Composites equilibrium kinetics

To examine the possibility that the different kinetic behaviors for series al, a3, and a5 may be due to differences in ionic strength, the solutions of series al and a3 were replaced by solutions of slighdy different compositions in which small amounts of sodium perchlorate were included. Extreme assumptions about the way that the sodium perchlorate would affect the equilibrium quotients for the bisulfate dissociation were used to calculate the detailed compositions of the new solutions in order to maintain constant ionic strength at 1.50, constant sulfate ion concentration at 0.276Af, and hydrogen ion concentrations close to those of solutions al and a3. No matter which of the assumptions was made in computing the compositions, the kinetic behaviors observed for the new solution series were very similar to those reported herein for series al and a3. 

The second part of Eqn. (4.136) is true if the attempt frequency v° (Eqn. (4.131)) is independent of the composition. This kinetic steady state condition is obviously equivalent to the thermodynamic equilibrium condition. 

Equilibrium, Kinetic, and Chromatographic Controls of the Solution Composition Obtained during the in situ Leaching of a Uranium Orebody... 

In order to be truly predictive, models for the complex leaching process must contain equilibrium, kinetic, and chromatographic parameters. Development of such comprehensive models will not only aid in the optimization of solution compositions for the most effective uranium recovery, but will also allow a more realistic environmental impact assessment. 

The above computations for the fiee energy changes for composite latex particles have shown that only core-shell, inverted core-shell and hemispherical particles are stable in a thermodynamic sense. All of the other reported morphologies (e.g. sandwich-like , raspberry or confetti-like particles or occluded domains) are non-equilibrium, kinetically controlled structures, prepared... 

The design equations would include, in addition to the usual heat and mass balances and vapor-liquid equilibria, equations for chemical equilibria and/or reaction kinetics. The occurrence of a chemical reaction can severely restrict the allowable ranges of temperatures and phase compositions by virtue of the additional equations for chemical equilibrium/kinetics. This effect can be quantitatively analyzed by constructing a residue curve map (RCM). It explicitly shows the shifting of distillation boundaries in the presence of reaction and defines the limits of feasible distillation column operation. We illustrate this (Venimadhavan et al., 1994) by considering the reaction... 

In general, the composition of polymer-salt complexes is a result of rather complicated equilibrium and some non-equilibrium, kinetically limited phenomena (Fauteux and Robitaille 1985 Fauteux et al. 1985 Lee and Crist 1986 Minier et al. 1984 Munshi and Owens 1986 Robitaille and Fauteux 1986 Stainer et al. 1984). According to the Gibbs phase rule (Gibbs 1870 Mindel 1962), for all the compositions ranging from pure polymer up to that of the thinnest crystalline complex, two phases should be present pure, crystalline PEO and pure PEO-salt crystalline complex. Nevertheless, polymeric materials are intrinsically impure , for example due to their polydispersity. Additionally, their crystallisation is kinetically limited, therefore in all polymeric materials there are always amorphous domains. Thus PEO-salt complexes usually consist of three phases (Fig. 2.4) - pure crystalline PEO, crystalline PEO-salt complex and amorphous PEO-salt complex the latter is of undefined composition (Wieczorek et al 1989). 

Therefore, thermodynamic equilibrium of a macroscopic system is an effect of local equilibration processes at a subsystem level (structural hierarchies). In other words, concentration equilibrium in a system (molecular composition equilibrium) doesn t correspond to the equilibrium number and size and especially inner structure (topological) distributions of supramolecular structures and completion of the phase separation process with stationary concentrations in contacting phases doesn t characterize phase solution layering because of possible further coalescence and morphology system changes. All these aspects are evidently stipulated by thermodynamical equilibration kinetics. Further information on equilibration kinetics at various structural organization levels is presented in Parts II and III of this book. 

There is a triple poinf at which a liquid phase (in the case of Pb-Sn, a liquid solution of Pb and Sn), and two solid phases coexist. This is known as the eutectic point. It occurs at a specific composition, the eutectic composition, and melts/freezes completely at a specific temperature, the eutectic temperature. In the case of Pb-Sn, the eutectic composition is 61.9 weight % Sn, and the eutectic temperature is 181 °C. Notice that the eutectic solid is a two-phase mixture. In spite of this, both phases melt at the same temperature for the eutectic alloy. The maxima of solid solubility for Sn in solid Pb and Pb in solid Sn occur at the eutectic temperature and are, 19.2 weight % and 97.5 weight % Sn, respectively. A solid mixture having the eutectic composition on average is the only mixture that melts entirely at a single temperature. All other compositions (except pure elements) in a eutectic system partially melt leaving a solid with altered composition (if kinetics allow the solids to maintain equilibrium) in contact with a liquid of different composition. 

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). 

The kinetics of spinodal decomposition is complicated by the fact that the new phases which are formed must have different molar volumes from one another, and so tire interfacial energy plays a role in the rate of decomposition. Anotlrer important consideration is that the transformation must involve the appearance of concenuation gradients in the alloy, and drerefore the analysis above is incorrect if it is assumed that phase separation occurs to yield equilibrium phases of constant composition. An example of a binary alloy which shows this feature is the gold-nickel system, which begins to decompose below 810°C. 

Product composition may be governed by the equilibrium thermodynamics of the system. When this is true, the product composition is governed by thermodynamic control. Alternatively, product composition may be governed by competing rates of formation of products. This is called kinetic control. 

Equilibrium between the various enolates of a ketone can be established by the presence of an excess of the ketone, which permits proton transfer. Equilibration is also favored by the presence of dissociating solvents such as HMPA. The composition of the equilibrium enolate mixture is usually more closely balanced than for kinetically... 

This paper surveys the field of methanation from fundamentals through commercial application. Thermodynamic data are used to predict the effects of temperature, pressure, number of equilibrium reaction stages, and feed composition on methane yield. Mechanisms and proposed kinetic equations are reviewed. These equations cannot prove any one mechanism however, they give insight on relative catalyst activity and rate-controlling steps. Derivation of kinetic equations from the temperature profile in an adiabatic flow system is illustrated. Various catalysts and their preparation are discussed. Nickel seems best nickel catalysts apparently have active sites with AF 3 kcal which accounts for observed poisoning by sulfur and steam. Carbon laydown is thermodynamically possible in a methanator, but it can be avoided kinetically by proper catalyst selection. Proposed commercial methanation systems are reviewed. 

Kinetic-molecular theory provides an explanation on a molecular level for this equilibrium. Evaporation from the liquid occurs as fast moving molecules on the surface escape from the liquid. In turn, molecules in the gas phase strike the liquid and condense, As the concentration (pressure) of gas molecules builds up in the gas phase, the rate of condensation increases. Eventually, a pressure is reached where the rate of condensation and rate of evaporation just balance, and equilibrium is achieved. The equilibrium pressure is denoted by p and is known as the vapor pressure. The magnitude ofp depends upon the substance, composition of the liquid, and any two of our thermodynamic variables such as temperature and total pressure. The criteria for equilibrium that we will now derive provide the thermodynamic relationships that will help... 

Thirdly, the multicomponent model was applied to the case of crystallization of a random A-B copolymer by Helfand and Lauritzen [127]. Their main result is that the composition of, 4 s and B s in the crystal is determined by kinetic, rather than equilibrium considerations the inclusion of excess B increases with growth rate. 

The failure to identify the necessary authigenic silicate phases in sufficient quantities in marine sediments has led oceanographers to consider different approaches. The current models for seawater composition emphasize the dominant role played by the balance between the various inputs and outputs from the ocean. Mass balance calculations have become more important than solubility relationships in explaining oceanic chemistry. The difference between the equilibrium and mass balance points of view is not just a matter of mathematical and chemical formalism. In the equilibrium case, one would expect a very constant composition of the ocean and its sediments over geological time. In the other case, historical variations in the rates of input and removal should be reflected by changes in ocean composition and may be preserved in the sedimentary record. Models that emphasize the role of kinetic and material balance considerations are called kinetic models of seawater. This reasoning was pulled together by Broecker (1971) in a paper called "A kinetic model for the chemical composition of sea water."... 

Many reactions show appreciable reversibility. This section introduces thermodynamic methods for estimating equilibrium compositions from free energies of reaction, and relates these methods to the kinetic approach where the equilibrium composition is found by equating the forward and reverse reaction rates. 

The kinetic equilibrium constant is estimated from the thermodynamic equilibrium constant using Equation (7.36). The reaction rate is calculated and compositions are marched ahead by one time step. The energy balance is then used to march enthalpy ahead by one step. The energy balance in Chapter 5 used a mass basis for heat capacities and enthalpies. A molar basis is more suitable for the current problem. The molar counterpart of Equation (5.18) is... 

Surface composition. The principle of surface segregation in ideal systems is easy to understand and to derive thermodynamically the equilibrium relations (surface concentration Xg as a function of the bulk concentration Xb at various temperatures) is also very easy (4,8). Even easier is a kinetic description which can also comprise some of the effects of the non-ideality (9). We consider an equilibrium between the surface(s) and the bulk(b) in the exchange like ... 

Polarization equations are convenient when (1) the measurements are made in solutions of a particular constant composition, and (2) the equilibrium potential is established at the electrode, and the polarization curve can be measured both at high and low values of polarization. The kinetic equations are more appropriate in other cases, when the equilibrium potential is not established (e.g., for noninvertible reactions, or when the concentration of one of the components is zero), and also when the influence of component concentrations on reaction kinetics is of interest. 

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