Equilibrium theories

Adsorbed-Solution Theory The common thermodynamic approach to multicomponent adsorption treats adsorption equilibrium in a way analogous to fluid-fluid equilibrium. The theory has as its basis the Gibbs adsorption isotherm [Young and Crowell, gen. refs.], which is 

Equation (16-36) with y, = 1 provides the basis for the ideal adsorbed-solution theory [Myers and Prausnitz, AIChE /., 11, 121 (1965)]. The spreading pressure for a pure component is determined by integrating Eq. (16-35) for a pure component to obtain 

Example 5 Application of Ideal Adsorbed-Solution Theory 

Consider a binary adsorbed mixture for which each pure component obeys the Langmuir equation, Eq. (16-13). Let n = 4 mol/kg, n2 = 3 mol/kg, Kipi = K2p2 = 1- Use the ideal adsorbed-solution theory to determine nx and n2. Substituting the pure component Langmuir isotherm 

The first step in the analysis of copolymer crystallization is the development of quantitative concepts that are based on equilibrium considerations. Subsequently, deviations from equilibrium and a discussion of real systems will be undertaken. Problems involving the crystallization and melting of copolymers cannot in general be uniquely formulated since two phases and at least two species are involved. The disposition of the species among the phases needs to be specified. It cannot be established a priori by theory. This restraint is not unique to polymeric systems. It is a common experience in analyzing similar problems that involve monomeric components.(2) Thus, in the development of any equilibrium theory a decision has to be made prior to undertaking any analysis of the disposition of the co-units between the phases. Theoretical expectations can then be developed based on the assumptions made. 

Two possibilities exist with respect to the disposition of the co-units. In one case the crystalline phase remains pure, i.e. the co-units are excluded from entering the crystal lattice. In the other, the co-unit is allowed to enter the lattice on an equilibrium basis. Typical examples of the latter would be akin to compound formation, or isomorphous replacement, where one unit can replace the other in the lattice. In either of these two main categories ideal conditions are first calculated and analyzed. Subsequently nonideal contributions to both phases can be considered while stiU maintaining equilibrium. There is an analogy here to solution theory and to gases, where equilibrium conditions are established first. In the next step, nonequilibrium effects in either or both phases can be brought to bear on the problem. It needs to be recognized that deviations from equilibrium in copolymers exist and are in fact important. 

In general, one can expect to observe the types of phase diagrams that are found with low molecular weight systems in crystal-liquid equilibrium. For polymeric systems the liquid composition can usually be determined in a straightforward manner. However, establishing the composition in the solid state is quite difficult and presents a major problem in properly analyzing phase diagrams. 

The theory for the case where the crystalline phase remains pure has a mature development and is rich in concepts. This case will be treated first, utilizing Rory s classical work.(3,4) A model copolymer is considered that contains only one type of crystallizable unit, designated as an A unit. The noncrystallizable comonomeric unit will be designated as a B unit. In the initial molten state the A units occur in a specified distribution that is determined by the copolymerization mechanism. Upon crystallization, with the exclusion of the B units from the lattice, the sequence 

Following Flory (4) these concepts can be formulated in a quantitahve manner so that a description of the ideal equilibrium crystalline state results. The probability that a given A unit in the noncrystalline amorphous region is located within a sequence of at least such units is dehned as. The probability that a unit chosen at random from the noncrystalUne region is an A unit, and also a member of a sequence of f A units that are terminated at either end by B units is represented by Wj. The probability that the specific A unit selected is followed in a given direction by at least — 1 similar units can be expressed as 

---

Stell G 1964 Cluster expansions for classical systems In equilibrium The Equilibrium Theory of Classical Fluids ed H L Frisch and J L Lebowitz (New York Benjamin)... 

Chandler D 1982 Equilibrium theory of polyatomic fluids The Liquid State of Matter Fluids, Simple and Complex ed E W Montroll and J L Lebowitz (Amsterdam North-Holland)... 

Outhwaite C W 1974 Equilibrium theories of electrolyte solutions Specialist Periodical Report (London Chemical Society)... 

When a system is not in equilibrium, the mathematical description of fluctuations about some time-dependent ensemble average can become much more complicated than in the equilibrium case. However, starting with the pioneering work of Einstein on Brownian motion in 1905, considerable progress has been made in understanding time-dependent fluctuation phenomena in fluids. Modem treatments of this topic may be found in the texts by Keizer [21] and by van Kampen [22]. Nevertheless, the non-equilibrium theory is not yet at the same level of rigour or development as the equilibrium theory. Here we will discuss the theory of Brownian motion since it illustrates a number of important issues that appear in more general theories. 

Flere, we shall concentrate on basic approaches which lie at the foundations of the most widely used models. Simplified collision theories for bimolecular reactions are frequently used for the interpretation of experimental gas-phase kinetic data. The general transition state theory of elementary reactions fomis the starting point of many more elaborate versions of quasi-equilibrium theories of chemical reaction kinetics [27, M, 37 and 38]. 

Ericksen J L 1976 Equilibrium theory of liquid crystals Adv. Liq. Cryst. 2 233... 

Developments in equilibrium theory in the late nineteenth century led to significant improvements in the theoretical understanding of acid-base chemistry and. 

QET. quasi-equilibrium theory (of mass spectrometric fragmentation)... 

Equilibrium constant Equilibriumline Equilibrium theory Equimate... 

Equilibrium Theory. The general features of the dynamic behavior may be understood without recourse to detailed calculations since the overall pattern of the response is governed by the form of the equiUbrium relationship rather than by kinetics. Kinetic limitations may modify the form of the concentration profile but they do not change the general pattern. To illustrate the different types of transition, consider the simplest case an isothermal system with plug flow involving a single adsorbable species present at low concentration in an inert carrier, for which equation 30 reduces to... 

Local equilibrium theory Shows wave character—simple waves and shocks Usually indicates best possible performance Better understanding Mass and heat transfer very rapid Dispersion usually neglected If nonisothermal, then adiabatic... 

In local equilibrium theory, fluid and sorbed phases are assumed to be in local equilibrium with one another at every axial position in the bed. Thus, because of uniform concentrations, the overbar on /i is not necessary and we have Cj Cj [note Eqs. (16-52) and (I6-II9)]. 

Multiple Transition System Local equilibrium theory for multiple transitions begins with some combination of material and energy balances, written... 

The Hertz theory of contact mechanics has been extended, as in the JKR theory, to describe the equilibrium contact of adhering elastic solids. The JKR formalism has been generalized and extended by Maugis and coworkers to describe certain dynamic elastic contacts. These theoretical developments in contact mechanics are reviewed and summarized in Section 3. Section 3.1 deals with the equilibrium theories of elastic contacts (e.g. Hertz theory, JKR theory, layered bodies, and so on), and the related developments. In Section 3.2, we review some of the work of Maugis and coworkers. 

The equilibrium theory of homogeneous fluids may be constructed by using the hierarchy of the direct correlation functions [48]. This approach has been of much utility for the development of the theory of inhomogeneous simple fluids. The hierarchy of the direct correlation functions is defined by the following relation... 

If the transition state theory is applied to the reaction of two hard spheres, the result is identical with that of simple collision theory. - pp Because transition state theory is an equilibrium theory, it can be inferred that collision theory is also an equilibrium theory. 

For nonlinear systems, however, the evaluation of the flow rates is not straightforward. Morbidelli and co-workers developed a complete design of the binary separation by SMB chromatography in the frame of Equilibrium Theory for various adsorption equilibrium isotherms the constant selectivity stoichiometric model [21, 22], the constant selectivity Langmuir adsorption isotherm [23], the variable selectivity modified Langmuir isotherm [24], and the bi-Langmuir isotherm [25]. The region for complete separation was defined in terms of the flow rate ratios in the four sections of the equivalent TMB unit ... 

Fig. 9-7. Region for complete separation under Equilibrium Theory. Linear adsorption isotherms.

The case with k = 0.4 s (open squares) is close to the situation where mass transfer resistance is negligible. These differences are due to mass transfer resistances as we can easily conclude by comparing the separation regions obtained for the cases with k = 0.4 and k = 1 s k If mass transfer resistance is important, the region of complete separation can be significantly reduced from the one obtained by the Equilibrium Theory. For example, for a mass transfer coefficient of k = 0.1 s there is no separation region where extract and raffinate are 99.5 % pure. 

Fig. 9-18. Separation regions in a Yuryu Equilibrium Theory (100 %, line), mass transfer eoeffieient = 1 s (99.5 %, elosed squares), k = 0.4 s (99.5 %, open squares).

Nonlinear case The calculation of the flowrates is much more complex, and it is beyond the scope of this chapter to present it in detail. However, as a useful tool, Mor-bidelli and coworkers [48-50, 63], applied the solutions to the equations of the equilibrium theory (when all the dispersion phenomena are neglected) to a four-zone TMB. 

The above conditions are not robust because they are at the limit of the complete separation zone. The equilibrium theory neglects the dispersion phenomena and therefore the purity obtained under these flowrate conditions would be less than 100 % on a TMB system. Complex simulation software, which takes into account the dispersion phenomena, gives a more robust system with higher purities [57]. 

Figure 10.7. Zone of complete separation according to the equilibrium theory.

Equilibrium Theory of Fluid Structure. In all the theoretical work reported herein, we assume that the particles Interact with pair additive forces whose pair potentials can be approximated by... 

---