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INSTITUT FRANCAIS DU PETROLE PUBLICATIONS

Jean VlDAL
Associate Director of Research at IFP
Professor at IFP School

THERMODYNAMICS
APPLICATIONS IN CHEMICAL ENGINEERING
AND THE PETROLEUM INDUSTRY
Translated from the French by Thomas S. Pheney
and Eileen M. McHugh, TCNY

2003

t Editions TECHNIP

27 rue Cinoux, 75737 PARIS Cedex IS,FRANCE

FROM THE SAME PUBLISHER

Rtroleum Refining Series
1. Crude Oil. Petroleum Products. Process Flowsheets J.P.
2. Separation Processes J.P. WAUQUIER, Ed.
3. Conversion Processes P. LEPRINCE, Ed.
4. Materials and Equipment P. TRAMBOUZE,Ed.
5. Refinery Operation and Management J.P. FAVENNEC, Ed.

WAUQUIER,

Ed.

Petrochemical Processes.Technical and Economic Characteristics.
1. Synthesis-Gas Derivatives and Major Hydocarbons
2. Major Oxygenated, Chlorinated and Nitrated Derivatives
A. CHAUVEL, C. LEFEBVRE

Technology of Catalytic Oxidations (The)
1. Chemical, Catalytic and EngineeringAspects
2. Safety Aspects
P. ARPENTINIER, F. CAVANI, F. TRlFlRb

Scale-up Methodology for Chemical Processes
J.P. EUZEN, P. TRAMBOUZE, J.P. WAUQUIER

Computational Fluid DynamicsApplied to Process Engineering
P. TRAMBOUZE,

Ed.

Petroleum Process Thermodynamics

E. BEHAR,

Ed.

Permeability of Gases in Polymer Materials M.H. KLOPFFER
Chemical Reactors. Design. Engineering. Operation
P. TRAMBOUZE, H. VAN LANDECHEM, J.P. WAUQUIER

Combustion and Flames. Chemical and Physical Principles
R. BORCHI, M. DESTRIAU

Principlesof Turbulent Fired Heat G.

MONNOT

Translation of Thermodynamique.Application
au g h i e chimique et B I’industrie p&amp;roli&amp;e. J. Vidal
0 1997, Editions Technip, Paris

0 2003, EditionsTechnip, Paris
All rights reserved. No part of this publication may be reproduced or
transmitted in any form or by any means, electronic or mechanical,
includingphotocopy, recording, or any informationstorage and retrieval
system, without the prior written permission of the publisher.

ISBN 2-7108-0800-5

Preface

The petroleum industry has such an extensive range of application for the principles of
thermodynamics that a study such as this one could not possibly cover the subject in its
entirety. For catagenesis,successive migration in the formation of petroleum fluids, reservoir exploitation, transport of natural gas or crude oils, refining and petrochemical
processes,or energy applications,we avail ourselves of existing equations for properties as
diverse as density, energy, and equilibrium conditions between phases. These equations
were developed during what is called the “golden age” of classic thermodynamics.We are
left with finding the most appropriate way to apply them.
We are faced with many obstacles.
First of all, the composition of petroleum fluids is poorly understood. At best, we are
aware of their complexity.The sheer number of components and the poorly defined structure of some of these components requires simplifications,which are more or less justifiable. The pressure and temperature conditions of some natural gas or light oil deposits
place these fluids close to their critical conditions. In addition, we note the particularly
high pressure levels of recently discovered fields.
The treatment of natural gas, the separation by extractive distillation or liquid-liquid
extraction, the synthesis of compounds such as ethers that are used in place of tetraethyl
lead in the formulation of gasoline,and the very diversified field of petrochemistry lead to
the treatment of mixtures in which hydrocarbons and compoundswith heteroatomic structures coexist, and give rise to more complex molecular interactions.
The simulation of processes and their optimization assumes that the properties of the
mixtures concerned are known. Although this knowledge is still based on experimental
measurements for the most part, it is also the result of calculation methods that have been
developed. These methods owe their value to the laws of thermodynamics, which assure
them a wide range of application. For example, an equation of state allows of course for
the density calculation of a fluid as a function of pressure, temperature, and composition,
but it also allows for the calculation of phase change conditions,and the energy exchanges
that result from the imposed transformations.However, these methods are still approximations, and remain to be perfected.The need for advancement makes thermodynamics a
living discipline. It is based on the relationships that exist between experimentation and
the notion of “models”.The engineer must keep abreast of progress and evaluate it intelligently.
For the most part, this study is dedicated to the description of our methods for the calculation and prediction of the thermodynamic properties of the processed fluids during
reservoir exploitation,refining, or petrochemical processing. The evolution of these methods that we have just emphasized, the limits of our experience, and the need to choose,
make for numerous deficiencies.We offer only an introduction to the literature in the field,
which is treated in a far from exhaustive manner. The proliferation of literature proposing

VI

Preface

new models, or sometimes real improvements to existing methods, is such that the bibliographical references provided are incomplete, or, even worse, the methods described in
this text probably will be abandoned. However, we believe that a discussion of these methods will facilitate understanding of future developments. At the end of this preface, we
mention a number of books that were the source of constant inspiration during our
research, and that have, more or less knowingly,influenced the writing of this study.
Some sections are merely necessary updates of an older text, published over twenty
years ago [Vidal, 1973,19741.The chapters dealing with phase equilibria duplicate, in part,
a text that is published in a more general work on petroleum refining. We have made no
attempt to introduce unrealistic modifications in this text, but we have, of course,
expanded it.
While preparing the text, we were very much inspired by our teaching experience at the
&amp;Cole Nationale Suptrieure du Pttrole et des Moteurs (ZFP-School),and by the questions
from students.We may hope that this very same teaching has benefited from our research
work. It is certainly at the source of some of its outcome.
Suffice it to say that the use of different conventionswithin the field gives rise to endless
discussions about nomenclature and notation. As much as possible, we have conformed to
the most common usage. For the units, we have taken some liberties with the International
System, using the Celsius and Kelvin scale, and preferring to express pressures in bar,
which is much more “day-to-day” than the pascal and its multiples. Only rarely have we
retained data in the Anglo-Saxon unit systems.
I would like to thank several of my colleagues who have contributed to this study to
varying degrees: MUeA. Boutrouille, MM. J. Ch. de Hemptinne, C1. Jaffret and B. Tavitian,
and especially M.L. Asselineau, whose knowledge, experience, and friendship I valued
each day.
Not to have dedicated a section, no matter how brief, to polymer solutions would have
been an important omission. I owe thanks to Mme G. Bogdanic (INA R&amp;D) for having
instructed me in this area and guided the writing of the chapter, which is dedicated to it.
Finally, I wish to acknowledge Professor Renon (&amp;Cole des Mines de Paris) who welcomed me into the research group that he formed at the Znstitut Franeais du Pttrole (ZFP),
and inspired my first works in the field of calculating thermodynamic properties, and
Professor PCneloux (Universitt d’Aix-Marseille ZZ) with whom I enjoyed constant and
fruitful exchanges.

Editions Technip would like to give special thanks to Mr. Jean-Charles de Hemptinne
from Division Chimie physique of Znstitut franeais du pttrole for participating in the
realisation of this book with his valuable comments, suggestions and corrections on the
text.

Preface

VI I

REFERENCES
Lewis GN, Randall M, Pitzer KS, Brewer L (1961) Thermodynamics,2nd edition. McGraw-Hill, New
York.
Prigogine I, Defay R (1950) Thermodynamique chimique. Desoer, LiBge.
Pkneloux A, Cours de Thermodynamique,Universitk d'Aix-Marseille.
Prausnitz JM, Lichtenthaler RN, de Azevedo EG (1986) Molecular Thermodynamics of Fluid-Phase
Equilibria. Prentice-Hall, Englewood Cliffs, New Jersey.
Reid RC, Prausnitz JM, Sherwood TZlK (1977) The Properties of Gases and Liquids, 3rd edition.
McGraw-Hill, New York.
Reid RC, Prausnitz JM, Poling BE (1987) The Properties of Gases and Liquids, 4th edition. McGrawHill, New York.
Rowlinson JS, Swinton FL (1982) Liquids and Liquid Mixmres. Butterworth, London.
Sandler SI (1989) Chemical Engineering Thermodynamics.Wiley, New York.
Tassios DP (1993) Applied Engineering Thermodynamics.Springer-Verlag,Berlin.
Vidal J (1973, 1974) Thermodynamique. Mtthodes appliqutes au raffinage et au gtnie chimique.
Editions Technip,Paris.

Table of Contents

Preface ..........................................................................................................................
Symbols ........................................................................................................................

.

1

V
XIX

.

Principles Thermodynamic Functions The Ideal Gas
1.1 Definitions .............................................................................................................
1.2 The First Law ........................................................................................................

1.2.1 The Energy of a System............................................................................................
1.2.2 Energy Exchanges during a Transformation....................................................
1.2.3 Statement of the First Law Applied to a Closed System..................................

..................................................................................................
.........................................................................
............................................................................
.........................................................
1.3 Application of the First Law to an Open System..............................................
1.2.3.1
1.2.3.2
1.2.3.3
1.2.3.4

General
IsochoricTransformations
Adiabatic Compression
Transformationsat Constant Pressure

1.3.1 General ...........................................................................................................
1.3.2 Steady-StateSystems ......................................................................................

1.4 The Second Law ....................................................................................................
1.4.1
1.4.2
1.4.3
1.4.4

Entropy ...........................................................................................................
Relationshipbetween Internal Energy and Entropy .......................................
Application of the Equilibrium Condition ......................................................
Statistical Significanceof Entropy ...................................................................

1.5 Helmholtz Energy and Gibbs Energy ................................................................
1.6 ThermodynamicFunction:Internal Energy, Enthalpy, Entropy,
Helmholtz Energy and Gibbs Energy ................................................................
1.6.1 Dependence on Temperature,Volume,or Pressure .........................................
1.6.2 CharacteristicFunctions..................................................................................

6

6
8
8
9
10
11
12
12
19
19

22

1.7 The Ideal Gas ........................................................................................................

23
23
25
26

References....................................................................................................................

26

1.7.1 Equation of State and Thermodynamic Properties ..........................................
1.7.2 Heat Capacity .................................................................................................
1.7.3 DataTables .....................................................................................................

X

Table of Contents

2

Properties of Pure Substances

.

..

2.1 The Relationship between Pressure.Volume and Temperature
Liquid-Vapor E q ~ i b rurn
i ...................................................................................
2.2 Vapor Pressure ......................................................................................................

2.2.1 Liquid and Vapor States ..................................................................................
2.2.2 Vapor Pressure Equations ...............................................................................

.................................................................................
.............................................................................
2.3 Enthalpy Diagram and Heat of Vaporization ...................................................
2.3.1 Dependence of Enthalpy on Pressure and Temperature..................................
2.3.2 Heat of Vaporization.......................................................................................
2.4 Calculation ofThermodynamic Properties .......................................................
2.4.1 Residual Enthalpy...........................................................................................
2.4.2 Residual Gibbs Energy ...................................................................................
2.4.3 Fugacity .........................................................................................................
2.4.4 Calculation of Thermodynamic Properties in the Liquid Phase ......................
2.4.5 General Equations .........................................................................................
Conclusion....................................................................................................................
References ....................................................................................................................
2.2.2.1 Clapeyron Equation
2.2.2.2 Empirical Correlations

3

29
34
34
36
36
37

44
44
46
49
50
51
53
54
58
59
59

.

Predicting Thermodynamic Properties of Pure Substances
General Principles Corresponding States
Group Contributions

.

.

3.1 Techniques of Molecular Simulation..................................................................

63

3.2 The Corresponding States Principle...................................................................

64
69
69
70
71
72
74
75
81
81
82
83

3.2.1 Correlations Using the Critical CompressibilityFactor ...................................

.......................................................................................
....................................................................................
3.2.2 Correlations Using the Acentric Factor ...........................................................
3.2.2.1 Prediction of the Second Virial Coefficient ..................................................
3.2.2.2 Properties at Liquid-Vapor Equilibrium......................................................
3.2.2.3 Lee and Kesler Method ............................................................................
3.2.3 Extensions to the CorrespondingStates Principle ...........................................
3.2.3.1 Extension to Polar Compounds .................................................................
3.2.3.2 Extension to Mixtures ..............................................................................
3.2.4 Conclusion Concerning the CorrespondingStates Principle............................
3.2.1.1 Watson Method
3.2.1.2 Rackett Equation

Table of Contents

3.3 Structure Property Correlations........................................................................

3.3.1 Properties of the Ideal Gas .............................................................................
3.3.2 Critical Coordinates ........................................................................................
3.3.3 Calculation of Molar Volume in the Liquid Phase ...........................................

3.4 Examples of the Relationships Between ThermodynamicProperties...........
3.4.1 Calculation of Critical Properties from Vapor Pressure and Density Data ......
3.4.2 Calculation of the Heat of Vaporization: Watson Equation .............................
3.4.3 Empirical Equations Developed from the Normal Boiling Point and Density .

Conclusion....................................................................................................................
References ....................................................................................................................

XI

84
85
88
90
90
90
92
93
94
95

4

Equations of State
4.1 Equations of State Derived from the Vial Development ..............................
4.1.1
4.1.2
4.1.3
4.1.4

Volume Virial Equation of State Truncated after the Second Term .................
Volume Virial Equation of State Truncated after the Third Term ....................
Pressure Virial Equation of State Truncated after the Second Term ................
The Benedict. Webb. and Rubin Equation .......................................................

4.2 Equations of State Derived from the Van Der Waals Theory .........................

4.2.1 The Soave-Redlich-Kwong and Peng-Robinson Equations of State ................
4.2.2 Recent Developments of Cubic Equations of State .........................................
4.2.2.1 Dependence of Attraction Parameter ‘a’ on Temperature ..............................
4.2.2.2 Modifications of the AttractionTerm
4.2.2.3 Application of the Concept of Group Contribution ......................................
4.2.2.4 Equations of State for Rigid Spheres and Hard Chains..................................

..........................................................

102
105
106
106
111
112
113
124
125
126
130
134

4.3 Specific Equations of State for Certain Pure Substances................................

136

4.4 The Tait Equation.................................................................................................
References ....................................................................................................................

138
139

5

Characterization of Mixtures
5.1 Partial Molar Values in the Homogeneous Phase ............................................

144
144
145

5.2 Chemical Potential................................................................................................

149
149
150

5.1.1 Definitions. Main Equations ............................................................................
5.1.2 Determination of Partial Molar Values ...........................................................
5.2.1 Definition .......................................................................................................
5.2.2 Equilibrium Condition Between Phases ..........................................................

XI1

Table of Contents

5.2.3 Relationshipsbetween the Chemical Potential and the Other Thermodynamic
151
Properties .......................................................................................................

.....................................................................

5.2.3.1 The Gibbs-Duhem Equation
5.2.3.2 Dependence of Chemical Potential on Pressure and Temperature
5.2.3.3 Relationships between the Chemical Potential and the Other Thermodynamic.
Functions

...................

...............................................................................................
5.3 Fugacity..................................................................................................................
5.3.1 Definition .......................................................................................................
5.3.2 Dependence of Fugacity on Temperature, Pressure. and Composition ............
5.4 Mixing Values Activity .........................................................................................
5.4.1 Definitions ......................................................................................................
5.4.2 Dependence of Activity on Temperature. Pressure, and Composition .............
5.5 The Ideal Solution ................................................................................................
5.6 Calculation of Fugacity ........................................................................................
5.7 Excess Values and Activity Coefficients............................................................
5.7.1 Definitions .......................................................................................................
5.7.2 Dependence of Excess Values on Temperature,Pressure. and Composition ....
5.7.3 Activity Coefficients .......................................................................................
5.7.4 Dependence of Activity Coefficients on Temperature,Pressure, and
Composition ...................................................................................................
5.8 Comparison of ' h o Methods for Calculating Fugacity....................................
5.9 Asymmetric Convention:the Henry Constant..................................................
Reference......................................................................................................................

151
151
152
152
152
153
154
154
155
156
157
159
159
160
160
161
163
165
166

6
Mixtures: Liquid-Vapor Equilibria
6.1 Description of the Vaporization or Condensation Phenomena......................

6.1.1 Isobaric Liquid-Vapor Equilibrium Diagram ..................................................
6.1.2 Isothermal Liquid-Vapor E uilibrium Diagrams.Evolution with Temperature
Critical Point and Retrogra e Condensation...................................................
6.1.3 Azeotropic Systems.........................................................................................

8

.

6.2 The Liquid-Vapor Equilibrium Condition The Equilibrium Coefficient .....

..

6.3 Dependence of the Equilibrium Conditions on Temperature,Pressure,
and Composibon...................................................................................................

6.3.1 Dependence of Bubble Pressure on Composition ...........................................
6.3.2 Dependence of Bubble Pressure on Temperature. Clapeyron Equation
Applied to a Mixture.......................................................................................
6.3.3 Coherence Tests ..............................................................................................
6.3.4 Stability and Critical Point Conditions ............................................................

168
168
171
176
179
184
184
186
187
188

Xlll

Table of Contents

6.4 Liquid-Vapor Equilibrium Problems .................................................................

190
192
192
192

6.5 Calculation Algorithms ........................................................................................

192
193
193
194
194
195
195
195
196
201
201
201
203

6.4.1 At Given Temperature (or Pressure) and Vaporized Fraction .........................
6.4.2 At Given Temperature and Pressure ...............................................................
6.4.3 Case Where One of the Data is a ThermodynamicProperty ...........................
6.5.1 Calculation of the Bubble Point ......................................................................

............................................................
.......................................................
6.5.2 Calculation of the Dew Point ..........................................................................
6.5.2.1 Calculation of the Dew Pressure ................................................................
6.5.2.2 Calculation of the Dew Temperature ..........................................................
6.5.3 Partial Vaporization ........................................................................................
6.5.4 Application to Ideal Solutions.........................................................................
6.5.5 Non-Ideal Solutions ........................................................................................
6.5.5.1 Non-Ideal Solutionsat Low Pressure ..........................................................
6.5.5.2 General Case ..........................................................................................
6.5.6 General Calculation Method of Liquid Vapor Equilibria ................................
6.6 Solubility of Gases in Liquids..............................................................................
References ....................................................................................................................
6.5.1.1 Calculation of the Bubble Pressure
6.5.1.2 Calculation of the Bubble Temperature

204
206

7

Deviations from Ideality in the Liquid Phase

..

7.1 Excess Quanbties..................................................................................................

7.1.1 Excess Volume. Excess Heat Capacity.............................................................
7.1.2 Heat of Mixing ................................................................................................
7.1.3 Excess Gibbs Energy and Activity Coefficients...............................................

210
210
212
214

7.2 Correlation of Liquid Vapor Equilibria at Low Pressure Coherence Test ...

218

7.3 Influence of Varying Molar Volume: the Combinatorial Term .......................

223

7.4 The Concept of Local Composition ...................................................................

227
229
229
230

7.5 Regular Solutions..................................................................................................

231

7.6 Empirical Models Based on the Concept of Local Composition....................

238
238
239
244
246

.

7.4.1 The.Lattice Model...........................................................................................
7.4.2 The Quasi Chemical Model.............................................................................
7.4.3 General Remarks............................................................................................

7.6.1
7.6.2
7.6.3
7.6.4

The Wilson Equation.......................................................................................
The NRTL Equation .......................................................................................
The UNIQUAC Model ...................................................................................
The Wilson, NRTL. UNIQUAC Models.Conclusion .......................................

XIV

Table of Contents

7.7 Group ContributionMethods .............................................................................

7.7.1 The ASOG Method .........................................................................................
7.7.2 The UNIFAC Method ......................................................................................
7.7.3 Group Contribution Method .Conclusion .......................................................

248
250
251
257

7.8 Associated Solutions ............................................................................................

258

7.9 Ionic Solutions.......................................................................................................

263

References....................................................................................................................

265

8

.

Application of Equations of State to Mixtures
Calculation of Liquid-Vapor Equilibria Under Pressure
8.1 Extensions ofthe Corresponding States Principle ...........................................

8.1.1 Calculation Rules for Pseudocritical Points .....................................................
8.1.2 Calculation of Thermodynamic Properties and Fugacity Coefficients in a
Mixture ...........................................................................................................

270
271
275

8.2 Virial Equations of State lhncated after the Second Term............................

277

8.3 Equations of State Derived from the vm der Wads The0ry...........................

279
279
282
283
286
290
290
291
292

8.3.1
8.3.2
8.3.3
8.3.4
8.3.5

The Classical Mixing Rules .............................................................................
Calculation of Chemical Potentials and Fugacity Coefficients .........................
Application Range and Results .......................................................................
The Binary Interaction Parameter ...................................................................
Alternatives on the Classical Mixing Rules .....................................................

.............................
.............................
8.3.6 Calculation of the Thermodynamic Properties of the Mixture .........................
Mixing Rules and Excess Functions ...................................................................
8.3.5.1 Dependence of the Attraction Parameter on Composition
8.3.5.2 Application of a Quadratic Mixing Rule to the Covolume

8.4

8.4.1 Calculation of Excess Quantities Using Equations of State: The Problem
of Reference States .........................................................................................
8.4.2 Mixing Rules Derived from Excess Gibbs Energy at Infinite Pressure ............
8.4.3 Mixing Rules and Excess Functions at Constant Packing Fraction ..................
8.4.3.1 Formulation of Equations of State Derived from the van der Waals Theory
in Terms of Packing Fraction
8.4.3.2 Calculation of the Helmholtz Energy A
8.4.3.3 Application to a Mixture and its Components
8.4.3.4 Results: Abdoul Group Contributions Method

8.4.4
8.4.5
8.4.6

.....................................................................
......................................................
..............................................
.............................................
The “MHV2”Method .....................................................................................
The Wong and Sandler Method .......................................................................
Advantages and Disadvantages of Mixing Laws Derived from Models and
Excess Functions .............................................................................................

292
293
298
302
302
303
304
307
309
313
315

XV

Table of Contents

8.5 Calculation of Liquid-Vapor Equilibria.............................................................

8.5.1 Newton Method ..............................................................................................
8.5.2 Tangent Plane Method ....................................................................................

316
319
321

Conclusion....................................................................................................................

324

References ....................................................................................................................

324

9

Liquid-Liquid and Liquid-Liquid-VaporEquilibria
9.1 Liquid-Liquid Equilibria and Deviations from Ideality...................................

330

9.2 General Description of Liquid-Liquid Equilibria.............................................

331
331
334

9.3 Selectivity ofthe Liquid-Liquid Equilibrium ....................................................

337

9.4 Liquid-Liquid-VaporEquilibria..........................................................................

339

9.5 Calculation Methods ............................................................................................

345

9.6 Water, Hydrocarbon Systems ..............................................................................

348
348

9.2.1 Binary Systems................................................................................................
9.2.2 Ternary Systems..............................................................................................

9.6.1 Total Immiscibility Hypothesis: Calculation of the Three-Phase Equilibrium ..
9.6.2 Application of Equations of State to the Calculation of Phase Equilibria for
Water Hydrocarbon Systems...........................................................................

351

Conclusion....................................................................................................................

352

References ....................................................................................................................

352

10

.

.

Fluid-Solid Equilibria Crystallization Hydrates
10.1 Liquid-Solid Equilibrium Diagram ..................................................................

356

10.2 Calculation of CrystallizationEquilibria .........................................................

359
359
363
365

10.3 Hydrates ..............................................................................................................

10.3.1 Generalities................................................................................................
10.3.2 Phase Diagrams..........................................................................................
10.3.3 Calculation of Hydrate Formation Equilibria .............................................

367
367
368
370

References ....................................................................................................................

372

10.2.1 General Equations .....................................................................................
10.2.2 Paraffin Crystallization...............................................................................
10.2.3 Fluid-Solid Phase Transition at High Pressure ............................................

XVI

Table of Contents

11

Polymer Solutions and Alloys
11.1 Polymers in Solution ..........................................................................................

377
377
379
382
383
386
389

11.2 Polymer Mixtures ...............................................................................................
Conclusion....................................................................................................................

389

References ....................................................................................................................

392

11.1.1 The Flory-Huggins Model ..........................................................................
11.1.2 The Influence of Free Volume ....................................................................
11.1.3 The Entropic-FV Model .............................................................................
11.1.4 The GC-Flory Model ..................................................................................
11.1.5 The GCLF Equation of State (Group Contribution Lattice Fluid) .............
11.1.6 Extension to Liquid-Liquid Equilibria........................................................

392

12

Multicomponent Mixtures
12.1 Pseudocomponents............................................................................................
12.1.1
12.1.2
12.1.3
12.1.4

Complex Mixture Analysis .........................................................................
Lumping.....................................................................................................
Thermodynamic Properties of Pseudocomponents.....................................
Representing the Heavy Fraction of Natural Gases ....................................

12.2 ContinuousThermodynamics...........................................................................

12.2.1 Definition ...................................................................................................
12.2.2 Chemical Potential, Fugacity Coefficient.and Equilibrium Condition
Between Phases ..........................................................................................
12.2.3 Application Examples ................................................................................

.......................................
...............................................................................
...............................................
Conclusion....................................................................................................................
References....................................................................................................................
12.2.3.1 Liquid-Vapor Equilibrium in an Ideal Solution
12.2.3.2 Excess Gibbs Energy of a Polymer Solution in Semicontinuous
Thermodynamics
12.2.3.3 Retrograde Condensation of a Natural Gas. Application of the
Soave-Redlich-KwongEquation of State

396
396
398
400
402
404
404
405
406
406
408
409
411
411

13

Chemical Reactions
13.1 ThennochemicalData .......................................................................................

13.1.1 Standard Enthalpy of Formation. Standard Gibbs Energy of Formation ....
13.1.2 Application of Group Contribution Methods .............................................

414
414
416

Table of Contents

13.1.3 “Coherent”Enthalpy Data .........................................................................
13.1.4 Standard Enthalpy and Gibbs Energy of Reaction .....................................
13.1.4.1 Definition and Calculation from Standard Enthalpies and Gibbs Energies
of Formation .....................................................................................
13.1.4.2 Dependence of Enthalpy of Reaction and Gibbs Energy of Reaction
on Temperature .................................................................................

XVI I

419
419
419
420

13.2 Heat of Reaction and Energy Balance ............................................................
13.2.1 Heat of Reaction ........................................................................................
13.2.2 Energy Balance of a Reactor or a Reacting Section....................................

421
421
424

13.3 Chemical Equhbria ...........................................................................................
13.3.1 The Equilibrium Condition ........................................................................
13.3.2 The Law of Mass Action .............................................................................
13.3.3 The Laws of Equilibrium Displacement .....................................................

425
425
426
429

13.4 Calculationof Simultaneous Chemical Equilibria .........................................

431

References....................................................................................................................

433

Appendix 1
Database .......................................................................................................................

435

Appendix 2
Lee and Kesler Method .Compressibility Factor. Residual Terms for Enthalpy.
Entropy. and Heat Capacity at Constant Pressure. and Fugacity Coefficient ......

443

Appendix 3
Surface Volume and Interaction Parameters Applied in the UNIFAC Method ..

477

Appendix 4
Properties of the Ethane (1) Propane (2) System at 45°Cand at 2.5 MPa as a
Function of Composition ...........................................................................................

479

Appendix 5
Detailed Analysis of a Straight-Run Gasoline Cut .................................................

483

Appendix 6
Units .............................................................................................................................

487

INDEX .........................................................................................................................

489

..

1

Principles
Thermodynamic Functions
The Ideal Gas

It would be inaccurate to say that the development of thermodynamics is not based on
experimentation. It is from observation that the concepts of quantity of heat, of temperature, of energy of a system, and of irreversibility have been developed. However, there was
an important turning point when the statements of the first and second laws allowed us not
only to “condense” the observations that produced them, but also to establish apriori laws
that subsequent observations have verified, to generalize from them and diversify the field
of their application. The greatest diversity exists, however, within the field of thermodynamics. Mechanical engineers, energy specialists, and chemists apply the same principles,
but they have constructed their own conventions reflecting their practical concerns.
Many statements of these principles can be set forth and it is not the goal of this chapter
to enumerate them or to demonstrate their equivalence. For the most part, they are familiar to the reader who may, if he so wishes, consult the general references listed in the bibliography. In this chapter, we will limit ourselves to a few topics emphasizing the points that
seem most important to us for the rest of this study.
First, we must review the terminology proper to thermodynamics. We derive our inspiration from the teachings of A. PCneloux [1992].

1.1

DEFINITIONS

Thermodynamics applies to a physical entity, the “system”,possibly composed of distinct
parts, or “subsystems”. The system is defined only if its physical limitations or “boundaries” are specified, as well as the nature of the exchanges that it may maintain with the
rest of the universe (“the surroundings”). The system is termed “closed”or “open”according to whether or not exchanges of matter are possible. For example, its properties may
change due to differences of temperature or pressure between the system and the surroundings. Its boundaries on the other hand may resist such transformations. The energy
exchanges with the surroundings will therefore have to be specified according to the
exchanges of matter. Similarly, certain changes may be excluded by virtue of internal

2

1. Principles. Thermodynamic Functions. The Ideal Gas

constraints; first and foremost, those changes defining the possible boundaries of subsystems that may be fixed or mobile, adiabatic or diathermic, impermeable or porous. It is also
well known that certain chemical reactions occur only with a catalyst or an initiator, that
other reactions may be inhibited and, for example, that we may apply the laws of thermodynamics to the solubility of air in hydrocarbons without being concerned about the possibility of combustion. As with boundaries, these constraints must also be specified in order
to describe a “system” and the changes that we may expect or exclude.
The most natural subsystems that we can define are made up of the phases into which
matter is organized. In particular, we shall study the equilibria between the liquid and
vapor phases. A “phase”forms a “homogeneous”physical entity in the sense that all the
parts of equal volume have the same properties (they have the same quantity of matter,
the same composition, etc.).
With a defined system, in principle we may describe its “state”by determining its “properties, or state values”, meaning the entirety of what is “observable”: temperature, pressure, volume, quantity of matter, and composition, for example. These properties are not
totally independent, and we know very well that the volume occupied by a system is fixed
from the moment we know the quantities of each component, the pressure, and the temperature.
Within the scope of this text, these properties can be defined only for “steady-state”systems whose state does not change over the course of time (and whose state does not
depend on its development). In fact, this steady-state is often due to the existence of
appropriate boundaries or constraints, and insofar as these boundaries are precisely
described, we will state that the system is in a state of “equilibrium”.This means that the
system will return to this state after any infinitesimal disturbance with respect to its boundaries. A state of equilibrium depends on the constraints imposed on the system; constraints
which we will take into account when defining and applying the conditions of equilibrium
stated in the laws of thermodynamics.
Some of these properties are “additive” in the sense that if we naturally or artificially
divide the system into several parts, such a property of the whole is calculated by using the
sum of the values of this property in each of the parts. For a homogeneous system, this
value is proportional to the size of the system, namely the quantity of matter. Volume is the
simplest example of such a property, but it is the same for internal energy, entropy, etc.,
which we will define later. Such properties are termed “extensive” and we shall consistently denote them by the capital letter symbols V U, S, etc.
On the other hand, other properties in a homogeneous system are independent of the
size of the system: density, pressure, temperature, etc. They are termed “intensive”and on
this subject, we recall the definition given by PCneloux [1992]: “phase refers to the entirety
of intensive properties”. Among these properties are some that govern the equilibrium
among the various parts of a system. If two parts are separated by a mobile boundary, the
pressures must be the same in each part. It is the same with temperatures if the boundary
is diathermic, and with the chemical potential (which we shall define in Chapter 5 ) of each
component if the boundary does not resist exchanges of matter. Such intensive properties
are “potentials” and intervene in conjunction with corresponding extensive properties:
volume, entropy, and quantities of matter. There are other intensive properties defined by

1. Principles. fiermodynamic Functions. The Ideal Gas

3

the value taken by an extensive property for the unit of matter (one mole): molar volume,
molar heat capacity,etc. These are sometimes referred to as “densities”.We shall give them
the same symbols,but in lower case: v, molar volume, u, internal molar energy, etc.
If certain constraints that assure the equilibrium of a system are removed, the system
generally undergoes a “transformation”, and changes via a series of intermediate states to
a new state of equilibrium. This change occurs by variation of its properties, and, in general, by “energy exchanges”with the surroundings, such as the work done by pressure, if its
volume changes, for example. In some cases, we can imagine that the opposite progression
might be possible. It would then be a “reversible” transformation, and in particular, it
might occur via a series of closely related states of equilibrium in which the properties of
the system do not change from one state to the next except in a finite manner. We are then
speaking about a “quasistatic”transformation. To accomplish this, the motive agents of this
transformation will continually adapt to the state of the system. Such is the case with a
compression process during which the external pressure remains practically equal to that
of the system, or with a transformation due to an infinitesimal difference in temperature.

1.2

THE FIRST LAW

1.2.1 The Energy of a System
The first law is based on attributing an extensive property to any closed material system,
namely energy, and on the establishment of an exact balance between the variations of this
property during the course of a transformation on the one hand, and on the other, the
work accomplished by the surrounding environment due to mechanical, electrical, and
magnetic forces, as well as so-called heat exchanges caused by differences in temperature.
Certain components of energy are familiar and independent of the internal structure of
the system. In the first place, they involve potential energy in the gravitational field EpOt,
which is expressed as a function of mass m,the acceleration of gravity g, and the calculated
elevation compared to an arbitrary reference 2.It is the same for the kinetic energy of the
entire system, Ekinr
that is related to mass, to the moment of inertia, and to the speed of the
system.
Mechanics is a function of these components. In thermodynamics we also take into
account “internal energy”, U,which originates from:
0 First and foremost, the kinetic energy that accompanies the random movement of the
molecules that make up the system (translation, rotation of all or part of the molecule, longitudinal or transversal vibrations of the interatomic bonds). It determines
the properties of the “ideal gas”.
0 Secondly, the intermolecular cohesive energy related to the forces of attraction and
repulsion between the molecules. Opposing the disordered kinetic agitation of molecules, they contribute to the organization of matter in different phases.
0 And finally, the interatomic cohesive energy that assures the stability of the molecular structure and is evident during chemical reactions.

4

1. Principles. ThermodynamicFunctions. The Ideal Gas

Of course, it would be appropriate to quantify the intra-atomic cohesive energies that
are much more numerous, but they do not occur in the phenomena studied here that
preserve the structure of the atom.
As with the potential energy in the gravitational field, internal energy can be calculated
only in relation to an arbitrary reference.
We therefore write:
(1.1)
E = Epot+ Ekin+ U
0

In the problems we present here, most often the variations of potential energy in the
gravitational field and in the kinetic energy of the system will be zero or negligible. The
variations of system energy will be reduced to variations of internal energy.

1.2.2 Energy Exchanges during a Transformation
When the system in question changes from one state of equilibrium to a new state of equilibrium due to the removal of certain constraints that insured the stability of the initial
state, this transformation is accompanied by energy exchanges with the surrounding environment. We shall now attempt to specify the nature of these exchanges. For the most part,
they are attributable to external forces acting on the system, and in particular, to the forces
the work supplied to the
of pressure. If the pressure acting on the system is designated Pext,
system can be expressed by the equation:

For this transformation to be reversible, it is necessary that the pressure acting on the
system Pextbe equal to the prevailing pressure within the system P, and the work dW,,,
will then be equal to:
6Wr,, = -P dV
(1.3)
If we take into account the sign of the variation of volume dV that is related to the Values of Pextand P respectively,we can easily convince ourselves that:

6Ws mY,,,

or

6Ws-PdV

(1.4)

It is similarly appropriate to point out, for example, the work due to the presence of
electrical and magnetic fields surrounding the system. In general they will be absent from
the transformations that we will study, and as a reminder we shall designate them here as
W’.
We also know that, due to the difference in temperature between the surrounding environment and the system, the state of the system may change. It is customary to state that
the surrounding environment “provides heat” to the system. Certainly this statement is
incorrect as neither the surroundings nor the system possesses heat. Nevertheless, we
retain this term as it has passed into practice and is unambiguous. These heat exchanges
shall be designated by Q, or for an infinitesimal transformation, by 6Q.

1. Principles. Thermodynamic Functions. The Ideal Gas

5

1.2.3 Statement of the First l a w Applied to a Closed System
1.2.3.1 General
The first law, applied to a closed system, establishes an exact balance between the variation
of the total energy of the system during any transformation, reversible or irreversible, and
the total work, and the quantities of heat absorbed by the system:

AE= A(E,,, + ELio+ U )= W + W + Q

(1.5)

the symbol A designates energy variation: E,, - Ehi~d.
Very often, when retaining only the internal energy variations, the work done by pressure and the thermal exchanges,we shall write the equation in the form:
AU=W+Q

(1.6a)

dU=@+6W

(1.6b)

or, for an infinitesimal change, as:

At this point, it is appropriate to consider a few particular cases that correspond to this
simplified version of the first law.

1.2.3.2

lsochoricTransformations

If the transformation in question takes place at constant volume and the work done by
pressure is equal to zero, the preceding equation is reduced to:

Auv = Qv

(1.7a)

dUv = @,

(1.7b)

or, for an infinitesimal change, to:
The thermal exchanges accompanying an isochoric transformation therefore correspond to the variation of the internal energy of the system.
If this transformation has as its only effect a variation in the temperature of the system
(there is neither a change in phase nor a chemical transformation),the heat exchangesmay
then be expressed as a function of the heat capacity at constant volume, Cv, and of the
temperature variation:
dUv = Cv dT
(1.8)

1.2.3.3 Adiabatic Compression
In this case, thermal exchanges equal zero, and the work done by adiabatic compression is
expressed as:
Wadiabatic = &quot;
(1.9)

6

1. Principles. Thermodynamic Functions. The Ideal Gas

1.2.3.4

Transformationsat Constant Pressure

If the external pressure is constant, and the system is in pressure equilibrium with the surroundings at both the onset and at the end of this transformation, the work done by pressure is expressed by the equation:
Wp=-Pext AV
(1.10)
and, if we take into account the pressure equilibrium conditions at both the beginning and
at the end of the transformation, we can write:
AV = A ( P V ) ,

cxt

where P stands for the system pressure, such that the first law allows us to calculate the
quantity of heat exchanged:
(1.11)
Q p= A( U + P V ) ,
The sum U + PV is the “enthalpy” of the system. It is an extensive property just like
internal energy and volume. It is more commonly used in practical calculations because
the previous equation relates these variations to isobaric heat exchanges. It is also used in
the expression of the first law applied to open systems, as we shall see subsequently. It is
designated by the symbol H
H=U+PV
(1.12)
such that we can write:
Qp

=m

p

(1.13)

Therefore the quantities of heat necessary for fusion, for the vaporization of a pure substance, or for a chemical reaction performed at constant pressure will often be called
“enthalpy of fusion” (negative of the enthalpy of melting), “enthalpy of vaporization” or
“enthalpy of reaction”. If the only effect of the transformation is to cause a variation in
system temperature, it would seem that the elementary enthalpy variation is expressed as
a function of heat capacity at constant pressure, and may then be written as:
dHp = C, d T

1.3

(1.14)

APPLICATION OF THE FIRST LAW TO AN OPEN SYSTEM

1.3.1 General
We have made it clear that the preceding equations relate to a closed system that does not
exchange matter with the surroundings. In practice, however, such exchanges are the rule
and it is appropriate to account for the energy contained within the flows of matter that
enter into or exit from the system in question, just as we will have to include their mass in
establishing a balance of mass.
Examples of open systems are varied: a balloon that we blow up, a reservoir that we fill,
a section of a pipeline, or a turbine. Several of these examples can be combined to make up
a less elementary open system. A distillation column, a chemical reactor, or a moving car

1. Principles. Thermodynamic Functions. The Ideal Gas

7

are other cases. In such systems, neither the pressure nor the temperature are uniform;
they change in space and time.
During a short interval of time, dt, the kinetic energy, the potential energy in the gravitational field, the internal energy of the system, and the mass of the system will vary. They
vary according to the type of content, mechanical work, and the heat and modifications of
the system properties (pressure, temperature, nature of components).
The system can be defined initially by the position of its boundaries (certain of which
are real, others imaginary). For example, a unit of pipeline will be marked between two
clearly indicated sections. We will denote flows of matter dmi, where m is mass, and assign
them a positive value if matter enters into the system, and a negative one if flows are leaving the system. We can also account for instantaneous outflows: dmi = Di dt. Each flow is
characterized by a certain number of properties: potential energy Epot,i,kinetic energy
Ekin,i,
temperature Ti,
pressure Pi,and, given here per unit of mass: volume wi , internal
energy ui,and enthalpy hi.
Heat transfers emanating from sources with temperature Ti will, however, be denoted
by Se,.
k will also be excluded, for example, the work done by presOther energy transfers, m
sure accompanying the possible variation in volume of the system (a balloon being blown
up), the energy supplied by the motor of a pump, etc. However, at this level we shall not
exclude the work accompanying the transfer of matter.
The total energy of the system Eo is:
EO

= EO,pot

(1.1)

+ EO,kin 0
+ '

Transfer of matter is broken down into two steps. Mass dmi occupies a volume w idmi ,
and possesses an internal energy ui dm, .
First, we shall unite the system itself with the mass dmi.This step therefore increases the
mass of the system by dmi, and its energy by:
dE, = epot,idmi + ekin,i dmi + uidmi
This step is not accompanied by work done by pressure; the volume of the system has,
however, varied by + v idmi.
In the second step, we will reduce the new system formed in this way to its initial boundaries. The variation in volume will therefore be -wi dm, and the work that we must supply
Piwi dmi. The transfer of matter therefore is accompanied by a total amount of energy
equivalent to:
epot,idmi + ekin,i
dmi + uidmi + Piwi dmi = (hi + epot,i+ ekin,i)dmi
In the total balance that describes the energy variation of the system, we will sum these
elementary transfers, as well as those corresponding to the work performed on the system
and the quantities of heat:
dEo =

c (hi+
i

epot,i+ ekin,j)dmi

+

c
i

@j

+

c

m
k

(1.15)

k

This very general expression of the first law obviously must be adapted to each particular case.

8

1. Principles. Thermodynamic Functions. The Ideal Gas

1.3.2 Steady-State Systems
We say that an open system is in a steady-state when there is no variation on any point in
the quantity of matter or energy, over time. Temperature, pressure, and composition may
vary from one point to the next within the system, but remain on all points invariable over
time.
The previous equation therefore may be integrated over any time period At with
dE, = 0. We denote by Q the quantity of total heat supplied to the system during this time
span, by W the sum of work (other than that corresponding to the transfer of matter), by
H2 the enthalpy of the flows of matter leaving the system, and by H I the enthalpy of the
flows of matter entering into the system:
H2 = - z h i D i At

if

D, d 0 (dm, G 0)

Hl = + x h i D i At

if

D, 3 0 (dm, 2 0)

However, we will exclude the variations in kinetic and potential energy of these flows of
matter. Under these conditions, the energy balance is written as:

H2-Hl=Q+W

or

AH=Q+W

(1.16)

Applied to an open system in steady-state, and, inasmuch as the variations of potential
energy in the gravitational field and the kinetic energy of the flow of matter may be
excluded,the first law establishes an exact balance between the variation of enthalpy from
the entry into, to the exit from the system on the one hand, and on the other hand the
quantities of heat and the work absorbed by the system excluding those accompanying
exchanges of matter.
This equation is closely related to the equation expressing the first law for a closed system. Enthalpy has been substituted for internal energy (to include the work of transfer).
Moreover, the operator A here corresponds to the difference produced between exit from
and entry into the system, while in a closed system, it corresponds to the difference
between the final state and the initial state. We must also recognize the importance of the
function of enthalpy. The energy balances of open systems in steady-state are enthalpic
balances. In this way, for example, the variation in enthalpy will correspond to the work of
adiabatic compression,or even to the heat balance of an item in the absence of work other
than transfer work. Finally,flow that takes place without exchange of heat or work is called
“isenthalpic”.

1.4

THE SECOND LAW

The first law in no way permits us to specify how the energy exchanges are distributed
(work done by pressure,heat, etc.) during the course of a transformation.It expresses only
the total energy exchange. However, we have seen that the work done by pressure is at
least equal to an attained limit in cases of reversible change. Are heat transfers also limited?

1. Principles. Thermodynamic Functions. The Ideal Gas

9

We know that the steady-state of a system is assured only by the existence of appropriate boundaries. If some are removed, the system will evolve toward a new state and the
reverse change will not be possible. Given two states, it is therefore important to know
which one is favored. It is equally important to specify the conditions that the properties of
the system must satisfy in order for its state to be favored in relation to all transformations
respecting certain conditions (transformations at constant temperature or pressure, for
example), and that it is in a state of equilibrium.
It is to such questions that the second law of thermodynamics responds. Its statement
varies considerably from one author to another, and an in-depth analysis shows that it is in
fact inseparable from the first law.

1.4.1 Entropy
The statement of the second principle, the definition of the function of entropy and that of
the absolute temperature scale, are inseparable:
Given a system described by its volume, internal energy, and the quantity of matter for
each of its components, there exists a property of the system, entropy S:
It is an extensive function.
0 During a transformation its variations are broken down into two terms:
ds=ds,+ds,
(1.17)

The term dSeis related to heat exchanges by the equation:
SQ
a,=T

(1.18)

where T is a property of the system that depends solely on its temperature, and we designate this property as “absolute temperature”.
dsi is a term related to internal modifications of the system. It is always positive for
spontaneous transformations, and zero for reversible transformations.

Consequently:

dSt20

(1.19)

ds&gt;-SQ

(1.20)

T

The previous definitions leave the signs undetermined (which are related) for absolute
temperature and for entropy. As a convention, we state T &gt; 0.
We see later on that the properties of low density fluid allow us to determine the relationship of two absolute temperatures. The absolute scale will therefore be fixed by adopting a fixed point; by convention we state:
at the triple point of water
T = 273.16

It appears that for an isolated system, and therefore in the absence of heat exchange
with the surroundings, entropy can increase only during irreversible processes and stabi-

10

1. Principles. ThermodynamicFunctions. The Ideal Gas

lizes itself at a maximum value when equilibrium is attained. Of course, in the case of heat
exchange, entropy may decrease.
Moreover, the Equation (1.20) may be written as:
6QcTdS

(1.21)

and thus corresponds to the limit mentioned earlier (Eq. 1.4) for the work done by pressure:

6W36Wre,

or

6Wa-PdV

(1.4)

1.4.2 Relationship between Internal Energy and Entropy
The variation of internal energy between two states may be arrived at by observing any
transformation connecting the two states. We may do this using a quasistatic transformation during which, if the work done to the system is limited to the work done by pressure,
we will always have:
6Wr=-PdV

and

6Qr=TdS

(1.22)

We therefore obtain the fundamental expressions that tie the first and second law
together:
dU=TdS-PdV
(1.23a)
1
P
dS=-dU+-dV
T
T

and:

(1.24a)

These equations generally pertain to one mole:
du = Tds - P dv
1
P
ds=-du+-dv
T
T

(1.23b)
(1.24b)

Of course, if other forms of energy transfer intervene (for example, due to the existence
of an electromotive force), it is appropriate to take them into account and we would write:
dU = T dS - P dV +

zFidli

(1.25)

Within these expressions appear the intensive values, or potentials, P, T, Fi, and the
extensive values with which they are associated S, V, and li .
These equations in no way allow for the calculation of work done on the system or the
heat transfers during the course of any process. We can state only:

Q.1

2
1

TdS

and

W s I 21 - P d V

(1.26)

1. Principles. Thermodynamic Functions. The Ideal Gas

11

1.4.3 Application of the Equilibrium Condition
This equilibrium condition shall be illustrated using an isolated system, namely one that
exchanges neither heat nor work (of any kind) with the surroundings:
dW= 0, dQ = 0
The internal energy and the volume are therefore constant:
dU = 0, dV= 0
and, according to the second law, entropy can only increase or be at maximum:
dS 3 0
We will split this system into two subsystemsA and B, using a partition that will be fixed
initially,adiabatic,and impervious to all flow of matter such that each of these two subsystems is also isolated. On either side of the partition temperature, pressure, and composition may be different.
Let us now suppose that the partition is diathermic,that is to say that it allows for heat
flow. As internal energy and the entropy are extensive functions and the system is isolated,
we have:
dU= dUA+ dUB= 0 and dS = dSA+ dS,
We shall express the entropy variations of each part A and B by applying Equation 1.24a. Since the volumes VA and V, are unchanged and the total internal energy is
constant, we obtain:

As entropy can only increase, the result is that dUAand TA- TBare opposite in sign. If
we now suppose that the partition is at the same time diathermic and mobile, we must take
into account the variations in volume dVAand dV,, which are related:

dVA+ dV, = 0
and would write:
and thus:

and, finally:

The maximum condition imposed on the system by the second law thus yields the
equality of temperatures and pressures.

12

1. Principles. Thermodynamic Functions. The Ideal Gas

Finally, if we had considered that the partition separating the two subsystems was not
impermeable,we would have had to use the derivatives of entropy as related to quantities
of matter, and would have established the condition of equality of chemical potentials.This
condition will be examined later (Chapter 5).

1.4.4 Statistical Significance of Entropy
In statistical thermodynamics we can calculate the number of configurations Wthat a system can assume, given the energy and volume. It is possible to show that this number is
related to entropy by the equation:
S=klnW
(1.27)
6 being the Boltzmann constant.
In this way entropy is a measure of the disorder of the system, of its indetermination.
This equation must be retained in order to interpret the sign of the variations of entropy
with volume, the entropy of mixing (Chapter 5), and to establish the expression of combinatorial entropy (Chapter 7).

1.5

HELMHOLTZ ENERGY AND GIBBS ENERGY

We have examined the condition of equilibrium and change in an isolated system. Any
possible change may be characterized as internal rearrangement, and is possible only with
an increase in entropy. In practice, we will deal with transformations during which the system exchanges heat and work with the surroundings. In order to take these exchanges into
account,we relate the two equations:
d U = 6Q+ 6 W + 6W'
in which 6W represents the work done by pressure and 6W' the other work (electrical,
etc.), and:
6QsTdS
(1.21)
Substituting 6Q we obtain:
4

6W + 6W' b dU - T dS

During the course of a change at constant temperature, this equation may be written as:
6W + 6W' 3 d( U - TS ),
or W + W' A( U - TS ) T
(1.28)
Furthermore, if the system changes at constant volume, then the work of pressure force
is zero and this equation becomes:
sW'&gt;d(U-TS),,
or W'bA(U-TS)
at constant temperature and volume
The function:
(1.29)
A=U-TS
is called the Helmholtz energy.

1. Principles. Thermodynamic Functions. The Ideal Gas

13

The preceding equations are therefore written as:

6W’3 dA,, or W’ 3 AA
at constant temperature and volume

(1.30)

A change at constant temperature and volume is therefore possible only if the
Helmholtz energy decreases. In this case, the system may give off a quantity of energy
(-W)to the surroundingsthat is at most equal to the decrease in the Helmholtz energy. If
the Helmholtz energy increases, for the change to be possible, it would be necessary to
supply to the system (in the form of electrical energy, for example) a quantity of energy
that is at least equal to the increase in the Helmholtz energy. Since we are presuming that
there are no such exchanges here, we shall retain the condition of change:
dAT,V

(1.31)

dAT,V=o

(1.32)

and the condition of equilibrium:
If the system changes at constant temperature and pressure, and in pressure equilibrium
with the surroundings,the work done by pressure is then expressed by the equations:

6W=-PdV

or

W=-PAV

such that we would write Equation 1.28 as:
or W’ 3 A(U + PV- TS)
at constant temperature and pressure

6W’3 d(U + P V - TS),,,
The function:

G = U + P V - TS = H- TS = A

+PV

(1.33)

6W’3dGT,,
or W’3AG
at constant temperature and pressure

(1.34)

is called the Gibbs energy.
The previous equations therefore may be written as:

Thus, a change at constant temperature and pressure is possible only if the Gibbs
energy decreases. In this case, the system may give off a quantity of energy (-W)to the
surroundings that is at most equal to this decrease. If the Gibbs energy increases, for the
change to be possible, it would be necessary to supply to the system (in the form of electrical energy, for example) a quantity of energy that is at least equal to the increase in the
Gibbs energy.
Since we presume that there are no such exchanges, we shall retain the condition of
change:
(1.35)
dGT3 &lt; 0
and the condition of equilibrium:
dG,

=0

or more precisely, minimum G at constant temperature and pressure.

(1.36)

14

a

1. Principles. Thermodynamic Functions. The Ideal Gas

EXAMPLE 1 .I

Condensation by adiabatic expansion
To illustrate the use of thermodynamic properties in the calculation of compression
work and heat exchanges, we will examine an ethylene condensation process that
functions according to the steps that follow.
Ethylene, available at atmospheric pressure and at 100°F (37.8&quot;C), is compressed in
an isothermal fashion to 50 bar. The fluid then undergoes adiabatic expansion to
return to atmospheric pressure. During this expansion, there is partial condensation;
the vapor phase is recycled.
It is necessary to calculate the proportion of liquefied ethylene, the work and heat
exchanges accompanying the compression step, and the work that may be collected
during expansion. To do this, we make use of a diagram (Fig. 1.1) that provides the
pressure variation, shown on the ordinate, as a function of enthalpy, shown on the
abscissa, along the isothermal or isentropic curves. These calculations will force us to
make certain hypotheses relative to the reversibility of each step. Most often, the laws
of thermodynamics do not allow us to estimate the exchanges of heat and work
except in the case of reversibility.

Figure 1.1 Diagram of ethylene enthalpy pressure
[Canjar and Manning, 19671.

15

1. Principles. Thermodynamic Functions. The Ideal Gas

First, a few brief comments on the diagram: given its origin [Canjar and Manning,
19671, the units are British and the necessary conversions will be done using the factors supplied in Appendix 6.
We note that on the diagram the two-phase envelope curve takes the shape of a
dome. It is made up of two parts that relate to the two phases, liquid and vapor in
equilibrium. At a given temperature, the enthalpy of the liquid phase is less than that
of the vapor, and we may therefore easily describe the two parts that meet tangentially at the critical point (T, = 282.4 K, P, = 50.3 bar).
In the vapor phase zone, at low pressure, the isotherms are practically parallel to the
pressure axis, conveying the fact that these vapors are close to the state of an ideal gas,
the state at which enthalpy does not depend upon pressure, as we shall see at the end
of this chapter. If the temperature is less than the critical temperature, the isotherm
curve drops upon crossing the two-phase envelope to a constant pressure plateau that
is equal to the vapor pressure at the temperature in question. In effect, under these
conditions, the system is monovariant.
As for the necessary hypotheses, we will assume that the compression and expansion
steps are reversible. As it is adiabatic, expansion is isentropic.
On the diagram we locate the points that are characteristic of the process. We note
that at the end of isentropic expansion, the point representative of the system is found
within the two-phase zone. The liquid and vapor phases in equilibrium are represented by two points situated on the isobaric and isothermal plateau, and on the twophase envelope. The table below gives the properties corresponding to each of these
points.
Table 1.1
Properties of ethylene during a liquefaction cycle
State
Vapor
Vapor
Two-phase
Liquid
Vapor

Temperature
(OF)

100
100
-154.6
-154.6
-154.6

Pressure
(Psi)

Enthalpy
(Btu/lb)

Entropy
(Btdlbl'R)

14.7
735
14.7
14.7
14.7

1100
1 060
990
815
1 020

1.89
1.55
1.55
1.02
1.70

We note that the properties of the saturated phases are related by the equilibrium
equation:
gvP-gL'J = 0 + (hVP- hLP) - T(sVK'- &amp;'J) = 0
(1.37)
The proportion of condensed ethylene will be determined by application of an
enthalpy balance to the two-phase state. Exponents Lo and Vo respectively denote
the liquid and vapor saturated states. For enthalpy and mass, we use the following balances:
m = mv,'J+ mL.a
(1.38)
m .h = mvPhv.0 + mL.'JhLP

16

1. Principles. ThermodynamicFunctions. The Ideal Gas

and thence derive:
mL.0

--

m

-

hv*a- h
1020 - 990
= 0.146
hv~a-hL,&quot;- 1020-815

(1.39)

To determine energy exchanges during the isothermal compression, we apply the first
law considering that the system is an open system in steady-state:
(1.16)

AH=Q+W
2

and the second law:

Q s j ' TdS

(1.40)

As compression is reversible and isothermal, we have:
and

Q=TAS

(1.41)

WT = AH- Q = AGT

(1.42)

Calculationswill be performed using the units provided in the diagram. In particular, we
note that the temperatures are expressed in degrees Rankine (&quot;R).Toconvert to Kelvin:
T(&quot;R) = 1.8. T(K) = @ ( O F )

+ 460

As the unit of energy is Btu/lb, we have:

1lb = 453.6 g,

1Btu = 252.cal= 1054 J

and

1Btu/lb = 2.324 J.g-'

So, we arrive at:
T = 560&quot;R
Q = 560*(1.55- 1.89) = -190.4 Btdlb = -442.5 J-g-'
W = (1060 - 1100) - 560 *(1.55- 1.89) = 150.4 Btu/lb = 350 J-g-'
In order for the expansion to be reversible, it is appropriate to recover the work of
expansion. We will apply the previous Equation (1.16) to calculate this work, while
taking into account that there is no exchange of heat, and therefore:
W=AH
(1.43)
for an adiabatic transformation
or:

W = 990 - 1060 = -70 Btdlb = -163 J-g-'

In reality, compression and expansion are irreversible to a certain degree, and the Values above represent the limits. Furthermore, compression is accomplished in two (or
more) adiabatic steps, and therefore is accompanied by an increase in temperature.
Each increase is followed by cooling to restore the initial temperature. Thus the first
step can be substituted with two isentropic compressions going from atmospheric
pressure to 7 bar, and then from 7 to 50 bar. The calculation shows that in this case the
work that must be provided is in the neighborhood of 400 J .g-l and the heat to be
released approximately -500 J * g-'.
Expansion may also take place without recovering work. We would have Q = 0 and
W = 0 at the same time. Expansion would therefore be isenthalpic. With the help of
the diagram, we may verify that it would not allow for ethylene condensation if the
compression step were not modified, and that the compression should reach approximately 120 bar to achieve the same rate of condensation.

1. Principles. Thermodynamic Functions. The Ideal Gas

17

EXAMPLE 1.2

Methanol battery
The electromotive force of a methanol battery operating in an alkaline environment
at atmospheric pressure and under reversible conditions is theoretically equal to
1.33V at 25°C.
In practice, the following results were observed in the laboratory under the same
conditions of temperature and pressure: a potential difference of 0.2 V and a heat
release of 721 kJlmol from oxidized methanol.
The reaction is described by the stoichiometry:

CH30H+80H--6e
3
-00,+3H20+6e
2
CH30H +

+

CO;-+6H20

+

60H-

3
0, + 2 OH-+
2

CO 3&quot;-

+ 3 H,O

The reactants and the products are in an aqueous liquid solution with the exception of
oxygen, which is gaseous.
These data allow us to determine the variations of the thermodynamic functions
accompanying the oxidation reaction, as well as the exchanges of heat in the case of
reversibility or free combustion.
First, we note that the variations of the thermodynamic functions will be the same
regardless of the degree of irreversibility of the process. Indeed, the initial and final
states are the same.
The work done by pressure can be easily calculated since the pressure of the surroundings is constant.
Wp = -P AV = -A(PV)
The variation in volume results from the consumption of oxygen.We apply the equation for the ideal gas state:
A ( P V ) = RTAv
where Av denotes the variation in the number of moles (-1.5 mol) and R
(8.3145 J-mol-l K-') denotes the ideal gas constant. We find that for a mole of
methanol oxidized: W, = 3.7 kJ.
Furthermore, we can calculate the electrical energy involved: If E designates the
electromotive force, N Avogadro's number (6.022 x
e the electron charge
(e = 1.602 x
C), and An the number of electrons transferred, on the basis of the
preceding stoichiometry (An = 6), then:
W' = N AneE = 96472 AnE

-

We thus find that W' = -770 kJ in the case of reversibility (E = 1.33 V) and
W' = -115.7 kJ from the results observed (E = 0.2 V). This energy is considered negative as the setup produces energy (electromotiveforce).

18

1. Principles. ThermodynamicFunctions. The Ideal Gas

By applying Equation (1.34) in a reversible case, we derive:
AG = W:ev = - 770 kJ
at constant temperature
In the case of an actual working battery, we first determine that the electrical energy
given off to the surroundings is effectively less than the decrease in the Gibbs energy:
-W’ &lt; -AG. We can further group the energy exchanges: W = 3.7 kJ, Q = -721 kJ and
-W’ = 115.7 kJ. We can therefore calculate the variation of internal energy:
A U = W + W’+Q=-833kJ
from which we derive the variation of enthalpy:
AH = A U

+ A(PV) = -836.7 kJ

and of the Helmholtz energy:
AA = AG - A(PV) = -766 kJ
The entropy variation is derived from the values of AH and AG:
AS =

AH - AG
= -223.7 J. K-’
T

It is from this value that we can calculate the heat exchanges in the case of reversibility:
Q = T AS = -66.7 kJ
Finally, if there is no production of electrical energy (free combustion):

Q = AH = -836.7 kJ
Table 1.2 below summarizes these results (units: kilojoule, kelvin).

Table 1.2
Variation of thermodynamic properties, heat exchanges,
and work during the oxidation of methanol

AU
AH
AS

AA
AG

Q
W
W’

Reversible Battery

Experimental Battery

Free Combustion

-833
-836.7
-223.7
-766
-770

-833
-836.7
-223.7
-766
-770

-833
-836.7
-223.7
-766
-770

-66.7
3.7
-770

-721
3.7
-115.7

-836.7
3.7
0

19

1. Principles. Thermodynamic Functions. The ideal Gas

1.6

THERMODYNAMIC FUNCTION: INTERNAL ENERGY, ENTHALPY,
ENTROPY, HELMHOLTZ ENERGY AND GIBBS ENERGY

1.6.1 Dependence on Temperature, Volume, or Pressure
Helmholtz energy,A, and Gibbs energy, G, are, like internal energy, enthalpy, and entropy,
extensive functions. A change in each one of these properties allows us to judge the possibility or impossibility of a transformation, in accordance with the constraints that are
maintained over the course of this transformation. Thermodynamic equilibrium is related
to an extreme condition of any one of these functions. It is for this reason that the functions may be called “thermodynamic potentials”. It is therefore essential to know their
variation as a function of the properties of the system.
As independent variables we will consider temperature, and either volume (for internal
energy, entropy, and Helmholtz energy), or pressure (for enthalpy, entropy, and Gibbs
energy). To arrive at the elementary variations of the thermodynamic functions we will use
the fundamental Equation 1.23:
dU=TdS-PdV
(1.23a)
Taking into account the definition of Helmholtz energy and Gibbs energy (Eqs. 1.29
and 1.33) one may write:
(1.44a)
dA = -P dV- S d T
dG = V d P - S d T
(1.45a)
or for one mole:

(1.44b)
(1.45b)

da = -Pdv --s d T
dg = v d P - s d T

Since the internal energy, the Helmholtz energy and the Gibbs energy are state functions, and dU, dA,and dG, are exact, total differentials, we derive the following equations
from them:
a2A - a2A
that is
(1.46)
aTav avaT
V

(g)T=
(g)

(1.47)
Similarly,we obtain the derivatives of internal energy and enthalpy as they relate to the
volume and to the pressure respectively:
U =A

+ TS

gives ( g ) T = ( g ) T + T ( $ ) T

and:
H = G + TS gives

au

andthus i a v ) , = T ( g )

(g)T=
(g)T+
T (g)Tand thus

aH

(%);

V- T

V- P

(g)

P

(1.48)

(1.49)

We have already seen that the elementary variations of internal energy at constant volume, or of enthalpy at constant pressure as a function of temperature, was expressed, for a
homogeneous mixture, using the corresponding heat capacities Cv or C, (Eqs. 1.8 and
1.14). The same is true of course, for the elementary variations of entropy.

20

1. Principles. ThermodynamicFunctions. The Ideal Gas

We list below (Table 1.3) the expressions for the thermodynamic function differentials
represented by TV for internal energy, entropy, and Helmholtz energy, and by T P for
enthalpy, entropy, and Gibbs energy. These expressions do not include the variable for
quantity of matter, which must be considered for systems of variable composition. They
will be given subsequently (Chapter 5 ) . Since these equations are all related to extensive
properties, if we apply them to a pure substance,or to a homogeneous mixture of constant
composition, these equations will be preserved while showing not the property of the system (U,H, S, A, G, V),but the molar property: u, h, s, a, g, v.
Table 1.3
Variation of thermodynamic functions with temperature and either volume or pressure

dT
dS=Cv---+(;)
T

V

dV

or

du=c,dT+

(1.50)

dP

or

dh=c,dT+

(1.51)

dV

or

dT
ds=c,- T

(1.52)

+&amp;(!),

(1.53)

dA=-PdV-SdT
dG=VdP-SdT

or
or

da=-pdv-sdT
dg=vdP-sdT

These equations are fundamental. Indeed, it is with the help of the derivatives of the
thermodynamic functions with regard to volume or pressure that we shall arrive at the
“deviations from the ideal gas law”. Here we invoke the Gibbs-Helmholtz equation
applied to Helmholtz energy and Gibbs energy:

(1.54)

(1.55)

Their proof is very simple.For example, for the second equation we can write:
a-G
H
- - G + - 1 aG
- = - - (1H - T S ) - - S =1- T (3T)p
T2
T
T2

(&amp;),=i2

1. Principles. Thermodynamic Functions. The Ideal Gas

21

They should come close to a very common graphical representation in physical chemistry: we use the variation of the logarithm of a value such as vapor pressure, the
liquidhapor equilibrium constant, the solubility of a gas, the equilibrium constant of a
chemical reaction, or the rate constant, as a function of the inverse of absolute temperature. In fact, for each of these values there exists an equation relating it to the variation of
Gibbs energy, AG, accompanying the described phenomenon: Gibbs energy of condensation, of reaction, or of formation of an activated complex, etc. and the graphical representation, allows us to obtain the variation of enthalpy, AH, of this same phenomenon.
This enthalpy variation itself is most frequently only slightly sensitive to temperature,
and the graph obtained is linear.
If the variation of temperature causes a phase change or a chemical reaction, it is then
no longer possible to state that dUv = C , dT or dHp = Cp dT and it is necessary.to introduce the “latent” heat associated with this phase change or with the heat of reaction. On
the other hand we may, with the help of Equations 1.51 or 1.55, calculate the variation of
the heat of reaction or the Gibbs energy of reaction with the temperature, while pressure
remains constant, provided that the systems formed by (1)the reactants, and (2) the products, do not themselves undergo a phase change. We will then write:

(1.56)

1

R

(1.57)

where AG and AH are, respectively, Gibbs energy and enthalpy of reaction, and AC, represents the difference between the heat capacity of the products and reactants at constant
pressure, taking into account the stoichiometry (see Chapter 13).
The heat capacities at constant volume and pressure are related, as are the internal
energy and the enthalpy.We may thus write:
dU = d(H- PV) = dH- P dV- VdP
which is to say that by application of Equation 1.50:

[ (31

dU=CvdT+ T - - P dV=dH-PdV-VdP
or:

CvdT+T - dV=dH-VdP
(

3

V

If we are looking at an isobaric process, we would write:
CvdT+T($)

and thus:

(E)P d T = d H p = C p d T

v aT

Cp-Cv=T($)

v (E)
P

(1.58)

22

1. Principles. ThermodynamicFunctions. The Ideal Gas

We can only use the derivatives at constant pressure or at constant volume by applying
the equations:

or:

dV= - d T + - d P
(zF)p

(zF)T

andthus

Thus is obtained:
(1.59)

1.6.2 Characteristic Functions
We return here to the particular role played by each of the functions that we have defined:
the second law may be expressed using any one function from among them, according to
the constraints applied to the system:
Internal energy, U:minimization at constant entropy and volume
Enthalpy, H:minimization at constant entropy and pressure
Entropy, S: maximization at constant internal energy and volume
Helmholtz energy, A: minimization at constant temperature and volume
Gibbs energy, G minimization at constant temperature and pressure
Furthermore, knowing one of these functions as a function of the variables thus related
to it, in fact provides all the necessary information.We shall use the example of Helmholtz
energy.
If the function A (T,V) is known, one disposes also by applying Equation 1.44 on the
pressure expressions (equation of state) and the entropy as a function of these variables:
(1.60)
(1.61)
and the internal energy, enthalpy, and Gibbs energy are obtained immediately:

(3,

U(T,V) = A + TS =A(T,V) - T -

(1.62)

1. Principles. Thermodynamic Functions. The Ideal Gas

23

H(T,V)=U+PV=A(TV)-T - -V -

(1.63)

V):(

( g ) T

MT

G(T,V)=A+PV=A(T,V)-V -

(1.64)

To obtain the heat capacity at constant volume, we can apply Equation 1.52:

Cv = T ( g )
V

and thus, according to Equation 1.61:
(1.65)
Heat capacity at constant pressure is obtained from the heat capacity at constant volume
by application of Equation 1.59,which utilizes only the derivatives of pressure expressed in
1.60, compared to volume at constant temperature or temperature at constant volume.
This calculation of all of the thermodynamicproperties from the characteristic function
A (T,V) may be used for other characteristic functions, G (T,P),for example.
In fact, more often than not, we do not use the specified characteristic function as a
function of its own variables, but rather the equation of state P (TV) or V (T,P).We may
also use the expression for the residual or configuration property that expresses the difference between the characteristicfunction and the value that this function would have if the
fluid were an ideal gas. As we shall see, these data (equation of state and residual property)
are equivalent,but insufficientin the sense that it is also necessary to know the variation of
heat capacity of the ideal gas with temperature.
Note that all the preceding equations, written for a system containing N moles and involving the extensive values U,H, S, A, G, Cp,etc, may be used for molar values: v, 4 h, s, etc

1.7

THE IDEAL GAS

1.7.1 Equation of State and Thermodynamic Properties
It is important to recall the definition and properties of the ideal gas. Firstly, it represents
the limit state of the real fluid when the density (mass/volume) approaches zero (often
misstated as when pressure approaches zero). It also makes up, as we shall soon see, a
constant step in evaluating the thermodynamic properties.
In the ideal gas state, the forces of intermolecular cohesion are null, and the effective
volume of the molecules is also null. Knowing this fact, the equation of state that will form
the definitive equation is especially simple.From now on, we shall denote the fluid properties in the ideal gas state by the exponent #, with N representing the number of moles, and
therefore would write:
PV' = NRT
(1.66a)

24

1. Principles. ThermodynamicFunctions. The Ideal Gas

or for one mole:

Pv' = RT

(1.66b)

R, the ideal gas constant, is expressed using the unit that is most appropriate to the calculation being performed:

R = 8.314 J-rnol-l*K-'
or:
R = 1.987 cal.rnol-l.K-l= 83.145 b a ~ c r n ~ . r n o l - ~ .=K82.058
- ~ atrn*~rn~*rnol-~.K-~
If we apply the equations in Table 1.3, we end up with the results listed in Table 1.4 (for
one mole of ideal gas).
Table 1.4
Variation of thermodynamic functions of the ideal gas
with temperature and either volume or pressure

du# = C; dT

(1.67)

= c! d T

(1.68)

dh'
&amp;#=c[-

da# = -RT

d T + R -dv =c&amp; d T -R- d P
T
v
T
P

dv

- - s# d T

(1.69)

or

da; = -RT d In v

(1.70)

or

dg:=-RTdInP

(1.71)

V

dg'=RT-

dP
-s#dT
P

In this way, internal energy and enthalpy of the ideal gas depend solely on temperature.
The same is true for heat capacities at constant volume or constant pressure between
which exists the Mayer equation:
C;-C:=
R
(1.72)
But entropy, Helmholtz energy, and Gibbs energy depend on both temperature and volume (or pressure). However, we note the particularly simple formula that relates Gibbs
energy and pressure. We know that for ideal gas, pressure provides us with a convenient
scale of Gibbs energy. Several simple expressions of equilibrium conditions result from
this equation; notably Raoult's law for liquid-vapor equilibria, and the Guldberg and
Waage law for chemical equilibria,where partial pressures play a role.
Thermodynamic properties in the ideal gas state are often related to an arbitrary reference temperature To,and a &quot;standard&quot; pressure of 1bar, designated as Po.
At this standard pressure we can write:

+I

T

ho(T,Po)= ho(To,Po)

TO

c;(T,Po) dT

(1.73)

25

1. Principles. Thermodynamic Functions. The Ideal Gas

s&quot;(T,P&quot;) =s&quot;(To,P0)+

c,&quot;(T,P&quot;)

I

To

and at pressure P:

dT

(1.74)

c,&quot;(T,P&quot;) dT

(1.75)

T

+I

T

h#(T,P) = h&quot;(T,P&quot;) =h&quot; (To,P&quot;)

TO

s#(T,P) = s&quot;(To,P&quot;) +

dT-Rln-

P
P&quot;

(1.76)

1.7.2 Heat Capacity
It is thus clear that the properties for the ideal gas can be known as soon as we have the
values for heat capacity c,&quot;. We know that the kinetic theory of gases yields particularly
simple results in the case of mono and diatomic compounds. Heat capacities are practically
invariant with temperatures close to 5 cal mol-' K-' for noble gases, and 7 cal mol-' * K-'
for diatomic compounds under ordinary conditions of temperature. For more complex
molecules, heat capacity depends on the nature of the compound in question and on temperature. In this calculation, we must take into account the rotational movement of the
groups composing the molecule (and possibly the barriers to rotation), and the longitudinal and transversal vibrations of the interatomic bonds. This calculation is supported by
spectroscopic data. It generally results in complex expressions that are difficult to integrate (which is a major inconvenience when calculating the variations of enthalpy and
entropy), and it is generally preferable to apply more empirical equations that permit us to
obtain the results of rigorous calculations with all the precision required.
Polynomial expressions:
c;= a + b T + cT2 + dT3 + ...
(1.77)
in which a, b, c, d are empirical variables dependent on the component under examination,
are quite often used. However, it is advisable to avoid all extrapolation beyond the temperature range for which these expressions have been formulated, a range that should be
known. In general, they may not be used at low temperature (below 25°C). Therefore, the
expression proposed by Aly and Lee [1981] is preferable:

&quot;=

-t

D/T
(D/T)

[ sinh

F/T

] [ cosh (F/T) ]

(1.78)

+

The expressions corresponding to enthalpy and entropy are:
h&quot;(T) = h&quot;(T,,)
s&quot;(T) = so( To) +

I. [s
In T + C

coth

+

(g)

-In sinh (;)]-E

(1.79)

;[

tanh (;)-In

cosh

(:)]Lo

(1.80)

26

1. Principles. Thermodynamic Functions. The Ideal Gas

In these equations, B, C, 0,
E, F a r e the empirical coefficients that depend on the nature
of the compound in question.
As an example, we can also cite the formula applied by Younglove and Ely [1987] to the
calculation of heat capacity for light hydrocarbons (methane, ethane, propane, n-butane,
and isobutane):
(1.81)

1.7.3 Data Tables
The literature supplies the values of thermodynamic properties for a large number of pure
substances in the “standard” state, that is, for most substances that we encounter, in the
ideal gas state, and at the standard pressure of one bar (0.1 MPa). The principal exceptions
are carbon, which is in the graphite state, and sulfur, whose state varies according to the
tables.
These properties are often supplied for a number of temperatures, and at least for
298.15 K. They contain molar heat capacity, enthalpy, and entropy. For enthalpy, the starting point is arbitrary: sometimes we find the values for h’+- hO, ,that is, the enthalpy relative to a temperature of 0 K. Any other starting point is acceptable. For entropy, most of
the tables adhere to the “third law of thermodynamics”, or to the Nernst theorem, according to which “the entropy of any pure, solid, crystallized substance is null at absolute zero”.
There is no disadvantage in selecting an arbitrary starting point as long as the calculations
are not for a chemical reaction (see Chapter 13).
Such data, for a very limited number of compounds, are found in Appendix 1.

REFERENCES
General Works
Bett KE, Rowlinson JS, Saville G (1975) Thermodynamics for Chemical Engineers. The Athlone
Press, London.
Callen HB (1960) Thermodynamics. John Wiley and sons, New York.
Dodt M (1956) Bases fondamentales et applications de la thermodynamique chimique. SociCtC d’Bdition d’enseignement supBrieur,Paris.
Model1 MM, Reid RC (1983) Thermodynamics and its Applications. Prentice-Hall, Englewood Cliffs,
New Jersey.
PBneloux A (1992) Cours de Thermodynamique, UniversitC d Aix-Marseille.

Specific References
Aly FA, Lee LL (1981) Self consistent equations for calculating the ideal gas heat capacity, enthalpy
and entropy. Fluid Phase Equilibria, 6,169-179.

1. Principles. Thermodynamic Functions. The Ideal Gas

27

Canjar LN, Manning FS (1967) Thermodynamic Properties and Reduced Correlations for Gases.Gulf
Publishing Corporation, Houston,Texas.
Younglove BA, Ely JF (1987) Thermophysical properties of fluids. 11. Methane, ethane, propane, isobutane and normal butane.J. Phys. Chem. Re$ Data, 16,577-797.

2

Properties of Pure Substances

The equations presented in the previous chapter show that the determination of equilibria
or of energy balances implies the evaluation of thermodynamic properties.
Due to the existence of intermolecular cohesion forces, the calculation of these properties in the ideal gas state is only a first step. Since the systems we deal with in real life are
themselves mixtures,a knowledge and understanding of the behavior of pure substances is
itself but a second step, and generally an essential one. To study the behavior of a mixture,
we often apply simple rules of weighting to the properties of its components.Any imprecision during this second step inevitably affects the final result. For example,we cannot hope
to correctly calculate the liquid-vapor equilibria of mixtures if the vapor pressures for pure
substances are in error.
Although sometimes merely a review of well known information,knowledge of some of
the values is almost indispensable to applying the methods that will be developed later on.
To a certain extent, these values make up the minimal “database”that, although limited,
nevertheless allows us to understand the thermodynamicproperties across a wide range of
temperature, pressure, and composition. They are the critical points, vapor pressure, and
heat of vaporization.
We also stress the continuity that exists between the liquid and gas states. This continuity warrants, sometimes even imposes, that the same calculation methods be applied to
these states; it helps to understand the behavior of fluids at high pressure that we sometimes find extraordinary.
In this chapter, we will repeatedly point out deviations from the ideal gas laws. For
properties such as volume or the thermodynamic functions, we will gather the expressions
that relate these deviations and allow us to calculate them.

2.1

THE RELATIONSHIP BETWEEN PRESSURE, VOLUME
A N D TEMPERATURE. LIQUID-VAPOR EQUILIBRIUM

If the distance between molecules is very large and the molar volume approaches infinity,
the cohesion forces become negligible and the properties of the real fluid approach those
of the ideal gas. The first step with calculation of the thermodynamic properties of the ideal
gas is then a reasonable approximationfor gases at low pressure.The extent of the range of

30

2. Properties of Pure Substances

application depends on temperature, and of course on the desired precision, as we shall see.
It is thus in particular at low density, that the pressure tends toward the value calculated for
the ideal gas and that, regardless of temperature, the ratio Pv/RT approaches 1:
if v + w

then P + -

RT

or

V

Pv

-+l
RT

This does not mean that the difference between the molar volume of the real fluid and
the molar volume of the ideal gas cancel each other out. When the molar volume
approaches infinity, the value limit of this difference is not zero. It is called the second virial coefficient, and increases from very negative values at low temperature to low, but positive values at high temperature, as shown in Table 2.1 and Figure 2.1. Knowing the values
of the second virial coefficient allows for an exact evaluation of the thermodynamicproperties at low density. These values were the subject of compilations [Dymond and Smith,
19801 and of predictive correlations that we shall discuss later (Chapter 3).
Table 2.1
Variation of the second virial coefficient, B, of ethane with temperature (units: cm3 . mol-’)
T
(K)
173.15
198.15
223.15
248.15
273.15
298.15
323.15

B
-539
408
-321
-260
-215
-180
-152

T
(K)
348.15
373.15
398.15
423.15
448.15
473.15
523.15

B
-130
-112
-97
-84
-73
-63
47

T
(K)
573.15
623.15
673.15
723.15
773.15
823.15
873.15

B
-34
-24
-15
-8
-2
3
8

By following,at constant temperature, the evolution of the pressure as a function of volume for a given quantity (one mole for example) of a compound, we can qualitatively
characterize the different ranges of “the pressure, volume, temperature space” and determine certain characteristic properties such as vapor pressure, molar volumes of saturated
phases, and critical points.
Let us consider the case of ethane, for example (Figs. 2.2 and 2.3,Table 2.2). When the
system is homogeneous and if the amount of matter is known, two variables, such as temperature and either pressure or volume, are necessary to determine its properties.At 20°C
(293.15 K) and at atmospheric pressure (0.101325 MPa), ethane is in a gaseous state and
the occupied volume is close to 23861 cm3 * mol-l, a value slightly less (24054 cm3 * mol-l)
than the volume obtained by applying the equation for the ideal gas. The “compressibility
factor” defined by the equation:
z = Pv
-=-v
RT v #
is slightly less than one. We state that “the deviation from the ideal gas” is negative, at least
as regards the calculation of the volume. It is the result of intermolecular attraction forces,
which will be more pronounced if we decrease the average distance between the mole-

31

2. Properties of Pure Substances

1oc

c

-1 00

-I

7

E

m

5

v

al -200

-300

400
200

400

600

800

Temperature (K)

Figure 2.1 Variation of the second virial coefficient of
ethane with temperature.

cules. Indeed, when we decrease volume and increase pressure, the compressibilityfactor
decreases.We follow on the 293.15 K isotherm in Figures 2.2 or 2.3 the curve AB. When the
pressure reaches 3.76 MPa and the molar volume 345 cm3* mol-’, we see that a more dense
phase appears, the liquid phase. We speak of a “dew” point. The state of the system is represented by point B with coordinates vvO and Pa on Figure 2.2, and Po and Zva on
Figure 2.3, point C with coordinates
(88.3 cm3/mol) and Pa on Figure 2.2, and Pa and
ZL*“on Figure 2.3 corresponding to the liquid phase. If we continue to decrease the volume, this liquid phase will accumulate at the expense of the vapor phase. The equilibrium
condition reduces the number of independent intensive variables. The specific or molar
volumes of each one of the phases vva and .‘,“as well as pressure Pa will remain constant
for the duration of the condensation process. This pressure, characteristic for a pure substance in a state of equilibrium between the two phases of liquid and vapor, is called vapor
pressure. Representative point M of the heterogeneous system will lie on segment BC of
Figure 2.2, marked by pressure Pa, and the molar volumes of the equilibrium phases vva

32

2. Properties of Pure Substances

*

Figure 2.2 Pressure, volume, temperature diagram for ethane.

-:isotherms; - - - - - :two-phase envelope;

:critical point.

0.25-

0

2.5

5

7.5

Pressure (MPa)

Figure 2.3 Variation of the compressibility factor for ethane with
pressure and temperature.
: isotherms;
:two-phase envelope;
:critical point.

- ----

*

33

2. Properties of Pure Substances

Table 2.2
Volumetric properties of ethane
T = 293.15 K

T = 298.1 K

P (MPa)

v (cm3.mol-')

Z

P (MPa)

v (cm3.mol-')

Z

0.101 325
3.763 4
3.763 4
5.5

23 861
345
88.3
82.5

0.990
0.533
0.136
0.186

0.101 325
4.1876
4.1876
5.5

24 278
283
95
87

0.992
0.478
0.160
0.193

T = 303.15 K

T = 305.34

P (MPa)

v (crn3.mol-')

Z

P (MPa)

v(cm3.mol-')

Z

0.101 325
4.650 8
4.6508
5.5

24 694
215
107.9
93.6

0.993
0.397
0.199
0.204

0.101 325
4.871 4
4.871 4
5.5

24 877
145.5
145.5
98

0.993
0.279
0.279
0.212

P (MPa)

v (cm3.mol-')

Z

0.101325
3.5
4.5
5.5

25 111
513.8
316.9
107.6

0.999
0.702
0.557
0.262

and vL,&quot;, and their compressibility factors Z&quot;&quot; and ZL,&quot; of Figure 2.3. For a volume balance, we can know the proportion of each of the phases by:

This equation is expressed by applying the lever rule to segment BC of Figures 2.2 or 2.3.
The last trace of vapor phase disappears when the volume is reduced to a value of vL'&quot;
(88.3 cm3/mol).We may then speak of a &quot;bubble&quot; point. It corresponds to point C with
coordinates vL&gt;&quot;(88.3 cm3/mol)and P&quot; on Figure 2.2, and pb and ZL,&quot; on Figure 2.3. If we
continue to decrease the volume, since the liquid phase is not very compressible,the pressure rises rapidly (curve CD on Fig. 2.2) and the compressibilityfactor of the liquid phase
is practically proportional to the pressure (curve CD on Fig. 2.3). At a higher temperature,
298.15 K for example,the change in the system state and in the pressure is qualitatively the
same. However, we observe that vapor pressure is higher and that the difference between
the properties of the two equilibrium phases diminishes: densities approach each other
and the liquid phase becomes more compressible.Beyond 305.4 K (at 308.15 K for example) and regardless of the volume, we cannot detect a phase change, and pressure increases
in a regular fashion (curve B'D' in Figs. 2.2 and 2.3) when the volume is decreased. The
temperature above which the phase change phenomenon disappears is called the critical
temperature, 305.4 K in the case of ethane. The &quot;threshold of condensation&quot; is then
reduced to a point of inflection, which is called the critical point. The coordinates are the
critical volume v, and the critical pressure P,. The points representing saturated vapor

34

2. Properties of Pure Substances

phase B and saturated liquid phase C describe the two curves of the “two-phase envelope”.These curves are joined tangentially at the critical point.
Through the “critical constraints”:

(E)
=(E)
=O
av
av2 T

for T = T , a n d P = P ,

knowing the T,, P,, and v, coordinates often will allow for the identification of the variables of the equation of state (see Chapter 4). We will need it to apply the predictive correlations based on the corresponding states principle (see Chapter 3). For this reason, these
data may be considered characteristic properties of the components and put into any database. They are not always accessible to experimental determination. Therefore in paraffin
hydrocarbons with a high number of carbon atoms (greater than approximately 16) the
phenomenon of thermal decomposition hinders their determination [Teja, 19891.They are
then calculated using structure correlation properties (Chapter 3), or from other measurable variables such as boiling temperature at atmospheric pressure, or density. Without
physical reality, the critical points in this case represent only characteristic parameters
related to the correlations that produced them, and to the models that make use of them.

2.2
2.2.1

VAPOR PRESSURE
liquid and Vapor States

The vapor pressure curve is limited at low temperature by the triple point that represents
the conditions at which the solid, liquid, and vapor phases coexist, and by the critical point
at high temperature.
Figure 2.4 shows the vapor pressure curves for the alkanes from methane through
decane,for carbon dioxide, and for methanol.Note that the hydrocarbon critical points follow a regular variation with the number of atoms in the paraffin chain, at least above
ethane: increase in critical temperature, decrease in critical pressure. We also note that the
critical pressures of more polar compounds (carbon dioxide has a quadropolar structure),
or compounds autoassociated by hydrogen bonding such as methanol, are clearly higher
than those of the hydrocarbons. The critical point for water is: T, = 647 K, P, = 21.7 MPa.
Familiarity with the vapor pressure curve helps evaluate the states of matter, liquid or
vapor in particular.
Identification is, however, not without ambiguity. For example, if we have a pressure
between the triple point pressure and the critical point pressure at a temperature that is
higher than the melting temperature, a rise in temperature will allow us to observe the
phenomenon of vaporization. We will have then correctly identified the corresponding
ranges pertaining to the liquid and vapor states. On the other hand, if the pressure is higher
than the critical pressure, we will then detect no change of phase, and the attribution of
qualifiers and limits to the ranges involved is subjective.It is the same for isothermal compression according to whether it lies above or below the critical temperature.
It is useful to recall the experiment termed “critical point tour” shown in figures 2.2
and 2.5. Departing from point B characteristicof a saturated vapor (for ethane,Pa = 3.76 MPa,

35

2. Properties of Pure Substances

10

*CH,OH
I
I

*co,
I
I
I
I

h

m

a

E
E

I
I
I
I

i 5

2

a

0
200

300

400

500

600

Temperature (K)

Figure 2.4 Vapor pressure curves and critical points ( *)of paraffins
C, - C,,, of carbon dioxide, and of methanol.

vv'=
345 cm3/mol),at a temperature of 293.15 K and constant pressure, we increase this temperature to 308.15 K (path BB'), and then at constant temperature we compress the system to
a pressure higher than the critical pressure along B'D. We then return to the initial temperature (293.15 K) at constant pressure along D'D, and, at this temperature, decrease the pressure to a value equal to the vapor pressure (P&quot;=3.76 MPa) along DC.We find ourselvesat the
saturated liquid state (v&quot;&quot; = 88.3 cm3/mol).Along this path BB'D'DC, we never crossed the
vapor pressure curve, but nevertheless went from the saturated vapor state to the saturated
liquid state. We must conclude that there is continuity between the two states no matter how
marked the property differences such as molar volume are, or how natural the distinction
between liquid and vapor is along the vapor pressure curve.As we have seen, these property
differences decrease when temperature increases, and disappear at the critical point where
liquid and vapor are identical.We conclude by saying that this distinction between the liquid
and vapor states makes sense only if both states coexist.
We must also point out certain features of the homogeneous fluid phase within the critical zone. As shown in Figure 2.2, the molar volume varies very quickly with pressure or
temperature. It is also the case with molecular interactions as well as with some thermodynamic properties such as enthalpy. Under these conditions, fluid, while remaining homogeneous, demonstrates intermediate properties between the liquid and vapor phases and its
properties are particularly sensitive to relatively small variations of temperature and pressure. Among these properties is solvent power, put to good use in supercritical solvent
extraction. We also know that it is particularly difficult to evaluate accurately within this
zone certain properties, such as heat capacity for example (see Chapter 4, Fig. 4.3).

36

2. Properties of Pure Substances

6

Dt- - - - - YD’
I
I
I
I
I
I
I
I

h

m

I
I
I
I

/ i

a

3
??

i 4
2

(I

B

9

2
280

300
Temperature (K)

320

Figure 2.5 Continuity of liquid and vapor states: the “criticalpoint tour”.

The vapor pressures of the most common hydrocarbons and a large number of compounds have been determined, and the results have been collected in many databases such
as the ones assembled by Reid et al. [1976,1987],Daubert and Danner [1986],or Boublik
et al. [1984]. However, for compounds with large molar mass, in particular of heavy hydrocarbons, experimental data are scarce and sometimes inaccurate. A real effort is being
made to determine them, motivated by the problems posed in calculating the thermodynamic properties of petroleum fluids.

2.2.2
2.2.2.1

Vapor Pressure Equations
Clapeyron Equation

Applying the equilibrium condition allows for the derivation of the Clapeyron equation.
The transfer of dn moles from the liquid phase to the vapor phase is accompanied by a



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