Some Thermodynamic Relationships

Consider an equilibrium state between species A (reactant) and species B (product)  

If reactant A and product B are introduced at unit activity (a), one of three conditions would be met  

AG° = negative, reaction will move from left to right to meet its equilibrium state 

AG° = positive, reaction will move from right to left in order to meet its equilibrium state (for the definition of AG or AG°, see Chapter 7). 

Dividing Equation I by RT and raising both sides to base e gives 

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Ogston, A.G. (1962). Some thermodynamic relationships in ternary systems, with special reference to the properties of systems containing hyaluronic acid and protein. Archives of Biochemistry and Biophysics, Supplement 1, 39-51. 

Thermodynamic integration is a method for free energy difference calculations based on some thermodynamic relationship for the derivative of the free energy with respect to the quantity X in which the two states differ. Simple integration of this derivative then gives the free energy difference AF between the two states, described by X = 0 and X = 1. Examples of these thermodynamic relationships are the volume derivative of the Helmholtz free energy. 

As noted earlier in this chapter, there is definite relationship between the disjoining pressure theory of adsorption and the x theory. In this section, some thermodynamic relationships for the spreading pressure are derived. It is questionable at this point how useful these relationships will be. They may be useful in extending the theory into the solution chemistry since these relationships are important in that area of research. 

There is probably no area of science that is as rich in mathematical relationships as thermodynamics. This makes thermodynamics very powerful, but such an abundance of riches can also be intimidating to the beginner. This chapter assumes that the reader is familiar with basic chemical and statistical thermodynamics at the level that these topics are treated in physical chemistry textbooks. In spite of this premise, a brief review of some pertinent relationships will be a useful way to get started. 

This chapter presents some basic thermodynamic relationships that apply to all compressors. Equations that apply to a particular type of compressor will be covered in the chapter addressing that compressor. In most cases, the derivations will not be presented, as these are available in the literature. The references given are one possible source for additional background information. 

Chapter 3 starts with the laws, derives the Gibbs equations, and from them, develops the fundamental differential thermodynamic relationships. In some ways, this chapter can be thought of as the core of the book, since the extensions and applications in all the chapters that follow begin with these relationships. Examples are included in this chapter to demonstrate the usefulness and nature of these relationships. 

This equation is coupled to the component balances in Equation (3.9) and with an equation for the pressure e.g., one of Equations (3.14), (3.15), (3.17). There are A +2 equations and some auxiliary algebraic equations to be solved simultaneously. Numerical solution techniques are similar to those used in Section 3.1 for variable-density PFRs. The dependent variables are the component fluxes , the enthalpy H, and the pressure P. A necessary auxiliary equation is the thermodynamic relationship that gives enthalpy as a function of temperature, pressure, and composition. Equation (5.16) with Tref=0 is the simplest example of this relationship and is usually adequate for preliminary calculations. 

The operation of a plant under steady-state conditions is commonly represented by a non-linear system of algebraic equations. It is made up of energy and mass balances and may include thermodynamic relationships and some physical behavior of the system. In this case, data reconciliation is based on the solution of a nonlinear constrained optimization problem. 

Calculated vibrational frequencies, along with calculated equilibrium geometries, may be employed to yield a variety of thermodynamic quantities. The most important of these from the present perspective are associated with bringing energetic data obtained from calculation into juxtaposition with that obtained in a real experiment. The former are energies of non-vibrating molecules at OK, while the latter are free energies at some finite temperature. Standard thermodynamic relationships provide necessary connections ... 

For some reacting mixtures, it is difficult to find physical property data. An alternative version of Leung s method331 makes use of the Clausius-Clapeyron thermodynamic relationship to give a formula-in which all. the data required can be measured experimentally. The Clausius-Clapeyron relationship (T(dPydT) = hfg/vfg) only holds, for ideal single-component systems, and so its use introduces the following additional conditions of applicability ... 

In some cases, it is possible to take advantage of various thermodynamic relationships to write some property as a fluctuation-dependent quantity. Thus, for example, the entropy change may be computed from... 

In 1923, Peter Debye and Erich Hiickel developed a classical electrostatic theory of ionic distributions in dilute electrolyte solutions [P. Debye and E. Hiickel. Phys. Z 24, 185 (1923)] that seems to account satisfactorily for the qualitative low-ra nonideality shown in Fig. 8.3. Although this theory involves some background in statistical mechanics and electrostatics that is not assumed elsewhere in this book, we briefly sketch the physical assumptions and mathematical techniques leading to the Debye-Hiickel equation (8.69) to illustrate such molecular-level description of thermodynamic relationships. 

Equations (11) and (13) are two of the most important thermodynamic relationships for biochemists to remember. If the concentrations of reactants and products are at their equilibrium values, there is no change in free energy for the reactions going in either direction. Living cells, however, maintain some compounds at concentrations far from the equilibrium values, so that their reactions are associated with large changes in free energy. We expand on this point in chapter 11. 

In the next two chapters, we use thermodynamic relationships summarized in Chapter 1 la to delve further into the world of phase equilibria, using examples to describe some interesting effects. As we do so, we must keep in mind that our discussion still describes only relatively simple systems, with a much broader world available to those who study such subjects as critical phenomena, ceramics, metal alloys, purification processes, and geologic systems. In this chapter, we will limit our discussion to phase equilibria of pure substances. In Chapter 14, we will expand the discussion to describe systems containing more than one component. 

We will always choose G so that it is a criterion for spontaneity under conditions of constant intensive variables, T, P, and conjugate driving forces, Lt. It should be realized, however, that new displacement coordinates introduce the possibility of a number (see Question 9) of other new energy-like functions, and consideration of some of these may provide useful thermodynamic relationships. In particular, we will have some use for the function... 

Some rate coefficients have been estimated by means of thermodynamic relationships between the rate coefficients of direct and reverse processes, or by considering ratios of rate coefficients in competing reactions, or by analogies between similar processes. 

Fig. 31. Illustration of some possible thermodynamic relationships between crystal, isotropic, and liquid-crystal phases.

The aforementioned macroscopic physical constants of solvents have usually been determined experimentally. However, various attempts have been made to calculate bulk properties of Hquids from pure theory. By means of quantum chemical methods, it is possible to calculate some thermodynamic properties e.g. molar heat capacities and viscosities) of simple molecular Hquids without specific solvent/solvent interactions [207]. A quantitative structure-property relationship treatment of normal boiling points, using the so-called CODESS A technique i.e. comprehensive descriptors for structural and statistical analysis), leads to a four-parameter equation with physically significant molecular descriptors, allowing rather accurate predictions of the normal boiling points of structurally diverse organic liquids [208]. Based solely on the molecular structure of solvent molecules, a non-empirical solvent polarity index, called the first-order valence molecular connectivity index, has been proposed [137]. These purely calculated solvent polarity parameters correlate fairly well with some corresponding physical properties of the solvents [137]. 

For polymorphic systems of a particular material we are interested in the relationship between polymorphs of one component. A maximum of three polymorphs can coexist in equilibrium in an invariant system, since the system cannot have a negative number of degrees of freedom. This will also correspond to a triple point. For the more usual case of interest of two polymorphs the system is monovariant, which means that the two can coexist in equilibrium with either the vapour or the liquid phases, but not both. In either of these instances there will be another invariant triple point for the two solid phases and the vapour on the one hand, or for the two solid phases and the liquid on the other hand. These are best understood in terms of phase diagrams, which are discussed below, following a review of some fundamental thermodynamic relationships that are important in the treatment of polymorphic systems. 

Stability of polymorphs in general. As noted in Section 2.2.2 the relative stability of polymorphs depends on the free energy (AG = AH — TAS) between them. The relative importance of the two terms on the right can be measured by the ratio between them (say TAS/AH). As seen in Fig. 2.5 at absolute zero T = 0, AG = AH and TAS/AH = 0. At a transition temperature between two polymorphic phases, AG = 0 so the ratio TAS/AH = 1. Above a transition temperature this ratio will be > 1. Applied to some of the polymorphs of 5-Xn, for example, for the pair Y-R at the melting point of R the ratio is 0.85, which means that while Y is the more stable form at that temperature, the entropy is an important contributor to the free energy. Other similar comparisons based on the data in Table 5.2 strengthen the notion of the importance of entropy in the consideration of thermodynamic relationships among polymorphs. 

The text is intended to take the reader through some of the topics covered in the first two years of an undergraduate course in chemistry. It assumes some basic knowledge of topics which should be covered in other courses at this level. These include atomic structure, simple quantum theory, simple thermodynamic relationships and electrode potentials. A knowledge of group theory is not explicitly required to follow the text, but reference is made to the symbols of group theory. The data in the text are based on published sources. However, it should not be assumed that data in the problems are based on actual measurements. Although reported data have been used where possible, some values have been calculated or invented for the purpose of the question. 

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