⊙ THE MECHANICAL THEORY OF HEAT. BY R.CLAUSIUS. RANSLATED BY WALTER R.BROWNE,M.A., LATE FELLOW OF TRINITY COLLEGE,CAMBRIDG卫， 道ondon: MACMILLAN AND CO. 1879 Digiized by Google
o THE MECHANICAL THEO·RY OF HEAT. ~ tt~:lj~~T; BY WALTER R,·'.l!ROWNE, M.A., UTIlI FELLOW OJ!' TRINITY COLLEGE, C.U(BRIDGE. f'l.onbon: MACMILLAN AND CO. 1879 Digitized by Coogle
TRANSLATOR'S PREFACE. THE following translation was undertaken at the instance of Dr T.Archer Hirst,F.R.S.,the translator of the first collected edition (mentioned in the Author's Preface below) of Professor Clausius'papers on the Mechanical Theory of Heat.The former work has however been so completely re- written by Professor Clausius,that Dr Hirst's translation has been found scarcely anywhere available;and I must there- fore accept the full responsibility of the present publication. I trust it may be found to supply a want which I have reason to believe has been felt,namely,that of a systematic and connected treatise on Thermodynamics,for use in Universities and Colleges,and among advanced students generally.With the view of rendering it more complete for this purpose,I have added,with the consent of Professor Clausius,three short appendices on points which he had left unnoticed,but which still seemed of interest,at any rate to English readers.These are,(1)The Thermo-elastic pro- ed by ole
TRANSLATOR'S PREFACE. THE following translation was undertaken at the instance Archer Hirst, translator of l.iW"",'cU'vu edition Author's Preface Professor Clausius' the Mechanical The former work been so written by Professor Clausius, that Dr Hirst's translation has been found scarcely anywhere available; and I must there- I fore accept the full responsibility of the present publication. I trust it may be found to supply a want which I have reason to believe has been felt, namely, that of a systematic connected treatise Thermodynamics, for and With the it more "V'''I'.no purpose, I have the' consent of Clausius, three short appendices on points which he had left unnoticed, but which still seemed of interest, at any rate to English readers. These are, (1) The Thermo-elastic pro-
vi TRANSLATOR'S PREFACE. perties of Solids;(2)The application of Thermo-dynamical principles to Capillarity;(3)The Continuity of the Liquid and Gaseous states of Matter.My best thanks are due to Dr John Hopkinson,F.R.S.,both for the suggestion of these three points,and also for the original and very elegant investigation from first principles,contained in the first Appendix,and in the commencement of the second.My thanks are also due to Lord Rayleigh,F.R.S.,E.J.Routh, Esq,and Professor James Stuart,for kindly looking through the first proof of the translation,and for various valuable suggestions made in connection with it. WALTER R.BROWNE. 10,VICTORIA CHA3:rBER月，WESTMIK8TEB, November,1879. poireavGOOgle
vi TRANSLATOR'S PREFACE. perties of Solids; (2) The application of Thermo-dynamical principles to Capillarity; (3) The Continuity of the Liquid and Gaseous states of Matter. My best thanks are due to Dr John Hopkinson, F.R.S., both for the suggestion of these three points, and also for the original and very elegant investigation from first principles, contained in the first Appendix, and in the commencement of the second. My thanks are also due to Lord Rayleigh, F.R.S., E. J. Routh, Esq., and Professor James Stuart, for kindly looking through the first proof of the translation, and for various valuable suggestions made in connection with it. WALTER R. BROWNE. 10, VICTORIA CUAlIBEBS, WZSTllIll'STEB, November, 1879. Digitized by Coogle
AUTHOR'S PREFACE. MANY representations having been made to the author from different quarters that the numerous papers"On the Me- chanical Theory of Heat,"which he had published at different times during a series of years,were inaccessible to many who, from the widespread interest now felt in this theory,were anxious to study them,he undertook some years back to publish a complete collection of his papers relating to the subject. As a fresh edition of this book has now become necessary, he has determined to give it an entirely new form.The Mechanical Theory of Heat,in its present development,forms already an extensive and independent branch of science. But it is not easy to study such a subject from a series of separate papers,which,having been published at different times,are unconnected in their form,although they agree in their contents.Notes and additions,however freely used to explain and supplement the papers,do not wholly over- come the difficulty.The author,therefore,thought it best so to re-model the papers that they might form a connected whole,and enable the work to become a text-book of the science.He felt himself the more bound to do this because Google
AUTHOR'S PREFACE. MANY representations having been made to the author from different quarters that the numerous papers CI On the Mechanical Theory of Heat," which he had published at different times during a series of years, were inaccessible to many who, from the widespread interest now felt in this theory, were anxious to study them, he undertook some years back to publish a complete collection of his papers relating to the subject. As a fresh edition of this book has now become necessary, he has determined to give it an entirely new form. The Mechanical Theory of Heat, in its present development, furms already an extensive and independent branch of science. But it is not easy to study such a subject from a series of separate papers, which, having been published at different times, are unconnected in their form, although they agree in their contents. Notes and additions, however freely used to explain and supplement tIle papers, do not wholly overcome the difficulty. The author, therefore, thought it best so to re-model the papers that they might form a connected whole, and enable the work to become a text-book of the science. He felt himself the more bound to do this because Digitized by Coogle
viⅲ AUTHOR'S PREFACE. his long experience as a lecturer on the Mecbanical Theory of Heat at a Polytechnic School and at several Universities had taught him how the subject-matter should be arranged and represented,so as to render the new view and the new method of calculation adopted in this somewhat difficult theory the more readily intelligible.This plan also enabled him to make use of the investigations of other writers,and by that means to give the subject greater completeness and finish.These authorities of course have been in every case duly recognized by name.During the ten years which have elapsed since the first volume of papers appeared,many fresh investigations into the Mechanical Theory of Heat have been published,and as these have also been discussed,the contents of the volume have been considerably increased. Therefore in submitting to the public this,the first part of his new investigation of the Mechanical Theory of Heat, the author feels that,although it owes its origin to the second edition of his former volume,still,as it contains so much that is fresh,he may in many respects venture to call it a new work. R.CLAUSIUS. Boxx,December,1875. Digitized by Google
viii AUTHOR'S PREFACE. his long experience as a lecturer on the Mechanical Theory of Heat at a Polytechnic School and at several Universities had taught him how the subject-matter should be 'arranged and represented, so as to render the new view and the new method of calculation adopted in this somewhat difficult theory the more readily intelligible. This plan also enabled him to make use of the investigations of other writers, and by that means to give the subject greater completeness and finish. These authorities of course have been in every case duly recognized by name. During the ten years which have elapsed since the first volume of papers appeared, many fresh investigations into the Mechanical Theory of Heat have been published, and as these have also been discussed, the contents of the volume have been considerably increased. Therefore in submitting to the public this, the first part of his new investigation of the Mechanical Theory of Heat, the author feels that, although it owes its origin to the second edition of his former volume, still, as it contains so much that is fresh, he may in many respects venture to call it a new work. R. CLAUSIUS. BONN, December, ]875. Digitized by Coogle
TABLE OF CONTENTS MATHEMATICAL INTRODUCTION. ON MECHANICAL WORK,ON ENERGY,AND ON THE TREATMENT OF NON-INTEGRABLE DIFFERENTIAL EQUATIONS. PAGE $1.Definition and Measurement of Mechanical Work 1 $2.Mathematical Determination of the Work done by variable com- ponents of Force. 2 3.Integration of the Differential Equations for Work done ￥ 54. Geometrical interpretation of the foregoing results,and obser- vations on Partial Differential Coefficients. 8 S5. Extension of the foregoing to three Dimensions 9 86. On the Ergal 11 §7. General Extension of the foregoing 13 $8.Relation between Work and Vis Viva. 1 s9.0 n Energy·· 19 CHAPTER I. FIRST MAIN PRINCIPLE OF THE MECHANICAL THEORY OF HEAT.- EQUIVALENCE OF HEAT AND WORK. §1. Starting Point of the theory 21 s2. Positive and Negative Values of Mechanical Work 22 s3. Expression for the First Main Prineiple 23 84 Numerical Relation between Heat and Work 4 24 s5. The Mechanical Unit of Heat 25 6. Development of the First Main Principle 27 §7. Different Conditions of the quantities J,W,and II 27 88. Energy of the Body 80 $9.Equations for Finite Changes of Condition-Cyelical Processes 32 10.Total Heat-Latent and Specific Heat 33 $11.Expression for the External Work in a particular case 35 poiredvCGOOgle
TABLE OF CONTENTS. MATHEMATICAL INTRODUCTION. ON MECHANICAL WORK, ON ENERGY, AND ON THE TREATMENT OF NON-INTEGRABLE DIFFERENTIAL EQUATIONS. PAGE §1. § 2. Definition and Measurement of Mechanical Work . • . 1 Mathematical Determination of the Work done by variable com- ponents of Force • . . . . .. 2 § 3. § 4. Integration of the Differential Equations for Work done . . 4 Geometrical interpretation of the foregoing results, and observations on Partial Differential Coefficients 8 § 5. § 6. § 7. §8. § 9. Extension of the foregoing to three Dimensions 9 On the Ergal • • • . 11 General Extension of the foregoing 13 Relation between Work and Vis Viva 17 On Energy • 19 CHAPTER I. FIRST MAIN PRINCIPLE OF THE MECHANICAL THEORY OF HEAT.- EQUIVALENCE OF HEAT AND WORK. § 1. Starting Point of the theory . • • . § 2. Positive and Negative Values of Mechanical Work § 3. Expression for the First Main Principle • § 4. Numerical Relation between Heat and Work § 5. The Mechanical Unit of Heat • • . . § 6. Development of the First Main Principle. . § 7. Different Conditions of the quantities J, W, and H . . § 8. Energy of the Body • • . • . . . . § 9. Equations for Finite Changes of Condition-Cyclical Processes § 10. Total Heat-Latent and Specific Heat . • . t 11. Expression for the External Work in a particular case Digitized by Coogle 21 22 23 24 25 27 27 30 32 33 35
X CONTENTS. CHAPTER II. ON PERFECT GASES. PAGE 81. The Gaseous Condition of Bodies 38 82. Approximate Principle as to Heat absorbed by Gases 42 83. On the Form which the equation expressing the first Main Prin- ciple assumes in the case of Perfect Gases 43 $4.Deduction as to the two Specific Heats and transformation of the foregoing equations 46 $5.Relation between the two Specifie Heats,and its application to calculate the Mechanical Equivalent of Heat 48 86. Various Formule relating to the Specifie Heats of Gases 52 §7. Numerical Calenlation of the Specific Heat at Constant Volume 56 s8. Integration of the Differential Equations which express the First Main Principle in the case of Gases 60 $9.Determination of the External Work done during the change of Volume of8Gas··· 61 CHAPTER III. SECOND MAIN PRINCIPLE OF THE MECHANICAL THEORY OF HEAT. $1.Description of a special form of Cyelical Process. 63 $2.Result of the Cyclieal Process 71 83. Cyelical Process in the case of a body composed partly of Liquid and partly of Vapour. 73 §生 Carnot's view as to the work performed during the Cyelical Proce88· 76 s5. New Fundamental Principle concerning Heat 78 $6.Proof that the Relation between the Heat carried over,and that converted into work,is independent of the matter which forms the medium of the change 79 87. Determination of the Function(TT). 81 §8. Cyelical Processes of a more complicated character 84 89. Cyclical Processes in which taking in of Heat and change of Temperature take place simultaneously . ·87 CHAPTER IV. THE SECOND MAIN PRINCIPLE UNDER ANOTHER FORM;OR PRIN- CIPLE OP THE EQUIVALENCE OF TRANSFORMATIONS. $1.On the different kinds of Transformations. 91 $2.On a Cyclical Process of Special Form 02 83. On Equivalent Transformations·· 6 pomedvCOOgle
§1. § 2. § 3. §4. § 5. § 6. § 7. § 8. § 9. CONTENTS. CHAPTER II. ON PERFECT GASES. The Gaseous Condition of Bodies • • • Approximate Principle as to Heat absorbed by Gases. . • On the Form which the equation expressing the first Main Principle assumes in the case of Perfect Gases • • • • Deduction as to the two Specific Heats and transformation of the foregoing equations • • • • • • • • Relation between the two Specific Heats, and its application to calculate the Mechanical Equivalent of Heat. • Various Formullll relating to the Specific Heats of Gases. • Numerical Calculation of the Specific Heat at Constant Volume . Integration of the Di1ferential Equations which /lxpress the First Main Principle in the case of Gases • . • • • Determination of the External Work done during the change of Volume of a Gas CHAPTER III. PAGFl Sf! 42 4S 46 48 62 56 60 64 SECOND MAIN PRINCIPLE OF THE MECHANICAL THEORY OF IIEAT. §1. § 2. § S. §4. § 5. § 6. § 7. § 8. § 9. Description of a special form of Cyclical Process • Result of the Cyclical Process • • • • • • . Cyclical Process in the case of a body composed partly of Liquid and partly of Vapour • • • • • • • . Carnot's view as to the work performed during the Cyclical Process. • • • • • • New Fundamental Principle concerning Heat • • • • Proof that the Relation between the Heat carried over, and that converted into work, is independent of the matter which forms the medium of the change. • • Determination of the Function .p (Tl Tt) • • Cyclical Processes of a more complicated character • • • Cyclical Processes in which taking in of Heat and change of Tempera.ture take place simultaneonsly • CHAPTER IV. 69 71 73 7/l 78 79 81 84 R7 THE SECOND MAIN PRINCIPLE UNDER ANOTHER FORM; OR PRINCIPLE OF THE EQUIVALENCE OF TRANSFORliATIONS. § 1. On the di1ferent kinds of Transformations § 2. On a. Cyclical Process of Special Form § 3. On Equivalent Transformations • Digitized by Coogle III !J:! 116
CONTENTS. xi PAGE $4.Equivalence Values of the Transformations. 07 $5.Combined value of all the Transformations which take place in a single Cyclical Process 101 6.Proof that in a reversible Cyclical Process the total value of all the Transformations must be equal to nothing 102 7.On the Temperatures of the various quantities of Heat;and the Entropy of the body 10G n the Function of Temperature,r.. 107 CHAPTER V. FORMATION OF THE TWO FUNDAMENTAL EQUATIONS. $1.Discussion of the Variables which determine the Condition of the body. 110 $2.Elimination of the quantities U and S from the two fundamental equations 112 3.Case in which the Temperature is taken as one of the Independent Variables 115 84. Particular Assumptions as to the External Forces 116 85. Frequently occurring forms of the differential equations 118 $6.Equations in the case of a body which undergoes a partial change 87. Clapeyron'Equation and Camota Funetion in its condition of aggregation 119 CHAPTER VI. APPLICATION OF THE MECHANICAL THEORY OF HEAT TO SATURATED VAPOUR. $1.Fundamental equations for Saturated Vapour 126 s2. Specific Heat of Saturated Steam 129 83. Numerical Value of h for Steam 133 84. Numerical Value of h for other vapours 135 85. Specifio Heat of Saturated Steam as proved by experiment. 189 86. Specifio Volume of Saturated Vapour 142 7..Departure from the law of Mariotte and Gay-Lussao in the case of Saturated Steam .143 8.Differential-coeficienta of pa .148 D8n $9.A Formula to determine the Specific volume of Saturated Steam, and its comparison with Experiment 150 $10.Determination of the Mechanical Equivalent of Heat from the behaviour of Saturated Steam,·· 155 时Google
§ 4. § 5. § 6. § 7. § 8. § 1. § 2. § 3. §4. § 5. § 6. § 7. CONTENTS. xi PAGB Equivalence Values of the Transformations. • • • • 97 Combined value of all the Transformations which take place in a single Cyclical Process. • • • • • • 101 Proof that in a reversible Cyclical Process the total value of all the Transformations must be equal to nothing • • • 102 On the Temperatures of the various quantities of Heat; and the Entropy of the body • • 106 On the Function of Temperature, T 107 CHAPTER V. FORHATION OF THE TWO FUlfDAMENTAL EQUATIONS. Discussion of the Variables which determine the Condition of the body • • • • • . • • • • . 110 EIimination of the quantities U and 8 from the two fundamental equations • • • • • • • . • . 112 Case in which the Temperature is taken as one of the Independent Variables • • • • • • •• 115 Particular Assumptions as to the External Forces .' 116 Frequently occurring forms of the differential equations • • 118 Equations in the ease of a body which undergoe8 a partial change in its condition of aggregation • • 119 Clapeyron's Equation and Carnot's Function 121 CHAPTER VI. APPLICATION OF THE MECHANICAL THEORY OF HEAT TO SATURATED VAPOUR. § 1. Fundamental equstions for Saturated Vapour 126 § 2. Specific Heat of Saturated Steam 129 § 3. Numerical Value of h for Steam. . 133 § 4. Numerical Value of h for other vapours . • .. 135 § 5. Specific Heat of Saturated Steam as proved by experiment. 139 § 6. Specific Volume of Saturated Vapour. . • . . • 142 § 7. • Departure from the law of Mariotte and Gay-LussaCl in the case of Saturated Steam 143 § 8. Differential-coefficients of ~ 148 PIn § 9. A Formula to determine the Specific volume of Saturated Steam, and its comparison with Experiment • • • • • 150 § 10. Determination of the Mechanical Equivalent of Heat from the behaviour of Saturated Steam. 155 Digitized by Coogle
xii CONTENTS. PAG县 11.Complete Differential Equation for Q in the case of a body composed both of Liquid and Vapour 156 s12. Change of the Gaseous portion of the mass 157 s13. Relation between Volume and Temperature 159 814. Determination of the Work as a function of Temperature .160 CHAPTER VII FUSION AND VAPORIZATION OF SOLID BODIES. 81. Fundamental Equations for the process of Fusion 163 s2. Relation between Pressure and Temperature of Fusion 167 3. Experimental Verification of the Foregoing Result 168 §4. Experiments on Substances which expand during Fusion 169 85. Relation between the Heat contained in Fusion and the Tempe- rature of Fusion 171 $6.Passage from the solid to the gaseous condition.. 172 CHAPTER VIII. ON HOMOGENEOUS BODIES. $1.Changes of Condition without Change in the Condition of Aggre- gation 175 §2. Improved Denotation for the differential coefficients 176 3.Relations between the Differential Coefficients of Pressure, Volume,and Temperature 177 $4.Complete differential equations for Q. 178 s5. Specific Heat at constant volume and constant pressure. 180 86. Specific Heats under other circumstances . 184 s7. Isentropic Variations of a body . 187 $8.Special form of the fundamental equations for an Expanded Rod 188 s9. Alteration of temperature during the extension of the rod.. 190 10.Further deductions from the equations ·,192 CHAPTER IX. DETERMINATION OF ENERGY AND ENTROPY. $1.General equations 195 $2.Differential equations for the case in which only reveraible changes take place,and in which the condition of the Body is determined by two Independent Variables 197 piotredrGOOgle
XlI COnENTS. PAOli § 11. Complete Differential Equation for Q in the case of a body composed both of Liquid and Vapour • 166 § 12. Change of the. Gaseous portion of the mass • •• 167 § 13. Relation between Volume and Temperature . .• 169 § 14. Determination of the Work as a funotion of Temperature. 160 CHAPTER VII. FUSION AND VAPORIZATION OF SOLID BODIES. § 1. Fundamental Equations for the process of Fusion. 163 § 2. Relation between Pressure and Temperature of Fusion 167 § 3. Experimental Verification of the Foregoing Result. 168 § 4. Experiments on Substances which expand during Fusion. . 169 § 6. Relation between the Heat contained in Fusion and the Temperature of Fusion . . • • • • 171 § 6. Passage from the solid to the gaseous condition . 172 CHAPTER VIII. ON HOMOGENEOUS BODIF.8. § 1. Changes of Condition without Change in the Condition of Aggregation • • . . • • . .• 175 § 2. Improved Denotation for the differential ooefficients. • . 176 § 3. Relations between the Differential Coeffioients of Pressure, Volume, and Temperature • • 177 § 4. Complete differential equations for Q. • •. 178 § 6. Specifio Heat at constant volume and constant pressure ISO § 6. Specific Heats under other oircumstances • 184 § 7. Isentropic Variations of a body. . • • • . • 187 § 8. Special form of the fundamental equations for an Expanded Rod 188 § 9. Alteration of temperature during the extension of the rod 190 § 10. Furthcr deductions from the equations 192 §1. § 2. CHAPTER IX. DETERMINATION OF ENERGY AND ENTROPY. General equations • • • • • • • • • Differential equations for the case in which only reversible changes take place, and in which the condition of the Body is determined by two Independent Variables Digitized by Coogle 195 Ill7
CONTENTS. xiii PAGE $3.Introduction of the Temperature as one of the Independent Variables 200 s4.Special case of the differential equations on the assumption that the only External Force is a uniform surface pressure 203 $5.Application of the foregoing equstions to Homogeneous Bodies and in particnlar to perfect Gases. 205 $6.Application of the equations to a body composed of matter in two States of Aggregation 207 $7.Relation of the expreasions Dzy and Ary CHAPTER X. ON NON-REVERSIBLE PROCESSES. $1.Completion of the Mathematical Expression for the Second Main Principle 。 213 $2.Magnitude of the Uncompensated Tran*formation 214 $3.Expansion of a Gas unaccompanied by Work 215 $4.Expansion of a Gas doing partial work .218 85.Methods of experiment used by Thomson and Joule. ·220 $6.Development of the equations relating to the above method 221 $7.Results of the experiments,and the equations of Elasticity for the gases,as deduced therefrom 221 $8.On the Behaviour of Vapour during expansion and under various circumstance8 223 CHAPTER XI. APPLICATION OF THE THEORY OF HEAT TO THE STEAM-ENGINE. $1.Necessity of a new investigation into the thoory of the Steam- Engine· 234 52.On the Action of the Steam-Engine. 236 $3.Assumptions for the purpose of Simplification. 233 §4. Determination of the Work done during a single stroke 239 85. Special forms of the expression found in the last section ,241 s6. Imperfections in the construction of the Steam-Engine ·241 $7.Pambour's Formula for the relation between Volume and Pres- gure 242 $8.Pambour's Determination of the Work done during a single 8troke. 244 $9.Pambour's Value for the Work done per unit-weight of steam 247 10.Changes in the Steam during its passage from the Boiler into the Cylinder,··· 249
§4. § o. § 6. § 7. Introduction of the one of the Independent Variables. _..... _ . Special CBBe of the di1ferential equations on the assumption that the only Enemal Force is a uniform surface pressure. • Application of the foregoing equations to Homogeneous Bodies and in particular to perfect Gases. . . • • • Application of the equations to a body composed of matter in two States of Aggregation • • Relation of the expressions Dzy and t.zll x ON PROCESSES. 203 205 207 209 § 1. Completion of the Mathematical Expression for the Second Main Principle • . • • • • 212 § 2. Magnitude of the Uncompensated TramCnrma.tion 214 § 3. Expansion of a Gas unaccompanied by Work 215 § 4. Expansion of a GBB doing partial work . • 218 § 5. Methods of experiment used by Thomson and Joule • 220 § 6. Development of the equations relating to the above method 221 § Results of the experiments, equations of EI&I!ticity the gases, as deduced •.•. On the Behaviour of expansion and under circumstances CHAPTER XI. APPLICATION OF THE THEORY 01' HEAT TO THE STEAM-ENGINE. § 1. Necessity of a new investigation into the theory of the SteamEngine. • 234 On the Action of the Dttlanl-L,ngme Assumptions for the I)etermination of the a single stroke Special forms of the the last section Imperfections in the the StelWl-Engine Pambour's Formulllll between Volume sure • • • PlWlbour's Determination stroke • • • • • . • • '.' 244 § 9. Pambour's Value for the Work done per unit-weight of stelWl • 247 § 10. Changes in the Steam during its passage from the Boiler into the Cylinder . 24'3