Equilibrium calculation, approximation

In vapor-liquid equilibria, it is relatively easy to start the iteration because assumption of ideal behavior (Raoult s law) provides a reasonable zeroth approximation. By contrast, there is no obvious corresponding method to start the iteration calculation for liquid-liquid equilibria. Further, when two liquid phases are present, we must calculate for each component activity coefficients in two phases since these are often strongly nonlinear functions of compositions, liquid-liquid equilibrium calculations are highly sensitive to small changes in composition. In vapor-liquid equilibria at modest pressures, this sensitivity is lower because vapor-phase fugacity coefficients are usually close to unity and only weak functions of composition. For liquid-liquid equilibria, it is therefore more difficult to construct a numerical iteration procedure that converges both rapidly and consistently. 

In a series of papers by Leung and coworkers (AlChE J., 32, 1743-1746 [1986] 33, 524-527 [1987] 34, 688-691 [1988] J. Loss Prevention Proc. Ind., 2[2], 78-86 [April 1989] 3(1), 27-32 [Januaiy 1990] Trans. ASME J. Heat Transfer, 112, 524-528, 528-530 [1990] 113, 269-272 [1991]) approximate techni ques have been developed for homogeneous equilibrium calculations based on pseudo-equation of state methods for flashing mixtures. 

The basic thermodynamic data for the design of such reactions can be used to assess the dissociation energies for various degrees of dissociation, and to calculate, approximately, tire relevant equilibrium constants. One important source of dissociation is by heating molecules to elevated temperamres. The data below show the general trend in the thermal dissociation energies of a number of important gaseous molecules. 

From such crude data as are to be found in the literature we can calculate approximate values of the equilibrium constants, and hence of the free energies of dissociation for the various hexaarylethanes. From our quantum-mechanical treatment, on the other hand, we obtain only the heats of dissociation, for which, except in the single case of hexaphenylethane, we have no experimental data. Thus, in order that we may compare our results with those of experiment, we must make the plausible assumption that the entropies of dissociation vary only slightly from ethane to ethane. Then at a given temperature the heats of dissociation run parallel to the free energies and can be used instead of the latter in predicting the relative degrees of dissociation of the different molecules. 

These four equations are perfectly adequate for equilibrium calculations although they are nonsense with respect to mechanism. Table 7.2 has the data needed to calculate the four equilibrium constants at the standard state of 298.15 K and 1 bar. Table 7.1 has the necessary data to correct for temperature. The composition at equilibrium can be found using the reaction coordinate method or the method of false transients. The four chemical equations are not unique since various members of the set can be combined algebraically without reducing the dimensionality, M=4. Various equivalent sets can be derived, but none can even approximate a plausible mechanism since one of the starting materials, oxygen, has been assumed to be absent at equilibrium. Thermodynamics provides the destination but not the route. 

The amount of lead that remains in solution in surface waters depends upon the pH of the water and the dissolved salt content. Equilibrium calculations show that at pH >5.4, the total solubility of lead is approximately 30 pg/L in hard water and approximately 500 pg/L in soft water. Sulfate ions, if present... 

If 1.0 mol of W and 3.0mol of Q are placed in a 1.0-L vessel and allowed to come to equilibrium, calculate the equilibrium concentration of Z using the following steps (a) If the equilibrium concentration of Z is equal to x, how much Z was produced by the chemical reaction (b) How much R was produced by the chemical reaction (c) How much W and Q were used up by the reaction (d) How much W is left at equilibrium (e) How much Q is left at equilibrium (/) With the value of the equilibrium constant given, will x (equal to the Z concentration at equilibrium) be significant when subtracted from 1.0 (g) Approximately what concentrations of W and Q will be present at equilibrium (/t) What is the value of x (/) What is the concentration of R at equilibrium (7) Is the answer to part (/) justified ... 

The processing of hydrocarbons always has the potential to form coke (soot). If the fuel processor is not properly designed or operated, coking is likely to occur (3). Carbon deposition not only represents a loss of carbon for the reaction but more importantly also results in deactivation of catalysts in the processor and the fuel cell, due to deposition at the active sites. Thermodynamic equilibrium calculations provide a first approximation of the potential for coke formation. The governing equations are ... 

Write the equations for each of the reactions shown below. Using the E° values below, calculate approximate Gibbs energies for each reaction, and show by the relative length of the arrows on which side of the reaction the equilibrium lies. 

Values of the activity coefficients are deduced from experimental data of vapor-liquid equilibria and correlated or extended by any one of several available equations. Values also may be calculated approximately from structural group contributions by methods called UNIFAC and ASOG. For more than two components, the correlating equations favored nowadays are the Wilson, the NRTL, and UNIQUAC, and for some applications a solubility parameter method. The fust and last of these are given in Table 13.2. Calculations from measured equilibrium compositions are made with the rearranged equation... 

In other cases, the equation for x in terms of K may be quite complicated. One approach to solving the equation is to use a computer or the equilibrium calculator on the CD that accompanies this text. The alternative is to look for an approximate solution. One approximation technique can greatly simplify calculations when the change in molar concentration (x) is less than about 5% of the initial concentrations. To use it, assume that x is negligible when added to or subtracted from a number. Thus, we can replace all expressions like A — x or A — lx, for example, by A. When x occurs on its own (not added to or subtracted from another number), it is left unchanged. So, an expression like (0.1 — 2x)2x simplifies to (0.1 )2x. At the end of the calculation, it is important to verify that the calculated value of x is indeed smaller than about 5% of the initial values. If it is, then the approximation is valid. If not, then we must solve the equation without making an approximation. The approximation procedure is illustrated in Example 9.10. 

Nonetheless, equilibrium considerations can greatly aid attempts to understand in a general way the redox patterns observed or anticipated in natural waters. In all circumstances equilibrium calculations provide boundary conditions toward which the systems must be proceeding, however slowly. Moreover, partial equilibria (those involving some but not all redox couples) are approximated frequently, even though total equilibrium is not approached. In some instances active poising of particular redox couples allows one to predict significant oxidation-reduction levels or to estimate properties and reactions from computed redox levels. 

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

As a major deficit, in both DH and MSA theory the Mayer functions fxfi = exp —f q>ap(r) — 1 are linearized in ft. This approximation becomes unreasonable at low T and near criticality. Pairing theories discussed in the next section try to remedy this deficit. Attempts were also made to solve the PB equation numerically without recourse to linearization [202-204]. Such PB theories were also applied in phase equilibrium calculations [204-206]. 

In this book, we will calculate approximate solubilities assuming that ionic solutes are completely dissociated [reaction (1)]. In the case of PbCl2, ignoring the second equilibrium gives a calculated solubility that is too low by a factor of about 2. 

The object of Chapter 4 was to provide an overview of phase equilibria concepts, which are more easily obtained through phase diagrams and the approximate, historical methods. With Chapter 4 as background, the subject of the present chapter is the phase equilibrium calculation method that is both most accurate and most comprehensive. 

The results for a 55 amp load change are shown in Figures 9.10 and 9.11. Figure 9.10 shows the temperature histories for the cell, interconnect, anode exit gas and cathode exit gas. As can be seen the thermal conditions reach their new equilibrium by approximately / = 800 s, resulting in an exponential time constant of r = 266 s. The temperature change for the gas stream is about 150°C. A calculation of the thermal time constant based on the fully lumped model (Equation (9.28) shows... 

Consider the case of a single collective solvent coordinate y. This coordinate is linearly coupled to the solute at the transition state by generalized Langevin theory [71-75], (It is not necessary to couple the solvent to the solute for the calculation of reactant properties because we retain the equilibrium-reactant approximation.) The form of the coupling is [60,61]... 

Because K, depends on concentrations and the product KyKx is concentration independent, Kx must also depend on concentration. This shows that the simple equilibrium calculations usually carried out in first courses in chemistry are approximations. Actually such calculations are often rather poor approximations when applied to solutions of ionic species, where deviations from ideality are quite large. We shall see that calculations using Eq. (47) can present some computational difficulties. Concentrations are needed in order to obtain activity coefficients, but activity coefficients are needed before an equilibrium constant for calculating concentrations can be obtained. Such problems are usually handled by the method of successive approximations, whereby concentrations are initially calculated assuming ideal behavior and these concentrations are used for a first estimate of activity coefficients, which are then used for a better estimate of concentrations, and so forth. A G is calculated with the standard state used to define the activity. If molality-based activity coefficients are used, the relevant equation is... 

Approximate procedures have been evolved which permit one to determine the state of the expansion process for a given system. In fact these procedures permit the performance to be calculated when the chemical rates are finite and thus do not correspond to frozen, essentially zero chemical rate or equilibrium, essentially infinite chemical rate,flow. As one would expect intuitively, the results of these finite rate determinations show that the flow remains nearly in chemical equilibrium at the beginning of the expansion process, and at a given temperature or point in the nozzle the composition becomes frozen and remains so throughout the expansion process. Finite rate performance calculations are very complex and are presently limited to only a few systems due to lack of kinetic data at the temperatures of concern. Thus most performance calculations are made for either or both equilibrium and frozen flow and it is kept in mind that the actual results must lie somewhere between the two. For most systems equilibrium calculations are very satisfactory. 

This is the same as Bronsted s theory which was designed particularly for solutions. The concentration of the activated complex can be expressed in terms of the reactants and the equilibrium constant K. Also the heat of the reaction, AH, to give the activated complex, can be calculated approximately from the quantum theory or from the Arrhenius theory. Since AF= —RT In K and AF = AII — TAS, and since K can, in some cases, be calculated from known, fundamental constants, the entropy term remains the only unknown. Rodebush has long pointed out that the unknown quantity 5 in the formula k = se E/RT is related to an entropy term. As a first approximation it has been related to a collision frequency in bimolecular reactions and to a vibration frequency in unimolecu-lar reactions. Combining the two thermodynamic equations23... 

The reformer route gives —990 — 817.9 = —1807.9 above (Section A.3.9). The mismatch in circulator powers is acceptable for two disparate calculations involving approximate equilibrium constants approximate concentrations and logs of very large pressure ratios. 

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