˛

(6.13)

where the first term is repulsive, like that of a “cold” gas, and the second is attractive for ˛ > 1 [cf. (6.3)]. N0 is the total number of particles at T D 0, and A and C are two positive constants. The condition for thermodynamic stability is

@p @V

T

.s C 1/ D sA V

N0 V

sC1

s C Œ.˛ 1/s 1C T ˛ V .˛1/s2 0: (6.14) ˛

If the repulsive contribution vanishes, A D 0, the condition of stability becomes s

1 ; ˛1

(6.15)

where the equality sign pertains to a phase equilibrium where the pressure becomes independent of the volume, as we will see in Sect. 6.2. On the strength of the Grüneisen relation (2.56), the expression for the pressure, (6.13), gives the internal energy as EDA

N0 V

s N0 C

C s ˛ V z ; ˛

(6.16)

which is an explicit expression for (6.7). Therefore, the heat capacity at constant volume is @E @S ˛1 D Cv D C z DT ; (6.17) @T V @T V Integrating (6.17) with respect to T gives the expression for the entropy S.z/ D

C ˛1 z ; ˛1

(6.18)

where we have set the arbitrary constant of integration equal to zero. Comparing (6.17) with (6.18) we can sympathize with Carnot’s confusion between “heat capacity” and “heat.”

150


The relation between the thermodynamic definition of pressure and that using (6.18) is p @T @S @S @S D C : (6.19) D T @V E @V T @T V @V E The entropy (6.18) has no knowledge of the repulsive contribution to the pressure. It is brought in by the last term in (6.19), viz.,

@T @V

E

sA D C z˛1

N0 V

sC1 s

˛1T : ˛ V

In the absence of repulsion, the condition that the entropy be concave with respect to volume, holding either E or T constant, is (6.14). Whereas the entropy is not a complete, or characteristic (Gibbs 1928), function, insofar as all other potentials cannot be derived from it, the Helmholtz free energy,

N0 F .V; T / D A V

s N0

CT z˛1 ; ˛.˛ 1/

(6.20)

is a characteristic function (Einbinder 1948). Its derivatives with respect to V and T ,

@F @V

(

T

N0 D s A V

and

@F @T

sC1

D V

C C z˛ V .sC1/ ˛

)

C ˛1 z ; ˛1

(6.21)

(6.22)

are the negative of the pressure (6.13) and negative of the entropy (6.18), respectively. This is on account of the Grüneisen relation (2.56), which is obtained by multiplying (6.21) by V and setting it equal to sE. Even in the absence of the mechanical repulsive term in the energy, (6.16), the derivative of the entropy with respect to V does not yield the ratio p=T , but, rather, is related to it by (6.19). The condition for the gas to be attractive is

@E @V

T

˛1 ˛ z AN0sC1 V .sC1/ > 0; Ds C ˛

(6.23)

while reversing the inequality means that it is repulsive. Quantum mechanically, we may attribute the repulsion between the particles to quantum exchange effects. The relation between (6.23) and the pressure (6.13) is:

@E @V

D

S

@E @V

C

T

@E @T

V

@T @V

: S

(6.24)


151

The last term is evaluated by differentiating T V s D const. thereby obtaining .@T =@V /S D sT =V . The term which multiplies this is the heat capacity, (6.17). Hence, the last term in (6.24) is ˛p. And since the first term is .˛ 1/p, according to (6.3), the left-hand side is indeed the negative of the pressure, (6.13). Multiplying both sides by V does give back the energy (6.16) when the left-hand side is evaluated with the aid of the Grüneisen relation (2.56). Also with the aid of the Grüneisen relation (2.56) we get the enthalpy as H D .1 C s/E D T S C N;

(6.25)

where is the chemical potential and N the particle number. Solving (6.25) for the Gibbs free energy results in N D .1 C s/A

N0 V

s N0 C

Œ.˛ 1/s 1 ˛ s Cz V ; ˛.˛ 1/

(6.26)

Œ.˛ 1/s 1 ˛1 Cz : ˛.˛ 1/

(6.27)

or, equivalently, . =T /N D .1 C s/AN01Cs z1 C

In the case of a degenerate electron gas at absolute zero, the second term vanishes and the limiting energy is essentially the chemical potential which varies as .N0 =V /s , where s D 23 . In the case of a nonrelativistic degenerate Bose gas, both terms in (6.26) vanish, the first because A D 0 and the second because the equality in (6.14) with s D 23 and ˛ D 52 . In the case where A D 0 and the stability condition (6.14) applies, N D const. From (6.27) we get N0 d . =T / D

1 fCv d ln T C R d ln V g ; ˛

(6.28)

which is, essentially, the expression for the chemical potential of a classical ideal gas, where WD s.˛ 1/. Thus, (6.27) offers us two possibilities: N D const., and =T D f .T /, or =T D const. and N D f .T /. The former conserves the particle number, the latter not. Quantum statistics gives distribution in the number of particles N./ in the energy range between and C i1 h ; N./=g./ D e . /=T ˙ 1

(6.29)

where g./ is the degeneracy, and the plus and minus signs refer to Fermi and Bose statistics, respectively. Integrating (6.29) over all energies, gives N D N.z/ and =T D const., or N D const., R 1 and =T D =T .z/. In other words, if the total number of particles is fixed 0 N./d is equated with the total number of particles,

152


N , and this makes =T a function of temperature (Mayer 1940, p. 379). If the total number of particles is not conserved, N D N.z/ and =T D const. In order to avoid a singularity in the case of Bose statistics, the chemical potential must be everywhere negative. This is insured by the stability condition (6.14) that the isothermal compressibility be positive. To gain some familiarity on how to apply these criteria let us look at some specific examples. Since a Bose gas will always have A D 0 and the equality in (6.14) at low temperatures, it suffices to consider degenerate electron gases. In the case of a nonrelativistic degenerate electron gas, the Grüneisen relation, p pV D 23 E, identifies s D 23 . Neglecting constants, we have pV V T 2 and E .N=V /1=3 V T 2 . From the Grüneisen relation it follows that .N=V /2=3 . The entropy S .N=V /1=3 V T , which means that the adiabatic invariant is V 2=3 T if N D const: From the expression for the chemical potential it follows that =T 1=V 2=3 T D const. So, in this case, we can have both N D const. and =T D const. Now, let us compare this to the relativistic degenerate electron gas. The Grüneisen equation of state at absolute zero is (Landau 1959, p. 157) p D 1 E=V .N=V /4=3 . Landau and Lifshitz claim that the Grüneisen relation holds 3 at all temperatures for the extreme relativistic gas and not only at absolute zero. We will see that also the third relation holds at all temperatures. Evaluating the Fermi integral, Landau and Lifshitz find pV . T /2 V .N=V /2=3 V T 2 . From this it follows that D .N=V /1=3 . Now the adiabats are V 1=3 T D const. so we can have =T D cont. and N D const:, just as in the nonrelativistic case. But, when we consider the heat capacity, Cv .N=V /2=3 V T , we find that N D const. would lead to a linear dependence upon the heat capacity, which is incorrect at low temperatures. Rather, if N V T 3 then Cv V T 3 . So, the same degenerate gas switches from one which conserves the particle number in the nonrelativistic limit to one that does not conserve particle number in extreme relativistic limit. The creation and annihilation of particles at high energies are responsible for the lack of particle conservation. Moreover, the relation p .N=V /4=3 , which was derived at T D 0, holds for all T when N V T 3 is introduced!1

1 Even in the nonrelativistic limit, we can have N as a function of temperature. Consider the rep lations pV V T 2 V T 5=2 if =T D const. Now, on the basis of the Grüneisen relation pV V T 5=2 .N=V /1=3 V T 2 . This last relation can be satisfied by setting N V T 3=2 . The heat capacity will then vary as T 3=2 instead of being proportional to the temperature at low temperatures in a degenerate Fermi gas (Lavenda 1995, p. 38). A possible example may be the instability in the conventional electronic state with respect to Cooper pair formation for temperatures T < Tc . Cooper made the interesting remark that if two electrons on the Fermi surface attract one another, unlike their repulsion elsewhere in free space, they will form a bound state (Brout 1964). The wavefunction describing the Cooper pairs is not an eigenfunction of N . The chemical potential is used to adjust the sum of expected number of P particles k hnk i to N . The initial gauge symmetry U.1/ is broken by a nonvanishing expectation of the pair creation or destruction operators, which would imply fluctuations in the total particle number. Fluctuations above Tc develop into collective modes below it.

6.2 Einbinder’s @-Theorem

153

6.2 Einbinder’s @-Theorem In what should have been recognized as a ground-breaking paper, Einbinder (1948) sets down his @-theorem, which, in its corrected version, reads: A system of mechanically non-interacting particles in equilibrium in a field-free space which individually satisfy the law of non- or completely relativistic mechanics has pV D sE; as T ! 0, then S ! C 0 and either it has a zero-point energy, E D C V s , .@E=@V /T < 0, or condenses p D C T 1C1=s , .@E=@V /T > 0.

The desire for extensive thermodynamic potentials is so ingrained that Einbinder [1948] introduced two conditions on the adiabatic exponents, one derived from thermodynamic stability, (6.14), and the other from the condition that “if the energy of the system: : : is to tend to zero, its volume must remain finite.” The only other condition we have is the positive definiteness of the coefficient of thermal expansion, 1 V

@V @T

D p

1 ˛ > 0; Œs.˛ 1/ 1 T

(6.30)

which, again, implies inequality (6.15) and not the reverse of that inequality. Somehow Einbinder obtains the reverse of inequality (6.15), and thereby concludes that if they are to be simultaneously satisfied only the equality can apply. Although the argument is concocted, the result is not: It is the condition for the pressure (6.13) be independent of the volume. This is the condition for the coexistence of more than one phase. It also happens to be the condition for the extensivity of the entropy, (6.18). Einbinder adds the caveat that “The @-theorem predicts the behavior of the system only in the neighborhood of absolute zero, having nothing to say about possible occurrences at higher temperatures.” Yet, he later adds that “blackbody radiation obeys the @-theorem at all temperatures.” Why is this? Einbinder is using power law expressions for the thermodynamic variables. As the entropy tends to zero it will do so as a power law. The third law requires that at absolute zero S D C 0 , which is what Nernst contends, while Planck argues S D 0, in which case the entropy is absolute, i.e., it contains no additive constants. A necessary condition that must be satisfied is that Cv ! 0 as T ! 0, which is clearly violated by an ideal classical gas for which .@E=@V /T D 0. Thus, in order to approach zero temperature, .@E=@V /T ¤ 0, which means that the gas is statistically attractive, if greater than zero, or statistically repulsive, if less than zero. According to Planck (1907) “these terms have no physical significance since only differences: : : of entropy: : : of the substance in different states play a part in natural processes.” He then goes on to discuss Nernst’s theorem and claims that “as the temperature diminishes indefinitely, the entropy of a chemical homogeneous body of finite density approaches indefinitely near to the value zero” (Planck 1907, p. 274). The latter statement is Planck’s formulation of the third law, and it appears to be in contradiction with his previous statement about the arbitrariness of the constant

154


of integration for the entropy (and not Nernst’s statement which claims that the entropy tends to a finite, constant value, independent of the volume as T ! 0). Einbinder does not know how to go from power laws to logarithmic expressions for the entropy, but knows that a power law is valid for the absolute entropy and energy in the case of blackbody radiation at all temperatures. This is undoubtedly his reason for adding the qualifying remark about the validity of his @-theorem “only in the neighborhood of absolute zero.” Moreover, the appearance of adiabatic exponents in the expression for the entropy appears to contradict our dictum that the entropy should not depend upon a parameter that is characteristic of a specific material or process. For, otherwise, there would be a different entropy for each material or process, which would make the entropies incomparable in a thermodynamic sense. However, the only exponents which appear in the expressions for the entropy are related to the dimensionality N of the space, where N D 2=s. Assertions to the contrary have appeared widely throughout the recent literature, such as “q-statistics,” where q is the exponent appearing in the entropy expression (Tsallis 1988) [cf. (6.54), where stands in for q]. However, these putative entropies can be nothing more than interpolation formulas which become true entropies in the limit where the characteristic parameter tends to one of its extreme values.2 The @-theorem applies only to extensive entropies, S D sC T 1=s V , where the equality sign applies in (6.15), leading to

@p @V

D 0:

(6.31)

T

It is also the condition for the validity of the Clausius–Clapeyron equation: It describes a phase equilibrium between vapor and liquid. Why then do we evaluate the Carnot cycle with the equation of state of an ideal gas? Precisely because the volume of gas is so much larger than that of the liquid so that the latter can be neglected for all intent purposes. That is, by varying the volume at constant temperature, the vapor pressure remains unchanged, and this can only come about by having the liquid either evaporate or condense so as to maintain the phase equilibrium. From (6.26) it is apparent that D 0, which is the condition of a degenerate gas, or one that does not conserve the particle number. It says that all degenerate gases possess extensive entropies. According to Einbinder, the condition ˛ D 1 C 1=s must hold in the case that the gas condenses. The criterion he claims is due to the thermodynamic stability criterion (6.14) together with reverse of the inequality which comes from the condition that it would be impossible for an electron to be confined to a volume of the order of a nucleus. The proof shows that such an electron would have to have a kinetic energy far greater than that found when an electron escapes from the nucleus as a ˇ-ray (Richtmyer 1942). In any case, the inequality 2 In information theory there are many interpolation formulas that connect the Hartley–Boltzmann entropy and the Shannon–Gibbs entropy. Most notably of these is the Rényi entropy, (Aczél 1975), which is also additive.

6.3 Unconventional Phase Equilibria

155

.˛ 1/s > 1 would contradict the thermodynamic stability criteria, the positive definiteness of the isothermal compressibility, (6.14), and the coefficient of thermal expansion, (6.30). But, what Einbinder is really after is the equality, and, this, as we have seen, applies to a phase equilibrium, where the entropy is always extensive. As a matter of fact, .˛ 1/ D 1=s is the criterion of extensivity of the thermodynamic potentials with ˛ 1. Now, in the limit as ˛ ! 1, the gas transforms into the classical ideal gas. In order to go to the limit we must consider entropy differences, S.z; z0 / D

C .˛1/ ln z e e.˛1/ ln z0 ; ˛1

(6.32)

for only entropy differences are measurable. Proceeding to the limit as ˛ ! 1, we find the indeterminate form 0=0 to which we apply l’Hôpital’s rule and obtain lim S.z; z0 / D Cv ln .z=z0 / D Cv ln .T =T0 / C R ln .V =V0 / ;

˛!1

(6.33)

where we have identified the arbitrary positive constant C as Cv and used s D R=Cv . Equation (6.33) is precisely the entropy difference of an ideal classical gas in which the arbitrary constant has cancelled out, meaning that only entropy differences are measurable.

6.3 Unconventional Phase Equilibria In general the number of particles, N , and the energy, E, will be functions of the adiabatic parameter, z, and the ratio =T . Expressing these functions as a power series in =T and eliminating =T from E.z; =T / with the aid of the expression for N.z; =T / give E.V; T / in the Grüneisen relation (2.56). As Einbinder (1948) tells us, this circuitous procedure (Mayer 1940) can be circumvented by expressing G as a function of z, viz., G=T D

E N D .z/ D .1 C s/ S T T .˛ 1/s 1 ˛1 D Cz : ˛.˛ 1/

(6.34)

Landau and Lifshitz (1958, p. 150) claim that for degenerate Bose and Fermi gases, the energy is calculated from E D N C T S pV D

@ @ T C ; @ @T

(6.35)

156


where the thermodynamic potential D pV D V T 5=2 f . =T /, and f is a function of a single variable and is equivalent to the Grüneisen equation of state (2.56) with s D 23 . This will be true provided

@f @f CT D0 @ @T

(6.36)

is satisfied. The auxiliary equation to (6.36) is dT d D ; T implying =T D const. Now, according to Landau and Lifshitz (1959), S D .@=@T /V; and N D .@=@ /T;V , but cannot be varied without varying T , and vice-versa, so only one variable is being held constant, and with these expressions (6.36) becomes N C T S 52 pV D 0: Comparing this with (6.34) gives s D 23 . This means if we set =T D .˛ 1/s 1;

(6.37)

in (6.34) then N.z/ D S.z/=˛;

(6.38)

where S.z/ is given by (6.18) with A D 0. In the event where the number of particles is not conserved, the entropy is proportional to it. The condition that (6.37) vanishes is not only the condition for a phase equilibrium, but, in addition, it renders the entropy extensive. In the limit where the condition of coexistence, (6.31), applies the chemical potential (6.37) vanishes, while in the limit as ˛ ! 1, N ! const., and =T will satisfy (Lavenda 1995, pp. 35–36) D S C S0 ; T T 0 as can be seen from comparing (6.28) and (6.33). Einbinder then goes on to solve the Gibbs–Helmholtz equation @ E=T @ F=T D .G sE/=T D ; @z @z z

(6.39)

where we have divided through by V s , since the volume is being held constant in differentiation with respect to T . Using (6.16), (6.34), and taking note of (6.6), (6.39) becomes (6.40)

0 .z/ s 10 .z/ D 1 .z/=z;

6.3 Unconventional Phase Equilibria

157

where

C ˛1 z ; ˛ and the prime stands for differentiation with respect to z. Einbinder (1948) gives the solution to (6.40) as: E=T D

1 .x/

D

(6.41)

Z 1 .z/ D z1=s B C (6.42)

.z/z.1=sC1/ dz s Z ˛ 1 1=s C z˛1 D z1=s B C C z˛1=s2 dz ; ˛.˛ 1/ ˛.˛ 1/

F=T D s

1 .z/

where B is an arbitrary constant. Since Einbinder was working under the phase equilibrium condition (6.15), he could only conclude that B D C =˛.˛ 1/. However, in the case where the inequality applies, integrating the second term in (6.42) gives the left-hand side of the equation, apart from an arbitrary constant of integration. Thus, we can set the arbitrary constant B D 0, for, otherwise it will cancel with the constant of integration. Taking the derivative of (6.23) with respect to V results in

@2 E @V 2

D s.˛ 1/ Œs.˛ 1/ 1 E=V 2 0:

(6.43)

T

If we impose the condition of a phase equilibrium, i.e., the vanishing of (6.37), then introducing the Grüneisen relation (2.56) into (6.43) with the equality sign requires

@2 p @V 2

D0

(6.44)

T

in addition to (6.31). This is precisely the condition for the critical point, O in Fig. 6.1, at which the gas–liquid coexistence curve terminates, to be stable. In other words, the critical state coincides with a point of inflection of the isotherm. Equations (6.31) and (6.44) constitute two equations in two unknowns and can only be satisfied at the critical point. In an ordinary phase transition, the transition point is determined by the equality of the chemical potentials, say for gas and liquid. In order to maintain such an equilibrium in the face of temperature and pressure variations, their differentials must also be equal. From the latter condition the Clausius–Clapeyron equation is usually derived (Frenkel 1946). For a degenerate gas, the condition for a “phase equilibrium” is the vanishing of the chemical potential (6.37) or equivalently (6.43). We have obtained condition (6.44) thermodynamically; it ordinarily is obtained using thermodynamic fluctuation theory. According to the principle of minimum work that must be done on a body to bring it from equilibrium to a neighboring state requires at constant temperature ıE T ıS C pıV D ıF C pıV 0;

(6.45)

158


Fig. 6.1 Phase equilibrium diagram

where ı represents small deviations in the quantity. Developing F in a Maclaurin series in ıV , we obtain 1 2

@p @V

T

ıV 2 C

1 3Š

@2 p @V 2

ıV 3 C

T

1 4Š

@3 p @V 3

ıV 4 C : : : 0:

(6.46)

T

The first term vanishes on account of the fact that the pressure is independent of the volume along the coexistence curves. These curves are located in the bell-shaped region XOY in Fig. 6.1. This region is determined where the isotherms terminate x 0 and y 0 in the figure. The solid curve is the gas–liquid curve which intersects the isotherms at x and y. Metastable states of superheated liquid and supercooled gas are found on the segments xx 0 and yy 0 , respectively. In order for the critical point O to be stable, (6.44) must hold: According to the thermodynamic fluctuation criterion (6.46), the inequality must hold for whatever the sign of the deviation of ıV ; thus, (6.44) follows for, otherwise, the criterion would be dependent on the sign of the fluctuation. The third derivative may also not be finite (Landau 1959, p. 267). The same equality (6.15) leads to the vanishing of these two terms as well as all higher derivatives. This is probably due to our approximating 2 .z/ in (6.7) by a single term. An ascending series in z would be more appropriate with the introduction of additional adiabatic exponents (Einbinder 1948). The play-off between attractive and repulsive terms can be observed from the equation of state (6.13). If we keep the repulsive term it will be necessary to consider > 1 in order to obtain the necessary, but not sufficient, critical condition (6.31). Rather, if we consider the first term attractive, by setting A D 1, and impose s D ˛ D 1, which is tantamount to considering an ideal gas, then (6.13) degenerates into pD

N2 CT 02 : V V

(6.47)

6.4 Nonextensive Adiabatic Potentials

159

This is easily recognized as the van der Waals equation without the excluded volume. The finite size of the molecules is taken into account by introducing an “excluded” volume, b, in the denominator of the first term of (6.47), which would then read CT : (6.48) pD V b This is known as Clausius’s equation of state. In the absence of the excluded volume parameter, the first term is the ideal gas term, which is neither attractive nor repulsive. The second term, being attractive, will cause the isothermal compressibility to become negative for volumes 2 0 =C T . The equality sign is the condition for (6.31). At this state, V 2 2N 2 @ p=@V T > 0 so that it corresponds to a minimum in the pV -plane. Although there is a point of inflection at V D 3N02 =C T , the curve will eventually slope downward, passing through a maximum, due to the absence of the repulsive term contained in the denominator of Clausius’s equation of state, (6.48). The critical volume is proportional to b, which must be finite in order for there to be a criti cal pressure and temperature. The energy will be given by E D V 1 C z N02 , making (6.47) equivalent to the Grüneisen relation with s D 1, and the entropy will have its usual characteristic logarithmic form S D C ln z C const: of an ideal gas, but where the adiabats are z D V T .

6.4 Nonextensive Adiabatic Potentials The open interval 1 < ˛ < 1 C 1=s contains nonextensive entropies. The classical ideal gas is the lower limit ˛ D 1, and the upper limit ˛ D 1 C 1=s is the criterion for a phase equilibrium, and both possess extensive entropies. There is no tendency for these gases to condense since ¤ 0 in (6.26), and the condition for (6.31) is not fulfilled. In fact, the Gibbs free energy, (6.26), can be used as a measure of the lack of extensivity of the entropy. Settinge z D z1=s , the nonextensive entropies are seen to be subadditive: z2 / < S .e z1 / C S .e z2 / ; S .e z1 Ce

(6.49)

where

sC e z ; (6.50) for 0 < D s.˛ 1/ < 1. The criterion of subadditivity (6.49) has also been referred to as the “triangle inequality” in mathematical circles (Beckenbach 1961). The reverse inequality > 1 would result in a convex function, which is unacceptable. In fact, the condition for the entropy to be concave is S.e z/D

z / D . 1/sCe z 2 < 0; S 00 .e

(6.51)

160


where the primes denote derivatives with respect to e z. The absolute entropy (6.50) also satisfies Planck’s formulation of the third law: S.0/ D 0

for T D 0:

(6.52)

Together (6.51) and (6.52) imply that S.e z /=e z is a decreasing function. Since the ratio S.e z /=e z is a decreasing function, z/ d S.e De zS 0 .e z / S.e z / 0; e z de z e z which implies z /: S.e z /=e z S 0 .e

(6.53)

This is a sufficient condition that the entropy obey (6.49) and be subadditive (Hardy 1952, Thm 103). From this we can conclude (Lavenda 1995, p. 29) Concavity C Planck’s Third Law ) Subadditivity: PN P zi = can be found by A lower bound to the sum of entropies N i Si D sC i e employing the inequality ln x x 1 for x > 0, x ¤ 1. For then 1 1

! N X e zi 1 i

! N X 1 ln e zi ; 1

(6.54)

i

where we have divided both sides by the positive constant .1 /. The right-hand side is the Rényi entropy, which is an additive entropy for all 2 Œ0; 1. The left-hand side is, apart from a different norming constant, the Havrda–Charvát entropy (1967), which has also become known as the Tsallis entropy (Tsallis 1988) [cf (6.117)]. This class of nonadditive entropies are known as “pseudoadditive entropies” (Lavenda 2004a). It is important to emphasize that the exponent depends only on the dimensionality of the system and not on the particular properties of the system, so it is the same entropy for all systems of a given dimension. In the limit as ! 0, the Rényi entropy transforms into the Hartley–Boltzmann entropy, ln N , and inequality (6.54) reduces to N 1 ln N , while in the limit as ! 1, both P reduce to the Shannon–Gibbs entropy, N i zi ln zi . Even if we impose extensivity on the entropy by taking the equality in (6.15), the adiabatic potential L.z/ D EV s D AN0s C

C ˛ sC 1C1=s 1Cs z D AN0s C T V ˛ 1Cs

(6.55)

6.4 Nonextensive Adiabatic Potentials

161

will still not be nonextensive. Unlike the entropy, the adiabatic potential (6.11) may also have a constant, repulsive term. Also, in contrast to the entropy, the adiabatic potential (6.11) is superadditive L .z1 C z2 / > L.z1 / C L.z2 / and

z1 C z2 L 2

1 fL .z1 / C L .z2 /g 2

(6.56)

(6.57)

so that it is also convex. Many an author has confused superadditivity (6.56) with the second law, especially in reference to a supposed “black hole area” theorem in which the combined area of two black holes must be greater than their sum. Either Nernst’s or Planck’s formulation of the third law apply to all adiabatic potentials which are increasing functions of the adiabatic parameter depending on whether there is or is not a zero-point energy. In particular, the adiabatic potential, L, satisfies Nernst’s form of the third law L.0/ D AN0s

for T D 0:

(6.58)

We may establish the superadditivity of L.z/ directly from convexity and the fact that L.z/=z is an increasing function. The latter implies that d z dz or

L.z/ z

0

L0 .z/ L.z/=z:

(6.59)

The convexity of L places the bounds on its derivative L.z C h/ L.z/ L.z/ L.z h/ L0 .z/ ; h h for h > 0. That is, we may take the inequality L.z C h/ L.z/ hL0 .z/ 0; as the criterion of convexity. We may also introduce the lower bound on the derivative L.z C h/ L.z C h/ L.z/ L.z/ ; z zCh h since either inequality will lead to L.1 z/ 1 L.z/;

(6.60)

162


where D z=.z C h/ < 1. Inequality (6.60) is a statement of superadditivity. Therefore, a sufficient condition for superadditivity is that the ratio L.z/=z increases. Although both S and L are adiabatic potentials, they are direct opposites.

6.5 The Second Laws The doctrine of latent and specific heats dQ D Lv dV C Cv dT;

(6.61)

was criticized by Clausius since Q cannot be a function of V and T , if these variables are independent of each other. For if it were, then by the well-known law of the differential calculus, that if a function of two variables is differentiated with respect to both of them, the order of differentiation is indifferent.

This just means that dQ is “d” of nothing in (6.61), and almost all writers in thermodynamics cross the d to show that it is not a true differential. The integrability condition for the internal energy, dE D dQ p dV D .Lv p/dV C Cv dT shows that the exactness condition for dQ is off by a factor @Lv @Cv dp D : dT @T @V

(6.62)

This led Planck to remark that dQ has frequently given rise to misunderstanding, for dQ has been repeatedly regarded as the differential of a known finite quantity Q. This faulty reasoning may be illustrated by the following example.

The example Planck gave resulted in (6.62), which Kelvin took as a statement of the first law. The statement that Kelvin took for the second law is Lv dp D ; dT C.T /

(6.63)

where, as before, C.T / is the Carnot function. It is this function that Kelvin showed to be equal to the absolute temperature. In doing so he showed that the ratio of the work done in an infinitesimal cycle, dp dV D .dp=dT /dT dV , to the heat absorbed, Lv dV , is a function solely of the temperature. Recalling Kelvin’s words: “The very remarkable theorem that dp=dT Lv must be the same for all substances was first given by Carnot,” although not precisely in those terms.

6.5 The Second Laws

163

For an ideal gas, Lv D p and Cv “can be a function of T only,” as Clausius realized. Clausius went on to conclude that “it is even probable that this magnitude, which represents the specific heat of the gas at constant volume, is a constant.” In contrast, for a generalized gas, 1 p D ˛p; Lv D 1 C s and [cf. (6.17)] E : T The ratio of the two is Lv =Cv D sT =V . The former invalidates Clausius’s conclusion that “a permanent gas, when expanded at constant temperature, takes up only so much heat as is consumed in doing external work during the expansion.” Clausius was thinking of an ideal classical gas, where pV is a measure of the absolute temperature, according to Mariotte’s law. The formula for the latent heat of a generalized gas shows that it absorbs more heat, since ˛ > 1, or that E is also a function of V . To convert dQ into a total differential of a state function, one can introduce the integrating factor and require the integrability condition Cv D ˛

@Cv @Lv D : @T dV Rearranging we get dp D dT

@Lv @Cv dT dV

D Cv

@ ln @ ln Lv : @V dT

(6.64)

Using a composite system argument, like that used by Carathéodory in 4.4, it can be shown that can only be a function of T or V . For consider two simple fluids in thermal contact. Such a system will have three independent variables, T , V1 , and V2 . The integrating factor can only be a function of the common variable T . Hence, .@Lv =@T @Cv =@V / d ln : D Lv dT On the strength of the exactness condition for the internal energy (6.62), this is equivalent to the Carnot–Clapeyron equation (6.63), and on the basis of Carnot’s theorem, C.T / D T is the integrating denominator for the increment of heat dQ. Their ratio gives the entropy (6.18) which is a function of V and T only through the combination z D T V s . Now, consider two simple fluids in mechanical contact with independent variables V , T1 , and T2 . As before, the integrating variable can only be a function of the

164


variable in common between the subsystems, in this case V . The exactness condition (6.64) now reduces to .dp=dT /dV D d ln .V /: (6.65) Cv Replacing isotherms by isochores, Cv dT is the quantity of heat absorbed in the first step of an equivalent Carnot cycle. As a corollary to Carnot’s very remarkable theorem, the product on the left-hand side of (6.65) is a product of dV and a function only of the volume. Thus, .dp=dT /=Cv is the same for all substances at the same volume. For a generalized gas, D V s , while D V .Cp Cv /=Cv for an ideal classical gas. This integrating factor now gives (6.11) as the function of state, rather than the entropy (6.18). Both functions depend upon V and T only through the combination dictated by the adiabatic variable, z D T V s . However, whereas S can be extensive, the adiabatic potential L cannot! Consider two states 1 and 2 with T1 > T2 . Any process connecting these states will be said to be irreversible if z1 > z2 . The inequality says that .V1 =V2 /s > T2 =T1 so that the Carnot efficiency, C , can never be inferior to the mechanical efficiency,

M , since T2 Vs

C D 1 1 1s DW M : T1 V2 If Q1 and Q2 are the quantities of heat absorbed and rejected to reservoirs at temperatures T1 and T2 , respectively, then there will be thermal equilibration if Vs T2 Q2 D 1s : T1 Q1 V2 Alternatively, if there is mechanical equilibration without thermal equilibration, then V1s Q2 T2 D : V2s Q1 T1 A greater amount of work can be accomplished when there is thermal equilibration than when there is mechanical equilibration because the engine reaches a lower final temperature, T2 . For an infinitesimal Carnot cycle we may replace the ratio of the heats, Q2 =Q1 , by a ratio of their differentials, dQ2 =dQ1 . Thermal equilibration over a cycle results when I dQ D0 (6.66) T and

I

V s dQ 0:

(6.67)

6.5 The Second Laws

165

Alternatively, mechanical equilibration over a cycle requires I

I

and

V s dQ D 0

(6.68)

dQ 0; T

(6.69)

the latter being Clausius’s inequality for spontaneous processes. This is a consequence of the fact that any irreversible cycle is necessarily less efficient than a Carnot cycle, since the final temperature is higher, dQ2 =dQ1 < T2 =T1 , and consequently, less work gets done. What are the first and second laws? We could argue for mechanical equilibration (6.68) expresses the first law and (6.69) the second law. But, for thermal equilibration, (6.66) is the criterion of the existence of the entropy function, so what is (6.67)? In Fig. 6.2 we have split the cycle up into two branches, A ! B and B ! A, such that the former contains all the irreversible elements. Then (6.67) gives Z

B

A

V s dQ > LB LA :

This would correspond to Z SB SA >

B

A

dQ ; T

Fig. 6.2 A cyclic path such that the transition from A to B is irreversible while from B to A is reversible

166


in the case of mechanical equilibration. Now, separate the heat absorbed and rejected to the reservoirs, dQ0 , and the sum of irreversible transfers made within the engine X dQi : dQ D dQ0 C i

Then for an isolated system, dQ0 D 0, and, according to (6.68) XZ i

B A

V s dQi > LB LA :

(6.70)

We can appreciate this inequality as analogous to the generalized Clausius inequalP H ity. Since i dQi =T D 0, it follows that SB SA >

XZ i

B A

dQi : T

(6.71)

Whereas inequality (6.71) is the well-known generalized Clausius inequality, inequality (6.70) appears novel. Again, the antipodal properties of the two adiabatic potentials have been displayed. For each individual process, the heat transfer dQi will appear twice, once as a positive quantity and once as a negative one. The second law (6.71) ensures that the integrating denominator in the former is larger than that in the latter, or “heat flows from hotter to colder bodies at constant volume.” In contrast, (6.70) demands that “as heat is absorbed the gas expands at constant temperature,” thereby making the integrating factor of the former smaller than that of the latter. Either (6.70) or (6.71) can be considered as statements of the second law, depending on the constraints that are imposed on the system. Remarkably, both criteria can be formulated in terms of comparable means, embellishing the fundamental property that the power means are monotonically increasing functions of their order.

6.6 An Equivalent Carnot Cycle A closed, reversible, cycle, entirely equivalent to Carnot’s cycle, can be constructed by replacing the isotherms by isochores. This will demonstrate a very remarkable theorem that the ratio .dp=dT /=Cv is the same for all substances at the same volume. In fact, this is precisely the analog to Carnot’s theorem that the ratio .dp=dT /=Lv is the same for all substances at the same temperature.

6.6 An Equivalent Carnot Cycle Fig. 6.3 A cycle equivalent to Carnot’s cycle

167 p p

2

Q1 > 0

Q2 < 0

p1

V V1

V2

The cycle consists of four branches shown in Fig. 6.3: 1. A quantity of heat, Q1 , is absorbed at constant volume, V1 , which causes the temperature to rise from T1 to T2 , and so, too, the pressure will rise from p1 to p2 . 2. An adiabatic expansion from V1 to V2 that lowers the temperature from T2 to T3 . 3. A quantity of heat Q2 is rejected by lowering the temperature from T3 to T4 , while keeping the volume constant at V2 . The pressure also decreases from p2 to p1 . 4. Finally, an adiabatic compression restores the system to volume V1 at temperature T1 > T4 . The adiabatic branches give the parameter of the ratio of volumes along the two isochores in terms of the temperatures as

V1 V2

s D

T3 T4 D D T2 T1

p1 p2

s=.sC1/ :

(6.72)

The ratio of the heat rejected, jQ2 j, to the heat absorbed Q1 , jQ2 j D Q1

gives an efficiency

V1 V2

M D 1

s

V1 V2

; s :

This will be the same as the Carnot efficiency provided the adiabatic conditions (6.72) hold. Otherwise, the inequality .V1 =V2 /s T3 =T2 will result making the Carnot efficiency C D 1 T3 =T2 superior.

168


6.7 Confines of Thermodynamics In this section we begin with a generalized gas whose specific heat is a nonlinear function of the temperature and show how, in a well-defined limit, the ideal gas is recovered. Realizing that the criterion for reversibility (4.77) always picks out the absolute temperature, we ask if there is any sense to the Curzon–Ahlborn criterion (4.50) if the intermediate temperatures are treated as empirical temperatures. Finally, we ask how close can we come to the Carnot efficiency by using the properties of mean values. If n isolated cells, or parts of an irregularly heated body, having heat capacities Cvi at constant volume and all at different initial temperatures Ti , are at some moment in time placed in thermal contact with one another with the whole system being isolated adiabatically, then, after an indeterminate amount of time, the system will reach a common mean temperature TM . This will be the final equilibrium temperature that will be common to all the cells. We now specialize to the case of an ideal gas so that (4.158) is n X i D1

T M0 Cvi ln Ti

or, equivalently T M0 D

n Y

D 0;

Cvi 1=

Ti

Pn

i D1

Cvi

(6.73)

:

(6.74)

i D1

This is Thomson’s (1853) result which identifies the final equilibrium temperature as the mean of order 0 or the geometric mean. When there are but two equal masses with the same heat p capacities, at temperatures T1 and T2 , the final mean temperature will be TM0 D .T1 T2 /. If we introduce this final mean temperature into (4.78), we obtain s Q2 T2 ; (6.75) D Q1 T1 which is the Curzon–Ahlborn result for the efficiency of their endoreversible engine. Adiabatic equilibration, (4.158), which leads to the geometric mean as the minimum common temperature of the composite system, will result, according to the first law, in the performance of work: W D

n Z X i

Ti

TM

Cvi dt:

(6.76)

For two cells with the same heat capacity, Cvi D Cv , (6.76) reduces to the work found by Tait [1868, p. 181], W D 2Cv .TA TG / D Cv

p p 2 T1 T2 ;

(6.77)

6.7 Confines of Thermodynamics

169

which corresponds to the Curzon–Ahlborn result, (4.59). Because the geometric mean temperature is the lowest common temperature attainable with the maximum amount of work being performed, it cannot be attained in a finite amount of time, contrary to what these authors have claimed. According to Curzon–Ahlborn, we would be tempted to write the optimal relation, corresponding to (4.50), as Q2 Q1 p Dp : T1 T2

(6.78)

Equation (6.78) would contradict the second law which, out of all possible empirical temperature scales, selects out the absolute one (Buchdahl 1966). What if the Ti were not absolute temperatures, but, rather, their square roots were? There is nothing unique about empirical temperatures. If an empirical temperature can be defined for one system, it can be defined for any system in thermal equilibrium with it. Were we to consider T1 and T2 as empirical temperatures, the resulting absolute temperature would increase as the square of the empirical temperature, contradicting the fact that the empirical temperature must increase at least as fast as the absolute temperature. The absolute temperature is the average kinetic energy of the system, in a nonrelativistic system of particles, and no empirical temperature can increase faster than it. In order to demonstrate that the empirical temperature increases with the absolute temperature, we write the doctrine of latent and specific heats as dQ D t ˇ dV C t ˇ 1 V dt:

(6.79)

It has been said that out of all the temperature scales the second law singles out the absolute temperature one. Thus in order that dS D

t ˇ 1 tˇ dV C V dt T T

be the derivative of a function of state, the following integrability condition must be satisfied dT dt : .ˇ 1/ D t T Hence, T / t ˇ 1 , and the requirement that T be an increasing function of t implies ˇ > 1. This is already implied in the doctrine of latent and specific heats, (6.79), since the specific heat must be an increasing function of temperature, or, at most, a constant, as in the case of an ideal gas. We also know that (6.79) misses being a total derivative by the factor @p D .ˇ 1/t ˇ 1 : @t

(6.80)

170


Integrating (6.80) and setting the arbitrary constant of integration equal to zero give pD

ˇ1 ˇ t < Lv : ˇ

Consequently, more heat is absorbed than work done which can be employed in areas other than performing external work. The integrability condition for the internal energy is satisfied identically, while that for the second law requires ˇ D s C 1, when V s is used for the integrating factor of heat. This clearly shows that ˇ has to be greater than unity and tends to the number of half degrees of freedom in the limit of a large number of degrees of freedom. Hence, the correct criterion of reversibility is (4.78) in which Q1 is the quantity of heat absorbed at temperature of the boiler, T1 , while Q2 is the heat that is rejected at temperature TM0 > T2 , which is the temperature of the condenser. And because TM0 > T2 , it follows that Q1 Q2 < 0; (6.81) T1 T2 giving rise to an apparent increase in entropy (4.55) (De Vos 1992). However, there is nothing irreversible about the Curzon–Ahlborn engine since Q2 Q1 D 0: T1 T M0 It is only when the final temperature TM0 is replaced by a still smaller temperature, T2 , that irreversibility appears, (6.81). In other words, it is only when the coldest part of the irregularly heated body be the receptacle for the rejected heat, which is at a lower temperature, does there appear to a positive entropy change given by (4.55). This possibility was already envisioned by Carnot and Kelvin. For according to Carnot, the more heat the body holds onto the greater will be its motive power, while according to Kelvin no heat should be given out to any auxiliary body at TM0 . Any other auxiliary body would be considered a condenser for the engine if it were below the temperature TM0 . How close can we actually come to the Carnot efficiency? In regards to means of order q, it is well known (Hardy 1952, p. 15) that lim TMq .T / D min T;

q!1

(6.82)

so that we might be tempted to write the second law as (Landsberg 1980) S D

n Z X i

TM Ti

Cvi dt; tı

(6.83)

6.8 Mathematical vs. Physical Inequalities

171

for some ı > 1 to obtain a mean of negative order that would correspond to a uniform temperature lower than the geometric mean temperature. Heat capacities with inverse power temperature dependencies do not vanish as T ! 0 and are therefore in disagreement with the third law (Sidhu 1980). Adiabatic equilibration would yield a final, common, mean temperature TM1ı . Although the entropy and 1ı , they are related like the integral is to heat capacity are both proportional to TM 1ı the differential. So if one is positive, the other will be negative for values of ı > 1. To avoid the possibility of negative entropy, or negative heat capacity, we, therefore, have to limit values of ı to the semiclosed interval .0; 1, and the inequality TM1ı TM0 follows from the property that means are monotonically increasing functions of their order, with the equality holding for ı D 1. Consequently, the geometric mean temperature is the lowest possible uniform mean temperature that the system is capable of achieving.

6.8 Mathematical vs. Physical Inequalities Since the arithmetic

geometric

harmonic

mean inequalities are particular examples of the fact that power means are increasing functions of their order, it was believed that the second law could be derived from this property without having recourse to any experiment whatsoever. For pure thermal conduction, if the first law defines the final mean temperature as XZ

M Tj

then the second law is

XZ

cvj .t/mj dt D 0;

M

Tj

cvj .t/mj dt=t 0;

precisely because the means are monotonically increasing functions of their order. That is, if Mt and Ms are means of order t and s, where t > s, then Mt Ms , where the equality applies if all the temperatures of the different cells, Tj , are equal. A thermodynamic derivation of the arithmetic-geometric means is given in Sommerfeld’s book (1956). If two bodies at temperatures T1 and T2 are placed in thermal contact, and no work is done, then the common temperature that will be reached is the average temperature. Consider two ideal gases with constant heat capacities Cv1 D m1 cv1 and Cv2 D m2 cv2 . With no work being done, the final temperature should be the weighted arithmetic mean T D a1 T1 C a2 T2 ;

172


where the weights are aj D Cvj =.Cv1 C Cv2 /. The changes in entropies of the ideal gases will be T T1

S1 D Cv1 ln

and

S2 D Cv2 ln

T : T2

Since the system is isolated from its surrounding, the effect of thermal interaction will be to increase the total entropy S D S1 C S2 D .Cv1 C Cv2 / ln T Cv1 ln T1 Cv2 ln T2 > 0: Dividing through by .Cv1 C Cv2 / and exponentiating give a1 T1 C a2 T2 T1a1 T2a2 ; showing that the arithmetic mean is greater than the geometric mean, except in the trivial case where T1 D T2 , in which case the equality sign holds. It is rather sterile to argue which is more general, thermodynamics or mathematical inequalities. It is like asking which came first, the chicken or the egg? Rather, the essential thing is that they do coincide! But what happens if the equilibration process is adiabatic XZ M Cvj .t/dt=t D 0‹ Tj

j

The final mean temperature would necessarily be less than the equilibrating temperature that would pertain to an energy equilibration. The lower mean temperature would imply that work has been extracted. This would provoke a negative change in the energy XZ M Cvj .t/dt < 0; j

Tj

making it comparable to an entropy evolution criterion. Again we see that the first and second laws can swap roles. We will now consider more such cases.

6.9 Comparable Means The first and second laws are only comparable in the case of pure thermal conduction because, for pure thermal conduction, the adiabatic potential L differs by a constant factor from the internal energy E. Means are said to be comparable if the following inequality holds 0 M˛ .z/ D ˛ 1 @

X j

1

0

pj ˛.zj /A Mˇ .z/ D ˇ 1 @

X j

1 pj ˇ.zj /A ;

6.9 Comparable Means

173

where x 1 is the inverse function and the pj are a positive, normalized set of weights (Hardy 1952, p. 13). A necessary and sufficient condition that this inequality holds is that the composite function ˇ ˇ ˛ 1 be convex. Moreover, the impossibility to distinguish between the Nernst and Planck formulations of the third law can be translated into the property of equivalent means. Means will be equivalent if (Hardy 1952, pp. 68–69) MB .z/ D Mˇ .z/; where B differs from ˇ by a mere translation, B D aˇ C b, where a and b are arbitrary constants with a ¤ 0. Consider a body formed from m cells which initially have rigid and adiabatic partitions. The initial and final states are zj and Zj , respectively. We want to inquire about the changes that ensue in the adiabatic potentials when the impregnable walls are removed. The changes in these potentials will be giving by L D

X

Z

Zj

dL.z/ D M˛˛ .Z/ M˛˛ .z/;

pj zj

j

and S D

X

Z

Zj

pj

dL=z D

zj

j

˛ ˛1 ˛1 M˛1 .Z/ M˛1 .z/ : ˛1

(6.84)

The vanishing of either relation would determine a unique mean value of the final state Z. The mean would be greater in the case of the vanishing of L than in the case that S would vanish. Consider first an L-equilibrating transition. Again we divide the initial states into two groups: those for which zj M˛ .z/ and those for which zj > M˛ .z/, where the mean M˛ .z/ is determined by L D 0. Now, it follows that X

Z

M˛

pj

dL=z > M˛1

zj

j

D M˛1

X

Z

dL.z/ zj

j

> X

Z pj

j

zj

dL.z/ > M˛

M˛

pj

> X

Z pj

j

zj

dL=z: M˛

Consequently, in an L-equilibration causes an increase in entropy S > 0. Moreover, X Z M˛ X Z Zj pj dL=z pj dL=z; j

zj

zj

174


since the last term in X

Z

Zj

dL=z D

pj

X

zj

j

Z

M˛

pj

dL=z

X

zj

j

Z pj

j

M˛

dL=z Zj

is positive. This is to say that the entropy achieves its maximum value in a state of uniformity, characterized by the common value M˛ .z/. Second, consider an entropy equilibrating transition. Again dividing the initial states into two groups depending on whether they are greater or less than the final mean M˛1 .z/, we get the string of inequalities M˛1

X

Z pj

dL
X

M˛1

Z

dL.z/=z < zj

j

M˛1

pj

dL.z/=z zj

j

M˛1

pj

Z

1 M˛1

> X

Z

M˛1

pj

dL: zj

j

This implies that L < 0 under an entropy equilibration. This is illustrative of the opposite natures of the two adiabatic potentials which we referred to in Sect. 6.4. That the change in L will be least when the final common state is M˛1 .z/ follows from the fact that the last term in X j

Z

Zj

pj zj

dL D

X j

Z

M˛1

pj zj

dL

X j

Z pj

M˛1

dL Zj

is always negative, regardless of the initial states zj . In other words, the mean M˛1 minimizes the adiabatic potential L. Even though L is nonextensive, it shares in common many properties of the complete, or characteristic, function of the Helmholtz free energy, which shows a spontaneous tendency to decrease in the face of irreversible processes (Lavenda 2005).

6.10 Bounds on Mean Temperatures and Volumes Carathéodory’s axiom, which we discussed in Sect. 4.4, states that there are always neighboring states to any given state that are inaccessible to it by an adiabatic process, whether it be quasistatic or not. This impossibility means that there are surfaces of constant i .z/ D const., where i stands for either of the adiabatic potentials, L or S . However, Carathéordory’s principle does not tell us what those states are. They must be determined by another principle.

6.10 Bounds on Mean Temperatures and Volumes

175

The principle we are searching for asserts that those states which are adiabatically accessible from a given state must increase the average internal energy: For pure thermal conduction this means 1 0 X Z tf X E D 1 .z/ pj dt r D 1 .z/ @tfr pj tjr A 0; ti

j

j

while for deformations E D

2 .z/

X j

Z pj

Vf

0 dV s D

Vi

s 2 .z/ @Vf

X

1 pj Vjs A 0;

j

since they are connected by the adiabatic constraint, z D const. In reference to the exponent appearing in the stability criterion (6.14), let us set ˛ D q=r 1, where r is the smallest exponent possible. The former inequality says that the final (empirical) temperature tf cannot be inferior to the minimum mean (kinetic energy) temperature, 11=r 0 X tf @ pj tjr A ; j

while the latter puts a lower bound on the final volume Vf . When q ! r or s ! 1, such that the product s.˛ 1/ D const: < 1, we find 0 Vf lim @ s!1

X

11=s pj Vjs A

D Vmin ;

j

which is the volume of the smallest cell. Thus, the larger the adiabatic index s, the more weight is given to cells of smaller size. Alternatively, when all the cells have reached a common empirical temperature t, the maximum work that can be performed when the final volume satisfies 0 1 X X Z Vf ˇs ˇs L D ct rˇ pj dV ˇs D ct rˇ @Vf pj Vi A 0; j

Vi

j

for arbitrary ˇ > 0, since, in the limit s ! 1, 0 Vf lim @ s!1

X

11=ˇs pj Vjˇs A

D Vmax :

j

This says the upper limit to the final mean volume is the volume of the largest cell.

176


Finally, in a process which does no work and all cells have the same volume V , 0 1 X X Z tf q q L D cV ˇs pj dt q D cV ˇs @tf pj tj A 0: j

ti

j

Since only pure thermal conduction is involved, the average change in energy will manifest the same willingness to decrease. The highest attainable temperature occurs in an L-equilibration, where the final temperature, 11=q 0 X q pj tj A D Mq .t/; tf D @ j

equals the largest mean temperature that is proportional to the internal energy. The difference between Mq and the minimum average temperature Mr .t/ not only determines the system’s capability of performing work, but, moreover, establishes a metric, in a thermodynamic space which would otherwise have none.

6.11 Geometry of Thermodynamics A holy grail of thermodynamics has been the search of a metric which would allow one to determine distances on a thermodynamic, or primitive, surface. Although Gibbsian thermodynamics has many geometrical features it lacks a definite metric. The fundamental relation, which expresses either the energy or the entropy in terms of all the other extensive variables has the form of either a convex, or concave, surface, respectively. The intensive variables are defined as the slopes of either the energy or the entropy with respect to the extensive conjugate variables, and the stability criteria are couched in the curvature of the primitive surface. Clerk-Maxwell was so impressed with these geometrical properties that he sent Gibbs a plaster model for the thermodynamic surface of water in 1875, which is shown in Fig. 6.4. Gibbs also determined the degrees of freedom of his system or the number of variables which can be assigned arbitrarily. If there are c components, there will be .c 1/ independent ones because the particle number is conserved. If there are f phases then there will be f .c 1/ components all together. Then there are the common values of the pressure p and temperature T throughout the phases. Thus, there will be a total of f .c 1/C2 variables in all. Now, a set of equations possess a solution when their number does not exceed the number of independent variables. Phase equilibrium is determined by the equality of the chemical potentials. There will be .f 1/ equations of the equalities of the chemical potential for each component. This makes a total of c.f 1/ equations. Consequently, c.f 1/ f .c 1/ C 2, and Gibbs’s phase rule follows: f c C 2;

6.11 Geometry of Thermodynamics

177

Fig. 6.4 The thermodynamic surface of water sent to Gibbs by Clerk-Maxwell

which, expressed in words, says that the number of phases coexisting in equilibrium cannot be greater than the number of components plus two. The difference between the number of variables and the number of equations is n D c f C 2;

(6.85)

or the number of thermodynamic degrees of freedom which can be arbitrarily assigned. Notwithstanding Gibbs’s intuitive approach, a metric in such a space is found wanting. Nearly half a century later, the mathematician Blaschke developed a theory of curvature, within the confines of affine geometry, that could be applied to Gibbs’s space. Affine geometry is essentially Euclidean geometry with congruence left out or the impossibility of making geometric figures coincide by a rigid transformation. The notion of perpendicularity is lacking, or an inner product, and when it is restored we get back the whole of Euclidean geometry. So it seems there is little to talk about in affine geometry. However, the quadratic form, which expresses the curvature of Gibbs’s space and thermodynamic stability is the affine fundamental form. This form is invariant under linear (affine)

178


transformations of determinant unity. An affine connection specifies a rule whereby a definite vector at one point is associated with a definite vector at a neighboring point. Within the group of all affine transformations, the quadratic form should be able to be brought into diagonal form. This, as Tisza (1966) points out, is unsatisfactory because the diagonal of the matrix of second-order derivatives already possesses a physical significance in terms of thermodynamic stability criteria, e.g., the adiabatic compressibility must be positive. But, even if this criterion is invalidated, it may not jeopardize the sign of the isothermal compressibility since the difference between the isothermal and adiabatic compressibilities is the positive term (3.20). It still remains that the theory of curvative which relies on a metric which specifies a line element is nonexistent in Gibbs space. The claim that “the laws of thermodynamics are nothing more nor less than the mathematical requirements of a metric geometry,” (Weinhold 1975) is inaccurate for conventional thermodynamics. For if the inner product could be suitably defined, the whole of Euclidean geometry could be resurrected which we know is not the case! Information theory too lacks a metric, where the so-called directed divergences, or negative of the Shannon–Gibbs entropy differences of different measures, satisfy all the topological properties of a metric except the triangle inequality and, therefore, cannot be termed a distance (Kullback 1959). This would correspond to the logarithmic differences in the fundamental relation for the entropy differences, and there is no hope of satisfying the triangle inequality which states that no one side of a triangle can exceed the sum of the other two sides. But, Weinhold, following Tisza’s failed attempt (Tisza 1966), wants to work with the “stiffness” matrix, or the Hessian, or what he calls the Gram matrix comprising all the second derivatives of the internal energy with respect to the entire set of extensive variables, and derive from it a “metric” space of dimension n given by (6.85). A set of base vectors, “kets,” jFi are assumed to span this space, and the inner product, defined by Weinhold with the “bra” hF j , is ˛ ˝ @Fi ; (6.86) Fi jFj D @Xj fXg0 where fX g0 stands for the complete set of extensive variables that are to be held constant in the differentiation other than Xj . The intensive variables Fi D .@E=@Xi /fXg0 are defined as in conventional thermodynamics, viz., the derivatives of the internal energy with respect to their conjugate extensive variables. Why definition (6.86)? Because (1) it is symmetric to the exchange of indices, which is guaranteed by the exactness conditions, @Fi @Fj D ; @Xj @Xi for the existence of the internal energy as a state function; (2) it is positive semidefinite, if the intensive variables are defined in a proper way; and (3) the distributive property holds.

6.11 Geometry of Thermodynamics

179

Weinhold then goes on to define length as: jFi j WD hFi jFi i1=2 :

(6.87)

However, this is what Tisza (1966, p. 236) categorically excludes: “Neither of these concepts (a metric based on length and/or the orthogonality of the basis vectors) is definable in Gibbs space spanned by such disparate variables as volume, mole numbers, and energy.” We may also ask why the definition (6.87)? The answer is to be found in the desire to define an angle between two vectors in the usual way ˛ı ˝ cos ij D Fi jFj jFi j jFj j:

(6.88)

Rather than being the definition of an “angle,” (6.88) is the definition of Pearson coefficient of correlation, %ij D

C.Fi ; Fj / ; D.Fi / D.Fj /

(6.89)

˛ ˝ between two variates Fi and Fj , where the correlation C is C.Fi ; Fj / D F˝i Fj˛ and D is the standard deviation, whose square is the variance D 2 .Fi / D Fi2 . From its definition, (6.89), it is obvious that %ij D %j i . As such it is a pure number with range from 1 to C1 inclusive, and vanishes when Fi and Fj are uncorrelated. Weinhold’s results are well known in the thermodynamic theory of fluctuations (Lavenda 1991, 3.6). As an example consider fluctuations in temperature, T , and pressure, p. His expression for the cosine of the angle between two “vectors” is simply the ratio (Lavenda 1991, p. 201) hT pi Œhp 2 i hT 2 i

1=2

D

T .@T =@V /S

ŒT .@p=@V /S T .@T =@S /V 1=2 s s Cv T s Cv T D ; Ds pV E

(6.90)

where we have introduced the Grüneisen equation of state, (2.56), into the penultimate expression. For an ideal gas, s D 23 and D 53 , while for a photon gas, s D 13 and D 43 . In the former case, represents the ratio of heat capacities, Cp =Cv , whereas in the latter case Cp D 1 since heat cannot vary with temperature at constant pressure, because the pressure is independent of the volume. But, as we mentioned earlier, everything works as if it were since the adiabatic Poisson p relation for a photon gas is V 4=3 p D const. Thus, we find the ratio (6.90) is 25 for an ideal gas and unity for a photon gas. We may, therefore, conclude that P and T are positively and perfectly correlated for a photon gas, while they are positively but imperfectly correlated for an ideal gas. Moreover, the triangle inequality is still found wanting and cannot be filled by considering the difference of the standard deviations to satisfy such an inequality (Weinhold 2009). In other words, there is no motivation

180


for the belief that the differences in standard deviations of two correlated variables satisfy the triangle inequality. Rather, it is well known (Cramér 1946, p. 180) that if two variates Fi and Fj are independent, then D 2 Fi C Fj D D 2 .Fi / C D 2 Fj : Had there been something new in regard to a metrical structure lodged into the quadratic form of the stiffness matrix would have meant that we could play the same game anytime we have a positive semidefinite quadratic form. In information theory this would make the Fisher information, I , times the mean-square error e 2 a metric space, viz., Ie 2 > 0, which we know is not. The inequality is known as the Cramér–Rao inequality, which is tantamount to the Heisenberg uncertainty relations in quantum mechanics for conjugate variables. Moreover, the relevant properties of quadratic forms are best recognized in their diagonal form. Although a symmetric matrix can be brought to diagonal form with real eigenvalues, this method does not apply to Gibbs space (Tisza 1966, p. 238), and diagonalization itself would lead to diagonal terms that have no clear stability properties as the original diagonal elements have. However, differences of power means have all the properties of a distance (or metric) as defined in topology including the triangle inequality. In other words they represent a distance on the set of all continuous and monotonic functions in the domain of the closed interval Œzc ; zh (Cargo 1969). Consider the change in the L potential 0 1˛=.˛1/ X X A L D c @ pj z˛1 c pj z˛j ; j j

j

due to an S -equilibration, represented by the first term on the right-hand side. Setting x D z˛ it becomes apparent that this difference is negative 0 L D c @

X

11= pj xi A

c

j

X

pj xj 0;

j

since D .˛ 1/=˛ < 1, and the inequality follows from the fact that power means are monotonically increasing functions of their order. Likewise, in an L-equilibration the average change in entropy is always positive 20 S D

c 4@

X j

1 pj yj1= A

X

3 pj yj 5 > 0;

j

unless all the yj are the same in which case the entropy shows no tendency to increase.

6.12 Thermodynamics of Coding and Fractals

181

Remarkably, a distance can be defined between the powers of the two means ˚

d.˛; ˇ/ WD sup jM˛ .z/ Mˇ .z/j ; z

for all z on the closed interval Œzc ; zh . For pure thermal conduction, the absolute value of the difference of the two means cannot be less than the temperatures between the hottest and coldest reservoirs, jM˛ Mˇ j zh zc : Consequently, the maximum distance between the two means divided by the highest temperature is bounded from above by the Carnot efficiency jM˛ Mˇ j C : zh It is easy to see that the thermodynamic distance d satisfies the triangle inequality ˇ 11= ˇˇ 0 ˇ ˇX ˇ X ˇ ˇ d.1; / D ˇ pj zj @ pj zj A ˇ ˇ ˇ j ˇ j ˇ ˇ ˇ ˇ ˇ 11= ˇˇ 0 ˇ ˇ ˇ ˇ X X X X ˇ ˇ ˇ ˇ ˇˇ pj zj pj zj ˇˇ C ˇ pj zj @ pj zj A ˇ : ˇ ˇ ˇ j ˇ ˇ j j j ˇ Thus, the supremum of the absolute difference between two comparable means of different order does constitute a bona fide metric.

6.12 Thermodynamics of Coding and Fractals In his seminal paper on information and entropy, Rényi (1961) laid down the basic tenets governing the relation between entropy and information theory. Rényi argued that any candidate for an entropy should be the negative of a logarithmic mean S .p/ D ln M .p/;

(6.91)

where the mean of the generating function is 1 0 N X pj .pj /A : M .p/ D 1 @ j

(6.92)

182


The otherwise arbitrary function, , must be monotonic, continuous, and possess an inverse, 1 . is often referred to as the Kolmogorov–Nagumo function. We have previously considered the weights, pj , to be independent of what they are weighting .pj /. One of the major difficulties in studying entropies related to information is that the “variables” .pj / are functions of the “weights” pj (Lavenda 2004b). Without any loss of generality we may consider the probaP bility distribution complete, N j pj D 1, for, otherwise, it would entail dividing through by the nonnormalized sum in the calculations. If q D .q1 ; q2 ; : : : ; qN / is another complete probability distribution, defined on the same probability space, then the direct product of p and q, denoted p ˝ q, consists of all numbers pi qj where the indices run over all values from 1 to N . The multiplicative property of the mean, M .p ˝ q/ D M.p/ M.q/;

(6.93)

is normally transcribed into the additivity property of the entropy, S .p ˝ q/ D S.p/ C S.q/:

(6.94)

This is the usual case in statistical physics where the joint probability of independent events reduces to the product of their individual probabilities for their occurrence. Due to the exponential relation between individual probabilities and their entropies, the latter are additive quantities. The multiplicative property of the means (6.93), implies that they be power means 0 11=.1˛/ N X pj pj.1˛/ A ; for ˛ ¤ 1; (6.95) M.1˛/ D @ j

and M0 .p/ D

N Y

p

pj j ;

for ˛ D 1:

(6.96)

j D1

Thus, the logic behind Rényi’s reasoning becomes clear: When (6.96) is introduced into (6.92) we come out with the Shannon–Gibbs entropy, S1 .p/ D

N X

pj ln pj ;

(6.97)

j

while if (6.95) is introduced into (6.92) there results

0 1 N X 1 S˛ .p/ D ln M.1˛/ .p/ D ln @ pj˛ A ; 1˛

j

which bears the name of its discoverer, Rényi.

(6.98)


183

Actually, the Rényi entropy is not an entropy at all, but, rather, an interpolation formula connecting two bona fide entropies, the Shannon–Gibbs entropy which is attained in the ˛ ! 1 limit and the Hartley–Boltzmann entropy that is recovered in the ˛ ! 0 limit, (6.99) S0 .N / D lim S˛ .p/ D ln N; ˛!0

which depends only on the size of the sample and not on the probabilities of its elements. And since the Rényi entropy is monotonic and continuous we get the hierarchy: (6.100) S0 .N / S˛ .p/ S1 .p/; as ˛ varies from 0 to 1. In contrast to the Rényi entropy, S˛ .p/, the Shannon–Gibbs entropy enjoys the property of recursivity. If a choice is broken down into two successive choices with probabilities ˇp1 and .1 ˇ/p1 , the Shannon–Gibbs entropy separates into: S1 .p1 ; p2 ; : : : ; pN / D S1 .ˇp1 C .1 ˇ/p1 ; p3 ; : : : ; pN / C p1 S0 .ˇ; 1 ˇ/: (6.101) Additivity (6.94) is a much weaker condition than recursivity, (6.101), so that by replacing recursivity by additivity, the class of entropy functions is broadened so as to include entropies, like the Rényi entropy, (6.98). We shall now study two examples of mean entropies that are related to coding theory and multifractals and strange attractors. The generating functions will be an exponential and power law functions, respectively. Moreover, we will show how the entropies that we introduced in this paragraph place bounds on the mean lengths of codes and volumes of multifractals, thus bringing out a definite similarity between the two phenomena which, seemingly, appeared to be completely unrelated.

6.12.1 Optimal Coding Lengths In the coding theorem for noiseless channels one minimizes the code length subject to the condition that it be uniquely decipherable (Feinstein 1958). The solution to this minimization procedure is that the best code length for an input symbol of probability pj is log pj . This has been criticized by the observation that when the probability of a symbol is very small the code length is very large. Specifically, suppose that we have an alphabet of D symbols into which the input symbols x1 ; x2 ; : : : ; xN are to be encoded. In this section the logarithm will be to the base D. Let each symbol have a probability pj , with j running from 1 to N , P and assume that the distribution is complete, viz., N j pj D 1. Let nj represent a sequence of xj characters from the alphabet. Kraft’s inequality tells us that there will be a uniquely decipherable code with lengths n1 ; n2 ; : : : ; nN if and only if N X j

D nj 1:

(6.102)

184


Suppose we are given another complete probability distribution, qj , and we want to discriminate between the two. Based on the concavity of the Shannon–Gibbs entropy [cf. (6.97)], N X S1 .x/ D xj log xj ; (6.103) j

we set xj D pj =qj and use the concavity of the entropy to establish: N X

qj S1

j

pj qj

0 1 0 1 N N X X pj A D S1 @ S1 @ qj pj A D 1: qj j

j

Taking logarithms of both sides and rearranging, we get N X

qj log pj

N X

j

pj log pj ;

(6.104)

j

which is Shannon’s well-known inequality. Using (6.104) we can establish a lower bound for the smallest average length L with the aid of the Kraft inequality (6.102). PN n ı PN n j j D 1, we have Since j D j D

N X

p.xj / log p.xj /

j

D log

N X j

N X

D nj C

j

N X

D nj p.xj / log PN nj j D

p.xj /nj

j

N X

p.xj /nj ;

j

from Shannon’s inequality (6.104), where the last inequality follows from Kraft’s P nj log 1 D 0. Thus, we arrive at the inequality inequality, i.e., log N j D La D

N X

p.xj /nj S1 .x/:

(6.105)

j

The equality is achieved in (6.105) when p.xj / D D nj ;

(6.106)

which, as we have mentioned, attributes long average lengths to extremely small probabilities. Can we do better than this? We have seen that it is usual practice to examine the mean length, N X j

nj p.xj /;


185

and to minimize it in order to compare different codes thereby selecting out the optimum code. But, there are times when the average cost, C , of using a code might be an exponential function of nj , say if the cost of encoding and encoding equipment are included (Campbell 1965). Therefore, we take as the average cost, C D

N 1 X nj D : N

(6.107)

j

Taking our lead from Campbell (1965) we employ Hölder’s inequality, 0 @

N X

11=a 0 xja A

j

@

N X

11=b yjb A

N X

j

xj yj

(6.108)

j

which is the reverse of the usual one because we set a D .1 ˛/ < 1. Since 1=a C 1=b D 1, we must set b D D .1 ˛/=˛. If xj D pj1= and yj D pj1= qj , where the probabilities, qj , are identified from (6.107) as, qj D D nj =CN , then Hölder’s inequality (6.108) becomes (Lavenda 1998) 0 0 11=.1˛/ 11= N N X X 1 @ @ pj˛ A pj D nj A : CN j

(6.109)

j

Taking the logarithm of both sides of (6.109) results in S0 S˛ S

(6.110)

where the following entropies have been defined: The Hartley–Boltzmann entropy [cf. (6.99)]:

S0 D log N C log C;

(6.111)

which is defined to within an arbitrary constant. The Rényi entropy (6.98). the entropy (Lavenda 1998)

0 1 N X 1 S .n/ D log @ pj D nj A

(6.112)

j

which is associated with the (exponential) mean length. We have already taken note that, on the open interval ˛ 2 .0; 1/, the entropies fall in the hierarchial order (6.100). Since the Hartley–Boltzmann entropy depends

186


only on the number of events and not on their frequencies, it was necessary to use a frequency-independent cost, (6.107), to get an upper bound on the logarithmic mean entropy (6.112). Let us observe that if we had used (6.106) for the probabilities qj in Hölder’s inequality we would have come out with a mean length, 0 1 N X 1 L D log @ (6.113) pj D n j A ; j

which Campbell [1965] assumes from the outset, based on his assumption of an exponential average cost function. He finds the Rényi entropy as a lower bound and concludes that “It is not possible to find a uniquely decipherable code whose average length of order is less than [S˛ ].” However, the length of order , (6.113), is a convex function of the nj and is not readily comparable to an entropy function. For that we need another entropy function, and (6.112) fits the bill since it is a concave function of the nj . In the limit as ! 0 we get the anticipated result S0 .n/ D lim S .n/ D !0

N X

p.xj /nj ;

(6.114)

j

which is analogous to the usual expression for the average code length, (6.105). However, in the limit as ! 1, the entire sum in (6.112) reduces to a single term S1 .n/ D lim log M .n/ D nmin ; !1

(6.115)

where nmin D min nj is the smallest of lengths, nj . Thus, as varies from 0 to 1, the average code length varies from (6.114) to (6.115).

6.12.2 Pseudoadditive Entropies The results of the last paragraph will throw some light on a property called “pseudoadditivity” that certain “entropies” have been found to possess. Consider the function D nj 1 ; (6.116) .nj / D which simply becomes nj in the ! 0 limit. Since (6.116) is a Kolmogorov– Nagumo function, its mean is equivalent to (6.112), for the exponential mean, and, consequently to the Rényi entropy (6.98), when (6.106) is imposed. Rather, if we introduce (6.106) into (6.116) to get .pj / D

pj 1

;


187

and form the weighted mean, we come out with the Havrda–Charvát entropy: e S .p/ D

N X

PN pj .pj / D

j

pi1 1

j

:

(6.117)

The Havrda–Charvát entropy (6.117), which is again more correctly considered an interpolation formula as ranges over all values from 0 to 1, is “pseudoadditive” in that e S .p ˝ q/ D e S .p/ C e S .q/ C e S .p/e S .q/: (6.118) It is this pseudoadditive property that replaces the additivity (6.94) of the Shannon– Gibbs (6.97), and Rényi entropies (6.98). The functional equation (6.118) arises from the first-order homogeneity property of the means, M .p/ D M .p/; for all positive . The generating function, , satisfies the functional equation (Hardy 1952, p. 69) .!1 !2 / D .!1 / C .!2 / C .!1 /.!2 /;

(6.119)

where is a separation constant. The solutions to (6.119) are: .!/ D c1 log ! C c2 ; .!/ D c1 ! C c2 :

(6.120a) (6.120b)

Now, if entropies are to be related to mean values, they must be translationally invariant because only entropy differences are measurable. The condition for translational invariance is 1 0 N X pj .nj C c/A M .n C c/ D 1 @ 0 D 1 @

j D1 N X

1 pj .nj /A C c D M .n/ C c;

(6.121)

j D1

where c is an arbitrary constant. The solutions to (6.121) are .n/ D a1 n C a2 ; .n/ D a1 D n C a2 ; where a1 ¤ 0 and a2 are arbitrary constants.

(6.122a) (6.122b)

188


The connection between the pairs of solutions (6.120a), (6.120b) and (6.122a), (6.122b), which relate the properties of first-order homogeneity to translational invariance is obtained by setting (Lavenda 2004b) !j D D nj :

(6.123)

The condition (6.123) gives the number of sequences of length nj taken from an alphabet of size D (Khinchin 1957). We could have taken the ratios of (6.123) and the total number of sequences of lengths nj as the a priori probabilities, in place of the probabilities qj we used in the Hölder’s inequality (6.109), which were obtained by introducing the cost function, (6.107) (Lavenda 2004a). Returning to the functional equation (6.118) and introducing a simple linear transformation, b S De S C 1; the pseudoadditive relation, (6.118), is converted into the functional equation: b S .p ˝ q/ D b S .p/ b S .q/:

(6.124)

Now, there is a complete loss of additivity since the functional equation (6.124) implies a power law solution. Hence, we can transform away the pseudoadditivity property (6.118) and convert it into a multiplicative relation (6.124) so that neither pseudoadditivity nor multiplicativity has any thermodynamic meaning regarding the lack of thermodynamic extensivity. We therefore conclude that the pseudoadditive entropies, e S , defined by (6.117), are weighted arithmetic means of generating functions and not the negative logarithm of the mean, as (6.92) indicates.

6.12.3 Multifractals to Strange Attractors Expression (6.105) appears reminiscent of the definition fractal dimensionality. Multifractals can be generated on a fractal support of the unit interval by removing a number of open intervals leaving behind line segments of length rj separated by voids, or “holes.” Each segment has a probability pj associated with it just as each symbol xj had a definite probability pj in coding theory. These probabilities express different growth rates as in diffusion-limited aggregation. After an infinite number of iterations, we are left with specks of dust having definite probabilities. These specks of dust, or density of points, are what characterize strange attractors. If we go to some point near the attractor and ask for the number of points on the orbit within a distance rj from the attractor, then the average volume of order containing these points, in the limit as the radius is allowed to shrink to zero, will play the same role as the code length of order , (6.113), in coding theory. Whereas in coding theory we are interested in optimizing code lengths, in multifractals


189

and strange attractors the relevant quantity to be optimized is the mean volume of order , since it is this quantity that will be related to the generalized fractal dimension. A self-similar Cantor set of equal lengths is characterized by one scaling exponent, the Hausdorff dimension, D. In contrast, multifractals, which are probabilistic objects, need two scaling exponents, one for the supporting fractal and another for the probabilities of the segments. Recall that the Hausdorff dimension is defined as a limiting process: If there are N pieces of length r that are required to completely cover the set then in the limit as r ! 0 the fractal dimension is defined as limr!0 N r D D C , where C is some positive constant. In this limiting process, it is precisely the Hausdorff dimension D that prevents the product N r D from exploding to infinity or shrinking to zero. Thus, the Hausdorff dimension is defined as D WD lim

r!0

log N log C ; log.1=r/

(6.125)

where the numerator looks like a Hartley–Boltzmann entropy, of the form (6.99). If the segments are not equal, then (6.125) must be replaced by N X

rjD D C;

(6.126)

j

whereas if each segment rj occurs with probability pj , then N X

pj˛ rjD˛ .1˛/ D C

(6.127)

j

is the condition that determines the generalized Hausdorff dimension D˛ , which becomes the usual definition D in the case the exponent ˛ D 0. The left-hand side of (6.126) is something that we might like to apply the Hölder inequality, (6.108). With xj D pj1= and yj D pj1= rjD˛ , and exponents a D .1 ˛/ and b D D .1 ˛/=˛, such that 1=a C 1=b D 1, we get

C D

N X

0 rjD˛ @

N X

j

where

11=.1˛/ p˛ A

V .r/ ;

(6.128)

j

0 V .r/ D @

N X j

11= pj rjD˛ A

(6.129)

190


is a mean volume of order . Instead of the exponent of coding theory, D nj , we now have a power law, r D , and the associated entropy is the logarithm of the volume (6.129), 1 0 N X 1 D S .r/ D log V .r/ D log @ pj r j ˛ A :

(6.130)

j

The entropy of mean volume (6.130) is analogous to the entropy of the microcanonical ensemble which is equal to the logarithm of the volume that the system occupies in phase space. The mean volume, (6.129), is a decreasing function of (Lavenda 1998): 0 V0 D @

0

1D˛

N Y

p rj j A

> V1 D @

N X

j D1

11 pj rjD˛ A

D˛ > V1 D rmin :

j

The first term, V0 is the geometric mean radius raised to the power D˛ , the second term, V1 is the harmonic mean volume, while the third term, V1 is the minimum volume possible, where rmin is the smallest, attainable radius. The larger becomes the more weight that is given to smaller values of the radius. Setting C D 1 in (6.128), as in the usual definition of the fractal dimension, and rearranging we get 0 1 log @

N X

1 pj rjD˛ A

j

N

X 1 log pj˛ ; 1˛

(6.131)

j

with 2 Œ0; 1. At the extremes of the interval we have D˛ PN j

S1 .p/ pj log.1=rj /

;

(6.132)

for D 0, or ˛ D 1, where S1 .p/ is the Shannon–Gibbs entropy (6.97), and D˛

S0 ; log .1=rmin/

(6.133)

for D 1 or ˛ D 0, where S0 is the Hartley–Boltzmann entropy, (6.99). In between these limits, where the Rényi entropy dwells, there are a whole host of other generalized Hausdorff exponents. Intermediate exponents may characterize nonlinear, chaotic phenomena. However, limiting the exponent to the interval Œ0; 1 naturally excludes exponents ˛ > 1, which imply negative values of . This does not mean that we can categorically exclude values ˛ > 1. For instance, the exponent ˛ D 2 is the so-called


191

“correlation” dimension, which is related to the correlation function of the fractal set or the probability of finding one member of the set at a given distance from another member (Schroeder 1991). But, exponents < 1 have no place in our analysis. For if we return to Hölder’s inequality, (6.109), and use the probabilities qj D D nj , we come out with Campbell’s (1965) result 1 1 0 0 N N X X 1 1 ˛ n pj A log @ pj D j A D L ; log @ 1˛ j D1

(6.134)

j D1

where ˛ D . C1/1 . Since 2 Œ0; 1, ˛ is constrained to the closed interval Œ0; 1, which makes sense since for D 0, or ˛ D 1, we get the noiseless coding theorem, while for D 1, or ˛ D 0, the mean code length coincides with nmax D max nj . The equality in (6.134) is satisfied for pj˛ D nj D PN

˛ j D1 pj

;

(6.135)

which has been referred to as “escort” probabilities in the literature. Taking the logarithm of both sides of (6.135), multiplying through by pj and summing over all values give (6.136) La D ˛S1 C .1 ˛/S˛ : For ˛ < 1, (6.136) expresses the average code length La as a “mixture” of the Shannon–Gibbs and Rényi entropies. Introducing inequality (6.105) into (6.136) gives the second inequality in (6.100) provided ˛ < 1. This interpretation is lost for values of ˛ > 1, or < 0, together with the possibility of interpreting (6.113) as a mean code length. Mean entropies whose orders are ˛ > 1 are inherently related to negative means which pose definite problems, as we have seen in the case of heat capacities (Sidhu 1980). We will now address such problems.

6.12.4 Exponential Entropies and the Correlation Dimension The Rényi entropy (6.98) for a mean of negative order is S .p/ D log .1=M .p// :

(6.137)

Exponentiating both sides of (6.137) gives the so-called “exponential” entropy D S D

N X j

pj1 :

(6.138)

192


The exponential entropy (6.138) is related to the “range” of the distribution (Campbell 1966); it gives the length of the interval on which the probability density is different from zero if the space is the real line. We will use it to show that (1) the means are monotonically increasing functions of their order, (2) entropies cannot be expressed as escort averages, and (3) derive the correlation dimensions. Differentiating (6.138) with respect to the order gives: N

X dS S D C S D S D pj1 log pj : d

(6.139)

j D1

If there exist any extrema, it would mean that the means are not monotonically increasing functions of their order. A necessary condition for the existence of an extremum is that dS =d D 0, the condition for which is given by (6.139), viz., PN S D

˛ j D1 pj log pj PN ˛ j D1 pj

;

(6.140)

where we have set ˛ D 1 . The entropy appearing in (6.140) is the “escort” average of log pj , and if it exists, would be a generalization of the Shannon– Gibbs entropy. Differentiating (6.139) again with respect to and evaluating it at the stationary point result in d2 S D d 2

PN

2 pj˛ log pj PN ˛ j D1 pj

j D1

PN

˛ j D1 pj log pj PN ˛ j D1 pj

!2 > 0:

(6.141)

The inequality is due to the fact that the right-hand side of (6.141) is the variance of log pj under escort averaging and so is a positive number. Thus, when > 0, d2 S =d 2 > 0, and when < 0, d2 S =d 2 < 0, so that there are two extrema: a local maximum for < 0 and a local minimum for > 0. This requires dS =d 0 at D 0, where the equality sign applies to the degenerate case with the extremes coinciding at the point of inflection at D 0. Rather, if we can show that dS =d > 0 as ! 0, then these extrema will be evanescent, and the means will be monotonically increasing functions of their order (Burrows 1986). If this will be the case then (6.140) does not exist so that S cannot be expressed as an escort average (Lavenda 2004b). Writing (6.139) in the form: 9 8 N = < X 1 dS D D S pj1 log pj C D S S ; ; d : j D1


193

it is apparent that the right-hand side is of the indeterminate form 0=0 in the limit as ! 0, where S ! S0 , the Shannon–Gibbs entropy, in that limit.3 Applying l’Hôpital’s rule of indeterminate forms we get X N dS 2 lim D S D pj .log pj /2 S02 .p/ > 0: !0 d j D1

Again, the inequality follows because the right-hand side is the variance of log pj . Consequently, dS =d > 0 as ! 0, and so it is for all . Expressed in words, this says that the exponential entropy, or the mean of order , is an increasing function of and that no stationary point whose entropy is given by (6.140) exists. In this regard, the Rényi entropy (6.137) will confirm that no stationary point exists. Differentiating (6.137) with respect to results in # " PN ˛ 1 dS j D1 pj log pj D ; S C PN ˛ d j D1 pj where, again, we have set ˛ D 1 . The stationary condition is again the escort average, (6.140). The product d2 S =d 2 > 0 so there is a local minimum for > 0 and a local maximum for < 0. This implies that the slope of the curve S vs. is negative as it passes through the origin. Negation of these statements are carried out in exactly the same way as before. As a bonus, the exponent is no longer restricted to positive values. In the particular case D 1, the Rényi entropy (6.137) becomes the entropy of the correlation dimension: P 2 log N j D1 pj : D2 D lim r!0 log.1=r/ Larger values of ˛ D 1 give still less prominence to smaller probabilities, i.e., those regions of the fractal which are rarely visited. With these examples, we have run the course of our journey. We began with the origins of thermodynamics, as they were conceived by their forefathers, and saw how thermodynamics developed into a mature science, with its uses and abuses. In this chapter, we went beyond the confines of classical thermodynamics by relying on information theory and the theory of means to guide our way. Let it suffice to say that the limits of classical thermodynamics have been surpassed, and new frontiers have been opened up for exploration.

Because we are considering the exponent D 1 ˛, and not ˛, D 0 corresponds to ˛ D 1, so that S0 is the Shannon–Gibbs entropy, which we previously designated as S1 .

3

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Index

acceleration, uniform, 98 accessibility, 78, 111, 118, 175 limitation of, 78 action at a distance, 5 adiabatic coefficient of expansion, 12 compressibility, 7 condition in endoreversible cycle, 85 constraints, 50, 91, 173, 175 exponent, 41, 43, 62, 147, 175 influencing, 140 invariance, principle of, 133, 136, 140 invariants, 46 method of quantization, 137 parameter, 79 path, 42, 46 potential, 42, 89, 148, 180 changes in, 173 convex, 161 non-extensivity of, 43, 44, 164 superadditive, 161 quasi-static transitions, 89, 148 slow variation, 134, 136 surfaces, 46 variable, 164 versus isentropic, 46 adiabats, 32, 49, 147 as a relation between integrating factors, 39 characteristic equations for, 61 coincidence with isotherms, 52 condition derived from, 60 conditions for, 43, 67, 79, 139, 167 equations for, 147 nonexistence of, 53 not crossing a neutral curve, 54 Poisson, 4, 9, 148 versus isotherms, 32, 135 Avogadro’s number, 3

Boltzmann’s principle, 142 Boyles’s law, see Mariotte’s law

caloric theory, 1, 4–6, 10, 11, 19–22, 26, 33, 38, 73, 78, 117–119 alternatives to, 5 canonical distribution, 128 Cantor set, 189 Carathéodory’s axiom, 41, 44, 174 defect in, 174 formulation, 42 Carnot’s analogy between waterfall and motive power of heat, 20 axiom, 21, 35, 39, 48–50, 55, 57, 58, 66, 72, 73, 75, 97, 117, 118, 141 modification of, 73, 79 possible violations of, 73, 74 special, 26, 39 efficiency, see efficiency Carnot formula, 7 function, 11, 23, 24, 28, 55, 162 Clapeyron’s assumption, 55 use for defining absolute scale, 28 heat function, 26 infinitesimal temperature differences, 10 knowledge of second law, 71, 111 law, see Carnot’s theorem negation of internal energy, 112 premise, 24 principal assumption, 24, 25 principle, 38 specific heats, 9 theorem, 26, 27, 29, 33, 34, 36, 37, 42, 163 corollary to, 39, 164 Carnot-Clapeyron equation, 23, 26, 28, 31, 35, 61, 136 appearance in Clapeyron’s work, 35

201

202 evaluation with ideal gas equation of state, 36 geometrical derivation of, 25 incompatible with Mariotte’s law, 35 integration of, 40 microscopic origin of, 135 Charles’s law, see Gay-Lussac’s law chemical potential, 145, 151 boson, 152 equality of condition for phase equilibrium, 176 fermion, 151 for ideal gas, 151 role of, 152 vanishing of, 156 Clausius’s assumption on work and heat, 35 criterion for reversibility kinetic analog of, 101 equation of state, 159 formulation of first law, 21, 72, 111, 122 formulation of second law, 21, 118, 120, 122 inequality, 63, 69, 112, 113, 165 generalized, 166 introduction of internal energy, 37 introduction of two state functions, 21 mechanical theory of heat, 71 modification of the Carnot-Clapeyron equation, 24 negation of entropy, 112 positive and negative transformations, 21 salvage of Carnot’s theory, 21, 141 thermoelectric theory, 109 uncompensated transformations, 63, 71 use of rejected heat to determine work, 75 Clausius–Clapeyron equation, 72, 154, 157 coding theorem, 183 coefficient of correlation, 179 coefficient of performance (COP), 102 coefficient of thermal expansion, 3, 31, 52, 153 coexistence curve, 157 compression ratio, 79 Compton wavelength, 138 conservation of energy, 11, 71, 122 constraints, 166 adiabatic, see adiabatic constraints contact mechanical, 90, 163 thermal, 114, 122, 163, 168, 171

Index Cooper pairs, 152 coordinates, constrainable versus unconstrainable, 109, 143 correlation dimension, 193 Cramér-Rao inequality, 180 critical point, 157 cycle Brayton, 80 comparison to Carnot, 82 Carnot, 74 additional step of condensation, 9 analogy with piston, 17 closing of, 5, 19 for photon gas, 56 for Stefan-Boltzmann law, 94 graphical representation of, 17 ideality of, 17 infinitesimal, 10, 164 maximum efficiency of, 20 relativistic, 96 strange, 53 comparison between reversible and irreversible, 101 equivalent to Carnot’s, 164, 166 irreversible, 69, 165 Otto, 74, 78 comparison to Carnot, 80 reverse, 69 reversible, condition for, 15

de Brogle relation, from the pressure of radiation, 139 degrees of freedom, 39, 43, 170 thermodynamic, 176, 177 dissipation, 111 doctrine of latent and specific heats, 2, 52, 55, 57, 72, 130, 169 Clausius’s criticism of, 162 Doppler shift, 139 longitudinal, 99

efficiency, 33 Carnot, 27, 35, 40, 50, 66, 76, 80, 95, 97 as upper bound, 51 as upper bound on mean differences, 181 reason for optimality, 123 complementary, 69, 101 concavity of, 77 endoreversible, 87 greater than Carnot’s, 50 isothermal engine, 90

Index mean temperature ratio, 91 mechanical, 164 non-Carnot, 60 non-maximal, 67, 76 Otto, 79 Ehrenfest’s theorem, see adiabatic principle Einstein’s addition law, see velocity addition formula elasticity, modulus of, 119, 132 electromotive force, 103, 104, 109 emissivity power, 96 energy balance equation, 73, 77, 78, 111 non-mechanical interpretation of, 118 ensemble canonical, 135, 139 limit of validity of, 137 microcanonical, 135 enthalpy, 30, 31, 55, 94, 97, 132, 151 effect of motion upon, 97 entropy, 21, 28 absolute, 154 additive, 182 as indicator of irreversibility, 71 average change in, 180 conservation of, 22 differences as measurable, 187 empirical, 89, 148 exponential, 191 extensivity of, 46, 164 Hartley-Boltzmann, 185, 190 Havrda-Charvát, 160, 187 increase of, 172 maximum of, 174 metrical, 89 nonadditive, see entropy, pseudo-additive of degenerate gas, 139 of ideal gas, 172 of mean length, 185 of mean volume, 190 production, 101, 130, 141 pseudo-additive, 186 Rényi, 160, 182, 185, 186, 191 relation between metrical and empirical, 89 relation to mean values, 181, 192 relation to means, 187 Shannon-Gibbs, 137, 182, 190 concavity, 184 recursive, 183 subadditive, 159 equation of state cold, 136 Grüneisen, see Grüneisen, equation of state ideal gas, 3, 9 separating repulsion and attraction, 136

203 equilibration adiabatic, 64, 89, 90, 115, 121, 168, 171–174 L, 180 S, 180 energetical, 122 mechanical, 164–166 thermal, 60, 87, 164, 165 escort probabilities, 191, 192 exactness condition, see integrability condition exponential family, 137, 145 exterior differential forms, 47

finite-time thermodynamics, 121 first law, 11, 34, 48, 50, 62, 77, 106, 113, 140, 142, 143, 165 Truesdell’s interpretation of, 67, 117 Fisher information, 180 fluctuation theory, 157 thermodynamic, 179 fluctuation-dissipation relation, 127, 128, 142 nonequilibrium, 142 fluctuations, 130, 135, 143 Fourier heat conduction, 103, 110 fractal dimension, 189 free energy, 143 rotational, 139 Gibbs, 123, 151 measure of non-extensivity, 159 Helmholtz, 123, 128 comparison with non-extensive adiabatic potentials, 174 fundamental inequality, 75, 81, 102

gas attractive, 147, 153 condition for, 150 constant, 2, 3 degenerate, 79, 154 at absolute zero, 151 bose, 151 fermi, 152 generalized, 50, 163, 168 photon, 50, 58, 61, 67 ideal, 13, 24, 38–40, 50, 60, 62, 67, 75, 79, 81, 83, 89, 92, 94, 114, 151, 153, 155, 163 condition for, 147 perfect, see gas ideal repulsive, 149 condition for, 150 Gauss’s error law, 137

204 Gay-Lussac’s law, 3, 29 generating function, 125 geometry affine, 177 Euclidean, 177 impossible for Gibbs’s space, 178 hyperbolic, 99 Gibbs’s phase rule, 176, 177 space, 177, 178 Gibbs-Helmholtz equation, 156 Grüneisen equation of state, 40, 49, 132, 133, 136, 139, 145, 147, 179 at absolute zero, 152 parameter, 41, 138

Hausdorff dimension, 189 generalized, 189, 190 heat, 2 absorbed, 13 as a state function, 22 caused by friction, 31 conduction, 5, 172, 176, 181 confusion with heat capacity, 14, 149 conservation of, see caloric theory current, 72, 81 death, ix, 67 field vector, 49, 55, 57 flow, 70, 83, 86 function, 23 integrating factors for, 25, 60, 147 Joule, 103 latent, 2, 22, 24 microscopic origins of, 131 relation between, 38 with respect to pressure, 38, 81 with respect to protocol, 135 with respect to volume, 31, 40, 50, 75, 163 motive power of, 38 practical value, 64, 112, 113 radiant, 6 rate of, 101 relation to gravity, 100 transfer, 83, 101 versus specific heat, 27 wasted, 71 heat and mass, equivalence of, 97 heat and work Carnot’s statement of, 20 equivalence of, 5 interconversion of, 14, 25, 27, 63

Index heat capacity, 4, 37 as logarithm of ratio of volumes, 5, 9 at constant pressure, 4, 7 in regard to Carnot-Clapeyron equation, 36 measurement of, 2 at constant protocol, 135 at constant volume, 4, 7 measurement of, 13 confusion in, 37 difference of, 7, 13, 38 relation to adiabatic exponent, 39 impossibility of negative, 171 limited by third law, 171 power laws, 40 ratio of, 2, 6, 22 relation between, 38, 52, 55 variation with temperature, 4, 23 heat content, see enthalpy Heaviside’s, statement of second law, 120 Holder’s inequality, 185, 189 Holtzmann’s hypothesis, 37, 38, 40, 50, 90, 147

inaccessibility, 41, 44, 174 incomparability of first and second laws, 89 information theory, 146, 180 and entropy, 181 lack of metric, 178 integrability condition, 134, 163 for entropy, 36, 38, 146, 147, 170 for internal energy, 23, 25, 62, 162, 163, 170 integrating factor, 25, 36, 45, 48, 81, 90, 163 existence for two independent variables, 41 role of in second law, 122 internal energy, 11, 21, 23, 25, 29, 49, 71, 89, 111, 113, 146, 149 absolute, 154 change in ordering of states, 71 complementary to Carathéodory’s principle, 175 condition for first-order homogeneity, 61 convexity of, 115 integrability condition for, 35 minimum change in, 115 of ideal gas, 90 ordering of states, 77 power law generalization of, 41 isobaric coefficient of expansion, 12 isothermal compressibility, 7, 38 condition for, 61

Index infinite for a photon gas, 58, 61 relation to adiabatic, 7 thermodynamic stability on, 61 isotherms, 32 flat for a photon gas, 58

Jensen’s inequality, 68, 77, 128, 130 Joule-Thomson experiment, 29

Kelvin’s absolute temperature scale, 31 challenge of caloric theory, 20 conjecture, 30 criterion for reversible processes, 88 derivation of second law, 27 graduating temperature scale, 33 heat dissipation, 112 inequality, 113 irregularly heated body, 122 postulate, 69 principle, 46 principle of minimum dissipation of energy, ix, 65 reversibility condition, 27 statement of second law, 117, 118, 120, 141 statements of first and second laws, 162 use of caloric theory, 118 Kolmogorov-Nagumo function, 186 Kraft’s inequality, 183

Le Châtelier principle, 135 Lorentz transform, 98, 99

Mariotte’s law, 3, 23, 29, 35, 163 Markov processes, 142 Maxwell relation, 8, 56, 131, 135 means arithmetic, 60 for temperature, 86, 171 arithmetic-geometric inequality, 27, 86, 171, 172 thermodynamic derivation, 171 bounds on ratios, 62 definition of, 121 equivalent, 173 geometric, 114, 168 for temperature, 85 for volume, 90, 175 homogeneity, 187 inverse harmonic

205 for temperature, 91 multiplicative property of, 182 of different orders, 89 comparability of, 90, 166, 172 properties of, 21, 171, 192 translational invariance, 187 mechanical equivalent of heat, 5, 9, 14 Carnot’s calculation of, 12 Joule’s calculation of, 15 Mayer’s calculation of, 13 method of characteristics, 4, 24, 49, 132, 133, 147 microscopic reversibility, principle of, 110, 142 moment of inertia, microscopic, 139 momentary rest frame, 99 motion hyperbolic, 98 non-uniform, 98 perpetual, 111, 117 used by Carnot as an argument for greater efficiency, 20

neutral curve, 53, 135

Onsager symmetry, 110 ordering of states, 118

Peltier force, 107, 109 heat, 103 Pfaffian forms, 41, 47 phase equilibrium, 24, 157, 176 piezotropic curve, 53 Planck’s quantization rule, 137 Poincaré’s addition law, see velocity addition formula power, 100 for uniform motion, 101 maximizing of, 84 maximum, 120 minimum consumption, 101 motive, 20, 66, 170 pressure as a relativistic invariant, 96 generalized, 147 kinetic interpretation of, 5 of a generalized gas, 61 of radiation, 93, 95, 138 of vibrations, 133

206 probabilities, a priori, 140, 188 probability density invariant, 142 joint, 142, 182 transition, 142 processes adiabatic, 22, 129 and isothermal, 8 versus isentropic, 122 combustion, modelling of, 79 irreversible, 45, 63, 68, 101, 111, 165, 174 loss of motive power, 20 origins of, 71 isentropic, 22 mechanical, 77 quasi-static, 94, 174 reversible, 12, 34, 63, 112 criterion for, 35, 97, 164 heating, 109 spontaneous, 165 protocol parameter, 124

radiation constant, 61 energy density of, 94 isotropic, 93 thermal, 96 rapidity, 98, 99 refrigeration effect, 101, 102 reversibility criterion for, 34, 60 Curzon-Ahlborn incorrect use of, 84, 169 rotating systems, statistical properties of, 139

scaling exponents, 189 second law, 12, 23, 48, 60, 72, 78, 90, 106, 120, 137, 141, 165, 166 derived from properties of means, 171 Fermi’s confusion in, 117 increase in entropy, 118 Truesdell’s interpretation of, 67 modification of, 170 refutation of, 53 Seebeck effect, 102, 108 Shannon’s inequality, 184 specific heat, see heat capacity of electricity, 104 Stefan’s law, Boltzmann’s derivation of, 96 strain, producing temperature differences, 102 strange attractors, 188 structure function, 140

Index system composite, 42, 89, 163 isolated, 45, 139, 143, 166, 172

Tait’s first law, 65 inequality, 64 second law, 65 temperature absolute, 28, 29, 169 empirical, 29, 43, 169, 175 zero point of, 29 lowest possible, 34 mean, 60 minimum mean, 175 neutral point, 108 relativistic decrease in, 97 zero-point of, 29, 31 tension, 125, 130–132, 134 thermal conductance, 83 effects irreversible, 103 reversible, 102, 103 resistance, principle of, 118–120 wavelength nonrelativistic, 138, 139 ultrarelativistic, 138 thermodynamic distance, see thermodynamic metric fields, 47 metric, 176, 181 lack of in classical thermodynamics, 47, 177 perturbation theory, 128 condition of applicability, 127 potential, 156 stability, 57, 130, 149, 177 thermoelectric circuit, 103 power, 105, 108 Thiele semi-invariants, 125, 126 third law, 153, 154, 171, 173 Nernst’s formulation, 161 Planck’s formulation, 160 Thomson heat, 103, 104, 107–109 time dilatation, 100 logarithmic, 98 relaxation, 142 switching, 128

Index van der Waals equation, 158 velocity Euclidean and hyperbolic measures of, 99 group and phase, 134 of sound Laplace’s adiabatic calculation of, 6, 22 Newton’s isothermal calculation of, 6

Wien’s displacement law, 137 formula, 137 work, 106 actual, 75, 113 as a second-order effect, 10 as motive power, 23 capacity for doing, 87, 176 Carnot’s expression for, 87

207 derived from Carnot’s principle, 56 dissipated, 143 done in finite-time, 126 external, 11, 14, 23, 24, 31, 37 from Carnot’s principle, 39 in adiabatic processes, 113 internal, 30 loss of, 75, 111 maximum, 24, 34, 86, 88, 91, 113, 122, 175 principle of, 85, 114, 121 minimum, 86 negative, 69 net, 26, 36, 117, 118 negative transfer, 101 non-maximal, 63, 71, 73, 75, 101 produced by change in mass, 100 upper bound to, 102 wasted, 71

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