One-Phase Reaction-Equilibrium Calculations

Note that for three components in three-phase equilibrium, the phase rule (9.1.14) requires only T = 2 properties to identify the state. So with T and P fixed, any set of overall compositions zj that leads to a three-phase situation will produce the same compositions as given in Table 11.6. However, different sets z will produce different distributions of material (L and V) among the three phases some sets z will produce only two equilibrium phases, and others will yield only a single phase. When only one or two phases are found, the compositions will differ from those in Table 11.6. 

To solve reaction equilibrium problems, we must combine material balances with the criteria for reaction equilibria. Consequently, such problems bear a superficial resemblance to isothermal flash calculations, though in the case of reaction equilibria the material balances are applied to elements, not species. For H independent reactions involving C species in a single phase at fixed T and P, the criteria for equilibrium were given in 7.6.1, 

Consider a closed reacting system containing C species in elements with the elemental balances given by 

Here A is an (m x C) formula matrix, N is a (C x 1) vector of mole numbers, and b is an (mg x 1) vector of constant elemental abundances. (Basics of linear algebra are reviewed in Appendix B.) The matrix A is known from the chemical formulae of the species present, and the abundances b are known from the amounts initially loaded into the reactor. But the mole numbers N are unknown. Moreover, the sets N that satisfy the balances (7.4.2) are not unique many different combinations of amounts of the given species (N) can produce the same elemental balances (b). This means that the formula matrix A is singular. 

The singularity of A can also be deduced in another way. Recall that the rank of A is related to the number of independent reactions by 

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Quantum-chemical (DFT, B3LYP/6-31G") calculations have been performed to determine the relative stabilities of the tautomer pairs 69/70 (gas-phase reaction enthalpy calculated 1.2 kcal mol , experimental value in diethyl ether 1.1 0.1 kcalmol ), 71/72 (gas-phase reaction enthalpy calculated 2.1 kcal mol ), and 73/74 (gas-phase reaction enthalpy calculated 2.6kcalmol , experimental value in dipentyl ether 1.9 0.1 kcal mol ). In all cases, the tautomer with the endo C=C bond is strongly favored over the one with the exo C=C bond. Because of the overwhelming stability of 2,5-dimethylfuran versus tetrahydro-2,5-bismethylenefuran 71 and 2,3-dihydro-5-methyl-2-methylenefuran 72, the equilibrium 71 72 cannot be observed experimentally <2001STC405>. ... 

We use 6.2 to emphasize the symmetries that exist among difference measmes and among ratio measures. Difference measures are commonly used to compute thermodynamic properties of single homogeneous phases, while ratio measmes are most often used in phase and reaction equilibrium calculations. In 6.3 we show that similarities among ratio measures extend to their physical interpretations. Then in 6.4 we collect in one place the five most important ratio measures that are used to compute values for fugacities. 

We will need values of conceptuals for two classes of problems (a) calculation of thermodynamic properties for one-phase systems and (b) calculation of multiphase and chemical reaction equilibria. For both kinds of problems, we use the same basic strategy (i) Compare raw or modeled experimental data with computed properties of an ideal substance to obtain measures for deviations from the ideality, then (ii) exploit the deviation measures to obtain expressions for the required conceptuals in terms of measurables. Calculations of one-phase properties are typically based on differences, while phase and reaction equilibrium calculations typically use ratios. In 6.2.1 and... 

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. 

When the kinetics are unknown, still-useful information can be obtained by finding equilibrium compositions at fixed temperature or adiabatically, or at some specified approach to the adiabatic temperature, say within 25°C (45°F) of it. Such calculations require only an input of the components of the feed and produc ts and their thermodynamic properties, not their stoichiometric relations, and are based on Gibbs energy minimization. Computer programs appear, for instance, in Smith and Missen Chemical Reaction Equilibrium Analysis Theory and Algorithms, Wiley, 1982), but the problem often is laborious enough to warrant use of one of the several available commercial services and their data banks. Several simpler cases with specified stoichiometries are solved by Walas Phase Equilibiia in Chemical Engineering, Butterworths, 1985). 

One word of warning when using the van t Hoff equation for reactions involving gases, the equilibrium constants must be K, not K(.. If we want a new value for Kc for a gas-phase reaction, we convert from K(. into K at the initial temperature (by using Eq. 12), use the van t Hoff equation to calculate the value of K at the new temperature, and then convert that K into the new Kc by using Eq. 12 at the new temperature. 

The present paper deals with one aspect of this problem the calculation of phase separation critical points in reacting mixtures. The model employed is the Soave-Redlich-Kwong equation of state (1 ), which is typical of several equations of state (2, 5) which have relatively recently come into wide use as phase equilibrium models for light gas mixtures, sometimes including water and the acid gases as components (4, . 5, 6). If the critical point contained in the equation of state (perhaps even for the mixture at reaction equilibrium) can be found directly, the result will aid in other equilibrium computations. 

A species frequently maintains phase equilibrium while it is reacting in one phase. An example is hydrocracking of heavy hydrocarbons in petroleum refining, where H2 from the vapor dissolves into the liquid hydrocarbon phase, where it reacts with large hydrocarbons to crack them into smaller hydrocarbons that have sufficient vapor pressure to evaporate back into the vapor phase. As long as equilibrium of the species between phases is maintained, it is easy to calculate the concentrations in the hquid phase in which reaction occurs. 

Solution There are three components in a single phase. One chemical reaction, 2NO + Br2 2NOBr, reduces the number of independent components to two. The number of degrees of freedom is 2 — 1 + 2 = 3, one of which is the fixed temperature of 298 K. The other two can be taken as the ratio of the initial pressure of NO to that of Br2 and the total final pressure. First, we calculate the equilibrium constant ... 

With adiabatic combustion, departure from a complete control of m by the gas-phase reaction can occur only if the derivation of equation (5-75) becomes invalid. There are two ways in which this can happen essentially, the value of m calculated on the basis of gas-phase control may become either too low or too high to be consistent with all aspects of the problem. If the gas-phase reaction is the only rate process—for example, if the condensed phase is inert and maintains interfacial equilibrium—then m may become arbitrarily small without encountering an inconsistency. However, if a finite-rate process occurs at the interface or in the condensed phase, then a difficulty arises if the value of m calculated with gas-phase control is decreased below a critical value. To see this, consider equation (6) or equation (29). As the value of m obtained from the gas-phase analysis decreases (for example, as a consequence of a decreased reaction rate in the gas), the interface temperature 7], calculated from equation (6) or equation (29), also decreases. According to equation (37), this decreases t. Eventually, at a sufficiently low value of m, the calculated value of T- corresponds to Tj- = 0, As this condition is approached, the gas-phase solution approaches one in which dT/dx = 0 at x = 0, and the reaction zone moves to an infinite distance from the interface. The interface thus becomes adiabatic, and the gas-phase processes are separated from the interface and condensed-phase processes. 

The pH of sea salt aerosol is an important property as many important aqueous phase reactions are pH dependent. For example, oxidation of S(IV) (SO2 + HSOs + SO ) by O3 is only important for pH of more than 6. Sea salt aerosol is buffered with HC03. Uptake of acids from the gas phase leads to acidification of the particles. According to the indirect sea salt aerosol pH determinations by Keene and Savoie (1998, 1999), the pH values for moderately polluted conditions at Bermuda were in the mid-3s to mid-4s. The equilibrium model calculations of Fridlind and Jacobson (2000) estimated marine aerosol pH values of 2-5 for remote conditions during ACE-1. Using a one-dimensional model of the MBL which includes gas phase and aqueous phase chemistry of sulfate and sea salt aerosol particles, von Glasow and Sander (2001) predicted that under the chosen initial conditions the pH of sea salt aerosol decreases from 6 near... 

We now pass to the explicit calculation of entropy production. We shall consider here the very important special case in which mechanical and thermal equilibrium are already established. Mechanical equilibrium excludes the production of entropy by viscous flow, while uniformity of temperature, which is necessary for thermal equilibrium, excludes the internal production of entropy arising from the transport of heat between two regions at different temperatures. Similarly we assume that diffusion equilibrium has been attained within each phase of the system. The only production of entropy which can take place in a system of this kind is that associated with chemical reactions, with the transport of matter from one phase to another, or in general with any change which can be expressed in terms of a reaction co-ordinate... 

Complex Chemical 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. (4-293). The treatment here is limited to gas-phase reactions for which the problem is to find the equilibrium composition for given T and P and for a given initial feed. 

The previous sections have attempted to provide some insight into the form of the microscopic expressions for the rate kernels and rate coefficients that characterize condensed-phase reactions. Although the equilibrium one-way flux rate coefficient ky is relatively easy to calculate and under certain circumstances may yield an adequate description of the rate, a variety of important dynamic effects are contained in the relaxing part of the rate kernel, In this section, we describe a kinetic theory that... 

Vapor-phase Cataljrtic Reactions. Since a catalyst does not change the equilibrium of a reaction, the calculated composition should be the same with a catalyst present as without. This is generally the case, but there are exceptions to bear in mind. One is the case of a possible competing reaction which is too slow to have any effect in the homogeneous system but which is speeded up by the catalyst to a greater extent than the main reaction. Consider, for example, the reactions... 

To calculate residue curve maps for the synthesis of TAME one has to proceed in the same manner as the MTBE example and calculate phase equilibria bet veen liquid and vapor phases, chemical equilibrium constants in the liquid phase, and kinetics of the chemical reactions. 

The heterogeneities of most concern to us are those that involve the presence of more than one phase. The analysis of multiphase systems can be important to the design and operation of many industrial processes, especially those in which multiple phases influence chemical reactions, heat transfer, or mixing. For example, phase-equilibrium calculations form the bases for many separation processes, including stagewise operations, such as distillation, solvent extraction, crystallization, and supercritical extraction, and rate-limited operations, such as membrane separations. 

These reaction-equilibrium criteria apply to each phase in which reactions are occurring. But we can often simplify a calculation by assuming reactions occur in only one phase. Such an assumption is legitimate because the affinities in (7.6.3) are merely particular combinations of fugacities, and the phase-equilibrium criteria (7.3.12) require the same value of the fugacity for each component in all phases. This means that... 

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