Conditions for equilibrium

FIG. 16-33 Elution curves under trace linear equilibrium conditions for different feed loading periods audN = 80. Solid lines, Eq. (16-172) dashed line, Eq. (16-174) for Tp= 0.05. 

For the equihbrium properties and for the kinetics under quasi-equilibrium conditions for the adsorbate, the transfer matrix technique is a convenient and accurate method to obtain not only the chemical potentials, as a function of coverage and temperature, but all other thermodynamic information, e.g., multiparticle correlators. We emphasize the economy of the computational effort required for the application of the technique. In particular, because it is based on an analytic method it does not suffer from the limitations of time and accuracy inherent in statistical methods such as Monte Carlo simulations. The task of variation of Hamiltonian parameters in the process of fitting a set of experimental data (thermodynamic and... 

Point (e) is the intersection of the equilibrium curve and the operating line and represents the equilibrium condition for tower oudet liquor, and the maximum liquor concentration. 

Curve A represents the equilibrium condition for tvater entering at 90°F, the gas entering saturated with water vapor at 90°F and isothermal tow er operation. 

The concentrations are contained in separate terms, and the equation (17) for the whole reaction can be divided into a number of separate equations expressing the equilibrium conditions for various possible partial reactions. 

The reader can easily show by substitutions from the equilibrium conditions for Eqs. (6-21) and (6-22) that the rate constant for Eq. (6-23) can be calculated from the experimental value in Eq. (6-20) by the equation... 

We then note that the equilibrium condition for reaction (3.7) now taking place not only at the tpb (three phase boundaries), but over the entire gas exposed Pt electrode surface is very similar to Eq. (3.28), i.e. 

Equilibrium conditions for the synthesis of intermetallic phases and compounds are summarized as a function of temperature and composition in the form of phase diagrams. Consequently, in the following subsections, phase relationships for group-IIA-group-IB metal systems are reviewed. Phase diagrams in ref. 1 are used as a baseline work published before this compilation is not specifically referred to, but that reported subsequently is used, as appropriate, to modify or replace these phase diagrams. 

Imai (1961) has observed that multivalent counterions are more strongly bound than are monovalent ones. This phenomenon can be demonstrated theoretically by considering equilibrium conditions for two counterions with valencies z and (Zj > z and degrees of dissociation and P -For a cylindrical model the equilibrium equations are... 

The equilibrium conditions for homogeneous systems with membranes were first formulated in this form by Frederick G. Donnan in 1911. Hence, such equilibria are often called Donnan equilibria, and the membrane potentials associated with them are called Donnan potentials. Sometimes these terms are used as well for the equilibria arising at junctions between dissimilar solutions (Section 5.3). 

For convenience, only four stages were used in this model. An iterative solution is required for the bubble point calculations and this is based on the half-interval method. A FORTRAN subroutine EQUIL, incorporated in the ISIM program, estimates the equilibrium conditions for each plate. The iteration routine was taken from Luyben and Wenzel (1988). The program runs very slowly. 

Henry and Fauske (1971) developed a model for critical flow in nozzles and short tubes, which allows for nonequilibrium effects and considers a two-phase mixture upstream of the break by using an empirical correlation to relate actual dXIdp to the value (flXJdp) under equilibrium conditions. For a dispersed flow, they assumed that... 

Equilibrium conditions for the adsorption of second, third, fourth, etc., segments from molecules already attached to the surface can be written down in similar fashion and lead to the following expression for the l step ... 

Allowing an open system to close and come to equilibrium implies an extent of reaction - a measure of how close the system has come towards the equilibrium condition. For a single species the extent of the reaction towards an equilibrium concentration is given by ... 

For a metal, the negative of the work function gives the position of the Fermi level with respect to the vacuum outside the metal. Similarly, the negative of the work function of an electrochemical reaction is referred to as the Fermi level Ep (redox) of this reaction, measured with respect to the vacuum in this context Fermi level is used as a synonym for electrochemical potential. If the same reference point is used for the metal s,nd the redox couple, the equilibrium condition for the redox reaction is simply Ep (metal)= Ep(redox). So the notion of a Fermi level for a redox couple is a convenient concept however, this terminology does not imply that there are free electrons in the solution which obey Fermi-Dirac statistics, a misconception sometimes found in the literature. 

Recall that the equilibrium condition for a closed system at constant T and p was given by eq. (1.41). For an open system the corresponding equation is... 

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence... 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

The equilibrium conditions for systems with curved interfaces [3] are in part identical to those defined earlier for heterogeneous phase equilibria where surface effects where negligible ... 

To establish the equilibrium conditions for pressure we will consider a movement of the dividing surface between the two phases a and [i. The dividing surface moves a distance d/ along its normal while the entropy, the total volume and the number of moles n, are kept constant. An infinitesimal change in the internal energy is now given by... 

At equilibrium (dt/) v n. = 0, which leads to the equilibrium condition for pressure expressed in terms of the two principal curvatures or alternatively in terms of the two principal radii of curvature ... 

Equation (6.27) is the Laplace equation, or Young-Laplace equation, which defines the equilibrium condition for the pressure difference over a curved surface. In Section 6.2 we will examine the consequences of surface or interface curvature for some important heterogeneous phase equilibria. 

For planar surfaces the pressure difference over the interface becomes zero and the equilibrium condition for pressure, eq. (6.27) reduces to... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

FIGURE 4.1. Equilibrium conditions for H2S in aqueous solution (Melbourne and Metropolitan Board of Works, 1989). 

The equilibrium condition for the distribution of one solute between two liquid phases is conveniently considered in terms of the distribution law. Thus, at equilibrium, the ratio of the concentrations of the solute in the two phases is given by CE/CR = K, where K1 is the distribution constant. This relation will apply accurately only if both solvents are immiscible, and if there is no association or dissociation of the solute. If the solute forms molecules of different molecular weights, then the distribution law holds for each molecular species. Where the concentrations are small, the distribution law usually holds provided no chemical reaction occurs. 

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