In all of the studies of thermodynamic equilibrium that have been presented in the previous chapters, we have neglected the effects of an external field on the equilibrium properties of a system. This has been justified because the field may be present only in specific cases, the effect of the field may be negligible, or the position of the system in the field may be unchanged. The conditions of equilibrium in the presence of a gravitational or centrifugal field, an electrostatic field, and a magnetic field are developed in this chapter.

A distinction must be made between gravitational effects for which the presence of material in the field does not change the intensity of the field, and the electrostatic and magnetic effects for which the presence of material within the field does alter the intensity. A complete treatment of electrostatic and magnetic effects would require a discussion of electromagnetic theory and the use of Maxwell's equations. However, we wish only to illustrate the thermodynamic effects of electric and magnetic fields. We therefore accept the results of a complete treatment and apply the results to simple systems.

THE GRAVITATIONAL AND CENTRIFUGAL FIELD

The gravitational and centrifugal potential

The gravitational field of the Earth is characterized by a potential, Φ, that has a definite value at each point in the field.

The thermodynamic equations for the Gibbs energy, enthalpy, entropy, and chemical potential of pure liquids and solids, and for liquid and solid solutions, are developed in this chapter. The methods used and the equations developed are identical for both pure liquids and solids, and for liquid and solid solutions; therefore, no distinction between these two states of aggregation is made. The basic concepts are the same as those for gases, but somewhat different methods are used between no single or common equation of state that is applicable to most liquids and solids has so far been developed. The thermodynamic relations for both single-component and multicomponent systems are developed.

The thermodynamic functions have been defined in terms of the energy and the entropy. These, in turn, have been defined in terms of differential quantities. The absolute values of these functions for systems in given states are not known. However, differences in the values of the thermodynamic functions between two states of a system can be determined. We therefore may choose a certain state of a system as a standard state and consider the differences of the thermodynamic functions between any state of a system and the chosen standard state of the system. The choice of the standard state is arbitrary, and any state, physically realizable or not, may be chosen. The nature of the thermodynamic problem, experience, and convention dictate the choice.

The energy and entropy functions have been defined in terms of differential quantities, with the result that the absolute values could not be known. We have used the difference in the values of the thermodynamic functions between two states and, in determining these differences, the process of integration between limits has been used. In so doing we have avoided the use or requirement of integration constants. The many studies concerning the possible determination of these constants have culminated in the third law of thermodynamics.

We can obtain a concept of the problem by referring to the discussion in Section 7.12. There we derive expressions for the thermodynamic functions for an ideal gas by expressing the energy and entropy in terms of the heat capacities and the required integration constants. The result is that the expression for the Gibbs energy contains the term (ẽ – T), where ẽ and are the required integration constants for the energy and entropy, respectively. When we consider the difference between the values of the Gibbs energy of the ideal gas at two temperatures, ẽ cancels but the quantity (T1 – T2 remains. Therefore, in order to make the calculation, we need to know the value of. The result is the same when we consider substances other than an ideal gas. We are, therefore, concerned with the possible determination of the absolute value of the entropy.

In this chapter we briefly review the ideas and equations relating to the important concepts of temperature, heat, work, energy, and enthalpy.

Temperature

The concept of temperature can be defined operationally; that is, in terms of a set of operations or conditions that define the concept. To define a temperature scale operationally we need: (1) one particular pure or defined substance; (2) a specific property of that substance that changes with a naive sense of ‘degree of hotness’ (i.e., temperature); (3) an equation relating temperature to the specific property; (4) a sufficient number of fixed points (defined as reproducible temperatures) to evaluate the constants in the equation in (3); and (5) the assignment of numerical values to the fixed points. Historically, many different choices have been made with respect to the five conditions listed above, and this, of course, has resulted in many temperature scales.

The ideal gas temperature scale is of especial interest, since it can be directly related to the thermodynamic temperature scale (see Sect. 3.7). The typical constant-volume gas thermometer conforms to the thermodynamic temperature scale within about 0.01 K or less at agreed fixed points such as the triple point of oxygen and the freezing points of metals such as silver and gold. The thermodynamic temperature scale requires only one fixed point and is independent of the nature of the substance used in the defining Carnot cycle.

The only two functions actually required in thermodynamics are the energy function, obtained from the first law of thermodynamics, and the entropy function, obtained from the second law of thermodynamics. However, these functions are not necessarily the most convenient functions. The enthalpy function was defined in order to make the pressure the independent variable, rather than the volume. When the first and second laws are combined, as is done in this chapter, the entropy function appears as an independent variable. It then becomes convenient to define two other functions, the Gibbs and Helmholtz energy functions, for which the temperature is the independent variable, rather than the entropy. These two functions are defined and discussed in the first part of this chapter.

Only closed systems have so far been considered. However, mass can be varied and is an important variable for all thermodynamic functions. The introduction of mass as an independent variable into the basic differential expressions for the thermodynamic functions yields the equations that Gibbs called ‘fundamental’. It is on these equations that much of the development of the applications of thermodynamics to chemical systems is based.

Many of the problems that are met in applications require the evaluation of various derivatives and the integration of differential quantities that involve the derivatives. To be of use, the derivatives must be expressed in terms of experimentally determinable quantities.

In the development of the second law and the definition of the entropy function, we use the phenomenological approach as we did for the first law. First, the concept of reversible and irreversible processes is developed. The Carnot cycle is used as an example of a reversible heat engine, and the results obtained from the study of the Carnot cycle are generalized and shown to be the same for all reversible heat engines. The relations obtained permit the definition of a thermodynamic temperature scale. Finally, the entropy function is defined and its properties are discussed.

Heat and work reservoirs

We define a heat reservoir as any body, used as part of the surroundings of a particular system, whose only interaction with the system is across a diathermic boundary. A heat reservoir is then used to transfer heat to or from a thermodynamic system and to measure these quantities of heat. It may consist of one or more substances in one or more states of aggregation. In most cases a heat reservoir must be of such a nature that the addition of any finite amount of heat to the system or the removal of any finite amount of heat from the system causes only an infinitesimal change in the temperature of the reservoir.

A work reservoir is similarly defined as any body or combination of bodies, used as part of the surroundings, whose only interaction with the system is one that may be described in terms of work.

The range and scope of thermodynamics is implied in Wigner's epigrammatic quote above, except for the fact that there are a great many phenomena for which thermodynamics can provide quite accurate calculations. Chemical engineering is to a large extent based on thermodynamic calculations applied to practical systems. The phase rule is indispensable to metallurgists, geologists, geochemists, crystallographers, mineralogists, and chemical engineers. Although students sometimes come away with the notion that thermodynamics is remote and abstract, it is actually the most practical of subjects. Part of what we endeavor to do in this book is to illustrate that practicality, the tie to everyday life and utility.

Thermodynamics comprises a field of knowledge that is fundamental and applicable to a vast area of human experience. It is a study of the interactions between two or more bodies, the interactions being described in terms of the basic concepts of heat and work. These concepts are deduced from experience, and it is this experience that leads to statements of the first and second laws of thermodynamics. The first law leads to the definition of the energy function, and the second law leads to the definition of the entropy function. With the experimental establishment of these laws, thermodynamics gives an elegant and exact method of studying and determining the properties of natural systems.

The study of phase equilibrium affords methods by which the excess chemical potentials of the components in a solution may be determined. It is concerned with the relations that exist between the intensive variables—temperature, pressure, and composition of the phases—when two or more phases are in equilibrium. In this respect, such studies also yield information on the colligative properties of a solution. For single-component systems without restrictions, the temperature and pressure are the only variables of interest. When restrictions are applied, an additional number of temperature or pressure variables must be introduced. In experimental studies, conditions are set so that any given system is univariant and it is possible to determine the dependence of one variable, a temperature or pressure, on one other variable, again a temperature or a pressure. Composition variables must be used for multicomponent systems, in addition to the temperature and pressure, and are introduced here through the chemical potentials. The total number of intensive variables and the number of independent intensive variables for a given system can be determined from the appropriate set of Gibbs–Duhem equations, consistent with any restrictions that may or may not be placed upon the system. The experimental and thermodynamic study of such a system then requires that a sufficient number of limitations be placed on the system so that it is univariant. Thus, we determine the dependence of one dependent variable on one independent variable, all other independent variables being held constant.

Introduction

Formation of polymeric cationic species in aqueous colloidal dispersions was described in section 4.4, while polymers obtained in hydrolysed tetraethyl silicate solutions have been referred to in section 5.5. The contents of this chapter, however, are concerned with synthesis and properties of polymeric ceramic precursors derived from chlorosilanes, namely polysilanes, polycarbosilanes and polysilazanes, together with their conversion, on pyrolysis, to ceramics mainly in the form of fibres and coatings on both the laboratory and industrial scale. This work, which lies at the organic–inorganic interface, illustrates the contribution that different branches of chemistry have made to ceramic synthesis, and also how an area of organosilicon chemistry, not immediately associated with ceramics, can make a significant impact on materials preparation.

Synthesis of polysilanes

Unlike conventional organic polymers, such as polystyrene, which contain a chain of carbon atoms or silicones consisting of alternating Si and O atoms, the polymer backbone in polysilanes comprises Si–Si link-ages (figure 7.1). Early work on aryl-substituted polysilanes was reported by Kipping and co-workers in a series of publications during the 1920s. As an example, diphenyldichlorosilane, (C6H5)2SiCl2, was reacted (Kipping & Sands, 1921) with molten sodium in an inert solvent, xylene, and several products were obtained. These included a sparingly soluble solid, Si4(C6H5)8, which they called an unsaturated compound and the major product, a soluble material with the same formula and referred to as a saturated compound; solvent evaporation left a resinous mass from the latter.