Simultaneous equilibria, calculation

For reaction systems consisting of several stoichiometrically independent reactions all values of Kp (or Kf or fCa) must be established (simultaneous equilibria). Calculation of the equilibrium composition is then not straightforward for more than two reactions (where a graphical solution is still possible based on the conversion of two reference reactants for both reactions as a function of Kp j and Kp ), see Example 4.2.8. 

Complex Clieinical-Reaction Equilibria When the composition of an equilibrium mixture is determined by a number of simultaneous reactions, calculations based on equilibrium constants become complex and tedious. A more direct procedure (and one suitable for general computer solution) is based on minimization of the total Gibbs energy G in accord with Eq. (4-271). The treatment here is... 

With a suitable equation of state, all the fugacities in each phase can be found from Eq. (6), and the equation of state itself is substituted into the equilibrium relations Eq. (67) and (68). For an A-component system, it is then necessary to solve simultaneously N + 2 equations of equilibrium. While this is a formidable calculation even for small values of N, modern computers have made such calculations a realistic possibility. The major difficulty of this procedure lies not in computational problems, but in our inability to write for mixtures a single equation of state which remains accurate over a density range that includes the liquid phase. As a result, phase-equilibrium calculations based exclusively on equations of state do not appear promising for high-pressure phase equilibria, except perhaps for certain restricted mixtures consisting of chemically similar components. 

This optional chapter provides tools to compute the concentrations of species in systems with many simultaneous equilibria.3 The most important tool is the systematic treatment of equilibrium from Chapter 8. The other tool is a spreadsheet for numerical solution of the equilibrium equations. We will also see how to incorporate activity coefficients into equilibrium calculations. Later chapters in this book do not depend on this chapter. 

The competition of metals for oxygen to form their oxides and for carbon to form their carbides is also a common problem for complex equilibrium calculations. In principle, multiple oxide and formation reactions can be considered simultaneously. This is the case for the reactions [6]... 

A frequent complication is that several simultaneous equilibria must be considered (Section 3-1). Our objective is to simplify mathematical operations by suitable approximations, without loss of chemical precision. An experienced chemist with sound chemical instinct usually can handle several solution equilibria correctly. Frequently, the greatest uncertainty in equilibrium calculations is imposed not so much by the necessity to approximate as by the existence of equilibria that are unsuspected or for which quantitative data for equilibrium constants are not available. Many calculations can be based on concentrations rather than activities, a procedure justifiable on the practical grounds that values of equilibrium constants are obtained by determining equilibrium concentrations at finite ionic strengths and that extrapolated values at zero ionic strength are unavailable. Often the thermodynamic values based on activities may be less useful than the practical values determined under conditions comparable to those under which the values are used. Similarly, thermodynamically significant standard electrode potentials may be of less immediate value than formal potentials measured under actual conditions. 

Reliable and fast equilibrium calculations (or so-called flash calculations) are the mechanism by which thermodynamic properties are used in industry. This area has received much attention in the past. Algorithms include successive substitution with acceleration and stability analysis,Inside-Out and Interval methods, Homotopy continuation methods with application to three-phase systems, and systems with simultaneous physical and chemical equilibrium. An area of recent focus is the flash algorithm for mixtures containing polydisperse polymers. However, many challenging problems remain. 

Recently, Comas et al.219 performed the thermodynamic analysis of the SRE reaction in the presence of CaO as a C02 sorbent. The equilibrium calculations indicate that the presence of CaO in the ethanol steam reforming reactor enhances the H2 yield while reducing the CO concentrations in the outlet of the reformer. Furthermore, the temperature range at which maximum H2 yield could be obtained also shifts from above 700 °C for the conventional steam reforming reaction without CaO to below 700 °C, typically around 500 °C in the presence of CaO. It appears that the presence of CaO along with ethanol reforming catalyst shift the WGS equilibrium in the forward direction and converts more CO into C02 that will be simultaneously removed by CaO by adsorption. 

The chemical equilibrium calculations are done by sophisticated computer codes, such as the CONDOR code [2], This code simultaneously considers the dual constraints of mass balance and chemical equilibrium. The operation of the CONDOR code and the general principles of chemical equilibrium calculations are best illustrated using a simplified version of iron chemistry in solar composition material. We define the total elemental abundance of iron as A(Fe). This is the atomic abundance of Fe relative to 106 Si atoms and is 838,000 Fe atoms [5]. The mole fraction (X) of total iron (XFe) in all Fe-bearing compounds is... 

One question that arises in phase equilibrium calculations and experiments is how many phases can be in equilibrium simultaneously, since this determines how many phases one should search for. Consider a one-component system. 

The algebra involved in solving for the molar extent of reaction in general chemical equilibrium calculations can be tedious, especially if several reactions occur simultaneously, because of the coupled, nonlinear equations that arise. It is frequently possible, however, to make judicious simplifications based on the magnitude of the equilibrium constant. This is demonstrated in the next illustration. 

Simultaneous Reactions, When two isothermal reactions take place simultaneously, the calculation is slightly more involved, but may be solved readily. The solution of this type of problem involves two unknowns. The procedure is to set up the equations involving the equilibrium constant for each reaction and to solve the simultaneous equations graphically. 

When buffer solutions were not used and the pH was not reported, we calculated the pH using the solution concentration and pK values for all dissociation reactions and assuming that the pH was 7.0 prior to solute addition. A general treatment of simultaneous equilibrium involving equations for all linearly independent reactions, the water dissociation reaction (K = 1.0 x lO" " ), a molecular balance on the active species, and an equation requiring solution electroneutrafity is required to calculate the natural pH (Brescia et al., 1975). A more detailed discussion of the adjustment for ionization and associated calculations is presented in Vecchia and Bimge (2002b). 

The fraction unionized was calculated using Equation 15.8 when one pK), was dominant. Otherwise, It was determined from a more rigorous solution of simultaneous equilibrium as described in Vecchia (1997) or Vecchia and Bunge (2002b). 

We consider esterification of ethanol with acetic acid to form ethyl acetate and water. This reaction has been much used for testing algorithms that perform simultaneous phase and reaction-equilibrium calculations. At ambient pressures, we assume the reaction occurs in a vapor phase but depending on the exact values for T and P, the mixture may exist as one-phase vapor, one-phase liquid, or a two-phase vapor-hquid system. The feed contains equimolar amounts of ethanol and acetic acid. The problem is to determine the equilibrium state the phases present and their compositions at 1.0133 bar and temperatures near 355 K. 

High-temperature equilibrium calculations (HTECs) are useful for studying complex chemical systems, and they enable the simultaneous investigation of condensed and gaseous phases. For the calculations general computer programs are used, and the equilibrium compositions in the solid, liquid, and ideal gas phases are calculated for the given amounts of the elements assumed to be present in the system (input... 

The extension of the austenite field in the C-Fe-Ni system below 900°C has been calculated [1977Uhr, 1978Uhr] and was reported in Fig. 5. The line with an arrow indicates the composition of the austenite (yFCjNi) in simultaneous equilibrium with (Fe,Ni)3C and ferrite (aFe). Slight modifications have been done according to the binary edges. 

An improvement can be achieved with the volume translation concept introduced by Peneloux et al [55]. The idea is that the specific volume calculated by the equation of state is corrected by addition of a constant parameter c. The volume translation has no effect on the vapor-liquid equilibrium calculation, as both the liquid and the vapor volume are simultaneously translated by a constant value. The procedure has also little effect on the calculated vapor volumes, as c is in the order of magnitude of a liquid volume far away from the critical temperature. 

In phase equilibrium calculations, three unknowns, namely the polymer concentration in the dilute phase Xg, the polymer concentration in the concentrated phase Xg, and the temperature T, occur. Specifying one of the unknowns, the two other ones can be calculated by solving Eq. (10.15) simultaneously. In order to find suitable initial values, the spinodal and/or the critical solution point can be helpful. The spinodal (see Figure 10.2) separates the metastable from the unstable region in the phase diagram and can be calculated with the help of the stability theory, where the necessary condition reads... 

Solving Eqs. (7.20) to (7.25) simultaneously permits calculation of the concentrations at equilibrium. If, as will frequently happen, the solutions are so dilute that Xaa and arc for all practical purposes equal to unity, Eqs. (7.20) and (7.21) reduce to... 

Phase and chemical equilibrium calculations are essential for the design of processes involving chemical transformations. Even in the case of reactions that cannot reach chemical equilibrium, the solution of this problem gives information on the expected behaviour of the system and the potential thermodynamic limitations. There are several problems in which the simultaneous calculation of chemical and phase behaviour is mandatory. This is the case, for example, of reactive distillations where phase separation is used to shift chemical equilibrium. Also, the calculation of gas and solid solubility in liquids of high dielectric constants requires at times the resolution of chemical equilibrium between the different species that are formed in the liquid phase. Several algorithms have been proposed in the literature to solve the complex non-linear problem however, proper thermodynamic model selection has not received much attention. 

The SMC also outputs the new bathymetry from simultaneous equilibrium in planform and beach profile, as illustrated in Fig. 29.15. Based on the results of this expected distribution of bottom contoms, the SMC then performs a series of numeric calculations on waves (for swells and storms) and longshore currents (not shown in this chapter). With the unique operational system inherited with the SMC and onscreen modification of bottom contoms via the tracking of the mouse on the screen, graphic outputs are usually available in a matter of minutes for most cases of bay beach design, once the bathymetry and wave conditions are pre-processed... 

There is no difficulty in calculating the chemical equilibrium of a system, in which a single chemical reaction takes place. The calculation, however, becomes increasingly difficult with the rising number of simultaneous reactions, until application of the same procedure to systems with more than three reactions proceeding simultaneously is practically impossible. Therefore, techniques have had to be worked out for more complicated chemically reacting systems, based on principles somewhat different from those of simple equilibrium calculation. The result are methods which allow equilibrium compositions to be calculated for systems of any degree of complexity whatever, in the ideal as well as real gas state. 

In this chapter we consider the simultaneous equilibrium of three phases. We shall see that there are circumstances in which the phases meet in a line of three-phase contact. Macroscopically, this locus of points in which they meet is one-dimensional and locally linear it is analogous to the macroscopically two-dimensional and locally planar interface between two phases. There is an excess free energy per unit length, or line tension, associated with the three-phase line, and we should in prindple be able to calculate that tension from a microscopic theory. Such a line of three-phase contact should have a predictable, and in principle discernible, three-dimensional structure at the molecular level, and its structure and tension should be related, just as are the structure and tension of the interface between two phases. 

The Lurgi gas cleanup system, shown in Figures 2, 3, and 4, is a good example of the problems involved. Each of the five counter-current exchangers represents a series of complicated, simultaneous equilibrium and heat transfer calculations for a polar mixture, with a three- and possibly four-phase system. (If the solid fines are considered, the system is four, possibly five phases but the solid phase, which most likely stays with the tar, is generally neglected.) Neither the available enthalpy data, nor the available equilibrium correlations, are really adequate for such mixtures, and the problems would be worse if the pressures were higher, as they may be in the future. This is not to say that... 

When only the total system composition, pressure, and temperature (or enthalpy) are specified, the problem becomes a flash calculation. This type of problem requires simultaneous solution of the material balance as well as the phase-equilibrium relations. 

The equilibrium ratios are not fixed in a separation calculation and, even for an isothermal system, they are functions of the phase compositions. Further, the enthalpy balance. Equation (7-3), must be simultaneously satisfied and, unless specified, the flash temperature simultaneously determined. 

The interfacial mole fractions yj and Xj can be determined by solving Eq. (5-252) simultaneously with the equilibrium relation = F(x,) to obtain y and Xj. The rate of transfer may then be calculated from... 

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