Equilibrium, true

Usually, when one deals with thixotropic systems, instead of equilibrium (true) values of rheological characteristics some reference (imaginary) values are measured, such as those recorded after a particular period of time had elapsed since a complete disintegration of thixotropically reversible structure. In some cases, in order to reach an equilibrium between... 

In practice, when working with thixotropic systems, one often uses some apparent values of the rheological characteristics, rather than the equilibrium ( true ) ones. The apparent characteristics can be evaluated at a particular time, for instance, after a complete degradation of the thixotropic-reversible structure. In some instances, a prolonged constant-rate deformation of the system is required for the equilibrium between the contact rupture and restoration to be established. Such conditions may not be always achievable in the laboratory. 

The estimated true values must satisfy the appropriate equilibrium constraints. For points 1 through L, there are two constraints given by Equation (2-4) one each for components 1 and 2. For points L+1 through M the same equilibrium relations apply however, now they apply to components 2 and 3. The constraints for the tie-line points, M+1 through N, are given by Equation (2-6), applied to each of the three components. 

In the highly nonlinear equilibrium situations characteristic of liquid separations, the use of priori initial estimates of phase compositions that are not very close to the true compositions of these phases can lead to divergence of iterative computations or to spurious convergence upon feed composition. 

However, when carboxylic acids are present in a mixture, fugacity coefficients must be calculated using the chemical theory. Chemical theory leads to a fugacity coefficient dependent on true equilibrium concentrations, as shown by Equation (3-13). ... 

CALCULATES THE CONSTRAINT FUNCTIONS FOP BINARY VAaOR-LIOUIO EQUILIBRIUM OATA PEOUCTION. THE CCNSTRAINT FUNCTIONS RELATING THE TRUE VALUES QP THE MEASURED VARIABLES ARE (1) PRESS = F(LI0 COMP,TEMP,PARAMETERS)... 

The true vapor pressure of hydrocarbons is expressed in bar and represents the vapor pressure directly above a saturated liquid in equilibrium with the vapor that rises over it. 

In Chapter III, surface free energy and surface stress were treated as equivalent, and both were discussed in terms of the energy to form unit additional surface. It is now desirable to consider an independent, more mechanical definition of surface stress. If a surface is cut by a plane normal to it, then, in order that the atoms on either side of the cut remain in equilibrium, it will be necessary to apply some external force to them. The total such force per unit length is the surface stress, and half the sum of the two surface stresses along mutually perpendicular cuts is equal to the surface tension. (Similarly, one-third of the sum of the three principal stresses in the body of a liquid is equal to its hydrostatic pressure.) In the case of a liquid or isotropic solid the two surface stresses are equal, but for a nonisotropic solid or crystal, this will not be true. In such a case the partial surface stresses or stretching tensions may be denoted as Ti and T2-... 

For liquids, the last term in Eq. VIl-5 is zero, so that r = G or t = 7, since we will use G and 7 interchangeably) the same would be true of a solid if the change in area dA were to occur in such a way that an equilibrium surface configuration was always maintained. Thus the stretching of a wire under reversible... 

The usual situation, true for the first three cases, is that in which the reactant and product solids are mutually insoluble. Langmuir [146] pointed out that such reactions undoubtedly occur at the linear interface between the two solid phases. The rate of reaction will thus be small when either solid phase is practically absent. Moreover, since both forward and reverse rates will depend on the amount of this common solid-solid interface, its extent cancels out at equilibrium, in harmony with the thermodynamic conclusion that for the reactions such as Eqs. VII-24 to VII-27 the equilibrium constant is given simply by the gas pressure and does not involve the amounts of the two solid phases. 

The adsorption of detergent-type molecules on fabrics and at the solid-solution interface in general shows a complexity that might be mentioned briefly. Some fairly characteristic data are shown in Fig. XlIl-15 [242]. There is a break at point A, marking a sudden increase in slope, followed by a maximum in the amount adsorbed. The problem is that if such data represent true equilibrium in a two-component system, it is possible to argue a second law violation (note Problem Xni-14) (although see Ref. 243). 

As is made evident in the next section, there is no sharp dividing line between these two types of adsorption, although the extremes are easily distinguishable. It is true that most of the experimental work has tended to cluster at these extremes, but this is more a reflection of practical interests and of human nature than of anything else. At any rate, although this chapter is ostensibly devoted to physical adsorption, much of the material can be applied to chemisorption as well. For the moment, we do assume that the adsorption process is reversible in the sense that equilibrium is reached and that on desorption the adsorbate is recovered unchanged. 

The course of a surface reaction can in principle be followed directly with the use of various surface spectroscopic techniques plus equipment allowing the rapid transfer of the surface from reaction to high-vacuum conditions see Campbell [232]. More often, however, the experimental observables are the changes with time of the concentrations of reactants and products in the gas phase. The rate law in terms of surface concentrations might be called the true rate law and the one analogous to that for a homogeneous system. What is observed, however, is an apparent rate law giving the dependence of the rate on the various gas pressures. The true and the apparent rate laws can be related if one assumes that adsorption equilibrium is rapid compared to the surface reaction. 

We have seen that equilibrium in an isolated system (dt/= 0, dF= 0) requires that the entropy Sbe a maximum, i.e. tliat dS di )jjy = 0. Examination of the first equation above shows that this can only be true if. p. vanishes. Exactly the same conclusion applies for equilibrium under the other constraints. Thus, for constant teinperamre and pressure, minimization of the Gibbs free energy requires that dGId Qj, =. p. =... 

Figure A2.1.10. The impossibility of reaching absolute zero, a) Both states a and p in complete internal equilibrium. Reversible and irreversible paths (dashed) are shown, b) State P not m internal equilibrium and with residual entropy . The true equilibrium situation for p is shown dotted.

An undesirable side-effect of an expansion that includes just a quadratic and a cubic term (as is employed in MM2) is that, far from the reference value, the cubic fimction passes through a maximum. This can lead to a catastrophic lengthening of bonds (Figure 4.6). One way to nci iimmodate this problem is to use the cubic contribution only when the structure is ,utficiently close to its equilibrium geometry and is well inside the true potential well. MM3 also includes a quartic term this eliminates the inversion problem and leads to an t". . 11 better description of the Morse curve. 

Effect of impurities upon the melting point. Let us take a specific example and examine the effect of the addition of a small quantity of naphthalene to an equilibrium mixture of pure solid and liquid a-naphthol at the temperature of the true melting point (95 5°) at atmospheric pressure. 

True equilibrium conditions can be established before any distillate is collected this is particularly important when the jacket temperature must be controlled. 

The true thermodynamic equilibrium constant is a function of activity rather than concentration. The activity of a species, a, is defined as the product of its molar concentration, [A], and a solution-dependent activity coefficient, Ya. 

The true thermodynamic equilibrium constant, Ksp, for the solubility of AglOa, therefore, is... 

A quantitative solution to an equilibrium problem may give an answer that does not agree with the value measured experimentally. This result occurs when the equilibrium constant based on concentrations is matrix-dependent. The true, thermodynamic equilibrium constant is based on the activities, a, of the reactants and products. A species activity is related to its molar concentration by an activity coefficient, where a = Yi[ ] Activity coefficients often can be calculated, making possible a more rigorous treatment of equilibria. 

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