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A concise texton thermodynamics

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Tore Haug -Warberg

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Boken er utgitt med støtte fra:Tematisk satsingsområde Materialer, NTNU.Fakultet for naturvitenskap og teknologi, NTNU.

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A concise text on thermodynamics aimsat a rigorous description of the classi-cal thermodynamics of macroscopic sys-tems. Using an axiomatic approach thatdeparts from the total di!erential of Eu-ler’s homogenous functions and their so-called Legendre transformations, the ba-sic structure is established with a mini-mum of e!ort. The theory is flanked by

relevant calculation examples where also the physical interpre-tation is important.The book is divided into 163 paragraphs. These can either beconsidered as independent sections, or as one heterogeneousset of problems. The topics are taken from a broad range ofapplications in physical chemistry, statistical mechanics, gasdynamics, heat engines, control volume theory, phase and re-action equilibria, and multicomponent mixtures. The diversityof the topics and the scientific level at which they are handledis suitable for graduate students at the mechanical engineering,physics og chemistry departments.Tore Haug-Warberg was educated from the Laboratory of Sil-icate and High Temperature Chemistry at NTH (Norway) in1982. He earned his doctor’s degree in thermodynamic equi-librium computations in 1988, and has since served manyyears as a teacher at Telemark University College and Nor-wegian University of Technology and Science. He has alsobroad experience from industrial research (Institute of EnergyTechnology, Norsk Hydro and Yara International), where hemostly has worked on the theoretical aspects of process sim-ulation tools (CADAS, Yasim) and world wide energy re-porting program (HERE). He is today an associate Professorin thermodynamics at the Department of Chemical Engineer-ing, NTNU (Norway). The main research topics are equi-librium calculations and algebraic representation of thermo-dynamic models using object-oriented programming in Ruby.

A concise text on thermodynamics

… for semiadvanced studies

Tore Haug-Warberg

Department of Chemical Engineering, NTNUE-mail address: haugwarb@ nt.ntnu.noURL: http://www.nt.ntnu.no/users/haugwarb

Contents

List of Figures ix

List of Tables xiii

List of symbols xv1. Latin symbols xv2. Greek symbols xvi3. Mathematical symbols (use of) xvi4. Thermodynamic states (superscript and subscript) xvii5. Phase labels and model contributions (superscript) xvii6. Process descriptions (subscript) xviii7. Vectors xviii8. Matrices xix

Foreword xxi

KP8108 Fall 2004 xxv

Chapter 1. Thermodynamic concepts 1

Chapter 2. Prelude 151. State functions 152. Mathematical operators 16

Chapter 3. The Legendre transform 231. Canonical potentials 252. Manifolds 313. Inversion 344. Maxwell relations 375. Gibbs–Helmholtz equation 39

Chapter 4. Euler’s Theorem on Homogeneous Functions 43

Chapter 5. Postulates and Definitions 57

Chapter 6. Equations of state 631. Ideal gas law 652. Intra-molecular degrees of freedom 743. Photons 82

v

vi CONTENTS

4. Phonons 835. Free electrons 856. Virial theorem of dilute gases 857. Van der Waals equation 888. Murnaghan’s equation 90

Chapter 7. State changes at constant composition 951. Non-canonical variables 962. Using volume as a free variable 983. Using pressure as a free variable 1034. The little Bridgman table 1045. Heat capacity 1086. Compressibility and expansivity 1107. Ideal gas 1108. Van der Waals equation of state 114

Chapter 8. Closed control volumes 1211. Thermostatic state change 1222. Thermal energy balance 1253. Mechanical energy balance 1294. Uniform flow 1325. Thermodynamic speed of sound 136

Chapter 9. Dynamic systems 1411. Emptying a gas cylinder 1422. Filling a gas cylinder 1473. Acoustic resonance 152

Chapter 10. Open control volumes 1571. Incompressible flow 1602. Compressible flow 166

Chapter 11. Gas dynamics 1731. Stagnation state 1732. Sonic flow 1753. Supersonic flow 1774. Thermodynamic shock 184

Chapter 12. Departure Functions 1891. Gibbs energy residual 1912. Helmholtz using pressure as a free variable 1943. Helmholtz energy using volume as a free variable 198

Chapter 13. Simple vapour–liquid equilibrium 2031. Saturation pressure 2062. Saturation volume 209

CONTENTS vii

3. Enthalpy of vaporization 2104. Critical exponents 2125. Maxwell equal area rule 216

Chapter 14. Multicomponent Phase Equilibrium 2211. Direct substitution 2232. Newton–Raphson iteration 2253. Chemical potential versus K-value 2284. Convergence properties 237

Chapter 15. Chemical equilibrium 2411. The elements 2422. Chemical reactions 2433. Equilibrium condition 2454. Equilibrium calculation 2495. Le Chateliérs principle 252

Chapter 16. Simultaneous reactions 2591. Reaction stoichiometry 2612. Minimum Gibbs energy 2633. Newton–Raphson–Lagrange iteration 2654. Newton–Raphson–nullspace iteration 2695. Combustion reactions 2716. Adiabatic temperature calculation 274

Chapter 17. Heat engines 2811. Thermodynamic cycles 2842. Power plants 2923. The steam engine 296

Chapter 18. Entropy production and available work 3031. The Clausius inequality 3032. The Gouy–Stodola theorem 3053. Available energy 308

Chapter 19. Plug flow reactor 3171. The mass balance 3172. The energy balance 3243. Modelling issues 334

Chapter 20. A thermodynamic allegory 349

Chapter 21. Material Stability 3651. Global stability 3652. Local stability 3673. The tangent plane test 371

viii CONTENTS

4. Intrinsic stability criteria 374

Chapter 22. Thermofluids 3791. Tensor versus matrix notation 3822. Reaction invariants 3833. Conservation laws 3874. Phenomenological models 3955. Governing equations 398

References 401

Biographies 403

Appendix A. T, s and p, v-Diagrams for Ideal Gas Cycles 405

Appendix B. SI units and Universal Constants 409

Appendix C. Derivative of integrals 413

Appendix D. Newton–Raphson iteration 415

Appendix E. Direct Substitution 419

Appendix F. Linear Programming 423

Appendix G. Nine Concepts of Mathematics 429

Appendix H. Code Snippets 4331. Interactive Matlab code 4332. Matlab functions 4783. Ruby scripts 494

Index 499

Paragraphs 505

List of Figures

2.1 Isotropic pressure in close vessel 15

2.2 The di!erential of y(x) 15

2.3 A thermodynamic cycle in T, p co-ordinates 18

2.4 A thermodynamic cycle in Q,W co-ordinates 20

3.1 Legendre transformation of internal energy 23

3.2 Legendre transformation of f (x) to "(#) 33

4.1 Iso-potential diagram of Gibbs energy 50

5.1 Formal relationships in thermodynamics 60

6.1 Second virial coe"cient 87

7.1 Integration steps for internal energy 100

7.2 Integration steps for entropy 102

7.3 Enthalpy and entropy for Ar, N2 og CH4 113

7.4 Van der Waals heat capacity 116

7.5 Van der Waals inversion curve 118

8.1 Time–temperature profile for the cooling of a thermos flask 127

8.2 Gas leak in a closed room. 128

8.3 Compression work for an ideal gas 131

8.4 Control volumes for pipe flow 133

9.1 Control volume for cylinder 141

9.2 Control volume for rocket engine 144

9.3 Emptying a gas cylinder 146

9.4 Temperature increase when filling a helium cylinder 149

9.5 Isolated system consisting of two gas cylinders 149

9.6 Joule’s eksperiment 151

9.7 Helmholtz resonance 154

ix

x LIST OF FIGURES

10.1 Open control volumes 15710.2 Parabolic velocity profile 15810.3 Bernoulli flow 16010.4 T, p, µ profiles for Bernoulli flow 16210.5 Boussinesq flow 16310.6 Money to burn 16410.7 Control volume over a water 16510.8 Isentropic work of compression 16610.9 Schematic diagram of a Linde machine 170

11.1 Flow rate versus Mach number 17611.2 Supersonic state changes 17811.3 Supersonic flow in a convergent–divergent nozzle 18111.4 Supersonic flow in a duct 184

13.1 Vapour pressure of CO2 20413.2 van der Waals vapour–liquid equilibrium 20713.3 p, $ diagram for CO2 20913.4 the van der Waals equation of state in pr, vr coordinates 21713.5 Minimum Helmholtz energy 218

14.1 Phase diagram of a synthetic natural gas 23014.2 Phase diagram of cyclohexane–cyclopentane–methanol 23114.3 Phase diagram of hexane–toluene 23414.4 Phase diagram of gold–copper 23614.5 First order Newton–Raphson iteration 237

15.1 van’t Ho!’s equation for CO2–H2–CO–H2O 25015.2 Gibbs energy for CO2–H2–CO–H2O 252

16.1 Combustion of methane 25916.2 CO, O2, H and OH content in exhaust gas 27316.3 Adiabatic temperature calculation 278

17.1 Optimal design of closed cycle 28417.2 Otto, Brayton and Carnot (ideal gas cycles) 28617.3 Stirling, Ericsson and Rankine (ideal gas cycles) 28717.4 The Stirling cycle projected onto T, s and p, v coordinates 28817.5 Overall e"ciency of a single-engine power plant 295

LIST OF FIGURES xi

17.6 E"ciency of a steam engine versus a steam turbine 29817.7 Rankine and Brayton steam cycles 299

18.1 The Clausius inequality 30418.2 Production of entropy by heat conduction 30618.3 Carnot engine 308

21.1 Exchange of phase sizes in a trial two-phase system 368

A.1 T, s diagrams 406A.2 p, v diagrams 407

D.1 Convergence of the Newton–Raphson iteration 418

List of Tables

6.1 Sackur–Tetrode standard entropy 77

7.1 the little Bridgman table 1057.2 Total di!erentials of u, h, s, v, T and p 106

8.1 Physical properties of selected compounds 124

13.1 van der Waals’ enthalpy of vaporization 21113.2 Critical exponents measured experimentally for selected

compounds 215

16.1 Thermodynamic data from JANAF 272

17.1 Thermodynamic e"ciency of the 6 basic cycles 29117.2 Thermodynamic work of the 6 basic cycles 292

18.1 Excerpts from the steam tables of water 313

22.1 Tensors versus matrices 38222.2 Taylor expansion of equilibrium state 386

xiii

List of symbols

1. Latin symbols

a1, a2, a3 unspecified model parameters –, –, –a energy parameter in Van der Waals equation J m3 mol-1

a, A Helmholtz energy J mol-1, JA cross sectional area m2

b1, b2 unspecified model parameters –, –b hard sphere volume in Van der Waals equation m3 mol-1

b Stefan–Boltzmann constant W m-2 K-4

B second virial coe"cient m3 mol-1

c speed of sound m s-1

cp,CP heat capacity (!h/!T )p, (!H/!T )p J mol-1 K-1, J K-1

cv,CV heat capacity (!u/!T )v, (!U/!T )V J mol-1 K-1, J K-1

e elementary charge Cek, EK kinetic energy J mol-1, Jep, EP potential energy J mol-1, Jg acceleration of free fall m s-2

g,G Gibbs energy J mol-1, Jh Planck constant J sh,H enthalpy J mol-1, Jk Boltzmann constant J K-1

k Euler homogeneity –K equilibrium distribution y/x, equilibrium constant –m,M mass (m is used to avoid conflict with Mach number) kg, kgM Mach number –Mw molecular weight kg mol-1

N mole number molNA Avogadro number –p pressure Paq,Q heat J mol-1, Jq,Q partition function –, –r,R intermolecular separation, radius m,mR universal gas constant J mol-1 K-1

s, S entropy J mol-1 K-1, J K-1

t time s

xv

xvi LIST OF SYMBOLS

T temperature Ku,U internal energy J mol-1, Jv,V volume m3 mol-1, m3

w,W work J mol-1, Jx, X Euler homogeneous variable, %x –, –x, y, z mole fractions –, –, –x, z horisontal, vertical space coordinate m, mz compressibility factor pv/RT –

2. Greek symbols

& thermal expansivity v-1 (!v/!T )p K-1

' isothermal compresibility !v-1 (!v/!p)T Pa-1

( heat capacity ratio cp/cv –)i j Kronecker delta (1 if i = j, 0 else) –* energy of same order of magnitude as kT J+ thermodynamic e"ciency W/Qh –,D Debye temperature K- Joule–Thomson expansion coe"cient (!T/!p)h K Pa-1

% scaling factor for Euler homogeneous functions –% Lagrange multiplier –µ chemical potential (!U/!N)S ,V J mol-1

# extent of reaction –# transformed Euler homogeneous variable –. negative thermodynamic pressure (!U/!V)S ,N Pa$ mass density kg m-3

/ hard sphere volume m3

0 thermodynamic temperature (!U/!S )V,N K1 flow velocity m s-1

" Legendre transform J" molecular pair potential J2 fugacity coe"cient exp(µr,p/RT ) –13 thermodynamic speed of sound (/$' m s-1

3. Mathematical symbols (use of)

A-1 inverted matrix#cp average heat capacity cp(T )dU eksakt di!erential of U!Q inexact di!erential of Q#A,BU di!erance UB ! UA between two states A and B

5. PHASE LABELS AND MODEL CONTRIBUTIONS (SUPERSCRIPT) xvii

#fH enthalpy of formation (the standard state must be specified)#rxH enthalpy of reaction (the stoichiometry must be specified)(!U/!S )V partial derivative of U with respect to S at constant Vui partial molar energy (!U/!Ni)T,p,Nj!i

4U (flow) rate of UU available energy (T ! T#)S ! (p ! p#)V + (µ ! µ#)NxT transposed vectorxD diagonalized vector (diagonal matrix)x-D inverted diagonal matrix

4. Thermodynamic states (superscript and subscript)

Note: There are more indices in use than those explained in this tableand the following one. For the most of it this applies to the chemical speciesindices i, j, k which are used in Ni, hi, µi etc. The index k does also find useas an iteration counter. Equally important are state specifications T , p, n, s,etc. which are used to denote partial di!erentiation and also to identify statechanges at constant temperature, pressure, composition etc. Finally, there area few composite symbols cp, cv, ep, ek etc. which are akin to the indexed ones.They are explained in the table of Latin symbols.

# standard H#, p#, h#i , p#i , etc.3 reference p3i , µ3i , etc.3 sonic (choked) flow T3, p3, v3 and 13c vapour–liquid critical point pc, Tc and vc

eq equilibrium condition Ueq, Aeq, etc.r reduced (dimensionless) p/pc, T/Tc and v/vc

sat vapour–liquid saturation point psati

, µsati

, etc.

5. Phase labels and model contributions (superscript)

&, ', ( equilibrium phases2.vır second virial equation2D, 3D two, three dimensionalC-C Clausius–Clapeyron equationDebye Debye equationEinstein Einstein equationelec electronic degree of freedomeos equation of stateex excess (non-ideal) contributionhs hard sphere potentialıd ideal mixture

xviii LIST OF SYMBOLS

ıg ideal gaslıq liquid phaseLJ Lennard–Jones potentialmur Murnaghan equationrad electro-magnetic radiationRK Redlich–Kwong equationrot rotational degree of freedomr,p residual using ideal gas at the same T and p as referencer,v residual using ideal gas at the same T and V as references solid phasesle solid–liquid equilibriumsw square well potentialtrans translational degree of freedomv vapour phaseVdW van der Waals equationvır virial equation of statevıb vibrational degree of freedomvle vapour–liquid equilibrium

6. Process descriptions (subscript)

a adiabatic (combustion) processc cold side in a thermal processe expansion processh hot side in a thermal processırr irreversible process (path)isolated isolated systemc-v applies to the control volume (storage tank, etc.)opt optimum value (minimum or maximum)rev reversible process (path)s shaft worktot total quantity of a system (energy typically)

7. Vectors

! Lagrange multipliers –µ, " chemical potentials J mol-1

# velocity vector m s-1

$ extent of reactions –b atom mole numbers An mole unity vector (1, 1, 1, · · · ) –

8. MATRICES xix

h partial molar enthalpies J mol-1

mw molecular weights kg mol-1

n mole numbers mols partial molar entropies J mol-1 K-1

v partial molar volumes m3 mol-1

x mole fractions (liquid phase) –y mole fractions (vapour phase) –

8. Matrices

A formula matrix (atoms $ species)G,H Hessian matricesN reaction stoichiometry (species $ reactions)

Foreword

This book is intended for PhD students coming from the departments ofchemical and mechanical engineering, material science, physics and chemistryalike—students who want to boost their thermodynamics knowledge and learnabout practical calculations to support their own research activities. The stu-dent should acquire both the theoretical insight which is needed and the nec-essary hands-on training to solve thermodynamic equilibrium problems in ageneric manner. Much emphasis is therefore put on the understanding of ba-sic topics like Euler functions and Legendre transformations, but besides thisa thorough understanding of fluid equations of state, residual functions andequilibrium calculations is also required.

To make the most out of this book it is important that the student has someprior knowledge in multivariable calculus, linear algebra and optimisation the-ory. He and she must also know how to computerise mathematical algorithmsinto modern script languages like Ruby, Matlab, Python, etc.

The study of thermodynamic system theory requires a good understandingof the so-called thermodynamic potentials (say energy functions) and their mu-tual connections springing out from the Legendre transform. Geometrically,the Legendre transform is the intersection between the tangent to the potentialsurface (along the abscissa of one free variable at the time) and the ordinateaxis. This construction is easily explained in words and its mathematical prop-erties are also quite straightforward, but the deeper physical meaning is lackingto many students. It is quite remarkable, then, that the text filling the space be-tween the covers of this book, solely written to shed light on the intricaciesof thermodynamics, can in itself be used to illustrate the Legendre transformwithout any knowlegde of physics.

The basic assumption is that the number of unique words u in a text ap-proaches a finite limit u3 when the total number of words w goes to infinity.This statement follows as a direct consequence from the fact that the Englishlanguage is in itself of finite size. Armed with this insight we can estimatethe size of the hypothetical dictionary, that would have been real if the bookwas infinitely long, by calculating the Legendre transform of y = log(w/u) ver-sus x = log(w). The key to our understanding is that the function graph y(x)has slope 1 in the limit x % & and that ! log(u3) is the intersection with theordinate when y(x) is extrapolated from x = & back to x = 0. But this is ex-actly the geometric definition of the Legendre transform calculated in the limit

xxi

xxii FOREWORD

!y/!x% 1! By analysing each single sentence in the book, and several randomchoices of the chapters, we can achieve a representative sampling of y(x) data.An empirical function is finally fitted to the data set and used for estimating thesize of the dictionary. The figure below shows how the construction looks indetail. The calculation of log(u3) is of course not exact, but it points neverthe-less to a quite extraordinary application of the Legendre transform, freed fromthe burdens of thermodynamic system theory and physical insight.

0

1

2

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x)There is a total number of w = 108634 words in the book. Ofthese u = 6301 are unique. The predicted size of the dic-tionary is given by the Legendre transform of log(w/u)with respect to log(w):

log(u3'14554) = limx%&

$y ! x ·

!y

!x

%

This means that the active vocabu-lary equals 43% of the esti-mated dictionary.

The figure legend tells us that the language used in writing this book is farfrom being saturated — only 43% of the hypothetically estimated dictionaryhas been used in practise which is considered to be a quite small proportion ofthe possibilities there exist. We could perhaps dare to say that there is not so

FOREWORD xxiii

much repetition of words and phrases in this book as it would have been if itwas a rhetoric text of perhaps commercial, political or religous character.

So, with certain reservations as to unforeseeable changes in the author’sattitude regarding pedagogical improvements and scientific endeavours, whichcertainly could impose large changes to the writing style, it is quite safe toclaim that a lot more (and then also di!erent) words might be used to expressthe contents of the current book or, alternatively, in explaining new but not toodistantly related subjects.

What comes as a surprise is the large impact any stipulated increase inthe language’s “saturation level” would mean to the book volume. Let’s sayit was my intention to develop a more verbose writing style and that I there-fore is seeking to increase the ratio of the active vocabulary compared to thesize of the hypothetical dictionary to 50%, 75% or even to 90%. The corre-sponding increase in the amount of written text corresponds to a staggering70000, 560000 or 2290000 additional words or, equivalently, to a 60%, 510%or 2100% increase in the book volume.

These numbers strongly underpins my humble opinion that the reading of asingle book on thermodynamics is not nearly enough to cover the subject in anygreat detail. They show in fact that it may take the reading of 4 – 6 additionalbooks before the sense of déjà vu comes to one’s mind. Take this as an excusefor seeking other literature sources and let it be a leading star in your study ofthermodynamics.

You should also know that thermodynamics is indeed a phenomenologicalscience, but that its postulates are still of the general kind and that the subjectitself is deeply rooted into a rich philosophical tradition. On these terms onemust learn to accept that there is not only one truth to any particular problembut many equivalent truths and that words are important to our understandingof the di!erent shades of reality and abstractions that we shall meet in ourcoming study of the subject.

KP8108 Fall 2004

This book has been prepared for colour printing in the hope that it willmake the material more accessible to the student. The text has also been di-vided into paragraphs in order to make important phrases stand out from themain text. Thus, black-and-white reproduction is tolerable, but only at the costof reduced readability (besides, of course, that the material will of get a some-what dull and conventional look).

The idea of collecting the theory into 163 short paragraphs, rather than intohundreds of pages of verbose text, is an educational hype from the author’sside, intended to make important results and concepts better segmented thanin most textbooks. Carefully sprinkled with margin notes ! in the main textthe segmentation o!ers a good overview of important definitions, observations,derivations and problems belonging to the course.

Each paragraph is made up of a short introductory text, maybe includingsome optional information, which thereafter is given a concise, theoretical, ex-position. It is not easy to be very lucid in an abstract field like thermodynamics,and many of the paragraphs are admittedly quite philosophic. You must there-fore be prepared to work through the course material several times with succes-sively more attention payed to the abstractions rather than to the mathematicaloperations.

§ 0 This is the zeroth out of 163 paragraphs found in the book. It is colouredto stand out from the main text and is given a quite narrow formulation. Not toomany practical issues are brought in at this point because the average studentlacks the requisite training it takes to appreciate any practicalities.

In some cases it is has been necessary to add some extra information rightafter the paragraph. Its purpose is to bring forward interdisciplinary considera-tions that would else blur the understanding of the problem.

Theory. The theoretical exposition has a style somewhat similar to a math-ematical proof. It is given a descriptive keyword and is ended with a QED(from Latin quod erat demonstrandum). Any graphs that are presented as partof the exposition are judiciously selected to emphasise important theoreticalaspects. The associated Matlab code found in Appendix H gives the reader anice opportunity to re-examine all the calculations at his or her wish.

xxv

xxvi KP8108 FALL 2004

There are a few mathematical cases, proofs, etc. of special importance thatare also emphasised by a Keyword and " . Whether they are given a specificcolour or not depends on the colour balance of the page. As a rule of thumb thecolours used in this book i.e. exercisecolor, proofcolor and bookcolor shouldbalance the visual impression of the text.

The syllabus for Advanced Thermodynamics KP8108 is simply ”read thisbook”. Since it is a PhD course primarily intended for students coming fromthe Department of Chemical Engineering it is to a certain extent directed to-wards the study of fluid phases. It should be stressed, however, that the prin-ciples of thermodynamics show no limits in this respect and consequently thatstudents coming from material sciences, mechanical engineering, chemistryand physics are equally welcome to participate.

Over the years I have also had a number of M.Sc. students following (parts)of the course with good results. It is required that they know (well) how to dealwith energy functions (internal energy, enthalpy, Gibbs energy and Helmholtzenergy), and that they are familiar with the role of chemical potentials in phaseand reaction equilibrium theory. A certain knowledge about partial di!erentia-tion, exact di!erentials, linear algebra and optimisation theory is also necessary.The same comment applies to the prior experience from computer program-ming in Matlab, Maple, Python, etc.

The examination in KP8108 takes place in the form of a written essay (PhDstudents) or oral questioning (M.Sc. students). The essay should be given aformat, length and thoroughness that makes it equivalent to a chapter or anappendix in a PhD thesis on the same topic. To reach this goal, which hasproved successful in most cases, it is required that each student has a projectof keen interest. It is impossible for me to suggest good projects for everyoneso I will therefore ask the students to identify private projects in collaborationwith their supervisors (and myself). Typical subjects over the last years havebeen: Equation of state implementation, theoretical equilibrium considerations,algorithms for equilibrium calculations, and model development. All-in-all thishas been a quite fruitful approach — the students have interesting subjects toconcentrate on and I get the challenge to learn something new!

KP8108 FALL 2004 xxvii

“Eine Theorie ist desto eindrucksvoller, je grösser die Einfachheit ihrerPrämissen ist, je verschiedenartigere Dinge sie verknüpft, und je weiterihr Anwendungsbereich ist. Deshalb der tiefe Eindruck, den die klassis-che Thermodynamik auf mich machte. Es ist die einzige physikalischeTheorie allgemeinen Inhaltes, von der ich überzeugt bin, dass sie im Rah-men der Anwendbarkeit ihrer Grundbegri!e niemals umgestossen wer-den wird (zur besonderen Beachtung der grundsätzlichen Skeptiker).”

“A theory is the more impressive the greater the simplicity of its premises,the more di!erent kinds of things it relates, and the more extended itsarea of applicability. Hence the deep impression that classical thermody-namics made upon me. It is the only physical theory of universal contentwhich I am convinced that, within the framework of applicability of itsbasic concepts, it will never be overthrown (for the special attention ofthose who are skeptics on principle).”

— Albert Einsteina

xxvi

a Albert Einstein: Philosopher–Scientist, The library of Living Philosophers, Vol. VII, OpenCourt Publishing Company, La Salle Illinois (1970) p. 32 [translated and edited by Paul ArthurSchilpp].

xxviii KP8108 FALL 2004

Norway spruce (Picea abies) at sunset, Trondheim (Norway) 2006.

CHAPTER 1

Thermodynamic concepts

163 Workouts in Thermodynamics, © 2001, 2010 – 2013 Tore Haug-Warberg

Language shapes our ability to think independently and to communicateand collaborate with other people. It has enabled us to go far beyond simplymeeting our basic needs associated with survival and social interaction. Ourability to observe, describe and record our thoughts about physical phenomenais vital to a subject like thermodynamics. For us it is particularly importantto reach a common physical understanding of abstract concepts such as en-ergy and entropya, but as languages are always evolving, the premises for ashared conception are constantly shifting. All authors have to contend withthat dilemma, and few of us have the good fortune to be writing for futuregenerations. The benefit of writing about a phenomenological subject such asthermodynamics is that as long as the observed phenomena remain unchanged,the subject will endure.

The challenge you face with a classical subject is combining the preserva-tion of an intellectual heritage with the injection of new ideas, and it is ratherutopian to believe that the latter is possible in our case. Since the rise of thermo-dynamics over the period 1840–1880, the original beliefs have been replacedby a more (ope)rational interpretation. Meanwhile, the content has become sowell-established and thoroughly tested that there is little room for innovation,although there is still scope to inject fresh life into old ideas.

Our understanding of thermodynamics depends on some vital, and pre-cisely defined, concepts that form part of a timeless vocabulary. Accurate useof language improves our insight into the subject, thereby reducing the riskof misunderstandings. However, an exaggerated emphasis on precision mayoverwhelm readers and thus be counterproductive. In this chapter we will re-vise and explain the key concepts in thermodynamic analysisb,c in accurate, butnevertheless informal, terms. Some of the concepts are concrete, whilst othersare abstract, and it is by no means easy to understand them on a first read-through. However, the aim is to convince readers that a good grasp of the basicconcepts is within reach, and that such a grasp is a prerequisite for masteringthe remaining chapters of the book.

a As illustrated by Professor Gustav Lorentzen’s article: Bør Stortinget oppheve 2. hovedset-

ning? (Should the Norwegian parliament abolish the second law of thermodynamics?), In-geniør-nytt, 23(72), 1987 — written as a contribution to the debate on thermal power stationsin Norway. b James A. Beattie and Irwin Oppenheim. Principles of Thermodynamics. Else-vier, 1979. c Rubin Battino, Laurence E. Strong, and Scott E. Wood. J. Chem. Eng. Educ.,74(3):304–305, mar 1997. 1

2 1. THERMODYNAMIC CONCEPTS

Thermodynamics describes natural phenomena in idealised terms. Thescope of the description can vary, and must be adapted to our needs at anygiven time. The more details we want to understand, the more information wemust include in the description. The abstract concept of a “system” is at theheart of our understanding. A thermodynamic system simplifies physical real-ity into a mathematical model. We can then use the model to perform thoughtexperiments that reveal certain characteristics of the system’s behaviour. Notethat the physical size and shape of the system is quite irrelevant — all simplethermodynamic systems have uniform properties and can always be describedby a single set of state variables regardless of the actual geometry. That is anabsolute principle.

syst

em

boundary

system

environment

§ 1 Explain in your own words what is meant bythe concepts: system, boundary and environment orsurroundings.

System. A system is a limited part of the uni-verse with a boundary that may be a mathematicalsurface of zero thickness, or a physical barrier be-tween it and the surroundingsa. A reservoir is asystem that can interact with other systems withoutundergoing any change in its state variables. Ourphysical surroundings such as the air, sea, lakes andbedrock represent unlimited reservoirs for individual human beings, but notfor the entire world populationb. Similarly, a hydropower reservoir represents athermodynamic reservoir for the turbine, but not for the power company. Thismay appear trivial, but it is still worth trying to justify the above statements toyourself or a fellow student. These reservoirs, and other simple systems, haveproperties that are spatially and directionally uniform (the system is homoge-neous with isotropic properties). A real-world system will never be perfectlyuniform, nor completely una!ected by its surroundings, but it is neverthelesspossible to make some useful simplifying assumptions. Thus, a closed system,in contrast to an open one, is unable to exchange matter with its surroundings.An isolated system can neither exchange matter nor energy. Note that the termcontrol volume is used synonymously with open system. The boundary is then

a We must be able to either actively control the system’s mass and composition, impulse (orvolume) and energy, or passively observe and calculate the conserved quantities at all times.All transport properties $ must satisfy limx%0+ $ = limx%0! $ so that $ is the same for the sys-tem and the surroundings across the control surface. In physics, this is called a continuumdescription. The exception to this is the abrupt change in state variables observed across a shockfront. Well-known examples of this include high-explosive detonations and the instantaneousboiling of superheated liquids. It is important to note that the concept of phase boundary inParagraph 4 does not satisfy the continuity requirement and is therefore unsuitable as a controlsurface. b The first proof that the oceans were polluted by man came from Thor Heyerdahl’sRa II expedition in 1970.

1. THERMODYNAMIC CONCEPTS 3

called a control surface. In this context an adiabatic boundary is equivalent toa perfect insulator, and a diabatic boundary is equivalent to a perfect conductor.

The state of a thermodynamic system is determined by the system’s prop-erties — and vice versa. The logic is circular, but experimentally the state isdefined when all of the thermodynamic properties of the system have beenmeasured. For example, the energy of the system is given by the formulaU = (!U/!X1) X1 + (!U/!X2) X2 + · · · once the state variables Xi and thederived properties (!U/!Xi) are known. This determines the value of U in onespecific state. If we need to know the value of U in a number of di!erent states,it is more e"cient to base our calculations on a mathematical model. (!U/!Xi)can then be calculated from the function U(X1, X2, . . .) through partial di!eren-tiation with respect to Xi. The measurements are no longer directly visible tous, but are instead encoded in the shape of model parameters (that describe theobservations to some degree at least). Normally it is only the first and secordderivatives of the model that are relevant for our calculations, but the study ofso-called critical points requires derivatives up to the third or maybe even thefifth order. Such calculations put great demands on the physical foundation ofthe model.

State variables only relate to the current state of the system. In other words,they are independent of the path taken by the system to reach that state. Thenumber of state variables may vary, but for each (independent) interaction thatexists between the system and its surroundings there must be one associatedstate variable. In the simplest case, there are C + 2 such variables, where C isthe number of independent chemical components and the number 2 representstemperature and pressure. The situation determines the number of chemicalcomposition variables required. For example, natural fresh water can be de-scribed using C = 1 component (gross chemical formula H2O) in a normalsteam boiler, or using C = 5 components involving 9 chemical compounds inan isotope enrichment planta.

The state of the system can be changed through what we call a process.Terms such as isothermal, isobaric, isochoric, isentropic, isenthalpic, isopiesticand isotonic are often used to describe simple physical processes. Without con-sidering how to bring about these changes in practice, the terms refer to variousstate variables that are kept constant, such that the state change takes place atconstant temperature, pressure, volume, entropy, enthalpy, (vapour) pressure orosmotic pressure respectively. Generically the lines that describe all of theseprocesses are sometimes called isopleths. Another generic expression, used insituations where the properties of ideal gases are of prime importance (aerody-namics, combustion, detonation, and compression) is the polytropic equationof state p2/p1 = (V1/V2)(. Here ( ( R can take values such as 0, 1 and cp/cv,

a Naturally occurring water consists of a chemical equilibrium mixture of 1 H 1 H 16 O+2 H 2 H 16 O = 2(1 H 2 H 16 O) and others, formed by the isotopes 1 H, 2 H, 16 O, 17 O and 18 O.


which respectively describe the isobaric, isothermal and isentropic transforma-tion of the gas.

§ 2 Explain in your own words what is meant by the terms: state, property,process and path.

The state concept. A thermodynamic state is only fully defined once all ofthe relevant thermodynamic properties are known. In this context, a thermody-namic property is synonymous with a state variable that is independent of thepath taken by the systema. In simple systems, the properties are (by definition)independent of location and direction, but correctly identifying the system’sstate variables is nevertheless one of the main challenges in thermodynamics.One aid in revealing the properties of the system is the process, which is usedto describe the change that takes place along a given path from one state to an-other. In this context, a path is a complete description of the history of the pro-cess, or of the sequence of state changes, if you like. Thermodynamic changesalways take time and the paths are therefore time-dependent, but for a steady(flow) process the time is unimportant and the path is reduced to a static statedescription of the input and output states. The same simplification applies to aprocess that has unlimited time at its disposal. The final state is then the equi-librium state, which is what is of prime concern in thermodynamic analysis.

y

x

A cycle is the same as a closed path. The cyclecan either be temporal (a periodic process) or spa-tial (a cyclic process). This choice greatly a!ectsthe system description. In a steady state, the vari-ables do not change with time, whereas in dynamicsystems they change over time. Between these twoextremes you have a quasi-static state: the state changesas a function of time, but in such a way that the sys-tem is at all times in thermodynamic equilibrium, asdescribed in greater detail in Paragraph 4. A state

that is in thermodynamic equilibrium appears static at the macroscopic level,because we only observe the average properties of a large number of particles,but it is nontheless dynamic at the molecular level. This means that we mustreformulate the equilibrium principle when the system is small, i.e. when thenumber of particles n % 0. At this extreme, intensive properties such as tem-perature, pressure and chemical potential will become statistical variables withsome degree of uncertainty.

a In simple systems, viscosity, conductivity and di!usivity also depend on the current thermo-dynamic state, in contrast to rheology, which describes the flow properties of a system withmemory. Two examples of the latter are the plastic deformation of metals and the viscosity ofpaint and other thixotropic liquids.


1±5 kg

1 kg+

!

In a theoretical reversible process, it is possibleto reverse any change of state by making a smallchange to the system’s interaction with its surround-ings. Thus, two equal weights connected by a stringrunning over a frictionless pulley will undergo a re-versible change of state if the surroundings add aninfinitely small mass to one of the weights. This isan idealised model that does not apply in practice toan irreversible system, where a finite force will berequired to reverse the process. Let us assume, for

example, that the movement creates friction in the pulley. You will then need asmall, but measurable, change in mass in order to set the weights in motion oneway or the other. The total energy of the system is conserved, but the mechan-ical energy is converted into internal (thermal) energy in the process — so it isirreversible. If the friction is suddenly eliminated the weights will accelerate.This change is neither reversible nor irreversible. Instead, it is referred to aslossless, which means that the mechanical energy is conserved without it nec-essarily being possible to reverse the process. For that to happen, the directionof the gravitational field would also need to be reversed.

In thermodynamics there are usually many (N > 2) state variables, as wellas an infinite number of derived properties (partial derivatives). Experimen-tally it has been shown that the size of a system is proportional to some of itsproperties. These are called extensive properties, and include volume, mass,energy, entropy, etc. Intensive properties, meanwhile, are independent of thesize of the system, and include temperature, pressure and chemical potential.Mathematically these properties are defined by Euler functions of the 1st and0th degree respectively, as described in the separate chapter on the topic. Thereare also properties that behave di!erently. For example, if you increase the ra-dius of a sphere, its surface area and volume increase by r2 and r3 respectively.This contrasts with the circumference, which increases linearly with the radius(it is extensive), and the ratio between the circumference and the radius, whichalways remains 2. (it is intensive).

The system’s mass is a fundamental quantity, which is closely related toinertia, acceleration and energy. An alternative way of measuring mass is bylooking at the number of moles of the various chemical compounds that makeup the system. A mol is defined as the number of atomsa which constituteexactly 0.012 kg of the carbon isotope 12 C, generally known as the Avogadroconstant NA = 6.022136(7)1023 . In this context, molality and molarity are

a A mole is defined in terms of the prototype kilogramme in Paris and not vice-versa. Atomicmass is measured in atomic mass units, where the mass of one atom of the 12 C isotope isdefined as 12 amu. The 1 amu was originally defined as being equal in mass to one hydrogenatom, which deviates a little from the current definition. The deviation is due to a di!erence innuclear energy between the two nuclei and is related to Einstein’s relation E = #mc2.


two measures of concentration stated in moles per kilogramme of solvent andmoles per litre of solution respectively. These measures are not of fundamentalimportance to thermodynamics, but they are important concepts in physicalchemistry, and are sometimes used in thermodynamic modelling.

Systems of variable mass contain a minimum number of independent com-ponents that together make up the chemical composition of the system. A com-ponent should here be considered a degree of freedom in Gibbs’ phase ruleF = C + 2 ! P, where C is the number of components and P is the number ofphases at equilibrium (see Paragraph 4). Only the composition at equilibriumcan be described in these simple terms, which is because we are forced to spec-ify as few as possible variables that depend on mass or the number of moles.This restriction influences our choice of components, and in practice the ques-tion is whether chemical reactions are taking place in the system, although theconventions of the applied discipline are equally important. Generally compo-nents are selected from the system’s chemical constituents or species, or fromits reactants, whereas for electrolytes and salt mixtures it is natural to use theions that make up the system’s chemical composition as componentsa.

§ 3 Explain in your own words what is meant by a property being intensiveor extensive. If you divide an extensive property by the number of moles in thesystem or its mass, you get a molar or specific property respectively. Show thatthe property obtained is intensive.

Size. In thermodynamics, size is not only a measure of the volume of asystem, but also of any properties related to its mass. This implies that twosystems with identical state descriptions become a system of double the sizewhen combined. Properties that can be doubled in this way, such as entropy S ,volume V and the number of moles N, are proportional to the size of the system,and are referred to as extensive variables. This means that all of the extensivevariables must be increased by the same factor. You cannot simply double thevolume while keeping the number of moles constant — entropy, volume andthe number of moles must all be doubled together. At the same time, otherderived extensive properties like energy, total heat capacity, etc. are scaled thesame amount.

Another group of properties is not a!ected by any change in the size of thesystem. These properties are referred to as intensiveb properties. Well-known

a Components do not necessarily represent physical constituents. One example is ternary re-cipocral systems (salt mixtures) of the type NaCl+KBr = NaBr+KCl, which have 4 possiblesubstances (salts), but only 3 independent components. For example, the salt NaCl can be de-scribed by the vector (1, 0, 0, 0), or (0,!1, 1, 1), or any linear combination of these two vectors.This illustrates the use of a seemingly non-physical (in this case negative) number of moles forone of the components. The specification does make sense, however, because the true amountsof each of the ions in the mixture are non-negative, and hence real physical quantitites. b Thevague definitions of extensive and intensive used here will later be replaced by ones involvingEuler’s homogeneous functions of the 1st and 0th degree respectively.


examples include temperature T , pressure p and chemical potential µ. Certainpairs of extensive and intensive variables combine to form a product with acommon unit (most commonly energy), and feature in important relationshipssuch as U = TS ! pV+µ1N1+µ2N2+ · · · . These pairs of T and S , p and V , andµi and Ni are called conjugate variables. A mechanical analogy is the flow rate$4V (extensive) and the gravitational potential g#z (intensive) at a hydropowerstation. The power generated by the turbine can be written g#z$4V , which isthe product of an intensive and extensive variable. Equivalent analogies can befound in electrical and mechanical engineering.

Dividing one extensive property by another gives you a new, intensive vari-able that represents the ratio between the two properties. Let e.g. f = axand g = bx be two extensive properties expressed as functions of x. Then$ = f /g = a/b is independent of x, in other words $ is intensive. These kindsof variables are widely used to describe systems in a way that does not referto their size, but many fields have di!erent practices, and it is often unclearwhether the definition is being expressed on a mass or mole basis (which is themost common source of misunderstanding). The di!erence between a specificand a molar quantity is that one is expressed per kilogramme and the other permole of the substance (or mixture). Common examples include specific andmolar heat capacity, specific and molar volume, etc.

Let us consider a thermodynamic system that does not change with time,which means that it must be in equilibrium. This minimises the degrees offreedom we have to specify. At high temperature chemical equilibrium, forexample, it is su"cient to state the quantity of each of the atoms present. Thedistribution of the atoms amongst the substances in the mixture is determinedby the principle of equilibrium (see below) and the system’s equation of state.In the case of phase equilibria, the chemical substances are similarly distributedacross the system’s phase boundaries and it is su"cient to specify the totalcomposition of the entire system. As a rule of thumb these problems are easy tospecifiy, but they do nevertheless require numerical solution by iteration whichin many cases is a challenging task. If the system has no internal degreesof freedom in the form of chemical reactions or phase transformations, theequilibrium state will be determined by an ordinary (but multivariate) function,generally without iteration.

The general principles of equilibrium mean that the energy of the systemwill be minimised with respect to all the degrees of freedom that form the basisfor the system description. Alternatively the entropy is maximised with respectto its system variables. The degrees of freedom are at all times controlled bythe physical nature of the system, and this determines which extremal principleto apply. The theorectical foundations for this topic are discussed in a laterchapter on Legendre transformations.

True equilibrium does not assert itself instantaneously, and from our per-


spective it is not possible to judge whether equilibrium can be attained, orwhether the system has kinetic limitations that prevent this. All that thermody-namics does is describe the state of the system at equilibrium, and not how longthe process takes. To describe in detail how the system changes over time weneed an understanding of kinetics and transport theory. Kinetics is the study offorces and the motion of bodies, whilst reaction kinetics looks specifically atrates of chemical reactions and phase transitions. Transport theory is a concepttaken from nonequilibrium thermodynamics that can be used to describe thechanges that take place in a system until it reaches equilibrium. As a generalrule, all transport problems must be formulated as partial di!erential equations,whereas thermodynamic equilibrium problems can always be expressed usingalgebraic equationsa. This di!erence in mathematical treatment reflects thegradients in the system. If they are not significant, it is su"cient to determineone representative value for each of the scalar fields of temperature, pressureand chemical potential. This distinguishes a thermodynamic problem from atransport problem, where the scalar fields must be determined simultaneouslythroughout the space.

syst

em

boundary

environment

phase b’ndary&

'(

§ 4 Explain in your own words the followingterms associated with equilibrium and equilibriumstates: phase, phase boundary, aggregate state, equi-librium and stability.

Equilibrium. A phase is defined as being a ho-mogeneous, macroscopic subsystem separated fromthe rest of the system by a phase boundary. It mustbe possible to separate a phase from other phasesthrough mechanical means alone. This is an impor-tant prerequisite. The system’s equilibrium phasesare often designated by the Greek letters &, ', (, etc. The entropy density,energy density and mass density are constant within the phases, but they varydiscontinuously across the phase boundaries. Temperature, pressure and chem-ical potential, meanwhile, are constant throughout the entire system (assumingequilibrium). Spatially a phase can be discretely distributed across the avail-able volume, cf. fog particles in air and drops of fat in milk. If the wholesystem only consists of one phase, it is said to be homogeneous. Otherwise, itis heterogenous. A phase is referred to as incompressible if the volume is de-pendent on the pressure, but in reality all phases are compressible to a greateror lesser extent (particularly gases). In principle a phase has two possible statesof matter: crystalline and non-crystalline (glass and fluids). For practical pur-poses a fluid may sometimes be referred to as a gas without a specific volume,

a This is the di!erence between distributed and lumped description.


a liquid with surface tension, or an electrically conductive plasma, but in ther-modynamics there is a gradual transition between these terms, and there are nostrict criteria as to what falls within which category.

stable

metastable

unstable

Equilibrium is the state attained by the system whent % &. If the system returns to the same equilibriumstate after exposure to a large random perturbation (distur-bance), the equilibrium is said to be stable. A metastableequilibrium is stable when exposed to minor perturbations,but it becomes unstable in the event of major displace-ments. An unstable equilibrium is a mathematical limitcase. It describes a material state that breaks down if ex-posed to infinitely small perturbations. It is impossible tophysically create this kind of equilibrium, but it is never-theless very important from a theoretical point of view, asit sets a limit on what can physically exist based on simple

laws of physics and mathematics. If the stability limit is exceeded, the systemwill split into two or more equilibrium phases (from the mechanical analogy inthe figure it is equally likely that the ball will fall to the right as to the left).

The assumptions and limitations set out in this chapter, and in the book asa whole, in relation to thermodynamic system analysis, are su"cient for thepurposes of analysing the e!ects of a large number of changes of state, butthey do not tell us anything about how the system relates to its surroundings.In order to analyse that, we must introduce two further terms, namely work Wand heat Q. These two properties control the system’s path as a function ofits physical relationship to its surroundings. On the one hand you have the ab-stract analysis of the system, which deals with state functions and mathematicalformulae, and on the other you have an equally idealised interpretation of thesurroundings. Within this context, heat and work are defined as two di!erentmodes of exchanging energy. It is important to note that heat and work are notstate variables of the system: they simply describe two di!erent mechanismsfor exchanging energy between the system and its surroundings.

State variables are variables that form part of a state function. A state func-tion always produces an exact di!erential, but not all di!erentials in physicsare exacta. One example from thermodynamics is (dU)n = !Q ! !W, whichdescribes the energy balance for a closed system. Here the energy U is a statefunction with the total di!erential (dU)n. For any change !Q ! !W there is a

a Let e.g. f (x, y) = xy + c be a state function with x and y as its state variables. Hered f = y dx + x dy is the total di!erential of the function. The right side of the equation isthen called exact. If as a pure thought experiment we change the plus operator on the rightside to minus, the di!erential becomes non-exact. It is possible to transform the left-hand sideinto y2 dg = y!x ! x!y, where y2 is an integrating factor for the di!erential and dg is the totaldi!erential of g(x, y) = xy-1 + c, but the new di!erential y !x ! x !y remains non-exact, since itcannot be expressed as the total di!erential of any known function.


unique value of (dU)n. However, the reverse is not true. For any change (dU)n

there are in principle an infinite number of combinations of !Q and !W, sinceonly the di!erence between heat and work is observable.

!W

!Q

dU=!

Q! !W

§ 5 Explain in your own words the meaning ofthe terms: heat, work and energy. Are all three statevariables?

Heat and work. Heat and work are two closelyrelated mechanisms for transporting energy betweenthe system and its surroundings. Transporting en-ergy a!ects the state of the system, but the heat andwork are not themselves accumulated in the system.Work involves moving a macroscopic mass, or ele-mentary particles, against an external force. A mov-ing piston, a rotating axle, electrons in an electric circuit and water flowingthrough a turbine are all examples of this. Heat results from large numbersof random, microscopic movements that do not result in any net movement ofmass. For heat to be converted into work, the microscopic movements mustfirst be coordinated. The 2nd law of thermodynamics then dictates that someof the heat will be lost to a thermal reservoir of the same temperature as thesurroundings. The term energy dissipation is used to emphasise the fact thatspontaneous processes always result in a reduction in available energy. Theconvention is for work done and heat supplied to be expressed as positive val-ues. Individually, neither heat nor work are state variables, but the di!erencebetween them gives us the change in the system’s energy, which is a state func-tion. In other words, any given change in energy can be produced in an infinitenumber of ways by varying the contributions made by heat and work.

Having defined the fundamental concepts, what we now must do is turntheory into practice. That will require a good understanding of the physicsinvolved in the problems that we will be looking at, some mathematics and inparticular di!erential equations and linear algebra, and a selection of relevantdescriptions of substances, known as equations of state. Last but not least, weneed to know the purpose of our analysis. It is often based in a wish to finda simple model to explain the changes in the system’s state. Thermodynamicsis, in short, a subject that combines most of the things taught in undergraduatechemistry, physics and mathematics at the university.

However, it is somewhat optimistic to believe that mechanical engineering,thermodynamics, electromagnetism and other calculation-heavy subjects canfully describe the world we live in, and that we are in a position to decidehow detailed an answer we want. Mathematics provides us with a useful tool,but that does not mean that models and reality are two sides of the same coin,


and hence that we, by carefully eliminating “all” assumptions, can reach anabsolutely true answer. Mathematical descriptions allow us to understand somecharacteristic events that surround us, but they do not give us the whole picture.

Duck or rabbit ?

Immanuel Kanta unified the most important strandsof rationalism and empiricism with his interpreta-tion of das Ding an sich. He stressed that we cannever be certain about the intrinsic, true nature ofanything, and that our understanding of the worldis limited by our subjective experiences in time andspace. Science is based on observable events takingplace in a world where time and space are assumedto be independent of us, but from a philosophicalpoint of view Kant argues that we may be unable to

see all the aspects of whatever we are observing. For example, does the pictureon the left represent a duck or a rabbit? With a bit of imagination we can spotboth animals, but we can only see one of them at any given time. The fullpicture is not available to us, and at first glance we don’t even realise that thereare two possibilitiesb.

32 J mol-1

h

x

#rxh or #mıxh ?

!

!

!!

! !

!

!

!

Thermodynamics is really a subject that describesdas Ding für unsc as opposed to das Ding an sich.The figure on the right illustrates this distinction ina rather subtle manner. The solid circles show calo-metric readings for the mixing enthalpy #mıxh of thesystem H2O–D2Od at varying compositions (molefractions x) of the two compounds. The continuousline has been fitted using the semi-empiric modelax(1 ! x), which gives a first-order approximationof #mıxh. The line fits the data points very well, andwe can therefore conclude that this simple model adequately represents thereadings taken. This posture is, in simple terms, what is meant by das Dingfür uns.

Based on our understanding of nature, isotopes are chemically identical,but in this case the mixing enthalpy measured is equivalent to a fall in temper-ature of 0.43 K for an equimolar mixture of the two isotopes, which is muchgreater than expected. The reason for this discrepancy is that the two com-ponents react endothermically to form HDO. The net reaction involves theprotonation of H2O " H+ +OH! and the deuteronation of D2O " D+ +OD!

which are fast reactions. The result is that H2O+D2O " 2 HDO reaches

a Immanuel Kant, 1724–1804. German philosopher and logician. b At first glance, aroundhalf of the population will see the duck, whilst the other half will see the rabbit. c Referred toas Erscheinung in Kant’s thesis. d D. V. Fenby and A. Chand. Aust. J. Chem., 31(2):241–245,1978.


equilibriuma in a virtually ideal mixture. The last statement appears to be self-contradictory, as the components react chemically at the same time as they takepart in an ideal equilibrium. This apparent contradiction arises because of theunclear boundary between a chemical reaction and a physical interaction. Fora thin gas there is an explanation for this, but for a liquid there is no entirelysatisfactory definition. The reaction product HDO is not stable either since itdecomposes instantaneously to H2O and D2O, which prevents us from observ-ing the substance in its pure state. Our understanding of nature is, in otherwords, based on evidence derived from indirect observations and mathematicalmodels of the physical properties in the water phase. In reality all scientificknowledge is based on theoretical models of one kind or the other, and there-fore does not imply that we have any exact understanding of das Ding an sich.

The foundations of phenomenological thermodynamics are too weak tostate hard facts about the true nature of the systems it describes, but strangelyenough thermodynamic theory can still be used to falsify claims that break thelaws upon which it is foundedb.

The theoretical basis is also su"cient for deriving important relationshipsbetween the state variables, but it does not constitute independent evidencein the mathematical sense. Thermodynamic analysis is capable of confirmingprior assumptions, or of demonstrating new relationships between existing re-sults, but the calculations are not necessarily correct even if the model appearsto correspond with reality. Firstly, the result is limited to the sample spacefor the analysis and to the underlying assumptions and secondly, there may beseveral equally good explanations for a single phenomenon (as shown in theenthalpy example above).

It is also worth remembering that even a small one-component system hasmaybe 1015–1020 microscopic degrees of freedom that are modelled with onlythree thermodynamic state variablesc. This means that a significant amount ofinformation about the system is lost along the way. It is therefore necessaryto develop our ability to recognise what is important for the modelling, so thatwe can perform the right calculations, rather than trying to look (in vain) forthe very accurate description. In keeping with that principle, we will mainly

a In the system CD3OD–CH3OD, covalent bonds form between C–D and C–H, preventingany isotopic reaction, meaning that the change in enthalpy is only 0.79 J mol-1 or 0.01 K; cf.T. Kimura et al. J. Therm. Anal. Cal., 64:231–241, 2001, shown as open circles in the figure.Subsituting OD to OH changes the chemistry to CD3OD–CH3OH which again allows for pro-ton–deuteron exchange and the equimolar change in enthalpy increases to an intermediate valueof 8.4 J mol-1 or 0.1 K. b The perpetuum mobile is the most famous example of this. In Norwayit is in fact impossible to apply for patent protection for a perpetual motion machine, cf. the Nor-wegian Industrial Property O"ce’s Guidelines for processing patent applications. The online(2010) version of Part C: Preliminary examination; Chapter II Contents of the patent applica-

tion, except requirements; 3.3.6 Insu!cient clarity excludes inventions that “. . . are inherentlyimpossible to produce, as doing so would require accepted physical laws to be broken — thisapplies e.g. to perpetual motion machines.” c Systems with many components are obviouslyeven more complex.


be looking at simple models such as ideal gas and the van der Waals equation,but that does mean that we will be very precise in our analysis of the ones thatwe do look at.

13


Bullfinch (Pyrrhula pyrrhula),© 2008 Jon Østeng Hov.

CHAPTER 2

Prelude

163 Workouts in Thermodynamics, © 2001 Tore Haug-Warberg

(2.1) pıg = NRTV.

Figure 2.1 The pressure of agas is the same in all direc-tions and on all surfaces —

inner and outer alike.

dx

dyy

y + #y

x x + #xFigure 2.2 The geometricinterpretation of the di!er-

ential of a function.

(2.2) dln pıg = dln N + dln R + dln T ! dln V .

(dln pıg)R = dln N + dln T ! dln V ,

(dln pıg)N,T = ! dln V

1. State functions

(2.3)!!ci

!x j

"xk! j

=!!c j

!xi

"xk!i

!!2 f!xi!x j

"xk!i, j

=!!2 f!x j!xi

"xk!i, j

,

x&

x#

d f ='i

xi&

x#i

ci dxi ,(2.4)

(d f = 0 ,(2.5)

f ! f# =x&

x#

d f .(2.6)

15

16 2. PRELUDE

Maxwella relation.!!T!V

"S ,N=!!!p!S

"V,N,

!!µ!V

"S ,N=!!!p!N

"S ,V,

!!µ!S

"V,N=!!T!N

"S ,V.

State function.

Cycle.

Reference point energy.

2. Mathematical operators

§ 6

Summation.

§ 7

Differential I.

dp =!!p!T

"V,N1...Nn

dT +!!p!V

"T,N1...Nn

dV +n'

i=1

!!p!Ni

"T,V,Nj!i

dNi

=!!p!T

"V,n

dT +!!p!V

"T,n

dV +n'

i=1

!!p!Ni

"T,V,Nj!i

dNi ,(2.7)

(2.8) dpıg = NRV dT ! NRT

V2 dV +n'

i=1

RTV dNi

dp = pT dT + pV dV +n'

i=1pi dNi ,

§ 8

a James Clerk Maxwell, 1831–1879. Scottish physicist.

2. MATHEMATICAL OPERATORS 17

Exact differential.

)!(NRT/V2)!T

*V,n

?=)!(NR/V)!V

*T,n

) ! NRV2 = ! NR

V2

)!(RT/V)!T

*V,n

?=)!(NR/V)!Ni

*T,V,Nj!i

) RV= R

V...

...

§ 9

Implicit differentiation.

(2.9)!dV ıg

dT

"p,n= V

T "!!V ıg

!T

"p,n= NR

p .

§ 10

Partial molarity I.

(2.10) (dV)T,p =n'

i=1

!!V!Ni

"T,p,Nj!i

dNi =n'

i=1vi dNi .

§ 11

Avogadroas law.

(2.11) ! NRTV2 (dV ıg)T,p +

RTV

(dN)T,p = 0

!dV ıg

dN

"T,p= V

N =RTp ,

§ 12

Kroneckerb delta.

(2.12) gi j =)! ln(Ni/N)!Nj

*Nk! j

= NNi

!)i j N!Ni

N2

"=)i j

Ni! 1

N .

18 2. PRELUDE

G = {gi j} =

+,,,,,,,,,,,,,,,,,-

1N1

0 · · · 00 1

N2· · · 0

....... . .

...0 0 · · · 1

Nn

./////////////////0! 1

N

+,,,,,,,,,,,,,,,-

1 1 · · · 11 1 · · · 1....... . .

...1 1 · · · 1

.///////////////0

= n-D ! N-1eeT ,(2.13)

§ 13

Vector notation.

!!2G!nT!n

"T,p=

n'i=1

!!2G!Ni!Ni

"T,p,

§ 14

Figure 2.3 A thermody-namic cycle in T, p co-ordinates. The theoreti-cal interpretation of cy-cles becomes more vis-ible in Chapter 17 deal-ing with heat engines.

p

T

D

A B

C

Finite difference.

§ 15

State functions II.

#A,BV ıg =VB&

VA

(dV ıg)n =TB&

TA

NRp dT !

pB&

pA

NRTp2 dp

= NRTp

1111pB

pA

= NRTA,B$ 1

pB! 1

pA

%.(2.14)


#B,CV ıg = NRpB,C

(TC ! TB) ,

#C,DV ıg = NRTC,D$ 1

pD! 1

pC

%,

#D,AV ıg = NRpD,A

(TA ! TD) .(2.15)

(2.16)(

(dV ıg)n = #A,BV ıg + #B,CV ıg + #C,DV ıg + #D,AV ıg = 0 .

§ 16

dU = T dS ! p dV +n'

i=1µi dNi ,

dS =CP

T dT !!!V!T

"p,n

dp +n'

i=1si dNi ,

Equation of state III.

(dU)n = T2

CP

T dT !!!V!T

"p,n

dp3! p dV

= T2

CP

TdT !

!!V!T

"p,n

dp3! p2!!V!T

"p,n

dT +!!V!p

"T,n

dp3.(2.17)

(dU ıg)n = CıgP

dT ! T NRp dp ! p NR

p dT + p NRTp2 dp

= (CıgP! NR) dT

= CıgV

dT .(2.18)

(2.19)(

(dU ıg)n =(

CıgV

dT =B&

A

CıgV

dT !C&

A

CıgV

dT = 0 .

§ 17

Differential form.

(2.20) (dU)n = !Q ! !W ,

20 2. PRELUDE

p

T

B

A,C

(dU = 0

W

Q impossible path

C B

A

!Q ! !W = 0

Figure 2.4 An isobaric cycle drawn side-by-side in T, p and Q,Wco-ordinates. In the right figure the path from C to A must followthe hatched diagonal. Drawn otherwise the state denoted A,C wouldvary along the path which is in contradiction to the assumption madein the left figure that A and C are really representing the same state.

(2.21) (dU)n = 4Q(t, x, T, p, n) dtmicroscopic work

! 4W(t, x, T, p, n) dtmacroscopic work

.

Integral form.

(2.22)B&

A

(dU)n =Q&

0!Q !

W&

0!W =

0&

0

4Q dt !0&

0

4W dt ,

(2.23) (#U)n = Q !W .

§ 18

The entropy.

(2.24)T,p&

T#,p#

!!Qrev

T

"Q,N= C

ıgV

ln!

TT#

"+ NR ln

!VV#

"= 0 .

(2.25)p#&

p

!!Qrev

T

"T,N=

p#&

p

pT (dV)T,N = NR ln

!pp#

".

T#&

T

!!Qrev

T

"p,N=

T#&

T

CıgV

T (dT )p,N +T#&

T

p#T (dV)p,N


= !CıgV

ln!

TT#

"! NR ln

!TT#

".(2.26)

( !!Qrev

T

"N=

T,p&

T#,p#

!!Qrev

T

"Q,N+

p#&

p

!!Qrev

T

"T,N+

T#&

T

!!Qrev

T

"p,N

= NR ln!

pp#

"!C

ıgV

ln!

TT#

"! NR ln

!TT#

"

= NR ln!

pp#

"+ NR ln

!VV#

"! NR ln

!TT#

"

= NR ln!

pVT#p#V#T

"

= 0 .

(!Qrev = NR ln

!pp#

"!C

ıgV

(T ! T#) ! NR(T ! T#)

= CıgP

ln!

TT#

"!C

ıgP

(T ! T#)

! 0

21

22 2. PRELUDE

Toadstool (Amanita muscara), Trondheim (Norway) 2002.

CHAPTER 3

The Legendre transform

163 Workouts in Thermodynamics, © 2001, 2012 Tore Haug-Warberg

H1

H2

H3

H4

Hi = Ui !!!Ui

!Vi

"Vi

= Ui + piVi

U1 U2 U3 U4

V1V2 V3 V4

Figure 3.1 Legendre transformationof internal energy to enthalpy.

The formal definition of the canoni-cal thermodynamic potentials is one of thethree fundamental pillars of thermodynamictheory,a together with the principle of equi-librium, and the somewhat opaque distinc-tion between heat and work.b,c. This is thebackground to what we will now discuss,but for the moment we will refrain from ex-amining to what extent mathematical formu-lae are of practical relevance to the appli-cations of the theory. Complications arisefrom the fact that there are several energyfunctions to choose from, and it can be di"-cult to know exactly which function is bestsuited for a particular problem. From apragmatic point of view it is convenient toremember that U is useful for dynamic sim-

ulation, H for stationary simulation, etc. If a feasible solution to the problemalready exists then this is perfectly adequate, but when seeking a new solutionwe need a deeper theoretical understanding.

Here, the Legendred transformation is the key, as it provides a simple for-mula that allows us replace the free variable of a function with the correspond-ing partial derivative. For example, the variable V in internal energy U(S ,V,N) can be replaced by (!U/!V)S ,N , see Figure 3.1. The new variable can beinterpreted as the negative pressure . and the resulting transformed function,called enthalpy H(S , .,N), is in many cases more versatile than U itself. Infact, as we will see later, H has the same information content as U. This is oneof the key reasons why the Legendre transformation is central to both thermo-dynamic and mechanical theory.

a I.e. the functions U(S ,V,N), H(S , p,N), A(T,V, N), G(T, p,N), etc. b Herbert Callen. Ther-

modynamics and an Introduction to Thermostatistics. Wiley, 2nd edition, 1985. c MichaelModell and Robert C. Reid. Thermodynamics and Its Applications. Prentice Hall, 2nd edition,1983. d Adrien-Marie Legendre, 1752–1833. French mathematician.

23

24 3. THE LEGENDRE TRANSFORM

Mathematically, the Legendre transformation "i of the function f is definedby:

"i(#i, x j, xk, . . . , xn) = f (xi, x j, xk, . . . , xn) ! #ixi ,

#i =!! f!xi

"x j ,xk ,...,xn

.(3.1)

As mentioned above, one example of this is the transformation of internal en-ergy to enthalpy:

H = UV (S , .,N) = U(S ,V,N) ! .V

. =!!U!V

"S ,N

= ! p

Note that the volume derivative of U is the negative pressure ., because Udiminishes when the system performs work on the surroundings — not viceversa. To begin to get an understanding of the Legendre transform we shallfirst write out the di!erential

d"i = d f ! xi d#i ! #i dxi ,

and then substitute in the total di!erential of the initial function f expressed as:d f = #i dxi +

'nj!i (! f /!x j)xi ,xk,...,xn

dx j. The simplification is obvious:

d"i = !xi d#i +n4

j!i

!! f!x j

"xi,xk ,...,xn

dx j .

If we now consider "i as a function of the derivative #i rather than of the originalvariable xi, the total di!erential of "i can be written

d"i =!!"i

!#i

"x j ,xk ,...,xn

d#i +n4

j!i

!!"i

!x j

"#i,xk ,...,xn

dx j .

Comparing the last two equations term-by-term gives us the important transfor-mation properties (!"i/!#i)x j ,xk ,...,xn

= !xi and (!"i/!x j)#i,xk ,...,xn= (! f /!x j)xi,xk ,...,xn

=

# j. The latter shows that further transformation is straightforward:

"i j(#i, # j, xk, . . . , xn) = "i(#i, x j, xk, . . . , xn) ! # jx j ,(3.2)

# j =!!"i

!x j

"#i,xk,...,xn

=!! f!x j

"xi ,xk,...,xn

.

1. CANONICAL POTENTIALS 25

Combining Eqs. 3.1 and 3.2 gives the alternative, and conceptually simpler,expression

(3.3) "i j(#i, # j, xk, . . . , xn) = f (xi, x j, xk, . . . , xn) ! #i xi ! # j x j ,

where the sequential structure of the Legendre transforms in 3.1 and 3.2 hasbeen replaced by a simultaneous transformation of two (or more) variables.The two alternative approaches are equivalent because # j is the same regardlessof whether it is calculated as (!"i/!x j)#i,xk,...,xn

or as (! f /!x j)xi ,xk,...,xnThis is

clear from the di!erential equation above, and from Paragraph 20 on page 27.Knowing the initial function f and its derivatives is therefore su"cient to defineany Legendre transform. Moreover, as the Legendre transform is independentof the order of di!erentiation, we know that "i j = " ji. Mathematically we saythat the Legendre operatora commutes. The three sets of variables (xi, x j, . . . ,xn), (#i, x j, . . . , xn) and (#i, # j, . . . , xn) are particularly important, and are oftenreferred to as the canonical variables of the functions f , "i and "i j = " ji.

1. Canonical potentials

The legal system of the Roman Catholic Church, also called canon law,has medieval roots, and is based on a collection of texts that the Church con-siders authoritative (the canon). Today the word canonical is used to refer tosomething that is orthodox or stated in a standard form. The latter definitionis particularly used in mathematics, where e.g. a polynomial written with theterms in order of descending powers is said to be written in canonical form. Inthermodynamics, we refer to canonical potentials, meaning those that containall of the thermodynamic information about the system. Here we will showthat the Legendre transforms of internal energy give us a canonical descriptionof the thermodynamic state of a system. Essentially, what we need to show isthat U(S ,V, n), A(T,V, n), H(S ,!p, n), etc. have the unique property that wecan recreate all of the available information from any single one of them. Thisis not trivial, as we will see that U(T,V, n) and H(T,!p, n), for instance, donot have this property.

§ 19 Derive all of the possible Legendre transforms of internal energy.State carefully the canonical variables in each case. Use the definitionsb 0 =(!U/!S )V,N, . = (!U/!V)S ,N and µ = (!U/!N)S ,V to help you.

a Mathematical operators are often allocated their own symbols, but in thermodynamics it ismore usual to give the transformed property a new function symbol. b Let 0 and . denotetemperature and negative pressure respectively. This is to emphasize that they are transformedquantities, like the chemical potential µ. In this notation, all intensive derivatives of internalenergy are denoted by lower case Greek letters.


Energy functions. For any given thermodynamic function with m = dim(n)+2 variables there are 2m ! 1 Legendre transforms. For a single-component sys-tem this means that there are 23 ! 1 = 7 possible transformations. By usingEq. 3.1 on each of the variables in turn we get three of the transforms:

A(0,V,N) = U(S ,V,N) !!!U!S

"V,N

S = U ! 0S ,(3.4)

H(S , .,N) = U(S ,V,N) !!!U!V

"S ,N

V = U ! .V ,(3.5)

X(S ,V, µ) = U(S ,V,N) !!!U!N

"S ,V

N = U ! µN .(3.6)

By using Eq. 3.3 on pairs of variables we can obtain three more transforms:

G(0, .,N) = U(S ,V,N) !!!U!V

"S ,N

V !!!U!S

"V,N

S

= U ! .V ! 0S ,(3.7)

Y(S , ., µ) = U(S ,V,N) !!!U!V

"S ,N

V !!!U!N

"S ,V

N

= U ! .V ! µN ,(3.8)

$(0,V, µ) = U(S ,V,N) !!!U!S

"V,N

S !!!U!N

"S ,V

N

= U ! 0S ! µN .(3.9)

Finally, by using Eq. 3.2 on all three variables successivly we can obtain thenull potential, which is also discussed on page 51 in Chapter 4:

O(0, ., µ) = U(S ,V,N) !!!U!V

"S ,N

V !!!U!S

"V,N

S !!!U!N

"S ,V

N

= U ! .V ! 0S ! µN* 0(3.10)

Several of the Legendre transforms of energy have their own names: Inter-nal energy U(S ,V,N) is used when looking at changes to closed systems, and isin many respects the fundamental relationship of thermodynamics. Helmholtzenergy A(0,V,N) is central to describing the properties of fluids. Gibbs energyG(0, .,N) has traditionally been the transform that is of most interest in chemi-cal thermodynamics and physical metallurgy. Enthalpy H(S , .,N) is importantfor describing thermodynamic processes in chemical engineering and fluid me-chanics. The grand canonical potential $(0,V, µ) is used when describing opensystems in statistical mechanics. Meanwhile, the null potential O(0, ., µ) hasreceived less attention than it deserves in the literature, and has no internation-ally accepted name, even though the function has several interesting properties,as we shall later see.


This just leaves the two functions X(S , ., µ) and Y(S ,V, µ), which are ofno practical importance. However, it is worth mentioning that X is the onlyenergy function that has two, and always two, extensive variables no matterhow many chemical components there are in the mixture. This function istherefore extremely well suited to doing stability analyses of phase equilibria.In this case the basic geometry is simple, because the analysis is done alonga 2-dimensional manifold (a folded x, y plane) where the chemical potentialsare kept constant. This topic is discussed further in a separate chapter on phasestability.

§ 20 The variable # j in Eq. 3.2 can be defined as either (!"i/!x j)#i,xk,...,xnor

(! f /!x j)xi ,xk,...,xn. Use implicit di!erentiation to prove that the two definitions

are equivalent.

Differentiation I. Note that the variables xk, . . . , xn are common to both "i

and f and may hence be omitted for the sake of clarity. We will therefore limitour current analysis to the functions f (xi, x j) and "i(#i, x j). It is natural to startwith Eq. 3.1, which we di!erentiate with respect to x j:

(3.11)!!"i

!x j

"#i=!!( f!#i xi)!x j

"#i,

where (!#i/!x j)#i is by definition zero. Hence, at constant #i:

d( f ! #i xi) #i =!! f!xi

"x j

dxi +!! f!x j

"xi

dx j ! #i dxi

= #i dxi +!! f!x j

"xi

dx j ! #i dxi

=!! f!x j

"xi

dx j

or, alternatively:

(3.12)!d( f!#i xi)

dx j

"#i=!! f!x j

"xi

.

The full derivative takes the same value as the corresponding partial derivative(only one degree of freedom). Substitution into Eq. 3.11 yields

(3.13)!!"i

!x j

"#i=!! f!x j

"xi

= # j

leading to the conclusion that di!erentiation of "i with respect to the untrans-formed variable x j gives the same derivative as for the original function f .

It is a matter of personal preference, therefore, whether we want to cal-culate # j as (! f /!x j)xi

before the transformation is performed, or to calculate


(!"i/!x j)#i after the transformation has been defined. Normally it is easiestto obtain all of the derivatives of the original function first — especially whendealing with an explicitly defined analytic function — but in thermodynamicanalyses it is nevertheless useful to bear both approaches in mind. The nextparagraph shows the alternative definitions we have for temperature, pressureand chemical potential.

§ 21 Use the result from Paragraph 20 on the previous page to show thatthe chemical potential has four equivalent definitions: µ = (!U/!N)S ,V =

(!H/!N)S ,. = (!A/!N)0,V = (!G/!N)0,.. Specify the equivalent alternativedefinitions for temperature 0 and negative pressure ..

Identities I. Let f = U(S ,V,N) be the function to be transformed. Thequestion asks for the derivatives with respect to the mole number N and it istacitly implied that only S and V are to be transformed. From the Eqs. 3.4 and3.5 we have "1 = A(0,V,N) and "2 = H(S , .,N), which on substitution intoEq. 3.13 give:

!!A!N

"0,V=!!U!N

"S ,V,(3.14)

!!H!N

"S ,.=!!U!N

"S ,V.(3.15)

The transform "12 = "21 = G(0, .,N) in Eq. 3.7 can be reached either via A orH. Inserted into Eq. 3.13 the two alternatives become:

!!G!N

"0,.=!!A!N

"0,V,(3.16)

!!G!N

"0,.=!!H!N

"S ,..(3.17)

Note that all the Eqs. 3.14–3.17 have one variable in common on the left andright hand sides (V , S , 0 and . respectively). For multicomponent systems thisvariable will be a vector; cf. xk, . . . , xn in Paragraph 20. So, in conclusion, thefollowing is true for any single-component system:

(3.18) µ =!!A!N

"0,V=!!H!N

"S ,.=!!G!N

"0,.=!!U!N

"S ,V.

By performing the same operations on temperature and negative pressure weobtain:

0 =!!H!S

".,N=!!X!S

"V,µ=!!Y!S

".,µ=!!U!S

"V,N,(3.19)

. =!!A!V

"0,N=!!X!V

"S ,µ=!!$!V

"0,µ=!!U!V

"S ,N

(3.20)


This shows very clearly that we are free to choose whatever co-ordinateswe find most convenient for the description of the physical problem we wantto solve. For example, in chemical equilibrium theory it is important to knowthe chemical potentials of each component in the mixture. If the external con-ditions are such that temperature and pressure are fixed, it is natural to useµi = (!G/!Ni)T,p,Nj!i

, but if the entropy and pressure are fixed, as is the casefor reversible and adiabatic state changes, then µi = (!H/!Ni)S ,p,Nj!i

is a moreappropriate choice.

§ 22 The Legendre transform was di!erentiated with respect to the orginalvariable x j in Paragraph 20. However, the derivative with respect to the trans-formed variable #i remains to be determined. Show that (!"i/!#i)x j ,xk,...,xn

=

!xi.

Differentiation II. As in Paragraph 20, the variables xk, . . . , xn are commonto both f and "i, and have therefore been omitted for the sake of clarity. Letus start with "i(#i, x j) = f (xi, x j) ! #ixi from Eq. 3.1 and di!erentiate it withrespect to #i. Note that the chain rule of di!erentiation

!! f!#i

"x j=!! f!xi

"x j

!!xi

!#i

"x j

has been used to obtain the last line below:!!"i

!#i

"x j=!! f!#i

"x j!!!(#i xi)!#i

"x j

=!! f!#i

"x j! xi ! #i

!!xi

!#i

"x j

=!! f!xi

"x j

!!xi

!#i

"x j! xi ! #i

!!xi

!#i

"x j.(3.21)

From Eq. 3.3 we know that (! f /!xi)x j= #i, which can easily be substituted

into Eq. 3.21 to produce

(3.22)!!"i

!#i

"x j= !xi ,

which leads to the following conclusion: The derivative of "i with respect toa transformed variable #i is the original variable xi, but with the sign reversed.In other words, there is a special relationship between the variables xi in f (xi,x j) and #i in "i(#i, x j)a. Due to the simple relationship set out in Eq. 3.22, thevariables (#i, x j) are said to be the canonical variables of "i(#i, x j).

The Eq. 3.22 is strikingly simple, and leads to a number of simplificationsin thermodynamics. The properties explained in the previous paragraph are ofprime importance. Before we move on, however, we should investigate what

a The variables x and # are said to be conjugate variables.


happens if we do not describe the Legendre transform in terms of canonicalvariables. Di!erentiating "i = f ! #i xi with respect to the original varibale xi

gives:!!"i

!xi

"x j!i=!! f!xi

"x j!i!!!#i!xi

"x j!i

xi ! #i = !!!2 f!xi!xi

"x j!i

xi ,

The expression on the right-hand side is not particularly complex, but un-fortunately it is not canonically related to f . It is nevertheless an importantresult, because "i must by definition be a function with the same variables as fitself. The change in variable from xi to #i is an abstract concept that will onlyrarely produce an explicit expression stated in terms of #i. Hence, if we wantto calculate the derivative of "i by means of an explicit function, we have to govia f and its derivatives — there is simply no way around it. A simple exampleshows how and why this is the case:

!!U!T

"V,N= !!!2A!T!T

"V,N

T =!!S!T

"V,N

T =CV

TT = CV (T,V,N) .

Here the internal energy is, at least formally speaking, a function of the canon-ical variables S , V and N, but there are virtually no equations of state thatare based on these variables. T , V and N are much more common in thatcontext, so to lure U into revealing its secrets we must use T rather S . Note,however, that in this case S = ! (!A/!T )V,N is a function to be di!erentiatedand not a free variable. This illustrates in a nutshell the challenges we face inthermodynamics: Variables and functions are fragile entities with changing in-terpretations depending on whether we want to perform mathematical analysesor numerical calculations.

§ 23 Use the results obtained in Paragraph 22 to prove that the derivatives(!H/!.)S ,N , (!G/!.)0,N and (!Y/!.)S ,µ are three equivalent ways of expressingthe volume V of the system. Specify the corresponding expressions for theentropy S and the mole number N.

Identities II. Let us start out once more from f = U(S ,V,N) and definea

the transform "2 = H(S , .,N) = U ! .V . Inserted into Eq. 3.22, this gives

a The symbol . = !p is used here as the pressure variable. This is quite deliberate, in order toavoid the eternal debate about the sign convention for p. As used here, 0, . and µ are subject tothe same transformation rules. This means that the same rules apply to e.g. (!H/!.)S ,N = !V

and (!A/!0)V,N = !S , whereas the traditional approach using (!H/!p)S ,N = V and (!A/!T )V,N =

!S involves di!erent rules for p and T derivatives. Note, however, that it makes no di!erencewhether !p or p is kept constant during the di!erentiation.

2. MANIFOLDS 31

us (!H/!.)S ,N = !V .Systematically applying Eq. 3.22 to all of the energyfunctions in Paragraph 19 on page 25 yieldsa:

!V =!!H!.

"S ,N=!!G!.

"0,N=!!Y!.

"S ,µ,(3.23)

!S =!!A!0

"V,N=!!G!0

".,N=!!$!0

"V,µ,(3.24)

!N =!!X!µ

"S ,V=!!Y!µ

"S ,.=!!$!µ

"0,V.(3.25)

Knowing the properties of the Legendre transform, as expressed by Eqs. 3.13and 3.22, allows us to express all of the (energy) functions U, H, A, . . . O thatwe have considered so far in terms of their canonical variables. The patternthat lies hidden in these equations can easily be condensed into a table of totaldi!erentials for each of the functions (see below).

§ 24 Use the results from Paragraphs 21 and 23 on pages 28 and 30 to findthe total di!erentials of all the energy functions mentioned in Paragraph 19.

Differentials I. The total di!erentials of the energy functions can be statedby taking the results from Eqs. 3.18–3.20 and 3.23–3.25 as a starting point:

dU (S ,V,N) = 0 dS + . dV + µ dN ,(3.26)dA (0 ,V,N) = !S d0 + . dV + µ dN ,(3.27)dH (S , .,N) = 0 dS ! V d. + µ dN ,(3.28)dX (S ,V, µ) = 0 dS + . dV !N dµ ,(3.29)dG (0 , .,N) = !S d0 ! V d. + µ dN ,(3.30)dY (S , ., µ) = 0 dS ! V d. !N dµ ,(3.31)d$ (0 ,V, µ) = !S d0 + . dV !N dµ ,(3.32)dO (0 , ., µ) = !S d0 ! V d. !N dµ .(3.33)

2. Manifolds

In what has been written so far we note that all of the state variables 0, S ,., V , µ and N appear in conjugate pairs such as 0 dS , !S d0, . dV, !V d., µ dNor !N dµ. This is an important property of the energy functions. The obvioussymmetry reflects Eq. 3.22, which also implies that the Legendre transform is

a Sharp students will note the absence of !S = (!O/!0).,µ, V = (!O/!.)0,µ and!N = (!O/!µ)0,p. These relations have no clear thermodynamic interpretation, however, be-cause experimentally 0, ., µ are dependent variables, see also Paragraph 26 on page 37.


“its own inverse”. However, this is only true if the function is either strictlyconvex or strictly concave, as the relationship breaks down when the secondderivative f (xi, x j, xk, . . . , xn) is zero somewhere within the domain of defini-tion of the free variables (see also Section 3). This is illustrated in Figure 3.2based on the transformation of the third order polynomial

f (x) = x(1 ! x2) ) #(x) =!! f!x

"= 1 ! 3x2 .

The Legendre transformation of f to " = f ! #x can be expressed in twodi!erent ways:

"(x) = 2x3 ) "(#) = ±2!

1!#3

"3/2.(3.34)

Moreover, both x and f can be expressed as functions of the transformationvariable #:

x = ±! 1!#

3

"1/2) f (#) = ±

!1!#3

"1/2 !2+#3

",

This means that in total we have to consider three functions involving x, andthree involving #: f (x), "(x), #(x), f (#), "(#) and x(#). In order to retain theinformation contained in f (x), we need to know either "(x) and #(x), or f (#)and x(#), or simply "(#). In principle, the latter is undoubtedly the best option,and this is the immediate reason why "(#) is said to be in canonical form. Nev-ertheless, there is an inversion problem when " is interpreted as a function of# rather than x. This is clearly demonstrated by the above example, where x isan ambiguous function of #, which means that "(#), and similarly f (#), are notfunctions in the normal sense. Instead, they are examples of what we call man-ifolds (loosely speaking folded surfaces defined by a function) as illustrated inFigures 3.2c–3.2da.

Finally, let us look at what di!erentiating "with respect to # implies. FromSection 22, we know that the answer is !x, but

5!"

!#

6= +!

1!#3

"1/2

initially gives us a manifold in # with two solutions, as shown in Figure 3.2.It is only when we introduce x(#) from Eq. 3.34 that the full picture emerges:(!"/!#) = !x. This does not mean that we need x(#) to calculate the valuesof (!"/!#). However, the relationship between x, # and (!"/!#) is needed forsystem identification. If the three values have been independently measured,then the relation (!"/!#) = !x can be used to test the experimental values for

a A function is a point-to-point rule that connects a point in the domain of definition with acorresponding point in the function range.

2. MANIFOLDS 33

Figure a–b). The func-tion f = x(1 ! x2) and itsLegendre transform " =2x3 defined as the inter-section of the tangent bun-dle of f and the ordinateaxis. Note that f showsa maximum and a mini-mum while " has no ex-trema.

x

f (x)

a)

x

"(x)

b)

#

f (#)

c)

#

"(#)

d)

Figure c–d). The sametwo functions shown asparametric curves with# = (! f /!x) = 1 ! 3x2

along the abscissa. Thefold in the plane(s) islocated at the inflectionpoint (!2 f /!x!x) = 0.

Figure 3.2 The Legendre transformation of f = x(1 ! x2) to " = f ! #x where # =(! f /!x) = 1 ! 3x2. The two domains of definition are x ( [!

,2/3,

,2/3] and # (

[!1, 1]. Together, the four graphs demonstrate how an explicit function of x, i.e. f (x)or "(x), is turned into an implicit manifold when expressed in terms of #, i.e. f (#) or

"(#).

consistency. The existence of these kinds of tests, which can involve a varietyof physical measurements, is one of the great strengths of thermodynamics.

The above example also has practical relevance. Many equations of state,and particularly those for the fluid state, are explicit equations of the formp = p(T,V,N). These equations of state can be used to calculate the Helmholtzenergy of the fluid. The problem in relation to manifolds becomes clear whenthis energy is transformed into Gibbs energy, which requires us to know theterm V = V(p) in the definition G(p) = A(V(p))+pV(p). Along the sub-criticalisotherms the relation V = V(p) is equivalent to a manifold with 3 roots as illus-trated in Figure 13.4 on page 217. This results in a characteristic folding of theenergy surface as examplified by the p, µ diagram in Figure 13.2 on page 207;where the molar volume v is used as a parameter for p(v) along the abscissaand for µ(v) along the ordinate axis. In view of these practical considerations,we cannot claim that Legendre transforms are globally invertible, but it is trueto say that they are locally invertible on curve segments where the sign of(!2 f /!x!x) remains unchanged. We shall therefore take the precaution of treat-ing each curve segment as a separate function. For instance, in vapor–liquidequilibrium calculations we shall designate one of the curve segments as be-ing vapour and the other as being liquid, although there is no thermodynamicjustification for this distinction, other than the fact that (!2A/!V!V)T,N = 0


somewhere along the isotherm.

3. Inversion

In this section we will explain how, and under what conditions, it is possi-ble to convert the Legendre transform " back to the original function f . Thisoperation will be based on the transformation rule set out in Eq. 3.1

(3.35) "(y) = f !!! f!x

"x ,

where the properties of the variable y are, as yet, unknown. What we need toknow is whether the same transformation rule applies to the inverse transformof "(y), such that

(3.36) f?= " !

!!"!y

"y ,

and, if so, what y is (we have an inkling that y = (! f /!x), but we need to proveit). Substituting Eq. 3.36 into Eq. 3.35, rearranging the expression slightly, andapplying the chain rule, gives us:

(3.37) !!!"!y

"y = !

!!"!x

" !!x!y

"y

?=!! f!x

"x .

Next we substitute Eq. 3.35 into Eq. 3.37:

!7!! f!x

"!!!2 f!x!x

"x !!! f!x

"8 !!x!y

"y

?=!! f!x

"x

!!2 f!x!x

"x!!x!y

"y

?=!! f!x

"x

!!x!y

"y

?=!! f!x

" !!2 f!x!x

"-1.(3.38)

For the inverse transform to exist, it must be true that (!2 f /!x!x) ! 0.This is a necessary condition. It is also worth adding that in general, partialdi!erential equations do not have an analytical solution. However, thermody-namic problems are often unusually simple, and that is also the case here. Letus define # = (! f /!x), which gives us

!!x!y

"y

?= #!!#!x

"-1.

3. INVERSION 35

Introducing the logarithmic variables dln y = dy /y and dln # = d# /# producesan elegant solution to the problem (note how the chain rule is used in reversein the second-last line):

!!x! ln y

"=!! ln #!x

"-1

!! ln #!x

" !!x! ln y

"= 1

!! ln #! ln y

"= 1(3.39)

The most general solution can be expressed as y = c# = c (! f /!x), wherec is an arbitrary factor. It is natural to choose c = 1, which means that y * #, aswe know from our earlier discussion of the Legendre transform, but any valueof c ( R+ will give the same result, as the inverse transformin Eq. 3.36 is in-sensitive to scaling. We say that x and # are the natural (or canonical) variablesof the functions f (x) and "(#), as they allow transformation in both directionswithout the loss of any information. As an aside, it should be mentioned thatthe function "(x) does not have this property, because y = x is not a solutionto Eq. 3.36.

In the context of thermodynamics, this means that e.g. A(T,V, n) can beinverse transformed to U(S ,V, n), provided that (!2U/!S !S )V,n ! 0. However,the transformation rule is symmetrical, and the same requirement must alsoapply in the other direction: U(S ,V, n) can only be transformed into A(T,V, n)if (!2A/!T!T )V,n ! 0. The two requirements may appear contradictory, but inactual fact they are overlapping, because:

!!2U!S !S

"V,n=!!T!S

"V,n=!!S!T

"-1V,n= !!!2A!T!T

"-1V,n

In states in which one of the derivatives tends to zero, and the other one tendsto infinity, the system is on the verge of being unstable. Depending on thevariables involved, this relates to either thermal (T , S ), mechanical (p, V) orchemical (µi, Ni) stability. These kinds of states are always buried inside a re-gion with two or more phases, being found in what is referred to as the spinodalregion of the system (from the gr. spinode, meaning “cusp”).

§ 25 Use the result from Paragraph 22 on page 29 to show that perform-ing two Legendre transformations on f , first with respect to xi and then withrespect to #i, returns the original function.

Inverse transform. Given Definition 3.1 and Eq. 3.22, the inverse Legendretransformation can be written

(3.40) "i !!!"i

!#i

"x j#i = "i ! (!xi) #i * f ,


but it is not entirely clear what variable set should be used to define f . Tounderstand the nature of the problem, let us consider the following example.From Eq. 3.4 it follows that:

(3.41) U(S ,V,N) !!!U(S ,V,N)!S

"V,N

S = U(S ,V,N) ! 0S = A(0,V,N) .

To approach it from the opposite direction, we have to calculate the Legendretransform of A(0,V,N) with respect to the variable 0:

(3.42) A(0,V,N) !!!A(0,V,N)!0

"V,N0 = A(0,V,N) ! (!S )0 * U(!S ,V,N) .

This clearly returns the original function U, but the set of canonical variableshas changed from S ,V,N to !S ,V,N. In order to get back to where we started,we must perform two further Legendre transformations:

U(!S ,V,N) !!!U(!S ,V,N)!(!S )

"V,N

(!S ) = U(!S ,V,N) ! (!0)(!S )

= A(!0,V,N) ,(3.43)

A(!0,V,N) !!!A(!0,V,N)!(!0)

"V,N

(!0) = A(!0,V,N) ! S (!0)

* U(S ,V,N) .(3.44)

In other words, performing repeated Legendre transformations reveals a closedcycle, where the original information contained in U is retained, as shown inthe figure below:a

U(S ,V,N)S!% A(0,V,N)

- !0 . 0

A(!0,V,N)!S/! U(!S ,V,N)

This example shows that the Legendre transform is (locally) invertible,which means that we can choose whatever energy function is most suitablefor our purposes. Moreover, Eq. 3.10 tells us that the Legendre transform ofU with respect to all of the state variables S , V and N is a function with veryspecial properties. The function, which is known as the null potential, is identi-cal to zero over the entire definition domain, and as such it has no inverse. Ourline of reasoning breaks down for this function, and we must expect O(T, p, µ)to have properties that di!er from those of U and the other energy functions.Note that this is not a general mathematical property of the Legendre transform;rather it follows inevitably from the fact that U and all of the other extensive

a In practice it may be easier to use the canonical variables (x, y) = (! (!g/!z)y , y) for theinverse transformation, rather than (x, y) = ((!g/!z)y , y) as has been done here.

4. MAXWELL RELATIONS 37

properties are described by first-order homogeneous functions to which Euler’stheorem can be applied. These functions display a kind of linearity that resultsin their total Legendre transforms being identical to zero. Hence, the di!eren-tials are also zero over the entire definition domain. In thermodynamics, thisis called the Gibbs–Duhem equation.

§ 26 Show that performing a Legendre transformation of internal energyU(S ,V,N) with respect to all the canonical variables S ,V,N gives the null po-tential O(0, ., µ) = 0. Next, show that the di!erential of O is identical to theGibbs–Duhem equation; see also Paragraph 34 in Chapter 4.

Null potential. The di!erential of the null potential in Eq. 3.10 is of course0. However, since this result is valid for the entire domain of definition it musthave a bearing on the degrees of freedom of the system. Mathematically, theO-function forms a hyperplane in dim(n) + 2 dimensions. This becomes clearif we di!erentiate O(0, ., µ) in Eq. 3.10: dU ! 0 dS ! S d0 ! V d. ! . dV !µ dN ! N dµ = 0. If we spot that 0 dS + . dV + µ dN is the total di!erential ofinternal energy, we can simplify the expression to:

(3.45) S d0 + V d. + N dµ = 0 .

This result is identical to the Gibbs–Duhem Eq. 4.23, which plays an importantrole in checking the thermodynamic consistency of experimental data. This isbecause Eq. 3.45 tells us about the relationship between 0, ., µ. If a seriesof measurements is taken in order to obtain experimental values for all of thevariables 0, ., µ, then the Gibbs–Duhem equation allows us to check the qualityof the measurements, given that Eq. 3.45, or Eq. 3.10 for that matter, must befulfilled. In practice, this is possible because the chemical potentiala, can beexpressed as a function of temperature and negative pressure, or alternativelyas f (0, ., µ) = 0.

4. Maxwell relations

We have in this chapter introduced seven energy functions: U, H, A, G, X,Y and O, and six state variables: S , V , N, t, p and µ. On reflection it is clear thatonly four of these variables are independent, because the discussion has reliedentirely on the fact that internal energy U is a function of S , V and N. All ofthe other varaibles have been defined at a later stage either as partial derivativesor as transformed functions of U. There is therefore a substantial degree ofredundancy in the variables that we are dealing with. This redundancy becomeseven clearer when the energy functions are di!erentiated twice with respect totheir canonical variables.

a For single component or multicomponent systems at fixed concentration.


§ 27 Set out all of the Maxwell relations that can be derived by applyingthe Leibniza rule !2 f /!xi!x j = !

2 f /!x j!xi to the functions U, A,H, Y,G,$, Xand O, restricting yourself to single-component systems.

Maxwell relations. Let us begin by illustrating what is meant by a Maxwellb

relation using U(S ,V,N) as our starting point. From the definitions of 0 and. it follows that:

!!2U!S !V

"N=!!2U!V!S

"N)2!!S

!!U!V

"S ,N

3

V,N=

2!!V

!!U!S

"V,N

3

S ,N

)!!.!S

"V,N=!!0!V

"S ,N.

Similarly, cyclic permutation of the variables S ,V and N yields:

!!2U!S !N

"V=!!2U!N!S

"V)

!!µ!S

"V,N=!!0!N

"S ,V,

!!2U!V!N

"S=!!2U!N!V

"S)

!!µ!V

"S ,N=!!.!N

"S ,V.

By systematically comparing the second derivatives of all of the Legendretransforms mentioned in Paragraph 19 on page 25, we get the results shownbelow. Also see Paragraph 24 on page 31 for a complementary table of totaldi!erentials and first derivatives. Note that the Maxwell relations which arederived from the null potential O(0, ., µ) have no physical interpretation, as ex-perimentally 0, ., µ are dependent variables. This means that e.g. (!0/!.)µ isunpredictable because Eq. 3.45 on integrated form is equivalent to a relationg such that g(0, ., µ) = 0. These relations have therefore been omitted fromthe table.

!!.!S

"V,N=!!0!V

"S ,N

!!µ!S

"V,N=!!0!N

"S ,V

!!µ!V

"S ,N=!!.!N

"S ,V

!!!S!V

"0,N=!!.!0

"V,N

!!!S!N

"0,V=!!µ!0

"V,N

!!.!N

"0,V=!!µ!V

"0,N

!!!0!.

"S ,N=!!V!S

".,N

!!0!N

"S ,.=!!µ!S

".,N

!!!V!N

"S ,.=!!µ!.

"S ,N!

!0!V

"S ,µ=!!.!S

"V,µ

!!!0!µ

"S ,V=!!N!S

"V,µ

!!!.!µ

"S ,V=!!N!V

"S ,µ!

!S!.

"0,N=!!V!0

".,N

!!!S!N

"0,.=!!µ!0

".,N

!!!V!N

"0,.=!!µ!.

"0,N

!!!0!.

"S ,µ=!!V!S

".,µ

!!!0!µ

"S ,.=!!N!S

".,µ

!!V!µ

"S ,.=!!N!.

"S ,µ

!!!S!V

"0,µ=!!.!0

"V,µ

!!S!µ

"0,V=!!N!0

"V,µ

!!!.!µ

"0,V=!!N!V

"0,µ

a Gottfried Wilhelm von Leibniz, 1646–1716. German mathematician. b James ClerkMaxwell, 1831–1879. Scottish physicist.

5. GIBBS–HELMHOLTZ EQUATION 39

5. Gibbs–Helmholtz equation

We are familiar with the key characteristics of Legendre transforms, hav-ing looked at topics like transformation geometry, di!erentiation rules, inversetransforms and Maxwell relations. We shall now focus on the simple rule usedin Figure 3.1, which shows that the Legendre transform is equal to the intersec-tion of the tangent bundle of the original function and the ordinate axis. Thesubject becomes even more interesting now, as we are going to prove anotherrule, which is even simpler, although not so easy to derive. The key to thisis a simple change in variables from y and x in (!y/!x) to y/x and 1/x in theexpression (!(y/x)/!(1/x)). This relationship is generally known as the Gibbs–Helmholtz equation.

§ 28 Verify the Gibbs–Helmholtz equation (!(G/0)/!(1/0)).,N * H. Seeif you can generalise this result.

Gibbs–Helmholtz. By di!erentiating and then substituting (!G/!0).,N =!S and G + TS = H, we can confirm that the equation is correct, but there ismore to it than that. From the mathematical identity

!(y/x)!(1/x) * y + 1

x!y!(1/x) * y ! x

!y!x

we can conclude that

(3.46)!!( f /xi)!(1/xi)

"x j ,xk,...,xn

= f ! #ixi = "i ,

cf. the Legendre transform in Eq. 3.1. The identity applies generally, includingto "i j and to any other derived transforms. This means that Eq. 3.46 providesan alternative way of calculating the value of a Legendre transform, withoutthis changing the original definition. One of the classic applications of theGibbs–Helmholtz equation is in calculating the temperature derivative of theequilibrium constant, as shown in Chapter 15, where ln K = !#rxG#/RT isshown to be an almost linear function of 1/T with a slope of #rxh#/R. Anequally useful equation is (!(A/0)/!(1/0))V,N * U, and in fact there are awhole series of similar equations that are based on the same principle.

It remains to be shown that it is straightforward to di!erentiate the Gibbs–Helmholtz equation. To illustrate this, let us use the derivative of the equationgiven in the worked example above:

!!H!.

"0,N=!!!.

!!(G/0)!(1/0)

".,N

"0,N=!!

!(1/0)

!!(G/0)!.

"0,N

".,N=!!(V/0)!(1/0)

".,N.

Note that the enthalpy has been di!erentiated with respect to negative pressure,which is a canonical variable; the result follows an easily recognisable pattern.


We will come back to this form of the Gibbs–Helmholtz equation in Chapter 7,but there it will be used to integrate equations of state at a fixed composition.Finally, it is worth adding that every time we derive an expression of the formy ! x (!y/!x) we can choose to replace it with (!(y/x)/!(1/x)). These twoalternatives look di!erent, but they are nevertheless mathematically identical.

5. GIBBS–HELMHOLTZ EQUATION 41

“There have been nearly as many formulations of the second law as therehave been discussions of it. Although many of these formulations aredoubtless roughly equivalent, and the proof that they are equivalent hasbeen considered to be one of the tasks of a thermodynamic analysis, Iquestion whether any really rigorous examination has been attemptedfrom the postulational point of view and I question whether such an ex-amination would be of great physical interest. It does seem obvious,however, that not all these formulations can be exactly equivalent, but itis possible to distinguish stronger and weaker forms.”

— Percy Williams Bridgmana

40

a The Nature of Thermodynamics, Harvard University Press (1941) p. 116.


Sea buckthorn (Hippophae rhamnoides), Gaulosen (Norway) 2003.

CHAPTER 4

Euler’s Theorem on Homogeneous Functions


The thermodynamic state of a so-called simple system is described mathe-matically by the mapping f : Rn+2 % R where n 0 0 stands for the number ofchemical componentsa in the system. For the system to be physically realisable,it must be macroscopically uniform. Apart from this constraint, the domain ofdefinition of a multicomponent system is infinitly large, but the mathematicaltreatment is considerably simplified by the fact that it can be verified experi-mentally that f is linear along all state vectors starting at the origin. The practi-cal consequences of this linearityb will be clarified when we start investigatingthe mathematical properties of f , but let us first formalise our statement bystating that the function f (x1, . . . , xn) is homogeneous of order k ( Z providedthat the parametrised function f (%x) is proportional to %k in the direction ofx = (x1, . . . , xn). Here, the parameter % ( R+ is a dimensionless measure of thedistance from the origin to the coordinate tuple %x ( Rn. More precisely, thefunction f (x1, . . . , xn, #n+1, . . . , #m) is homogeneous of order k in the variablesx1, . . . , xn if the following criteria are satisfied:

F(X1, . . . , Xn, #n+1, . . . , #m) = %k f (x1, . . . , xn, #n+1, . . . , #m) ,(4.1)Xi = %xi .(4.2)

X

F k=!1

k=0

k=1

k=2

Note that F has exactly the same function defi-nition as f . Hence, it is only by convention that wedistinguish the two forms. We may therefore write:

Fx#X

f

It is then assumed that #n+1, . . . , #m do not take partin the homogeneity of f . They are of course usedin the function descriptions of F, and in all the par-tial derivatives of F, but they do not take part inthe scaling of free variables in Eq. 4.2. That means

a An empty chamber has no chemical components, but it nevertheless constitutes a thermody-namic system of electromagnetic radiation with two degrees of freedom. b Herbert Callen.Thermodynamics and an Introduction to Thermostatistics. Wiley, 2nd edition, 1985.

43

44 4. EULER’S THEOREM ON HOMOGENEOUS FUNCTIONS

the scaling law in Eq. 4.1 is valid for all choices of#n+1, . . . , #m. Strange at first sight, maybe, but thegrouping of variables into disjoint subsets is quitenatural in physics. For example the kinetic energyof an n-particle ensemble EK(m, #) = 1

2'n

i mi12i is

homogeneous of order 1 with respect to the masses mi and homogeneous oforder 2 with respect to the velocities 1i. The same function is homogenousof order 3 if mass and velocity are considered simultaneously as independentvariables. Taking this a step further, one can say that f = xyz is homogeneousof order 1 in x if y, z are taken to be constant parameters (the same argumentholds circularly for y and z), and homogeneous of order 3 in x, y, z if all of thequantities are treated as free function variables.

Of particular interest to us are the energy functionsa U, A, . . . ,O with statevariables belonging to S ,V,N or 0, ., µ. The energy functions, and entropy, vol-ume and mole number, are homogeneous functions of order 1, while temper-ature, pressureb and chemical potential are homogeneous functions of order 0.In the context of thermodynamics these quantities are referred to as extensiveand intensive state variables respectivelyc.

We are not initially aiming to embark on a general discussion of the multi-component functions in Eq. 4.1. Instead, we will start with a detailed analysisof the simpler two-variable functions f (x, #) and F(X, #). The results will laterbe generalised in Chapter General Theory, and we lose nothing by taking an

a In its most general form homogeneity means that F(X) = %k f (x) where X = %x indicatesa scaling of the system by the dimensionless factor %. Here, k = 1 is an important specialcase, under which the function F is considered extensive. In thermodynamics it is commonto write F = X1 f 2 in this case, where X is a physical quantity, i.e. a quantity with an associ-ated unit of measurement. How this can be understood in terms of Euler’s theorem becomesclearer if we consider a system with one single group of homogeneous variables. We couldlook, for example, at the total number of moles for all components in the system while it iskept at a constant composition (constant mole fractions). In this case F(X) = % f (x) whereX = %x and X = {X} [X]. Here, {} denotes the magnitude of X and [] denotes its unit ofmeasurement. Let us now choose x = X/{X} i.e. % = {X}. It is then possible to writeF(X) = {X} · f (X/{X}) = {X} [X] · f (X/{X})/ [X] = X · f (X/{X})/ [X]. The most commonexamples in thermodynamics are X = N [mol], X = M [kg] and X = V [m3]. Specifically,when X = N [mol] we also get F(N) = N f (N = 1 mol)/ [mol] = N1 f 2. The expression is thesame as F = % f except that N is a physical quantity whilst % is a dimensionless number. Thismeans that f has the unit of [F] while 1 f 2 is the associated molar quantity [F mol-1]. Subtle,yes, but nevertheless essential because in our day-to-day work we use 1 f 2 rather than f . It is forinstance quite common to write the total Gibbs energy for the system as G(0, .,N) = Ng(0, ., x)where x is a vector of mole fractions, or, alternatively, as G = Ng(0, .) if x = [1]. The functiong = 1 f 2 denotes molar Gibbs energy in both cases. Correspondingly, we can write U = Nu(s, v),H = Nh(s, .) etc. for the other molar energies. Our Euler notation does not have one-to-oneequivalents in other writings on the subject, but it is hard to avoid conflicts when the establishedstandard does not fully reflect the physical-mathematical reality. b Here, 0 and . are usedfor temperature and negative pressure respectively, as already introduced in Chapter 3. In thiscontext we want to stress that the properties (in common with the chemical potential µ) are in-tensive quantities. c A physical quantity is extensive if it is proportional to the system size andintensive if it is insensitive to the system size.

4. EULER’S THEOREM ON HOMOGENEOUS FUNCTIONS 45

easy approach here. In order to exploit the function properties, it makes senseto use the total di!erential of F expressed in X and #-coordinates:

dF =!!F!X

"#

dX +!!F!#

"X

d# .

The variable X is defined as a function of x and % in Eq. 4.2, and by substitutingthe total di!erential of X, or more precisely dX = % dx + x d%, we obtain

(4.3) dF =!!F!X

"#

x d% +!!F!X

"#% dx +

!!F!#

"X

d# .

An alternative would be to make use of F = %k f from Eq. 4.1 as thestarting point for the derivation:

dF = k%k!1 f d% + %k d f .

Substitution of the total di!erential of f expressed in x and #-coordinates gives:

(4.4) dF = k%k!1 f d% + %k!! f!x

"#

dx + %k!! f!#

"x

d# .

Note that Eq. 4.3 and Eq. 4.4 are two alternative expressions for the samedi!erential dF (%, x, #). Comparing the equations term-by-term reveals threerelations of great importance to thermodynamic methodology:

Case d%. Comparing the d% terms reveals that (!F/!X)# x = k%k!1 f . Mul-tiplying both sides by % gives (!F/!X)# %x = k%k f , which if we substitute inEq. 4.1 and Definition 4.2 can be transformed into:

(4.5)!!F!X

"#

X = kF .

The closed form of Eq. 4.5 indicates that there is a general solution to theindefinite integral F(X, #) =

&(dF)# =

&(!F/!X)# dX. The result is known

as Euleras first theorem for homogeneous functions, or simply as the Eulerintegration of F.

Case dx. Comparing the dx terms reveals that (!F/!X)# % = %k (! f /!x)#.Dividing each side by % leads to:

(4.6)!!F!X

"#= %k!1

!! f!x

"#.

Because F has the same function definition as f (it is only the names of the freevariables that di!er) it follows that !F/!X and ! f /!x are equivalent functions.

a Leonhard Euler, 1707–1783. Swiss mathematician.


Hence, (!F/!X)# is a homogeneous function of order k ! 1. Di!erentiationwith respect to X therefore reduces the order of homogeneity of F by one.

Case d#. Comparing the d#-terms reveals that the homogeneity of the de-rivative with respect to # is unchanged:

(4.7)!!F!#

"X= %k!! f!#

"x,

Using the same arguments as in the case above this implies that the derivative ofF with respect to # is a new homogeneous function of degree k. Di!erentiationwith respect to # therefore conserves the homogeneity in X.

Note. It should be stressed that the Euler integration in Eq. 4.5 is not lim-ited to one particular interpretation of X. In fact, any scaled variable x = %-1

X satisfies the equation:

!! f!x

"#

x = k f .

The validity of this statement is confirmed by first combining Eqs. 4.5 and 4.6,and then substituting in Eq. 4.1 and Definition 4.2. This result emphasises thepractical importance of Euler’s theorem as outlined in Eq. 4.5.

Extended notation I. The general properties of homogeneous functions willbe explained further in Chapter General Theory, but to get a sense of the overallpicture we shall briefly mention what changes are required in Eqs. 4.5–4.7 tomake them valid for multivariate functions:

n'i=1

Xi

!!F!Xi

"X j!i,#l

= kF ,(4.8)!!F!Xi

"X j!i,#l

= %k!1!! f!xi

"x j!i ,#l

,(4.9)!!F!#k

"X j,#l!k

= %k!! f!#k

"x j ,#l!k

.(4.10)

§ 29 Internal energy U = U(S ,V,N) is an extensive function in the vari-ables S ,V and N. The total di!erential of U is dU = 0 dS + . dV + µ dN.Explore the homogeneity associated with the functions for 0, . and µ.

Intensive functions. The variables 0, . and µmentioned above must first bedefined. Mathematically, the total di!erential of U(S ,V,N) is:

dU =!!U!S

"V,N

dS +!!U!V

"S ,N

dV +!!U!N

"S ,V

dN .


Comparing this with the di!erential given in the text implies that 0 = (!U/!S )V,N,. = (!U/!V)S ,N and µ = (!U/!N)S ,V , see also Paragraph 19 on page 25 inChapter 3. Substitution of k = 1, Xi = S , X j ( {V,N} and #l = !a in Eq. 4.9yields:

(4.11) 0(S ,V,N) =!!U(S ,V,N)!S

"V,N= %0

!!u(s,v,n)!s

"v,n= 0(s, v, n) .

The function 0(S ,V,N) is obviously homogeneous of order 0 in the variablesS ,V,N because it is independent of the scaling factor %. The temperature iscommonly referred to as an intensive variable and is taken to be independentof the system sizeb. Similarly, di!erentiation of U with respect to V and Nyields

.(S ,V,N) =!!U(S ,V,N)!V

"S ,N= %0

!!u(s,v,n)!v

"s,n= .(s, v, n) ,(4.12)

µ(S ,V,N) =!!U(S ,V,N)!N

"S ,V= %0

!!u(s,v,n)!n

"s,v= µ(s, v, n) ,(4.13)

where the (negative) pressure and the chemical potential are intensive variablesas well.

§ 30 Internal energy U = U(S ,V,N) is an extensive function in the vari-ables S ,V and N. The total di!erential of U is dU = 0 dS + . dV + µ dN.What is the correct integral form of U? Explain the physical significance ofthis integral.

Euler integration. Substituting k = 1, Xi ( {S ,V,N} and #l = ! into Eq. 4.8,together with the definitions from Paragraph 29 on the facing page i.e. 0 =(!U/!S )V,N, . = (!U/!V)S ,N and µ = (!U/!N)S ,V yields Euler’s equationapplied to internal energy:

(4.14) U = 0S + .V + µN .

We are not being asked about multicomponent mixtures, but Eq. 4.14 is quitegeneral and can (by analogy with Eq. 4.8) be extended to:

(4.15) U = 0S + .V +n'

i=1µiNi = 0S + .V + µ

Tn

Remark. The total di!erential of internal energy for a single componentsystem is dU = 0 dS + . dV + µ dN. The di!erential can be integrated without

a Actually an empty set. Here it is used to denote a missing variable or an empty vector.b Valid only when the system is su"ciently big i.e. when the number of particles is large enoughto fix the temperature.


actually solving any partial di!erential equations because 0, ., µ are intensive(size-independent) variables. Physically, this means that the system can bebuilt up from zero sizea in a manner that keeps 0, ., µ constant during the pro-cess. This is achieved by combining a large number of small systems at fixedtemperature, (negative) pressure and chemical potential. When these subsys-tems are merged into one big system there will be no changes in the intensiveproperties, because the requirements for thermodynamic equilibrium are auto-matically fulfilled, see also Chapter 21 on page 365.

§ 31 Gibbs energy G(0, .,N) is an extensive function of the mole numberN at a given temperature 0 and (negative) pressure .. The total di!erential ofG is dG = !S d0 ! V d. + µ dN. Explore the homogeneity associated with thefunctions S and V .

Extensive functions. We will apply the same procedure as was used in Para-graph 29 on page 46. First of all the functions S ,V and µ must be defined.Mathematically, the total di!erential of G(0, .,N) is:

dG =!!G!0

".,N

d0 +!!G!.

"0,N

d. +!!G!N

"0,.

dN .

Comparing this with the di!erential in the problem formulation yields !S (0,.,N) = (!G/!0).,N , !V(0, .,N) = (!G/!.)0,N and µ = (!G/!N)0,.. FromEq. 4.10 we can deduce that both volume and entropy are homogeneous func-tions of order 1 in the mole number at specified temperature and (negative)pressure, i.e. they are extensive variables. From Eq. 4.9 it can be seen that thechemical potential is (still) a homogeneous function of order 0, see also Eq. 4.13.

Any further discussion of homogeneity should ideally be based on the prop-erly identified canonical state variables of the function. However, this is not anabsolute condition, as the thermodynamic state of a system is uniquely deter-mined when any of its independent variable sets is fully specified. For exampleU is extensive in S ,V,N while G is extensive in N at a given 0 and ., but wecan alternatively state the opposite and say that U is extensive in N at a given 0and . and that G is extensive in S ,V,N. There are plenty of alternative ways todescribe the system, but a combination of three arbitrary state variables is notalways su"cient. To demonstrate this, let us look at the specification H, 0,N.For example, the ideal gas enthalpy can be written Hıg = H(0,N). At a given0,N the system is underspecified, because there is no way we can determinethe pressure. If we try to specify H in addition to 0 and N, this will give us

a By definition, a system of zero size has zero internal energy, i.e. U = 0. In this context, “zerosize” refers to zero volume, zero mass and zero entropy. Note, however, that it is not su"cientto assume zero mass, because even an evacuated volume will have radiation energy proportionalto T 4!


a redundant (or even contradictory) statement. To avoid problems of this kindwe shall therefore make use of canonical variables unless otherwise stated.

§ 32 The Gibbs energy of a binary mixture is given as a homogeneousfunction of order 1 in the mole numbers N1 and N2 (at fixed T and p). Makea contour diagram illustrating the function G = aN1x2 + N1 ln x1 + N2lnx2where N2 is plotted along the ordinate axis and N1 along the abscissa. Let x1 =

N1/(N1+N2) and x2 = N2/(N1+N2). Show that the isopleths corresponding toconstant G (the contour lines) are equidistant for a series of evenly distributedGibbs energy values. Use a = 2.4 in your calculations.

Gibbs energy I. The function is nonlinear in N1 and N2, and the isoplethsmust be calculated iteratively using e.g. Newton–Raphson’s method: N2,k+1 =

N2,k ! (Gk ! G)/µ2, where µ2 = ax21 + ln x2 is the partial derivative of G with

respect to N2. A fixed value is selected for N1, and N2 is iterated until Gk%&has converged to G, see the Matlab-program 1.4 in Appendix H. The resultobtained is shown in Figure 4.1 on the next page. Note that each isopleth de-fines a non-convex region, which can be interpreted as a fundamental thermo-dynamic instability. The corresponding two-phase region (the symmetry of themodel reduces the phase equilibrium criterion to µ1 = µ2) can be calculatedfrom the total di!erential of G, rewritten here into the tangent of the isopleth:

!dN2dN1

"T,p,G

= ! µ1µ2.

It can be proved (do it!) that the tangent intersects with the y and x axes at G/µ2and G/µ1 respectively. This indicates that the phase equilibrium condition isfulfilled whenever two points on the same isopleth have common tangents (re-member that G takes constant values along each isopleth such that the criterionis reduced to µ1 = µ2).

§ 33 Homogeneity causes a whole range of remarkable results. One for-mula obtained by di!erentiating Eq. 4.5 is:

X d!!F!X

"#! k!!F!#

"X

d# = (k ! 1)!!F!X

"#

dX ,

For k = 1 this implies that (!2F/!X!X)# = 0 and (!F/!#)X = (!2F/!X!#) X.Verify these results and give a physical explanation for them.

Homogeneity. The left-hand side of Eq. 4.5 is di!erentiated and the right-hand side is replaced by the total di!erential of F:

X d!!F!X

"#+!!F!X

"#

dX = k!!F!X

"#

dX + k!!F!#

"X

d# .


Figure 4.1 Contour diagramof Gibbs energy (solid lines).Equidistant values (open cir-cles) are made visible along3 rays (dotted lines) fromthe origin. The two-phaseregion is spanned by the twooutermost rays. One of theisopleths indicates the phaseequilibrium condition in theshape of a convex hull con-struction.

0 2 4 60

1

2

3

4

5

6

N1

N2

Gµ1

Gµ2

For k = 1 the expression reduces to X d(!F/!X)# = (!F/!#)X d#. Note that Xand d# are arbitrary. To proceed we need the di!erential of !F/!X, and becauseF is a function of X and # it can be written as the total di!erential

d!!F!X

"#=!!2F!X!X

"#

dX +!!2F!X!#

"d# ,

where dX is also arbitrary. Substituted into the equation above (letting k = 1),this yields the intermediate result:

(4.16) X!!2F!X!X

"#

dX + X!!2F!X!#

"d# =

!!F!#

"X

d# .

The trick is to recognise that X, dX and d# are independent variables. Thisimplies that two non-trivial relations follow from Eq. 4.16 (one equation inthree variables gives two non-trivial relations), irrespective of the actual valuesof X, dX, # and d#:

!!2F!X!X

"#= 0 ,(4.17)

X!!2F!X!#

"=!!F!#

"X

(4.18)

From a physical point of view any extensive function F(X, #) can be ex-pressed in the form F = '(#)X. This stems from the fact that F(0, #) = 0 inaddition to !2F/!X!X = 0, see Eq. 4.17a The derivative of F with respect to #is therefore an extensive function '3(#)X where the second derivative of F with

a It should be noted that it is the second derivative function which is zero. It is not su"cient tosay that the second derivative is zero at a given point.


respect to both X and # is equal to the intensive parameter '3(#). The numeri-cal value of !F/!# can easily be obtained by integrating !2F/!X!# using theEuler method as shown in Eq. 4.18.

Extended notation II. In the general case, x and $ would be vector quan-tities. If F(x, $) is extensive in x it can be shown that the di!erential in Para-graph 33 on page 49 takes the form:

(4.19) xT d!!F!x

"$!!!F!$

"Tx

d$ = 0 ,

where the di!erential of !F/!x is written:

(4.20) d!!F!x

"$=!!2F!x!x

"dx +

!!2F!x!$

"d$ .

The two quantities dx and d$ are independent, and by substituting the di!eren-tial 4.20 into Eq. 4.19 it follows that:

!!2F!x!x

"$

x = 0 ,(4.21)!!2F!$!x

"x =!!F!$

"x.(4.22)

From a logical point of view Eq. 4.22 is a true generalisation of 4.18, whereasEq. 4.17 represents a specialisation of 4.21. This means that the second deriva-tive of F(X, #) with respect to X is zero for all single-variable functions, whilethe corresponding Hessian F(x, $) for multivariate systems is singular in the di-rection of x, where x is proportional to one of the eigenvectors of the Hessian.This is Euler’s second theorem for homogeneous functions.

§ 34 Substitute U = U(S ,V,N) into Paragraph 33 on page 49 and showthat S d0 + V d. + N dµ = 0. Do you know the name of this equation inthermodynamics? Does it make any di!erence if you plug in G = G(0, .,N)rather than U(S ,V,N)?

Gibbs–Duhem. Substituting F = U(x, $) into Eq. 4.19 where xT = (S ,V,N) and $ = !, reduces the expression to xT d(!U/!x) = 0. From the definitionsof 0, . and µ in Paragraph 29 on page 46 we can write

(4.23) S d0 + V d. + N dµ = 0 ,

which is better known as the Gibbs–Duhem equation. Alternatively, if F =G(x, $) where x = N and $T = (0, .), then Eq. 4.19 takes the form:

N d!!G!N

"0,.!!!G!0

".,N

d0 !!!G!.

"0,N

d. = 0 .


The partial derivatives of G with respect to N, 0 and . have been identifiedas µ,!S and !V in Paragraph 31 on page 48. The expression can thereforebe reformulated as N dµ + S d0 + V d. = 0 which is identical to the Gibbs–Duhem equation. In fact, all Legendre transforms of U end up giving the sameGibbs–Duhem equation.

We have not been asked for any extensions, but by analogy to Eqs. 4.8 and4.15 the Gibbs–Duhem equation may be extended to a multicomponent form:

(4.24) S d0 + V d. +n'

i=1Ni dµi = S d0 + V d. + nT dµ = 0 .

Note that the Gibbs–Duhem equation follows inevitably from the properties ofextensive functions when di!erentiated, which in turn reflect their homogene-ity. This is true for all functions of this kind, and not only thermodynamicones. The homogeneity, which e!ectively removes one degree of freedom inthe function expression, shows up as a mutual dependency in the nth derivatives(of U). As a result, only n ! 1 of the intensive state variables are independent.In single component systems this means that any arbitrary intensive variablecan be expressed as a function of (at most) two other intensive variables, seealso Paragraph 39 on page 54.

. It is important to realise that the information contained in U is conservedduring the Legendre transformation to H, A, . . . ,O. For example, knowing theGibbs energy G(0, .,N) really implies full knowledge of U(S ,V,N), and viceversa. The Gibbs–Duhem equation can therefore be derived from any of theenergy functions. In particular this also applies to the di!erential of the null-potential O(0, ., µ) which is identical to Eq. 4.23, see Paragraph 26 on page 37.

§ 35 Use the result from Paragraph 33 on page 49 to determine the secondderivatives (!2G/!N!N)0,., (!2Y/!S !S ).,µ and (!2$/!V!V)0,µ, where G =G(0, .,N) and Y = Y(S , ., µ) and $ = $(0,V, µ).

Linear potentials. The three functions G, Y and $ are extensive in N, Sand V respectively, i.e. in one variable each. This makes Eq. 4.17 valid and thesubstitution of the definitions for 0, . and µ yields:

!!2G!N!N

"0,.=!!µ!N

"0,.= 0 ,(4.25)

!!2Y!S !S

".,µ=!!0!S

".,µ= 0 ,(4.26)

!!2$!V!V

"0,µ=!!.!V

"0,µ= 0 .(4.27)


§ 36 Show that the Hessian of A = A(0,V,N) has only one independent ele-ment when the temperature 0 is taken to be a constant parameter. Use Eq. 4.21as your starting point.

Helmholtz energy is extensive in V,N at any given temperature 0. Bearingin mind Eq. 4.21 the Hessian of A can be expressed as a function of only oneindependent variable because the matrix, in addition to being symmetric, mustsatisfy the relation:

!!2A!x!x

"0

x =

+,,,,,,,,-

!!.!V

"0,N

!!.!N

"0,V!

!µ!V

"0,N

!!µ!N

"0,V

.////////0

+,,,,,,,-

V

N

.///////0 =

+,,,,,,,-

0

0

.///////0 .

The fact that the matrix, like all Hessians, must be symmetric, in this caseleads to (!./!N)0,V = (!µ/!V)0,N . Altogether there are 4 matrix elements,with 3 associated relations.

Hessians. The solution of the homogeneous (!) system of equations can beformulated in an infinite number of ways, one of which is:

(4.28) NV

!!µ!N

"0,V

5NV!1

!1 VN

6 5VN

6=

500

6.

Note that the second derivative of Helmholtz energy with respect to the molenumber is non-zero in Eq. 4.28, while the corresponding second derivative ofGibbs energy is zero in Eq. 4.25:

fra 4.25:!!µ!N

"0,.= 0 ,

og fra 4.28:!!µ!N

"0,V! 0 .

This emphasises the fact that it is important to have a proper understanding ofthe many peculiarities of the energy functions, and it also highlights that it isessential to know which variables are held constant during di!erentiation.

§ 37 Find analytical expressions for (!G/!0).,N and (!G/!.)0,N by usingthe Euler method to integrate the partial molar entropy and partial molar vol-ume.

Partial molarity. Replace F by G($, n) in Eq. 4.22, where $T = (0, .) andnT = (N1, . . . ,Nn). This gives, without much di"culty:

+,,,,,,,,-

!!G!0

".,N!

!G!.

"0,N

.////////0 =

+,,,,,,,-

!!2G!0!nT

".!

!2G!.!nT

"0

.///////0 n = !

+,,,,,,,,-

!!S!nT

"0,.!

!V!nT

"0,.

.////////0 n = !

+,,,,,,,-

sTn

vTn

.///////0 .


The partial derivatives of S and V on the right hand side are called the par-tial molar entropy and partial molara volume respectively. These and otherpartial molar quantities occur so frequently in the thermodynamic theory ofmixtures that they have been given their own di!erential operator, see Para-graph 10 on page 17. A more compact notation is therefore S = nTs and V = nTv.

§ 38 Show that G(S ,V,N) is really an extensive function of the variablesS ,V and N, thus verifying the conjecture set out in the introduction to thischapter (on page 48).

Proof. From the definition G(0, .,N) = U ! 0S ! .V , where U = U(S ,V,N), 0 = (!U/!S )V,N and . = (!U/!V)S ,N, it follows that we can express Gibbsenergy as a function of S ,V and N, because the right-hand side of the equationonly includes U and functions derived from U:

G(S ,V,N) = U(S ,V,N) ! 0(S ,V,N)S ! .(S ,V,N)V .

We already know that 0 and . are intensive variables, whereas U, S and V areextensive variables. From this it follows that

G(S ,V,N) = %u(s, v, n) ! %0(s, v, n)s ! %.(s, v, n)v= %g(s, v, n) ,

which clearly demonstrates that G is homogeneous, in accordance with the con-jecture. Note that the last equation tacitly assumes the homogeneity relationsS = %s,V = %v and N = %n.

§ 39 Starting from Paragraph 35 on page 52, can you tell how many ther-modynamic variables are needed to determine the intensive state of a system?

The state concept. A thermodynamic system is normally described by n+2independent state variables. However, the intensive state can be determinedonce n+1 (intensive) variables are known. This fact is illustrated by Eqs. 4.25–4.27, where the derivative of an intensive property with respect to an extensivevariable is zero if the other variables are intensive, and are held constant duringdi!erentiation. In fact, the (single) extensive variable determines the system

a A partial molar quantity is defined as f = (!F/!n)0,., irrespective of whether or not F has 0,.,N1, . . . ,Nn as canonical variables (it is only for Gibbs energy that there is a correspondencebetween the two sets of variables).


size but has no influence on 0, . and µ. For a single-component system we candescribe the intensive state in three di!erent ways:

0 = 0(., µ) ,. = .(0, µ) ,µ = µ(0, .) .

55


Dragonfly (Aeschna grandis), Trondheim (Norway) 2004.

CHAPTER 5

Postulates and Definitions


The unifying theory which embraces this book depends on four postulatesonly, but as in every axiomatic system there is a considerable degree of free-dom in how the postulates are stated (according to the incompleteness theoremsof Gödela). First of all we must select a merit function among the four basicquantities U, S , V and N. The volume is familiar to all of us while the molenumber is getting a little more vague (although molecular entities are no doubtcountable). On the other hand energy (converted work) and entropy (convertedheat) are pure abstractions. Among the latter two concepts most people find iteasier to accept energy as a state variable rather than entropy because macro-scopic work is more directly quantified than heat. The pros and cons of themerit functions are summarised below:

U(S ,V,N). The variables are strictly positive (+), but S is a rather di"cultconcept (-). Equilibrium is described as a minimum energy state (++).

S (U,V,N). The variables have simple physical interpretations (+), but arenot strictly positive (-) because U has no absolute zero point. Equilibrium isdescribed as a state of maximum disorder (-).

V(S ,U,N). The variables are not strictly positive (-), S is an abstract con-cept (-) and the equilibrium state has no clear physical interpretation (-). How-ever, all the variables tell something about the internal state of the system (+),while the function value is a measure for an external property (+).

N(S ,V,U). No contemplations — neither positive nor negative.

The thermodynamic equilibrium corresponds with a state of minimum en-ergy which justifies the choice of U(S ,V,N) as the basis function. The four

a Kurt Gödel, 1906–1978. Austrian–American logician.

57

58 5. POSTULATES AND DEFINITIONS

postulates can then be stated as:

Internal energy: U = U(S ,V, n)Heterogenous system: Utot =

'i

Ui(S i,Vi, ni)

Equilibrium state: Ueq = minS i,Vi,ni

Utot

Absolute entropy: limT%0

S = 0

We can build upon these postulates by defining four closely related energyfunctions from the Legendre transform of internal energy:

Internal energy: U(S ,V, n) =U = TS ! pV +n'

i=1µiNi

Helmholtz energy: A(T,V, n) =U!TS = !pV +n'

i=1µiNi

Enthalpy: H(S ,!p, n) =U + pV = TS +n'

i=1µiNi

Gibbs energy: G(T,!p, n) =U!TS + pV =n'

i=1µiNi

and two Massieu functions dedicated isolated and open stationary systems re-spectively (the functions have no special symbols):

S (U,V, n) = T -1U + T -1 pV ! T -1n'

i=1µiNi

S (H,!p, n) = T -1H ! T -1n'

i=1µiNi

Some extra definitions with roots in physical chemistry follow:

Temperature: T =!!U!S

"V,n

=!!H!S

"p,n

= . . .

Pressure: !p =!!U!V

"S ,n

=!!A!V

"T,n

= . . .

Chemical potential: µi =!!U!Ni

"S ,V,Nj!i

=!!A!Ni

"T,V,Nj!i

= . . .

Isochoric heat capacity: CV = T!!S!T

"V,n=!!U!T

"V,n

Isobaric heat capacity: CP = T!!S!T

"p,n=!!H!T

"p,n

5. POSTULATES AND DEFINITIONS 59

Isothermal expansivity: & =V -1!!V!T

"p,n= v-1

!!v!T

"p,n

Isobaric compressibility: !' =V -1!!V!p

"T,n= v-1

!!v!p

"T,n

Maxwell relations: !2F!Xi!X j

= !2F!X j!Xi

e.g.)

!!2U!S !V

"n=!!2U!V!S

"n

def)!

!!p!S

"V,n=!!T!V

"S ,n

Partial molarity: f i =!!F!Ni

"T,p,Nj!i

e.g.) gi = µi

) µi = hi ! T si

Figure 5.1 shows in a diagram how the connections are made.Finally, there is a table of all the Legendre transforms of internal energy

for a single component system (summarises much of what has already beenstated). The variables S ,V,N, 0, ., µ in column 2 are the kanonical variables ofthe functions listed in column 1. Placed together with the partial derivativesin column 3 they conduct three pairs of conjugated variables S , 0 and V, . andN, µ. These variables come in pairs because their products have the same units(of energy) as U itself. Note that the Euler form of each energy function isequivalent to the Legendre form of the same function albeit their functinalexpressions are totally di!erent:

Symbol Variables Derivatives Euler form Legendre formU S , V , N 0, ., µ 0S+.V+µN UA 0 , V , N !S , ., µ .V+µN U!0SH S , . , N 0,!V, µ 0S+ µN U !.VX S , V , µ 0, .,!N 0S+.V U !µNG 0 , . , N !S ,!V, µ µN U!0S !.V$ 0 , V , µ !S , .,!N .V U!0S !µNY S , . , µ 0,!V,!N 0S U !.V!µNO 0 , . , µ !S ,!V,!N U!0S !.V!µN

In some cases it will be natural to expose the Legendre form whilst in othercaser it may be smarter to use the Euler form. For the unexperienced stu-dent this liberty is bewildering — more than an opportunity — but to the expe-rienced thermodynamisist it conducts an extra degree of fredom in the problemformulation.

59

60 5. POSTULATES AND DEFINITIONS

Energy

EntropydU = dWtot

1st law 2nd law 3rd lawdU = !Q ! !W !Qrev = T dS

dU = T dS ! p dV +'iµi dNi

Chain rule Euler

Stoichiometry

Closed systemAn = b

Reactionµ = AT!

U = TS ! pV +'iµiNi

limT%0

S = 0

A(T,V, n)S (U,V, n)

O(T, p, µ)O( 1T, p

T, µ

T) L

egen

dre

Mas

sieu

Identities

Maxwell

...

...

...

......

Jacobi

!!U!S

"V,n= T

!!V!S

"p,n=!!T!p

"S ,n

!!T!p

"H,n

Opt

imiz

atio

n

Global stability

Equilibrium

Local stabilityµ 0 AT!

µ& = µ' · · ·AT!

!!2U

!(S ,V,n)!(S ,V,n)

"0 0

Gibbs–Duhem!S dT + V dp !

'i

Ni dµi = 0F%

G

Extended Gibbs–DuhemFT dT + Fp dp +

'i

Ni d f i = 0

F ='i

f iNi

Partial molar properties

Eul

er

dF = FT dT + Fp dp +'i

f i dNi

Figure 5.1 Formal relationships in thermodynamics

5. POSTULATES AND DEFINITIONS 61

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