The Thermodynamic Basis of Chemical Equilibrium

A knowledge of the position of chemical equilibria is of the utmost importance to us in water chemistry. By knowing the position of chemical equilibria, we can determine whether it is possible for certain reactions between reactants at given concentrations to proceed. For example, we can provide answers to questions such as Will calcium carbonate tend to precipitate or dissolve in this water Can I possibly oxidize sulfide with nitrate and so on. There are two general ways to answer questions like these. The first is to do an experiment and the second is to calculate the answer using previously determined equilibrium data. Although the first way may be more enjoyable to those who like puttering around in the laboratory, the second approach is far superior if time is of the essence. 

To explore the various techniques that can be used to answer the 

Most of the reactions with which we are concerned in water chemistry take place in a closed system or can be analyzed as though they take place in a closed system. For thermodynamic purposes a closed system is one to which matter cannot be added or removed. Energy, however, may flow across its boundaries. Since, in addition to working with closed systems, we usually are interested in systems at constant temperature and pressure, we can make extensive use of the thermodynamic expression for free energy, 

G = Gibbs free energy, kcal T = absolute temperature, 

For closed systems at constant pressure and constant temperature, the criterion for equUibxium is that the total free energy of the system Gj) is a minimum. For example, consider the reversible reaction previously examined in Eq. 3-1, 

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Why Do We Need to Know This Material The dynamic equilibrium toward which every chemical reaction tends is such an important aspect of the study of chemistry that four chapters of this book deal with it. We need to know the composition of a reaction mixture at equilibrium because it tells us how much product we can expect. To control the yield of a reaction, we need to understand the thermodynamic basis of equilibrium and how the position of equilibrium is affected by conditions such as temperature and pressure. The response of equilibria to changes in conditions has considerable economic and biological significance the regulation of chemical equilibrium affects the yields of products in industrial processes, and living cells struggle to avoid sinking into equilibrium. 

The historical development of chemical equilibrium has been described in several reviews (e.g., Berger, 1997 Laidler, 1985 Lindauer, 1962 Lund, 1965, 1968). The concept of chemical equilibrium was introduced in the 1860s in the context of empirical studies of incomplete and reversible chemical conversions. Explanations for these phenomena were formulated on the basis of two essentially different theoretical perspectives, a kinetic framework and a thermodynamic framework. 

The thermodynamic basis of the X ernst law is supplied by the equality of chemical potentials of the component in the two phases at equilibrium ... 

Another approach is also possible we can indeed consider the problem in a mathematical framework suitable for the discontinuous solutions we will look for a solution c(x,t) in the sense of distributions ("weak solutions"), which must be, on the basis of chemical equilibrium, directly that given in the preceding section. It can be shown that, in the new framework, the solution of the equation is no longer unique. Mathematicians use to add a so-called "entropy condition" that makes the selection of the solution with physical meaning [ll]. In the following sections, we will see the correspondence between the approach of mathematicians and that we can follow in thermodynamics. As may seem obvious for a phenomenon that involves strong gradients, the diffusion will be considered as the perturbative phenomenon used for the choice of the "physical" solution [ l]. 

The introductory Section 3.1.2.5 in Chapter 3 identifies the negative chemical potential gradient as the driver of targeted separation, and the relevant species flux expression is developed in Section 3.1.3.2 (see Example 3.1.9 also). Section 3.1.4 introduces molecular diffusion and convection and basic mass-transfer coefficient based flux expressions essential to studies of distillation and other phase equilibrium based separation processes. Section 3.1-5.1 introduces the Maxwell-Stefan equations forming the basis of the rate based approach of analyzing distillation column operation. After these fundamental transport considerations (which are also valid for other phase equilibrium based separation processes), we encounter Section 3.3.1, where the equality of chemical potential of a species in all phases at equilibrium is illustrated as the thermodynamic basis for phase equilibrium (Le. = /z ). Direct treatment of distillation then begins in Section 3.3.7.1, where Raouit s law is introduced. It is followed by Section 3.4.1.1, where individual phase based mass-transfer coefficients are reiated to an overall mass-transfer coefficient based on either the vapor or liquid phase. 

Chapter 15 (Principles of Chemical Equilibrium) has been significantly revised to emphasize the thermodynamic basis of equilibrium and to de-emphasize aspects of kinetics. There is an increased emphasis on the thermodynamic equilibrium constant, which is expressed in terms of activities, along with an updated discussion of Le Chatelier s principle to emphasize certain limitations associated with its use (e.g., for certain reactions and initial conditions, the addition of a reactant may actually cause net change to the left). Several new worked examples are included to show how equilibrium constant expressions may be simplified and solved when the equilibrimn constant is either very small or very large. 

For a polydisperse polymer, analysis of sedimentation equilibrium data becomes complex, because the molecular weight distribution significantly affects the solute distribution. In 1970, Scholte [62] made a thermodynamic analysis of sedimentation equilibrium for polydisperse flexible polymer solutions on the basis of Flory and Huggins chemical potential equations. From a similar thermodynamic analysis for stiff polymer solutions with Eqs. (27) for IT and (28) for the polymer chemical potential, we can show that the right-hand side of Eq. (29) for the isotropic solution of a polydisperse polymer is given, in a good approximation, by Eq. (30) if M is replaced by Mw [41],... 

Many additional consistency tests can be derived from phase equilibrium constraints. From thermodynamics, the activity coefficient is known to be the fundamental basis of many properties and parameters of engineering interest. Therefore, data for such quantities as Henry s constant, octanol—water partition coefficient, aqueous solubility, and solubility of water in chemicals are related to solution activity coefficients and other properties through fundamental equilibrium relationships (10,23,24). Accurate, consistent data should be expected to satisfy these and other thermodynamic requirements. Furthermore, equilibrium models may permit a missing property value to be calculated from those values that are known (2). 

The thermodynamic treatment of systems in which at least one component is an electrolyte needs special comment. Such systems present the first case where we must choose between treating the system in terms of components or in terms of species. No decision can be based on thermodynamics alone. If we choose to work in terms of components, any effect of the presence of new species that are different from the components, would appear in the excess chemical potentials. No error would be involved, and the thermodynamic properties of the system expressed in terms of the excess chemical potentials and based on the components would be valid. It is only when we wish to explain the observed behavior of a system, to treat the system on the basis of some theoretical concept or, possibly, to obtain additional information concerning the molecular properties of the system, that we turn to the concept of species. For example, we can study the equilibrium between a dilute aqueous solution of sodium chloride and ice in terms of the components water and sodium chloride. However, we know that the observed effect of the lowering of the freezing point of water is approximately twice that expected for a nondissociable solute. This effect is explained in terms of the ionization. In any given case the choice of the species is dictated largely by our knowledge of the system obtained outside of the field of thermodynamics and, indeed, may be quite arbitrary. 

It must be emphasized that the conditions of chemical equilibrium can be derived and explained most exactly on the basis of thermodynamics, that is without involving reaction rates at all. Textbooks of physical chemistry will of course contain the thermodynamical interpretation (cf. W. J. Moore s Physical Chemistry. 4th edn., Longman 1966, p. 167 et f.)... 

One way to recognize the significance of this equation is to remember that the ultimate objective of chemical thermodynamics is to calculate the equilibrium composition of a system of reactions. A chemical reaction system has R independent equilibrium constant expressions and C conservation equations, and this is just enough information to calculate the equilibrium concentrations of N species. Equation 7.1-9 is useful because it makes it possible to calculate a conservation matrix from a stoichiometric number matrix. In doing this with the operation NullSpace we will see again that it yields a basis for the conservation matrix. 

Chapters 2-5 deal with chemical engineering problems that are expressed as algebraic equations - usually sets of nonlinear equations, perhaps thousands of them to be solved together. In Chapter 2 you can study equations of state that are more complicated than the perfect gas law. This is especially important because the equation of state provides the thermodynamic basis for not only volume, but also fugacity (phase equilibrium) and enthalpy (departure from ideal gas enthalpy). Chapter 3 covers vapor-liquid equilibrium, and Chapter 4 covers chemical reaction equilibrium. All these topics are combined in simple process simulation in Chapter 5. This means that you must solve many equations together. These four chapters make extensive use of programming languages in Excel and MATE AB. 

As the fundamental concepts of chemical kinetics developed, there was a strong interest in studying chemical reactions in the gas phase. At low pressures the reacting molecules in a gaseous solution are far from one another, and the theoretical description of equilibrium thermodynamic properties was well developed. Thus, the kinetic theory of gases and collision processes was applied first to construct a model for chemical reaction kinetics. This was followed by transition state theory and a more detailed understanding of elementary reactions on the basis of quantum mechanics. Eventually, these concepts were applied to reactions in liquid solutions with consideration of the role of the non-reacting medium, that is, the solvent. 

The rate is actually given by /[S20g ][I ] (a second-order reaction) and not k/[S20g ] [ 1 ], as might be expected from the balanced chemical reaction (a fourth-order reaction would be predicted). The only sound theoretical basis for the equilibrium constant comes from thermodynamic arguments. See Gibbs free energy in Section 6.3 for the thermodynamic computation of.equilibrium constant values. 

Separation operations are interphase mass transfer processes because they involve the creation, by the addition of heat as in distillation or of a mass separation agent as in absorption or extraction, of a second phase, and the subsequent selective separation of chemical components in what was originally a one-phase mixture by mass transfer to the newly created phase. The thermodynamic basis for the design of equilibrium staged equipment such as distillation and extraction columns are introduced in this chapter. Various flow arrangements for multiphase, staged contactors are considered. 

The techniques we will develop are based on the branch of science known as thermodynamics. The theoretical aspects of thermodynamics are extremely precise and orderly its mathematical basis is complex. We, however, are only interested in what thermodynamics can do for us as a tool in solving problems of chemical equilibrium. We are in a situation similar to the automobile driver using a road map. Not many drivers thoroughly understand the principles of geometry and plane trigonometry that were used to draw the map. However, most know how to read a map and in doing so could manage reasonably well to get from Urbana to Berkeley. 

Within a thermodynamic framework, a qualitative explanation for equilibrium phenomena was first put forward by Horstmann (1873). He used the Second Law of Thermodynamics as a starting point to reason that, in a state of chemical equilibrium, the entropy of a system was at a maximum. In his view, molecular processes merely influenced the time it takes to reach a state of equilibrium. Horstmann discussed his explanation with Pfaundler (1867), the two accepting the validity of each other s theories, but differing in their view of the importance of these theories to provide a causal explanation for chemical equilibrium (Snelders, 1977). In later years quantitative formulations for chemical equilibrium were derived on the basis... 

This completes our description of the thermodynamic basis functions in terms of the configurational and momenta density functions obtained directly from the equilibrium solution to the Liouville equation. As will be shown in the next chapter, the nonequilibrium counterparts (local in space and time) of the thermodynamic basis functions can also be obtained directly from the Liouville equation, thus, providing a unified molecular view of equilibrium thermodynamics and chemical transport phenomena. Before moving on, however, we conclude this chapter by noting some important aspects of the equilibrium solution to the Liouville equation. 

Then why do we put so much stock in equilibria For one thing, a system at equilibrium is a system we can understand using thermodynamics. Also, though almost all chemical systems of interest aren t at equilibrium, the idea of equilibrium is used as a starting point. The concept of chemical equilibrium is the very basis for understanding systems that are not at equilibrium. An understanding of equilibrium is a central part of understanding chemistry. 

In accordance with the aforesaid, there may arise the problem of the correct calculation of chemical potential difference between inside and outside bulk phases of small vesicles. For macroscopic vesicles this problem is trivial, and can be easily solved using the well-known thermodynamic formula of the Gibbs-Nernst type for transmembrane difference in chemical potentials, Afip = k Tln(Cin/Cout) where and are the concentrations of a component P inside and outside the vesicle, respectively. Under conditions of chemical equilibrium these concentrations can be calculated on the basis of the mass action law. For small enough vesicles, however, the problem of the adequate estimation of mean concentrations of particles inside a vesicle becomes more complicated. There are three reasons why the conventional thermodynamic approach to the calculation of particles concentrations could be misleading ... 

Two major groups of performance models have been proposed. The first group considers the membrane as a homogeneous mixture of ionomer and water. The second group involves approaches that consider the membrane as a porous medium. Water vapor equilibrates with this medium by means of capillary forces, osmotic forces resulting from solvated protons and fixed ions, hydration forces, and elastic forces. In this scenario, the thermodynamic state of water in the membrane should be specified by (at least) two independent thermodynamic variables, namely, chemical potential and pressure, subdued to independent conditions of chemical and mechanical equilibrium, respectively. The homogeneous mixture model is the basis of the so-called... 

When van t Hoff received his Nobel prize in 1901, the study of chemical equilibrium thermodynamics was almost complete. Kinetics, however, belongs to the field of non-equilibrium thermodynamics, a subject for which the principles still had to be formulated. The 1931 work of Lars Onsager marks the beginning of the linear non-equilibrium thermodynamics. This discipline provides a firm basis for the kinetics of the steady state, which applies to many catalytic processes. Onsager received the Nobel prize in 1968. Recently, oscillating reactions have... 

The thermodynamic basis for calculating chemical reaction equilibrium was developed in Section 12.5. There we showed that if a system of componentsA2,. .., A/, reacting according to the equation ... 

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