Some Thermodynamic Notation

The PDT focuses on the chemical potentials pa that compose the Gibbs free energy 

For our problems here these chemical potentials are cast as 

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V 

Here = kBT, AUa is the binding potential energy of a molecule of type a to 

The potential distribution theorem has been around for a long time [13-17], but not as long as the edifice of Gibbsian statistical mechanics where traditional partition functions were first encountered. We refer to other sources [10] for detailed derivations of this PDT, suitably general for the present purposes. 

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These lists contain the symbols and abbreviations most frequently used in this volume, but they are not expected to be exhaustive. Some specialized notation is only defined in the relevant chapter. An attempt has been made to standardize usage throughout the volume as far as is feasible, but it must be borne in mind that the original research literature certainly is not standardized in this way, and some difficulties may arise from this fact. Trivial use of subscripts etc. is not always mentioned in the symbols listed below. Some of the other symbols used in the text, e.g. for physical constants such as h or tt, or for the thermodynamic quantities such as H or S, are not included in the list since they are considered to follow completely accepted usage. 

Piezoelectric solids are characterized by constitutive relations among the stress t, strain rj, entropy s, electric field E, and electric displacement D. When uncoupled solutions are sought, it is convenient to express t and D as functions of t], E, and s. The formulation of nonlinear piezoelectric constitutive relations has been considered by numerous authors (see the list cited in [77G06]), but there is no generally accepted form or notation. With some modification in notation, we adopt the definitions of thermodynamic potentials developed by Thurston [74T01]. This leads to the following constitutive relations ... 

Because the focus is on a single, albeit rather general, theory, only a limited historical review of the nonequilibrium field is given (see Section IA). That is not to say that other work is not mentioned in context in other parts of this chapter. An effort has been made to identify where results of the present theory have been obtained by others, and in these cases some discussion of the similarities and differences is made, using the nomenclature and perspective of the present author. In particular, the notion and notation of constraints and exchange with a reservoir that form the basis of the author s approach to equilibrium thermodynamics and statistical mechanics [9] are used as well for the present nonequilibrium theory. 

Much of this chapter will be a review for those who have had courses in chemical kinetics. In this chapter we will also review some aspects of thermodynamics that are important in considering chemical reactors. For students who have not had courses in kinetics and in the thermodynamics of chemical reactions, this chapter will serve as an introduction to those topics. This chapter will also introduce the notation we will use throughout the book. 

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

The introduction of affinity by De Donder marks the birth of the Brussels school the first publication appeared around 1922, but it took some years to make these concepts more precise.4 What was the reaction of the scientific community When we go through the proceedings of the Belgian Royal Academy, we see that De Donder s work indeed aroused much local interest. Verschaffel from Ghent and Mund from Louvain were among the people who became active in this newborn nonequilibrium chemical thermodynamics. However, one has to say that elsewhere De Donder s approach met with skepticism and even with hostility. His introduction of affinity was thought of as merely a different notation. 

Our foremost goal here is to establish enough notation and a few pivotal relations that the following portions of the book can be understood straightforwardly. The following sections identify some basic thermodynamics and statistical thermodynamics concepts that will be used later. Many textbooks on thermodynamics and statistical mechanics are available to treat the basic results of this chapter in more detail students particularly interested in solutions might consult Rowlinson and Swinton (1982). 

The use of this symbol to denote standard thermodynamic quantities is then permissible only when these refer to the pure substance as standard state. Some other convention is needed to denote the more general standard quantities defined in equation (7.51), where the standard state may for example refer to an infinitely dilute solution. After much consideration the symbol has been introduced. This symbol is based on the circle, and is thus closely related to the more common (but occasionally confusing) notation for standard quantities. It seems useful, however, to associate it with the plimsoll mark which, appropriately enough, refers to a reference state of loading of a ship this use of an ideogram has, we believe, some value in keeping the essential nature of standard states in prominence. 

A working knowledge of elementary calculus is presumed as is some acquaintance with elementary differential equations. Section 5.1 is a thumbnail sketch of some particularly important equations. A thorough course in thermodynamics is one of the staples of a chemical engineer s diet and should precede a course on reactors. Chapter 3 is therefore a bare outline of familiar thermochemistry in a notation conformable to the rest of the book. It is impossible to avoid duplications in notation and a list has been provided at the end of each chapter. 

The way in which the separation of the terms of the right hand side of the entropy equation into the divergence of a flux and a source term has been achieved may at first sight seem to be to some extent arbitrary The two groups of terms must, however, satisfy a number of requirements which determine this separation uniquely First, one such requirement is that the entropy source term totai must be zero if the thermodynamic equilibrium conditions are satisfied within the system. Another requirement the source term must satisfy is that it should be invariant under a Galilean transformation (e.g., [147]), since the notations of reversible and irreversible behavior must be invariant under such a transformation. The terms included in the source term satisfy this requirement [32]. 

In this chapter, we first present some of the notation that we shall use throughout the book. Then we summarize the most important relationship between the various partition functions and thermodynamic functions. We shall also present some fundamental results for an ideal-gas system and small deviations from ideal gases. These are classical results which can be found in any textbook on statistical thermodynamics. Therefore, we shall be very brief. Some suggested references on thermodynamics and statistical mechanics are given at the end of the chapter. 

Values extracted and in some cases rounded off from those cited in Rabinovich (ed.), Thermophysical Properties of Neon, At on, Krypton and Xenon, Standards Press, Moscow, 1976. v = specific volume, mVkg h = specific enthalpy, kj/kg s = specific entropy, kJ/(kg-K). This source contains an exhaustive tabulation of values. The notation 7.420.-4 signifies 7.420 x 10". This book was published in English translation by Hemisphere, New York, 1988 (604 pp.). The 1993 ASHRAE Handbook—Fundamentals (SI ed.) has a thermodynamic chart for pressures from 1 to 2000 bar, temperatures from 90 to 700 K. Saturation and superheat tables and a chart to 50,000 psia, 1220 R appear in Stewart, R. B., R. T. Jacobsen, et al.. Thermodynamic Properties of Refrigerants, ASHRAE, Atlanta, GA, 1986 (521 pp.). For specific heat, thermal conductivity, and viscosity see Thermophysical Properties of Refrigerants, ASHRAE, 1993. 

Abstract Fluctuation Theory of Solutions or Fluctuation Solution Theory (FST) combines aspects of statistical mechanics and solution thermodynamics, with an emphasis on the grand canonical ensemble of the former. To understand the most common applications of FST one needs to relate fluctuations observed for a grand canonical system, on which FST is based, to properties of an isothermal-isobaric system, which is the most common type of system studied experimentally. Alternatively, one can invert the whole process to provide experimental information concerning particle number (density) fluctuations, or the local composition, from the available thermodynamic data. In this chapter, we provide the basic background material required to formulate and apply FST to a variety of applications. The major aims of this section are (i) to provide a brief introduction or recap of the relevant thermodynamics and statistical thermodynamics behind the formulation and primary uses of the Fluctuation Theory of Solutions (ii) to establish a consistent notation which helps to emphasize the similarities between apparently different applications of FST and (iii) to provide the working expressions for some of the potential applications of FST. 

We now consider some other thermodynamic quantities of transfer for the processes mentioned above. First, we introduce a notation that will be useful later. We refer to process I as the process of transfer described immediately following relation (4.190). Likewise, the process of transfer described immediately following relation (4.191) will be referred to as process II. The free energy changes corresponding to these two processes are denoted by and zl/ °(II), respectively. From (4.190) and... 

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