External state variables

Thermodynamics is concerned with the state of a system when left alone, and when interacting with the surroundings. By ‘system’ we shall mean any portion of the world that can be defined for consideration of the changes that may occur under varying conditions. The system may be separated from the surroundings by a real or imaginary wall. The properties of the wall determine how the system may interact with the surroundings. The wall itself will not usually be regarded as part of the system but rather as part of the surroundings. We shall first consider two kinds of interactions, thermal and mechanical, and we may regard the name ‘thermodynamics’ as an indication that these interactions are of main interest. Secondly, we shall introduce interactions by exchange of matter in the form of chemical species. The name ‘thermochemistry’ is sometimes used as an indication of such applications. The term ‘thermophysical properties’ is sometimes used for thermodynamic properties which do not primarily involve changes in the content of various chemical species, e.g. heat capacity, thermal expansivity and compressibility.

One might imagine that the content of matter in the system could be varied in a number of ways equal to the number of species. However, species may react with each other inside the system. It is thus convenient instead to define a set of independent components, the change of which can accomplish all possible variations of the content.

Experimental conditions

There will be a driving force for a phase transformation if the conditions of a system are changed in such a way that the system moves from one phase field into another in the phase diagram. In this chapter we shall examine the character of such phase transformations and we shall find that they depend upon the experimental method of controlling and changing the conditions. It is important first to realize that the possibility of efficiently controlling the various state variables is very different. For gaseous and liquid phases it is comparatively easy to control the pressure. It can be kept constant or it can be changed gradually according to an experimental programme. At any moment it is very uniform in the system apart from effects due to the surface energy of curved phase interfaces. For solid systems it is more difficult to control the pressure, in particular during a phase transformation resulting in a volume change. This may give rise to local deformation and internal stresses. On the other hand, solid phases are usually so dense and rigid that the thermodynamic effect of pressure differences and stresses can often be ignored. From a practical point of view we may often regard the pressure as an experimental variable which can be reasonably well maintained at a low enough level to have a negligible effect.

Transformations and transitions

In Chapter 12 we were mainly concerned with the question whether a transformation is sharp or gradual. The difference between those cases is very practical and straight-forward. It is based on a one-dimensional phase diagram where the only axis represents the quantity that is being varied. If that diagram shows a two-phase field of some extension between the two one-phase fields, then the transformation between the two phases will be gradual. If the two-phase field has no extension, then the transformation will be sharp. For a unary system with a transformation this will happen if one varies a potential, e.g. T. The Gibbs energy is a continuous function of T across the sharp transformation but its derivatives, yielding S and V, show discontinuous jumps. This is why the phase boundaries separate when a molar axis is introduced (see Figs 9.1 and 9.2). In other cases there is no such separation because the first-order derivatives are zero. A typical example is found in a ferromagnetic substance, which gradually loses its magnetization as the temperature is increased. At the Curie temperature it reaches zero and the substance has thus become paramagnetic. There is no temperature where ferromagnetic and paramagnetic regions coexist in a pure substance, not even if one varies a molar quantity.

General principles

By ‘modelling’ we shall understand the selection of some assumptions from which it is possible to calculate the properties of a system. Sometimes it is possible to obtain a close mathematical expression giving a property as a function of interesting variables. In this chapter and the following ones we shall mainly concern ourselves with such models. However, in many cases the model cannot be expressed in a closed mathematical form but results can also be obtained by numerical calculations using some iterative method. When the iteration in some way resembles the behaviour of a real physical system one talks about ‘simulation’. Such methods are becoming increasingly more powerful thanks to access to more and more powerful computers.

The purpose of modelling is two-fold. From a scientific point of view one likes to learn how nature functions. One way of gaining knowledge is to define some hypothesis resulting in a model and test it by comparing the predictions from the model with experimental information. Then, it does not matter much if the predictions are made by an analytical calculation or by some numerical method. From a more technological point of view one likes to predict the properties of a particular system in order to put it to efficient use in some practical construction or operation. Then it is often most convenient to have a model which yields an analytical expression.

Thermodynamic treatment of kinetics of internal processes

In Chapter 1 we considered spontaneous processes inside a system when discussing the second law but later in that chapter we only considered equilibria. We shall now discuss the thermodynamic treatment of the kinetics of such processes. This field of thermodynamics is often called irreversible thermodynamics but the full term should rather be thermodynamics of irreversible processes. The word irreversible is often replaced by the word spontaneous. A process occurring inside a system may be caused by a change imposed upon the system by some external action, but it will here be regarded as a spontaneous result of the new conditions inside the system. All processes inside a system that actually occur will thus be regarded as spontaneous. It would really be unnecessary to use either of the terms irreversible and spontaneous processes if it were not for the need to distinguish them from the limiting case of a cyclic process, e.g. the Carnot cycle, when it is carried out in such a way that the internal processes it gives rise to produce a negligible amount of entropy. Since a cyclic process is controlled by actions from the outside and they could be performed in the reverse direction, it is possible to run the cycle in the reverse direction. All the internal processes it gives rise to will also reverse and if their entropy production is again negligible the two cases will be identical in the limit, except for the sign.

The requirement of the second law that the internal entropy production must be positive for all spontaneous changes of a system results in the equilibrium condition that the entropy production must be zero for all conceivable internal processes. Most thermodynamic textbooks are based on this condition but do not discuss the magnitude of the entropy production for processes. In the first edition the entropy production was retained in the equations as far as possible, usually in the form of Dd ξ where D is the driving force for an isothermal process and ξ is its extent. It was thus possible to discuss the magnitude of the driving force for a change and to illustrate it graphically in molar Gibbs energy diagrams. In other words, the driving force for irreversible processes was an important feature of the first edition. Two chapters have now been added in order to include the theoretical treatment of how the driving force determines the rate of a process and how simultaneous processes can affect each other. This field is usually defined as irreversible thermodynamics. The mathematical description of diffusion is an important application for materials science and is given special attention in those two new chapters. Extremum principles are also discussed.

A third new chapter is devoted to the thermodynamics of surfaces and interfaces. The different roles of surface energy and surface stress in solids are explained in detail, including a treatment of critical nuclei.

Thermodynamics is an extremely powerful tool applicable to a wide range of science and technology. However, its full potential has been utilized by relatively few experts and the practical application of thermodynamics has often been based simply on dilute solutions and the law of mass action. In materials science the main use of thermodynamics has taken place indirectly through phase diagrams. These are based on thermodynamic principles but, traditionally, their determination and construction have not made use of thermodynamic calculations, nor have they been used fully in solving practical problems. It is my impression that the role of thermodynamics in the teaching of science and technology has been declining in many faculties during the last few decades, and for good reasons. The students experience thermodynamics as an abstract and difficult subject and very few of them expect to put it to practical use in their future career.

Today we see a drastic change of this situation which should result in a dramatic increase of the use of thermodynamics in many fields. It may result in thermodynamics regaining its traditional role in teaching. The new situation is caused by the development both of computer-operated programs for sophisticated equilibrium calculations and extensive databases containing assessed thermodynamic parameter values for individual phases from which all thermodynamic properties can be calculated. Experts are needed to develop the mathematical models and to derive the numerical values of all the model parameters from experimental information.

Introduction

In this chapter we shall model the thermodynamic effect of some physical phenomena. In each case we shall start by defining an internal variable representing the extent of the physical phenomenon to be discussed. We shall proceed by deriving an expression for one of the characteristic state functions in terms of the internal variable together with a set of external variables. The choice of characteristic state function depends upon what set of external variables is most convenient. Then we shall calculate the equilibrium value of the internal variable by putting the driving force for its change equal to zero. Finally, we shall try to eliminate the internal variable by inserting the expression for its equilibrium value in the characteristic state function.

Our derivation of an expression for the characteristic state function will usually be based upon two separate evaluations, one concerned with the entropy due to the disorder created by the physical phenomenon and the other concerned with what may be called the non-configurational contribution. The entropy will be evaluated from Boltzmann's relation which is here preferred because it is felt that it gives a better physical insight than the more general and elegant method of statistical thermodynamics based upon the use of partition functions. The purpose of statistical thermodynamics is to model the thermodynamic properties of various types of systems from statistical considerations on the atomic level. The relation proposed by Boltzmann can be derived from such considerations.