Phase Transformations

By Michel Soustelle

This book is part of a set of books which offers advanced students successive characterization tool phases, the study of all types of phase (liquid, gas and solid, pure or multi-component), process engineering, chemical and electrochemical equilibria, and the properties of surfaces and phases of small sizes. Macroscopic and microscopic models are in turn covered with a constant correlation between the two scales. Particular attention has been given to the rigor of mathematical developments.

This fifth volume is devoted to the study of transformations and equilibria between phases. First- and second-order pure phase transformations are presented in detail, just as with the macroscopic and microscopic approaches of phase equilibria.

In the presentation of binary systems, the thermodynamics of azeotropy and demixing are discussed in detail and applied to strictly-regular solutions. Eutectic and peritectic points are examined, as well as the reactions that go with them. The study of ternary systems then introduces the concepts of ternary azeotropes and eutectics.  For each type of solid-liquid system, the interventions of definite compounds with or without congruent melting are taken into account. The particular properties of the different notable points of a diagram are also demonstrated.

• Language: English
• Publisher: Wiley
• Release date: Mar 14, 2016
• ISBN: 9781119178590

Book preview

1

Phase Transformations of Pure Substances

In this chapter we shall examine the transformations undergone by a definite pure compound, with no chemical alteration. These transformations belong to the category of phase transitions, or phase transformations. Hence, this chapter excludes the transformations of isomerization and decompositions which are accompanied by a chemical alteration – i.e. a modification of the molecule.

1.1. Standard state: standard conditions of a transformation

The standard state of a substance at temperature T is defined as the state of the pure substance at that temperature, at a pressure of 1 bar and as its stable state of aggregation in these conditions (solid, liquid or gas). If the substance is a gas, its behavior is perfect. If the pure substance is a crystalline solid, its stable state of aggregation determines the crystalline system. A transformation takes place in standard conditions if it occurs with the components in their standard state in the final state. This means that at the start of the reaction, the substance is in a non-standard state, which is unstable, and that it returns to its stable standard state at the end. Thus, only the state of aggregation may possibly have been modified by the transformation. This is indeed a phase transformation, in that the state of aggregation defines a phase.

1.2. Classification and general properties of phase transformations

Phase transitions are classified into different types. The advantage of classifying definite compound transformations depending on their order lies in the fact that a series of relations characterizes each order.

The first classification of these transitions is attributable to Ehrenfest, in which we say that a transformation is of order n when at least an nth derivative of the characteristic function in relation to its canonical variables undergoes a discontinuity for certain values of those variables, when the derivatives of order less than n are continuous. The most important are the first- and second-order derivatives which, by their discontinuities, respectively give us the first- and second-order transformations.

This classification has been abandoned, because it does not allow for the possibility of divergences other than discontinuities of derivatives of the Gibbs energy. However, numerous models allow for such divergences within the thermodynamic limit – i.e. when the system’s dimensions increase indefinitely. For example, a derivative such as the specific heat capacity (second derivative of the Gibbs energy in relation to temperature) is thought to be divergent during the ferromagnetic transition.

The current classification still distinguishes between first- and second-order transformations, but the definitions used are different.

First-order transformations are those which involve latent heat. In other words, they are transformations accompanied by an associated enthalpy value. During the course of these transformations the system absorbs or emits a certain fixed amount of energy, which is usually fairly large; as that energy cannot be transferred instantaneously between the system and the external environment, the transformations take place over extended periods of time, during which not all parts of the system undergo the transformation at the same moment.

During these transformations the systems are heterogeneous, meaning that at each moment of the transformation they involve the simultaneous presence of multiple phases, and therefore interfaces between those phases. Such is the case with numerous transitions between the solid, liquid and gaseous phases.

image007.jpg

Figure 1.1. Shape of the curves representative of the different values during a first-order transformation

Figure 1.1 shows the profiles of the modifications of first-order transformation functions; discontinuity of H (associated enthalpy), volume with temperature and pressure, entropy with temperature and continuity but with a change in the slope of the Gibbs energy. For simplicity’s sake, curved segments have been represented by linear segments which do not accurately reflect reality.

Second-order transformations are continuous-phase transitions: there is no latent heat associated with the transition, the system is homogeneous, the transition takes place within the phase at every point and no interfaces manifest themselves. These transformations are also known as continuous transformations. There are a number of substances which exhibit second-order transitions; let us cite the following examples:

– the transition between the two forms I and II of liquid helium;

– ferromagnetic substances whose transition point is the Curie temperature, at which they cease to manifest ferromagnetism;

– certain alloys which exhibit so-called order–disorder transitions1 in a fixed composition:

- superconductive substances at the point of disappearance of the property,

- certain crystals, such as ammonium salts which, at a low temperature, undergo what is known as a lambda transformation.

Unlike with the first-order phase-transition, in a second-order transition, the two states are not localized separately in space, and thus there is no interface: the two states constitute a single, unique phase.

image011.jpg

Figure 1.2. Shape of the different functions in a second-order transformation

The transition between a superfluid, liquid and gas near to the critical point is a second-order phase-transition.

Figure 1.2 shows the profile of the variations of the different functions at the transition point. To simplify, the curves have been represented by linear segments. We see the continuity of entropy with temperature but with a change of slope. The same is true of enthalpy and volume. The curve of Gibbs energy takes place with no change in slope and the variations in expansion coefficient, isothermal compressibility and specific heat capacity at constant pressure exhibit discontinuities at the transition.

1.2.1. First-order transformations and the Clapeyron relation

According to the definition given by Ehrenfest, at least one first derivative of the characteristic function in relation to a variable undergoes a discontinuity, though the characteristic function itself (the zero-order derivative) remains constant. We first examine it in the case of a system with two physical variables: temperature and pressure. The characteristic function is the Gibbs energy, such that:

[1.1a] image136.jpg

and:

[1.1b] image137.jpg

The first-order transformation is characterized by a discontinuity of entropy or volume at a given pressure and temperature, at which the system can exist in two states: 1 and 2. Such is the case, in particular, with phase changes of pure substances (melting, vaporization, sublimation, and polymorphic transformation of a solid).

Let us now establish the Clapeyron relation, which governs these first-order transformations.

Consider states 1 and 2, both stable simultaneously in the same thermodynamic conditions (i.e. same pressure and temperature). The Gibbs energy is constant, which is expressed by the following:

[1.2] image138.jpg

If the temperature and pressure are modified by an infinitesimal amount, the Gibbs energy takes on the new value G + dG, and the new continuity condition is written thus:

[1.3] image139.jpg

which, in view of equation [1.2], gives us:

[1.4] image140.jpg

Relation [1.4] is simply the application of the general equilibrium condition to the balance equation of the phase change (stoichiometric coefficients equal to +1 and -1).

However, in state 1, the differential of the Gibbs energy is:

[1.5] image141.jpg

A similar expression can be written for state 2. The continuity condition [1.4] becomes:

[1.6]

image142.jpg

For simplicity’s sake, let us set the differences:

[1.7a] image143.jpg

and:

[1.7b] image144.jpg

These are the changes in volume and entropy associated with the transformation image145.gif

By feeding these values back into expression [1.6], we find:

[1.8] image146.jpg

This relation is known as the Clapeyron equation. In particular, it shows that if one of the intensive values conjugal to one of the variables (e.g. the entropy) undergoes a discontinuity, the intensive value conjugal to the other variable (in this case the volume) also undergoes a discontinuity, and these two discontinuities are linked to one another by the Clapeyron equation.

Often, the term "latent heat L of the transformation between states 1 and 2" is used to denote the amount of heat involved in reversible conditions, and therefore:

[1.9] image147.jpg

Hence, there is equivalence between the first order as understood by Ehrenfest and the first order in the new classification. Transformations which are accompanied by latent heat exhibit a discontinuity of entropy.

By applying the Clapeyron equation [1.8], we obtain:

[1.10] image148.jpg

The Clapeyron equation can be generalized to apply to a system defined by the set of p physical variables Zi, whose conjugal extensive variables are Xi. The characteristic function, then, is a function image149.gif . By thinking about that function in the same way as we did with the Gibbs energy, we find the