Equilibrium Calculations Some Illustrative Examples

A catalyst has no effect on the condition of equilibrium in a reversible reaction. 

In Chapter 20, we will investigate in more detail the factors affecting reaction rates, including the way in which a catalyst is able to speed up a reaction and reduce the time it takes for the reaction to reach equilibrium. 

Two students are performing the same experiment in which an endothermic reaction rapidly attains a condition of equilibrium. Student A does the reaction in a beaker resting on the surface of the lab bench while student B holds the beaker in which the reaction occurs. Assuming that all other environmental variables are the same, which student should end up with more product Explain. 

We are now ready to tackle the problem of describing, in quantitative terms, the condition of equilibrium in a reversible reaction. Part of the approach we use may seem unfamiliar at first—it has an algebraic look to it. But as you adjust to this new look, do not lose sight of the fact that we continue to use some familiar and important ideas— molar masses, molarities, and stoichiometric factors from the balanced equation, for example. 

The five numerical examples that follow apply the general equilibrium principles described earlier in the chapter. The first four involve gases, while the fifth deals with equilibrium in an aqueous solution. (The study of equilibria in aqueous solutions is the principal topic of the next three chapters.) Each example includes an assessment that summarizes the essential features of equilibrium calculations exemplified by that type of problem. You may find it helpful to return to these assessments from time to time as you encounter new equilibrium situations in later chapters. 

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Altering Equilibrium Conditions Le Chatelier s Principle 15-7 Equilibrium Calculations Some Illustrative Examples... 

Equilibrium Calculations Some Illustrative Examples—For quantitative equilibrium calculations, a few basic principles and algebraic techniques are required. A useful method employs a tabular system, called an ICE table, for keeping track of the initial concentrations of the reactants and products, changes in these concentrations, and the equilibrium concentrations. 

There have been many attempts to calculate AH independent of the equilibrium constant. The difficulty of a complete theoretical treatment of the H bond unfortunately requires approximations. The uncertainties thus introduced deprive the calculations of predictive value. Briefly, the usual approximations are based on some sort of electrostatic model, with computation of electrostatic, dispersion, and repulsive contributions by the methods of classical physics. Of course, the calculations require knowledge or estimation of such quantities as molecular arrangement, charge distribution, potential function, etc. Only a few systems have been treated. Reference 1327, for HF dimers 25, for carboxylic acids and 1561b, for ice furnish illustrative examples. Many other references are listed in Section 8.3, where a more complete discussion of the theoretical treatments is given. 

Few liquid mixtures are ideal, so vapor-liquid equilibrium calculations can be more complicated than is the case for the hexane-triethylamine system, and the system phase diagrams can be more structured than Fig. 10.1-6. These complications arise from the (nonlinear) composition dependence of the species activity coefficients. For example, as a result of the composition dependence of y, the equilibrium pressure in a fixed-temperature experiment will no longer be a linear function of mole fraction. Thus nonideal solutions exhibit deviations from Raoult s law. We will discuss this in detail in the following sections of this chapter. However, first, to illustrate the concepts and some of the types of calculations that arise in vapor-liquid equilibrium in the simplest way, we will assume ideal vapor and liquid solutions (Raoult s law) here, and then in Sec. 10.2 consider the calculations for the more difficult case of nonideal solutions.. ... 

First, we distinguished between an T and an T identification of state. Less information is provided by an IF-spedfication than by an F -specification, but in particular situations one or the other may be more appropriate. For example, in vapor-liquid equilibrium calculations, an IF-specification is sufficient to close a bubble-T problem, but an F -spedfication fails to dose an isothermal flash problem. Furthermore, most reaction-equilibrium problems are not dosed by F-spedfications they require F -specifications. We have also illustrated that in some situations an F-specification may be suffident, but an F -spedfication may lead to a more advantageous problem formulation and solution technique. The prindpal pitfall is to apply an F-specification to a problem that demands an F -spedfication, for then the problem is ill-posed. 

Equilibrium Compositions for Single Reactions. We turn now to the problem of calculating the equilibrium composition for a single, homogeneous reaction. The most direct way of estimating equilibrium compositions is by simulating the reaction. Set the desired initial conditions and simulate an isothermal, constant-pressure, batch reaction. If the simulation is accurate, a real reaction could follow the same trajectory of composition versus time to approach equilibrium, but an accurate simulation is unnecessary. The solution can use the method of false transients. The rate equation must have a functional form consistent with the functional form of K,i,ermo> e.g., Equation (7.38). The time scale is unimportant and even the functional forms for the forward and reverse reactions have some latitude, as will be illustrated in the following example. 

These considerations illustrate why it is easy to mistake lack of agreement between calculated and experimental values of kn, due to the assumption of an incorrect reaction mechanism, for a medium effect. If the model of a reaction to which the simple equilibrium theory is applied is in error, the solvent isotope effect expression (56) will contain some incorrect factors of the form (1 — n + mf)). Suppose, for example, that the expression (48) or (49)—applicable to a reaction of A-l mechanism—is used in conjunction with experimental data for an A-2 mechanism. Analysis of the results should lead to the conclusion that a factor of the form (1—n + mf>)2 (cf. equation (50)) has been omitted from the required theoretical equation. However, alternatively the conclusion might be drawn that equation (100) ought to have been used in place of (48), and the lack of agreement would then be ascribed to the presence of the factor Y. But Fg is itself a quotient of transfer... 

Thermodynamic calculations have also been used to determine the equilibrium products (Mamyan and Vershinnikov, 1992 Shiryaev et ai, 1993), and to illustrate new possibilities for controlling the synthesis process, even for complex multicomponent systems. Correlating these calculations with the equilibrium phase diagram for each system provides a basis for predicting possible chemical interactions and even limits of combustion during CS of materials. Some examples are discussed in the following subsections. 

Obviously, a conceptually meaningful pe cannot be defined for a nonequilibrium—that is, nonstable or nonmetastable—system. Based on some of the observed activities of redox components of a seawater system (atmosphere, hydrosphere, and sediments), different pe values can be calculated. The examples in Figure 11.4 illustrate immediately that the various redox components are not in equilibrium with each other and that the real system cannot be characterized by a unique pc. 

Numerous applications of standard electrode potentials have been made in various aspects of electrochemistry and analytical chemistry, as well as in thermodynamics. Some of these applications will be considered here, and others will be mentioned later. Just as standard potentials which cannot be determined directly can be calculated from equilibrium constant and free energy data, so the procedure can be reversed and electrode potentials used for the evaluation, for example, of equilibrium constants which do not permit of direct experimental study. Some of the results are of analjrtical interest, as may be shown by the following illustration. Stannous salts have been employed for the reduction of ferric ions to ferrous ions in acid solution, and it is of interest to know how far this process goes toward completion. Although the solutions undoubtedly contain complex ions, particularly those involving tin, the reaction may be represented, approximately, by... 

From the principles of thermodynamics and certain thermodynamic data the maximum extent to which a chemical reaction can proceed may be calculated. For example, at 1 atm pressure and a temperature of 680°C, starting with 1 mole of sulfur dioxide and mole of oxygen, 50% of the sulfur dioxide can be converted to sulfur trioxide. Such thermodynamic calculations result in maximum values for the conversion of a chemical reaction, since they are correct only for equilibrium conditions, conditions such that there is no further tendency for change with respect to time. It follows that the net rate of a chemical reaction must be zero at this equilibrium point. Thus a plot of reaction rate [for example, in units of g moles product/(sec) (unit volume reaction mixture)] vs time would always approach zero as the time approached infinity. Such a situation is depicted in curve A of Fig. 1-1, where the rate approaches zero asymptotically. Of course, for some cases equilibrium may be reached more rapidly, so that the rate becomes almost zero at a finite time, as illustrated by curve B. 

Based on the results of this illustration, if we are interested only in computing the pH of a solution containing a weak acid (or base), it may be possible to neglect the ion activity coefficients. However, if our interest is in the extent of dissociation of the weak acid (or base), the activity coefficients of the ions should be included. What happens in the calculation here (and in other examples later in this chapter) is that there is some cancellation between the effect of the ion nonideality on the calculation of the equilibrium and on the calculation of the pH. This is especially true for a 1 1 acid (that is, an acid that on ionization produces a cation of charge +1 and an anion of charge -I). 

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