Equilibrium ideal case

An ideal gas obeys Dalton s law that is, the total pressure is the sum of the partial pressures of the components. An ideal solution obeys Raoult s law that is, the partial pressure of the ith component in a solution is equal to the mole fraction of that component in the solution times the vapor pressure of pure component i. Use these relationships to relate the mole fraction of component 1 in the equilibrium vapor to its mole fraction in a two-component solution and relate the result to the ideal case of the copolymer composition equation. 

In the o.s. reaction, the ion pair A+ - B is formed in a first step. The corresponding equilibrium constant can usually be obtained from simple electrostatic models. In this "ideal" case specific chemical interactions can be neglected and the rate constant of the E.T. step follows the theory of R.A. Marcus (see for example Marcus, 1975, or Cannon, 1980). In the i.s. reaction each of the three steps in reaction (9.2) may determine the reaction rates. The lability of the coordinated ligands at the... 

Dense membranes are used for pervaporation, as for reverse osmosis, and the process can be described by a solution-diffusion model. That is, in an ideal case there is equilibrium at the membrane interfaces and diffusional transport of components through the bulk of the membrane. The activity of a component on the feed side of the membrane is proportional to the composition of that component in the feed solution. 

Parameters describing a particular thermodynamic equilibrium system are derived from experimental quantities obtained by a variety of methods, for example, calorimetry, potentio-metry, and solubility studies. In the ideal case, critical examination of well-studied systems reveals high-quality experimental data that lead to a unique set of thermodynamic constants, which are internally consistent, not only formally, but also from a chemical point of view. In the course of our reviews, however, we encountered several cases of conflicting experimental data that resisted any attempt to cast them into a unique set of thermodynamic parameters. The following summarizes the conflicting data and our pragmatic solutions. 

D experiments are devised in the assumption that the various times involved in the cycle of Fig. 8.1 (with the exception of when present) are small with respect to the nuclear relaxation times. When the latter are short for any reason, e.g. in the case of paramagnetic molecules because of the presence of unpaired electrons, the system of spins may have reached the equilibrium, or almost reached the equilibrium, before the detection pulse. Under these circumstances no memory is left for the state of the spins during the preceding steps. As a consequence, cross peaks may be decreased in intensity until below detectability. It is necessary, therefore, to match all the time intervals with the nuclear relaxation times, in order to detect the maximum possible cross peak intensities. The ideal case is that t ... 

This equation indicates that equilibrium will be attained only after an infinite time, but it will be found that the equilibrium value will be approached very closely within a short time. It must be remembered that this is an idealized case. In reality there will be elastic waves and damping, so that true equilibrium will be reached in a finite time. 

Subsequently a well-defined area at the surface is depleted from the adsorbate layer by a focused laser pulse. Since thermal equilibrium at the surface is rapidly recovered, the bare spot can be refilled only by surface diffusion of adsorbates from the surrounding areas [31]. A second laser impulse is applied to desorb the transported adsorbates after a time interval t from the first pulse. The corresponding amount of material can be quantified by mass spectrometry. For the idealized case of a circular depletion region, with a step-like coverage gradient and a concentration-independent diffusivity, the time-dependent refilling from Fick s first law is [32,33] ... 

In the ideal case, when one considers the network of chains of equal lengths, the stresses under the given deformation can be obtained in a very simple way. In virtue of the speculations of the previous section, free energy of the whole network can be represented as the sum of free energy of all the chains, while each of the equal chains of the network can be characterised by the same equilibrium distribution function W(s), where s is the separation between adjacent junctions. In the state without deformation, the function has the form (1.5), while in a deformed state, it depends on the displacement gradients (1.39). The free energy of the whole network can be written down simply as... 

Note that the relative variance of the timing probability distribution is considerably larger in the equilibrium case than in the physiological case. However, even in the physiological case, the system behavior is far from that of a perfect timer. In this near ideal case, r.v. 1 /2, which is the minimal value obtained by Equation (5.37) when 2,i 2,2. 

The question of the ( -potential value at the electrolyte solution/air interface in the absence of a surfactant in the solution is very important. It can be considered a priori that it is not possible to obtain a foam film without a surfactant. In the consideration of the kinetics of thinning of microscopic horizontal foam films (Section 3.2) a necessary condition, according to Reynolds relation, is the adsorption of a surfactant at both film surfaces. A unique experiment has been performed [186] in which an equilibrium microscopic horizontal foam film (r = 100 pm) was obtained under very special conditions. A quartz measuring cell was employed. The solutions were prepared in quartz vessels which were purified from surface impurities by a specially developed technique. The strong effect of the surfactant on the rate of thinning and the initial film thickness permitted to control the solution purity with respect to surfactant traces. Hence, an equilibrium thick film with initial thickness of about 120 nm was produced (in the ideal case such a film should be obtained right away). Due to the small film size it was possible to produce thick (100 - 80 nm) equilibrium films without a surfactant. In many cases it ruptured when both surfaces of the biconcave drop contacted. Only very precise procedure led to formation of an equilibrium film. 

Before we get into an outline of the theory of pharmaceutical economics, we need to establish pure competition as a competitive process. Traditional microeconomics has assumed implicitly that the natural state is one that is depicted by pure competition. Deviations from the natural state occur as a disequilibrium, by the establishment of monopoly power, or through other often cited market failures. In cases of disequilibrium, the tatonne-ment will bring us to the equilibrium ideal of pure competition. Interestingly, the model of pure competition never really describes the process of the tatormement (equilibration) but only the conditions necessary for the process to operate and the final equilibrium to result when the process has worked itself out. 

In this section we introduce the matter of equivalent mechanieal circuits on an elementary level. First we restrict ourselves to linear viscoelastic behaviour. Second, to show the basic elements, idealized cases will be emphasized (mainly strain retardation and stress relaxation), ignoring for the time being the problem of how to carry out such experiments. As a rule, however, we keep in mind that stress-wise the monolayer is always at equilibrium, and strain has to adjust to it. In this section only dilational rheology will be considered, but this is not a real restriction because for shear the formalism is the same mutatis mutandis. 

Two idealized cases may be distinguished (Garrels and Christ, 1965) (i) During the process of CaC03 dissolution, the water remains in contact and equilibrium with a relatively large reservoir of CO2 of fixed partial pressure (Figure 4.2a,c). (ii) An initially C02-rich water becomes isolated from the... 

Excepting these really few ideal cases and regardless of the fact that the BET method is routinely used for the determination of the monolayer volume even for systems which are poorly described by the BET equation, the above equations are not found to describe accurately adsorption equilibrium on most adsorbents of chemical interest. 

Summary We have derived a deceptively simple equation, AG°= -Rl n K, which enables us to calculate equilibrium constants from AG° data. Having derived and discussed the concept of activity, we can easily take into account the non-ideal behaviour of solutions, and of gases at significant pressures. However, in cases where solutions are dilute, or gas pressures low, calculations for the ideal case are also easily made, by putting all activity coefficients equal to unity. 

The three topics of ion pairing, complex formation and solubilities are typical aspects of equilibrium in electrolyte solutions, and are handled in precisely the same manner as acid-base equilibria. As in the calculations on acid-base equilibria, only the ideal case is considered. Discussion of corrections for non-ideality are deferred until Sections 8.22 to 8.28. In this chapter pay special attention to ... 

In the ideal case, where no contamination is present in the system, relaxations enable the study of dynamic behaviour of the adsorption layer. Such investigations yield information about adsorption mechanisms as well as interfacial interactions and transitions of co-existing phases. Due to the small deviation from equilibrium, theories of relaxation experiments are usually easier to derive, because linearisations are justified. Thus, complex processes are better studied by relaxations than by adsorption kinetics. 

Figure 12.5 depicts schematically the gas- and aqueous-phase concentrations of A in and around a droplet. The aqueous-phase concentrations have been scaled by HART, to remove the difference in the units of the two concentrations. This scaling implies that the two concentration profiles should meet at the interface if the system satisfies at that point Henry s law. In the ideal case, described by (12.45), the concentration profile after the scaling should be constant for any r. However, in the general case the gas-phase mass transfer resistance results in a drop of the concentration from cA(oo) to cA(Rp) at the air-droplet interface. The interface resistance to mass transfer may also cause deviations from Henry s law equilibrium indicated in Figure 12.5 by a discontinuity. Finally, aqueous-phase transport limitations may result in a profile of the concentration of A in the aqueous phase from [A(/ ,)J at the droplet surface to [A(0)] at the center. All these mass transfer limitations, even if the system can reach a pseudo-steady state, result in reductions of the concentration of A inside the droplet, and slow down the aqueous-phase chemical reactions. 

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