Multiple chemical equilibrium

Certain Multiplicative Chemical Equilibrium Systems to Mathematically Equivalent Additive Systems, The RAND Corporation, P—2419, August lybl. 

The first question in this treatment is the meaning of the polymer-active center. The polymer chain contains many centers available for interaction with surfactant. Insofar as the centers are identical and indistinguishable, the polymer-surfacttmt complex formation can be treated as a multiple chemical equilibrium ... 

In the preceding chapter, the choice of reactor type was made on the basis of the most appropriate concentration profile as the reaction progressed, in order to minimize reactor volume for single reactions or maximize selectivity (or yield) for multiple reactions for a given conversion. However, there are still important effects regarding reaction conditions to be considered. Before considering reaction conditions, some basic principles of chemical equilibrium need to be reviewed. 

Advances continue in the treatment of detonation mixtures that include explicit polar and ionic contributions. The new formalism places on a solid footing the modeling of polar species, opens the possibility of realistic multiple fluid phase chemical equilibrium calculations (polar—nonpolar phase segregation), extends the validity domain of the EXP6 library,40 and opens the possibility of applications in a wider regime of pressures and temperatures. 

Predictions of high explosive detonation based on the new approach yield excellent results. A similar theory for ionic species model43 compares very well with MD simulations. Nevertheless, high explosive chemical equilibrium calculations that include ionization are beyond the current abilities of the Cheetah code, because of the presence of multiple minima in the free energy surface. Such calculations will require additional algorithmic developments. In addition, the possibility of partial ionization, suggested by first principles simulations of water discussed below, also needs to be added to the Cheetah code framework. 

Most industrial catalysts are heterogeneous catalysts consisting of solid active components dispersed on the internal surface of an inorganic porous support. The active phases may consist of metals or oxides, and the support (also denoted the carrier) is typically composed of small oxidic structures with a surface area ranging from a few to several hundred m2/g. Catalysts for fixed bed reactors are typically produced as shaped pellets of mm to cm size or as monoliths with mm large gas channels. A catalyst may be useful for its activity referring to the rate at which it causes the reaction to approach chemical equilibrium, and for its selectivity which is a measure of the extent to which it accelerates the reaction to form the desired product when multiple products are possible [1],... 

The statement applies not only to chemical equilibrium but also to phase equilibrium. It is obviously true that it also applies to multiple substitutions. Classically isotopes cannot be separated (enriched or depleted) in one molecular species (or phase) from another species (or phase) by chemical equilibrium processes. Statements of this truth appeared clearly in the early chemical literature. The previously derived Equation 4.80 leads to exactly the same conclusion but that equation is limited to the case of an ideal gas in the rigid rotor harmonic oscillator approximation. The present conclusion about isotope effects in classical mechanics is stronger. It only requires the Born-Oppenheimer approximation. 

The performance of propints is a unique function of the temp of the hot reaction products, their compn and their pressure. The pro-pint bums at constant pressure and forms a set of products which are in thermal and chemical equilibrium with each other. The multiplicity of the reaction products requires that the combustion chamber conditions be calcd from the solution of simultaneous equations of pressure and energy balances. This calcn is best performed by computer, although the manual scheme has been described well by Sutton (Ref 14) and Barr re et al (Ref 10). The chamber conditions determine the condition in the nozzle which in turn characterizes the rocket engine performance in terms of specific impulse and characteristic exhaust velocity... 

Ung S, Doherty MF. Vapor-liquid equilibrium in systems with multiple chemical reactions. Chem Eng Sci 1995 50 23 18. 

This bulk state of secular equilibrium applies to the total amount of the U-series nuclides, but does not necessarily say where the different elements reside within the system. If the bulk system has a single phase (such as a melt or a monomineralic rock) then that phase will be in secular equilibrium. If the material has multiple phases with different partitioning properties, however, the individual phases can maintain radioactive dis-equilibria even when the total system is in secular equilibrium. There are two basic sets of models that exploit this fact, the first assumes complete chemical equilibrium between all phases and the second assumes transient diffusion controlled sohd exchange. 

Gas-sensing electrodes are examples of multiple membrane sensors these contain a gas-permeable membrane separating the test solution from an internal thin electrolyte film in which an ion-selective electrode is immersed. For example, for the ammonia sensor, the pH of the recipient layer is determined by the Henderson-Hasselbach equation [Eq. (18)], derived from the chemical equilibrium between solvated ammonia and ammonium ions ... 

Because reactions among ionic species in solution are rapid, thermo-d5mamic calculations are used to constrain the activities of dissolved chemical species at equilibrium. Garrels and Thompson (1962) were the first to calculate the speciation of the major ions in seawater by determining the extent to which each species is involved in ion pairing with each counter-ion. This information is necessary to establish the percentages of free major ions available in chemical equilibrium calculations. This section presents an example of how such multiple equilibrium systems can be constrained. 

Other workers [112, 113] have shown that a chemical equilibrium model of hydrocarbons based on an exponential-6 fluid model using Ross s soft-sphere perturbation theory is successful in reproducing the behavior of shocked hydrocarbons. Our model of the supercritical phase includes the species H2, CH4, C2H6, and C2H4. We have chosen model parameters to match both static compression isotherms and shock measurements wherever possible. The ability to match multiple types of experiments well increases confidence in the general applicability of our high-pressure equation of state model. 

To prove this assertion, it is first useful to consider the mathematical technique of Lagrange multipliers, a method used to find the extreme (maximum or minimum) value of a function subject to constraints. Rather than develop the method in complete generality, we merely introduce it by application to the problem just considered equilibrium in a single-phase, multiple-chemical reaction system. 

By definition, a formal potential describes the potential of a couple at equilibrium in a system where the oxidized and reduced forms are present at unit formal concentration, even though O and R may be distributed over multiple chemical forms (e.g., as both... 

In this section we illustrate the calculation of chemical equilibrium when there are multiple phases as well as a chemical reaction taking place. The following example illustrates the important issues. 

The earliest application of the XPS technique to the study of CT interactions in modem electroactive polymers appears to be that of Hsu et al. [144] on the iodine-doped (CH) films. The presence of multiple chemical states for the iodine dopant, such as the and Is species, has been revealed by the combined XPS core-level and Raman scattering studies. The Is species was postulated to have residtcd from the equilibrium process of the type U + Is Is. with the 13ds/2 BEs for the I2 and Ij species lying at about 620.6 and 619 eV respectively. Similar results were... 

As a common analysis tool, residue curve mapping (RCM) is well established. Fien and Liu [4] published a comprehensive review of the synthesis and shortcut design of non-reactive separation processes based on RCMs. Barbosa and Doherty [5] developed RCMs for RD processes with single chemical equilibrium reaction. Ung and Doherty [6] extended this method to systems with multiple equUibrium reactions. 

First simulation results on steady state multiplicity of etherification processes were obtained for the MTBE process by Jacobs and Krishna [45] and Nijhuis et al. [78]. These findings attracted considerable interest and triggered further research by others (e. g., [36, 80, 93]). In these papers, a column pressure of 11 bar has been considered, where the process is close to chemical equilibrium. Further, transport processes between vapor, liquid, and catalyst phase as well as transport processes inside the porous catalyst were neglected in a first step. Consequently, the multiplicity is caused by the special properties of the simultaneous phase and reaction equilibrium in such a system and can therefore be explained by means of reactive residue curve maps using oo/< -analysis [34, 35]. A similar type of multiplicity can occur in non-reactive azeotropic distillation [8]. 

In all cases, kinetic multiplicity can be avoided by an increase of the Damkohler number, that is an increase of the number of active sites on the catalyst, or a decrease of the feed rate. Moreover, multiplicity will vanish if the column pressure is increased. In all cases the column gets closer to chemical equilibrium. This is consistent with previous experimental studies for the MTBE process at 7 bar and low feed rates [103] where no multiplicity was found. 

Figure 11.7 Algorithm for computing equilibrium mole fractions that result from multiple chemical reactions occurring in a single phase at fixed T and P. SVD means the singular value decomposition described in 11.2.1 and 11.2.2.

S. Ung, M. F. Doherty, Vapor-Liquid Equilibrium in Systems with Multiple Chemical Reactions, Chem. Eng. Sci., 1995, 50, 23-48. 

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