Table Of ContentEquilibrium Between Phases of Matter
Equilibrium Between
Phases of Matter
Phenomenology and Thermodynamics
H.A.J. Oonk
Utrecht University, The Netherlands
and
M.T. Calvet
University of Barcelona, Spain
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TABLE OF CONTENTS
INTRODUCTION vii
LIST OF FREQUENTLY USED SYMBOLS xi
LEVEL 0: an introduction to phase diagrams
§ 001 equilibrium 3
§ 002 variables 10
§ 003 the rules of the game 19
§ 004 pure substances 28
§ 005 binary and ternary systems 41
§ 006 distribution and separation 61
§ 007 chemical equilibrium 76
LEVEL 1: an introduction to thermodynamics and phase theory
§ 101 differential expressions 89
§ 102 work heat energy 93
§ 103 heat capacity and enthalpy 97
§ 104 the ideal gas, expansion and compression 101
§ 105 chemical energy 107
§ 106 entropy 113
§ 107 characteristic functions 121
§ 108 Gibbs energy and equilibrium 128
§ 109 data and tables 139
§ 110 pure substances 151
§ 111 chemical reactions and equilibrium 168
vi Table of contents
LEVEL 2: phase theory: the thermodynamics of equilibrium between phases
§ 201 mixtures and partial quantities 183
§ 202 the open system, chemical potentials 190
§ 203 change to molar quantities, molar Gibbs energy 197
§ 204 the ideal mixture 206
§ 205 non-ideal behaviour, excess functions 211
§ 206 magic formulae 222
§ 207 dilute solutions 231
§ 208 the solvent laws 237
§ 209 the solute laws 245
§ 210 ideal equilibria 252
§ 211 non-ideal systems - geometrically 263
§ 212 non-ideal systems - analytically 279
§ 213 non-ideal systems - numerically 299
SOLUTIONS OF EXERCISES
level 0 319
level 1 330
level 2 356
REFERENCES 377
SUBJECT INDEX 389
SUBSTANCES AND SYSTEMS INDEX 401
INTRODUCTION
About the book, the project
Equilibrium Between Phases of Matter – Phenomenology and Thermodynamics
is a textbook, in which the phenomenology, the thermodynamic theory, and the
practical use of phase diagrams are presented in three levels that diverge in
nature – in particular as regards the role of thermodynamics. The book has been
written from a chemical and geological teaching background. Each of the three
levels of the book is representative of a particular course in a curriculum.
Level 0: an introduction to phase diagrams
The philosophy behind the ground level is that most of the characteristics of
equilibrium between phases can be understood without the use of
thermodynamics, realizing that, in a common-sense manner, the experimental
observations on equilibria and spontaneous changes, and elementary notions
about interactions, indicate the way to go. In spite of all this, the central figure in
level zero, right from the beginning, is the chemical potential – a concept firmly
rooted in thermodynamics. Equilibrium conditions in terms of chemical potentials,
and the variables necessary to define a system in equilibrium are, are the basic
elements of the system formulation.
The first three sections deal with the characteristics of a system in equilibrium;
the variables necessary to define the system; the conditions for equilibrium and
the rules – the phase rule and the lever rule. Pure substances and their forms are
the subject of the fourth section. In the fifth section the phenomenology of binary
phase diagrams is developed in terms of the nature of the interaction between
the entities of the components; and, subsequently ternary phase diagrams are
developed from the diagrams of the binary subsystems. Much of the use of binary
phase diagrams is based on the fact that phases in equilibrium have different
compositions – sixth section. The last section is devoted to homogeneous and
heterogeneous chemical equilibria.
Level 1: an introduction to thermodynamics and phase theory
The intermediate level is an introduction to classical thermodynamics -
culminating in the Gibbs energy as the arbiter in equilibrium matters - followed by
the thermodynamic treatment of equilibrium between phases for a number of
elementary cases.
The thermodynamics part starts with differential expressions; work, heat, and
energy; heat capacity and enthalpy. After the ideal gas and its expansion and
compression; and a section on chemical energy; entropy and Gibbs energy make
their appearance – in particular their role in spontaneous changes and matters of
equilibrium. In the last sections the principles of equilibrium are applied to pure
substances – where molar Gibbs energy has the status of chemical potential –
and to homogeneous and heterogeneous chemical equilibria.
viii Introduction
Level 2: phase theory: the thermodynamics of equilibrium between phases
In the upper level the step is made to mixtures with their ‘puzzling’ partial molar
properties, revealing the identity between chemical potential and partial Gibbs
energy. After the introduction of ideal mixtures, a ‘magic formula’ for the Gibbs
energy of mixing is used to explain the phenomenon of demixing, and to derive
the properties of the ideal dilute solution. Ideality and non-ideality, in a certain
sense, give rise to the existence of two sub-levels. In the first sub-level the
function recipes of the chemical potentials are such that explicit relationships can
be formulated for systems where the phases are pure substances and/or ideal
mixtures or ideal dilute solutions. The second sub-level, the last part of the work
is devoted to systems where the phases are non-ideal mixtures, whether or not,
in combination with phases of fixed composition – and where the leading role of
partial Gibbs energy, read the chemical potential, is taken over by the integral
Gibbs energy.
A priori and a posteriori
To assist the reader in assessing her/his own level of understanding the ins and
outs of equilibrium between phases, each of the sections is provided with an a
priori, in which the subject is introduced, and an a posteriori, in which the main
conclusions are summarized.
Besides, owing to the level structure of the work, there is a certain, small overlap
between parts of different levels. Or, in other terms, some overlap has not been
kept out of the authors their way.
Exercises
Exercises are given at the end of each section, and the solutions of most of them
at the end of the book. Their main function, of course, is to interest and stimulate
the reader or a group of students in the class room. Occasionally, exercises are
used as a vehicle to introduce issues that are not treated in the main text.
Follow-up
Equilibrium Between Phases of Matter – Phenomenology and Thermodynamics,
is the first volume of a two-volume project. The second volume, by M.H.G.
Jacobs and H.A.J. Oonk, Equilibrium Between Phases of Matter –
Thermodynamic Analysis and Prediction, more than the first, has a postgraduate
and professional character – and is written from a research background in
materials science and geophysics.
In terms of thermodynamic properties, the distinction between the two volumes is
in the role of the temperature and pressure derivatives of the Gibbs energy. In the
first volume a dominating role is played by the Gibbs energy itself and its first
derivatives, which are entropy and volume. The three together account for most
of the phenomenology of equilibrium between phases – but not all of it; and one
Introduction ix
can think of retrograde phenomena. And when it comes to the real
thermophysical properties of a given system, and the prediction of its behaviour
under circumstances far from ambient temperature and pressure, one is
completely lost without the incorporation/availability of the second and higher-
order derivatives.
About languages
Writing a text in a foreign language is a delicate undertaking – in the sense that
one cannot express oneself as clearly and linguistically correctly as one would
like to. It is also extra time-consuming: the time needed to consult dictionaries
and books of synonyms is commensurate with the time needed for the real work.
The manuscript of the senior author’s 1981 book Phase Theory was read and
corrected by Dr Philip Spencer, who was rather satisfied with the English. On the
other hand, an American reviewer of the book, found the English somewhat
clumsy.
The authors of this book have been working together for more than twenty years
as participants in the REALM (Réseau Européen sur les Alliages Moléculaires) –
a European network on molecular alloys, as may be clear. French is the
REALM’s official language – in the family circle one of the authors speaks
Catalan, and the other Dutch.
Anyhow, dear reader, the message is clear. Please, enjoy the science and wink
at the linguistic shortcomings.
Not only communication in terms of real languages, but also the communication
between thermodynamicists from different branches of materials science has its
typical difficulties. Just to give an idea, in certain fields a prominent role is played
by activities and fugacities, whereas in other fields these terms virtually are non-
existent.
And, of course, the relationship a physical chemist has with thermodynamics is
quite different from the one built up by a geologist. A chemist creates his own
systems out of pure chemicals, and having the phases of his own choice. The
geologist, on the other hand, focuses his attention to stones created by nature
and often having a multitude of phases, which, sometimes, reflect the history of
the material.
The first volume of Equilibrium Between Phases of Matter has been written in the
tradition of the Dutch School - in the footsteps of Josiah Willard Gibbs’s disciples
J.D. van der Waals, H.W. Bakhuis Roozeboom, F.A.H. Schreinemakers, J.J. van
Laar, J.L. Meijering, and others. In this tradition a prominent role is given to
chemical potentials – and, when it comes to deal with deviation from ideal
behaviour, the use of activities and fugacities is avoided.
x Introduction
Acknowledgements
We should like to take the opportunity of expressing our gratitude to Mercedes
Aguilar, Jan den Boesterd, Ingrid van Rooijen, and Aloys Lurvink for fine artwork;
to Petra van Steenbergen, Hermine Vloemans, and Tamsin Kent for most
valuable suggestions related to publishing; and to Koos Blok for thorough
bibliographic research.
We are grateful to our students, who, through their remarks and probing
questions, have kept us alert – thermodynamics is tricky business after all.
Dedication
We dedicate this book to our spouses and the other members of our families, and
to our friends, who all, every now and then, remind us of the fact that equilibrium
between faces is also a matter of great importance.
Utrecht, The Netherlands, Harry A.J. Oonk
Barcelona, Spain M. Teresa Calvet
Spring 2007
LIST OF FREQUENTLY USED SYMBOLS
Latin letters
A Helmholtz energy; magnitude parameter of AB(cid:300) model
A first component in binary and ternary system
α activity
a, b, c system-dependent parameters
B general for substance; second component in binary and ternary
system
B asymmetry parameter in AB(cid:300) model
C third component in ternary system
c number of components
d ordinary differential
C heat capacity at constant pressure
P
C heat capacity at constant volume
V
E electromotive force
e electric charge
f variance, number of degrees of freedom; fugacity; activity
coefficient
G Gibbs (free) energy
g acceleration of free fall; parameter in excess Gibbs energy
H enthalpy
h parameter in excess enthalpy; altitude
ln natural logarithm
LN defined as LN(X) = (1-X)ln(1-X) + XlnX
M set of variables necessary to define an equilibrium system, and
number of elements in it; molar mass
m mass; molality
N set of equilibrium conditions, and number of elements in it
N Avogadro’s number
Av
n amount of substance
P pressure
p number of phases
Q, q heat
R gas constant
S entropy
s parameter in excess entropy
T thermodynamic temperature
t Celsius temperature
U energy
V volume
W, w work
W number of configurations
X variable in general; mole fraction; mole fraction of second
component
