Equilibria calculations

Complete calculations of chemical equilibria in natural waters and soil solutions are complicated because such a large number of solutes, solids and gases are 

Electron configuration of inert gas, low polarizability, hard spheres  

One to nine outer shell electrons, not spherically symmetric 

Electron number corresponds to Ni°, Pd° and Pt° (10 or 12 outer shell electrons), low electronegativity, high polarizability, soft spheres  

Source Stumm and Morgan (1996). Reproduced by permission of Wiley, New York. 

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. 

High-pressure systems in the vicinity of critical points, such as synthesis gas and air separation systems, remain a challenge. Our flash algorithm has difficulty in identifying the correct phase state, or converging to the correct vapor-Uquid solutions. This problem may be exacerbated by the difficulty in obtaining the equation of state volume root in the vicinity of the critical points. Further work to improve the algorithm and the equation of state volume root determination is required. It is believed that the homotopy continuation methods are probably better suited for calculations near the critical points. 

Liquid-liquid equilibrium calculations remain a problem area especially for systems containing non-volatile species such as strong electrolytes or high polymers. These species have negligible or no fugacities, and, as a result, many flash algorithms cannot properly account for them. 

Many important industrial systems make extensive use of surfactants for various reasons. Surfactants may dissolve in the bulk liquid phases, form distinct micelles, or preferentially concentrate on the interfaces. Existing flash algorithms do not address micelles or interfaces as possible reservoirs for the surfactants. The flash results are simply not valid for systems with surfactants. 

The pharmaceutical and life science industries often deal with large, complex molecules, and separation via crystallization is an important practice. Robust flash algorithms for solid-liquid equilibrium, particularly systems with multiple polymorphs, are highly desirable. 

Henry s, Raoult s, and Nernst s laws. If always present during a reaction (always in excess), the activities of pure solids and liquids may be assumed equal to unity, or a, = 1. For solid or liquid mixtures we can define ideal solutions for which a, = the mole fraction of i in the mixture. In a binary solution, for example, the mole fraction of component 1 in a solution with component 2 is given by 

Henry s law also applies to trace constituents or components in two coexisting solutions at equilibrium, for example, to two minerals in a rock or to a mineral in an aqueous solution. In general, the chemical potential of component i, /z, must be the same in two phases (solutions) at equilibrium. For example, in a fluid phase such as groundwater (phase jc), we may write 

In a coexisting mineral (phase y) at equilibrium the corresponding expression is 

At a particular temperature and pressure the exponential term is a constant, and the mole fraction ratio in the two coexisting phases will equal a constant, or 

In other words, the mole fraction ratio of / in the coexisting phases at equilibrium for a given T and P should be constant. This is Nernst s law (cf. Lewis and Randall 1961). K is also called the distribution coefficient, often symbolized by , and is used in the study of trace element partitioning between coexisting mineral solid solutions. 

Chemistry, the study of chemical reactions, would be a much simpler (and less interesting) science if all chemical reactions went to completion. This is not the case, however chemistry cannot be based on stoichiometry alone. Many reactions are reversible and do not go to completion. In such mixtures, the final composition is quite different from that which could be expected from stoichiometric considerations. A detailed knowledge of the composition, i.e., the concentration of each species present, of such mixtures is essential for the understanding of their chemistry. Short of actually measuring the amount of each and every species present, the only way to obtain this knowledge is from a consideration of the appropriate equilibria. 

In trying to interpret the physical, chemical, or biological properties of solutions, it is equally essential toknow their detailed composition since each species that is present may well make a unique contribution to these properties. 

The determination of equilibrium constants is naturally of utmost importance for the solution of problems of the t)q)e outlined above. In addition, the equilibrium constant provides a means of characterizing a chemical reaction and is usefiil in theoretical considerations. A study of equilibrium constant values provides a basis for evaluation of the influence on chemical reactions of the many factors related to chemical structure. 

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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. 

Figure 17 shows results for the acetonitrile-n-heptane-benzene system. Here, however, the two-phase region is somewhat smaller ternary equilibrium calculations using binary data alone considerably overestimate the two-phase region. Upon including a single ternary tie line, satisfactory ternary representation is obtained. Unfortunately, there is some loss of accuracy in the representation of the binary VLB (particularly for the acetonitrile-benzene system where the shift of the aceotrope is evident) but the loss is not severe. 

In modern separation design, a significant part of many phase-equilibrium calculations is the mathematical representation of pure-component and mixture enthalpies. Enthalpy estimates are important not only for determination of heat loads, but also for adiabatic flash and distillation computations. Further, mixture enthalpy data, when available, are useful for extending vapor-liquid equilibria to higher (or lower) temperatures, through the Gibbs-Helmholtz equation. ... 

The most frequent application of phase-equilibrium calculations in chemical process design and analysis is probably in treatment of equilibrium separations. In these operations, often called flash processes, a feed stream (or several feed streams) enters a separation stage where it is split into two streams of different composition that are in equilibrium with each other. 

Large errors in the low-pressure points often have little effect on phase-equilibrium calculations e.g., when the pressure is a few millitorr, it usually does not matter if we are off by 100 or even 1000%. By contrast, the high-pressure end should be reliable large errors should be avoided when the data are extrapolated beyond the critical temperature. 

A vapor-liquid equilibrium calculation shows that a good separation is obtained, but the required product purity of butadiene <0.5 wt% and sulphur... 

Hence it is necessary to measure the heat capacity of a substance from near 0 K to the temperature required for equilibrium calculations to derive the enthalpy as a fiinction of temperature according to equation (B1.27.15f... 

Fig. 1. Trends in effects of 4- and 5-substituenls (expressed as an a variation of R) on the proiomeric equilibrium calculated using the HMO method. When curves do not cross no inversion of protomeric equilibrium is expected to be induced by electronic substituent effects, 4-R-(----) 5-R-(-----). F,E formation energy (see Table 1).

Chemical Potential. Equilibrium calculations are based on the equaHty of individual chemical potentials (and fiigacities) between phases in contact (10). In gas—soHd adsorption, the equiHbrium state can be defined in terms of an adsorption potential, which is an extension of the chemical potential concept to pore-filling (physisorption) onto microporous soHds (11—16). 

In apphcatious to equilibrium calculations, the fugacity coefficients of species iu a mixture are required. Given au expression for G /RT as aetermiued from Eq. (4-158) for a coustaut-compositiou mixture, the corresponding recipe for In is found through the partial-property relation... 

These are general equations that do not depend on the particular mixing rules adopted for the composition dependence of a and b. The mixing rules given by Eqs. (4-221) and (4-222) can certainly be employed with these equations. However, for purposes of vapor/liquid equilibrium calculations, a special pair of mixing rules is far more appropriate, and will be introduced when these calculations are treated. Solution of Eq. (4-232) for fugacity coefficient at given T and P reqmres prior solution of Eq. (4-231) for V, from which is found Z = PV/RT. 

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. 

At the time Meijering published his research, Larry Kaufman was working for his doctorate at MIT with a charismatic steel metallurgist. Professor Morris Cohen, and they undertook some simple equilibrium calculations directed at practical problems of the steel industry. From the end of the 1950s, Kaufman directed his... 

Chemical reaction equilibrium calculations are structured around another thermodynamic term called tlie free energy. Tliis so-callcd free energy G is a property that also cannot be defined easily without sonic basic grounding in tlicmiodynamics. However, no such attempt is made here, and the interested reader is directed to tlie literature. " Note that free energy has the same units as entlialpy and internal energy and may be on a mole or total mass basis. Some key equations and information is provided below. 

Many equilibrium calculations are accomplished using tlie plrase equilibrium constant K,. Tliis constant has been referred to in industry as a coiiiponential split factor, since it provides the ratio of the mole fractions of a component in two equilibrium pluises. The defining equation is... 

A typical intramitochondrial concentration of malate is 0.22 mM. If the [NAD ]/[NADH] ratio in mitochondria is 20 and if the malate dehydrogenase reaction is at equilibrium, calculate the intramitochondrial concentration of oxaloacetate at 25°C. 

Haman, S. E. M. et al, Generalized Temperature-Dependent Parameters of the Redlich-Kwong of State for Vapor-Liquid Equilibrium Calculations, Ind. Eng. Chem. Process Des. Dev. 16, 1, (1977) p. 51. 

The molten system KC1 - K2TaF7 was analyzed using the same method and the ionic equilibrium calculated is as follows ... 

This is because the concentrations of solid copper and solid silver are incorporated into the equilibrium constant. The concentration of solid copper is fixed by the density of the metal—it cannot be altered either by the chemist or by the progress of the reaction. The same is true of the concentration of solid silver. Since neither of these concentrations varies, no matter how much solid is added, there is no need to write them each time an equilibrium calculation is made. Equation (21) will suffice. 

The starting point in any quantitative equilibrium calculation is the Equilibrium Law. For a generalized reaction,... 

Brdnsted-Lowry theory, 194 contrast definitions, 194 indicators, 190 reactions, 188 titrations, 188 Acids, 183 aqueous, 179 carboxylic, 334 derivatives of organic, 337 equilibrium calculations, 192 experimental introduction, 183 names of common, 183 naming of organic, 339 properties of, 183 relative strengths, 192, 451 strength of, 190 summary, 185 weak, 190, 193 Actinides, 414 Actinium... 

Whatever the exact form of carbon deposition may be, it must be recognized and taken into account in future calculations. The deposited material is called Dent carbon, and equilibrium constants based on its free energy are included in Figure 1. The deposition of carbon as Dent carbon was confirmed qualitatively by Pursley et al. (4). In other recently reported equilibrium calculations (5, 6, 7), it was assumed that... 

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. 

Several authors, notably Leland and co-workers (L2), have discussed vapor-liquid equilibrium calculations based on corresponding-states correlations. As mentioned in Section II, such calculations rest not only on the general assumptions of corresponding-states theory, but also on the additional assumption that the characterizing parameters for a mixture do not depend on temperature or density but are functions of composition only. Further, it is necessary clearly to specify these functions (commonly known as mixing rules), and experience has shown that if good results are to be obtained, these... 

Later, we will make equilibrium calculations that involve activities, and we will see why it is convenient to choose the ideal gas as a part of the standard state condition, even though it is a hypothetical state/ With this choice of standard state, equations (6.94) and (6.95) allow us to use pressures, corrected for non-ideality, for activities as we make equilibrium calculations for real gases.s... 

We are now prepared to use thermodynamics to make chemical equilibrium calculations. The following examples demonstrate some of the possibilities. 

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