Equilibrium systems existence/uniqueness

Countercurrent chromatography (CCC) is a support-free liquid-liquid partition system in which solutes are partitioned between the mobile and stationary phases in an open-column space. The instrumentation, therefore, requires a unique approach for achieving both retention of the stationary phase and high partition efficiency in the absence of a solid support. A variety of existing CCC systems may be divided into two classes [1] (i.e., hydrostatic and hydrodynamic equilibrium systems). The principle of each system may be illustrated by a simple coil as shown in Fig. 1. 

In the elementary theory of H2, it is considered as a simple system in which vibrational, electronic and rotational motions can be separated (the Born-Oppenheimer principle) and fully analytic solutions exist (uniquely for a molecule) which show that the molecule is stable. This, however, is not the complete story. In fact, as is separated into H and H+, one encounters an additional shallow minimum near the dissociation limit, at much larger internuclear distances than its equilibrium separation. This second minimum, which arises from a dipole in the neutral fragment induced by the presence of the charged fragment, is capable of supporting... 

It is obvious from the graphical presentation in Fig. 7 that the cation [2S ] and the anion [2 ] exist in equilibrium with the covalent hydrocarbon [28-2] as well as the radical [28-] and [2-] as formulated in (35). In other words, the THF solution is a unique system in which one can observe four elemental species of organic compounds, i.e. covalent molecule, cation, anion and radical, at the same time. 

As follows from the previous analysis for quasi and ordinary particles gases there exists a critical value of parameters a and b for which the least value of the distribution function for observable frequencies is observed. From the physical point of view this is in agreement with the absolute minimal realization of the most probable state. As in any equilibrium distribution, there is an unique most probable state which the system tends to achieve. In consequence we conclude that the observable temperature of the relic radiation corresponds to this state. Or, what is the same, the temperature of such radiation correspond to the temperature originated in the primary microwave cosmic background and the primitive quantum magnetic flow. 

Among the oxyacids of sulfur the predilection to form an anhydride with a sulfur-sulfur bond, rather than one with an oxygen bridge between the two sulfurs, is not restricted to sulfenic acids. We will see in a subsequent section that sulfinic acids also do this. Their anhydrides have the sulfinyl sulfone structure. RS(0)S02R, rather than RS(0)0S(0)R. What is unique about the sulfenic acid-thiolsulfinate system, however, is the fact that the anhydride (thiolsulfinate) is strongly preferred thermodynamically over the acid at equilibrium. With any other type of common acid the reverse is true, of course. The uniqueness of the sulfenic acid-thiolsulfinate situation can perhaps best be appreciated by realizing that, if the same stability relationship between acid and anhydride were to exist for carboxylic acids, acetic acid would spontaneously dehydrate to acetic anhydride ... 

The hydrates of sodium carbonate and vapour furnish a system with two independent components, and, if one hydrate be present, the system is bivariant, and at a given temp., the hydrate can exist in equilibrium with water vapour at different press. if two hydrates be present the system is univariant, and it is stable only when the press, at a given temp, has one unique value so that if the... 

When the system is heterogeneous—i.e., the temperature and pressure are such that solid carbon exists in equilibrium with its vapor, the value of Pci is uniquely determined by the temperature and can be calculated directly from the equilibrium constant Ki. Hence in a heterogeneous system, the partial pressure of cyanogen radicals and of cyanogen depend only on the temperature... 

Since the state of a crystal in equilibrium is uniquely defined, the kind and number of its SE s are fully determined. It is therefore the aim of crystal thermodynamics, and particularly of point defect thermodynamics, to calculate the kind and number of all SE s as a function of the chosen independent thermodynamic variables. Several questions arise. Since SE s are not equivalent to the chemical components of a crystalline system, is it expedient to introduce virtual chemical potentials, and how are they related to the component potentials If immobile SE s exist (e.g., the oxygen ions in dense packed oxides), can their virtual chemical potentials be defined only on the basis of local equilibration of the other mobile SE s Since mobile SE s can move in a crystal, what are the internal forces that act upon them to make them drift if thermodynamic potential differences are applied externally Can one use the gradients of the virtual chemical potentials of the SE s for this purpose ... 

The behavior of a mixture is determined by a system of ordinary differential equations, while the required state, either equilibrium or stationary, is determined by a time-independent system of algebraic equations. Therefore, at first glance one would not expect any qualitative difference between the equilibrium and stationary states. Ya.B. shows that in the equilibrium case, even for an ideal system, a variational principle exists which guarantees uniqueness. Such a principle cannot be formulated for the case of an open system with influx of matter and/or energy. 

For any pure chemical species, there exists a critical temperature (Tc) and pressure (Pc) immediately below which an equilibrium exists between the liquid and vapor phases (1). Above these critical points a two-phase system coalesces into a single phase referred to as a supercritical fluid. Supercritical fluids have received a great deal of attention in a number of important scientific fields. Interest is primarily a result of the ease with which the chemical potential of a supercritical fluid can be varied simply by adjustment of the system pressure. That is, one can cover an enormous range of, for example, diffusivities, viscosities, and dielectric constants while maintaining simultaneously the inherent chemical structure of the solvent (1-6). As a consequence of their unique solvating character, supercritical fluids have been used extensively for extractions, chromatographic separations, chemical reaction processes, and enhanced oil recovery (2-6). 

The standard method of solving the problem by the equilibrium constant approach is to use linearized matrix inversion. Convergence assumes, of course, that the solution not only exists but that it is unique. If a system can have several thermodynamically metastable states (local minima in the Gibbs function) then several nonunique solutions are possible. 

Since 0 is a function of q, Eq. (50) by itself does not exclude the possibility of the existence of more than one equilibrium value q. However, should more than one q exist, the Gibbs free energy hypersurface would necessarily include a spinodal region, and this is impossible for systems that are homogeneous over the whole composition space. Hence, in such systems the equilibrium composition is guaranteed to be unique. 

In summary, because many redox reactions in natural waters do not couple with one another readily, different apparent redox levels exist in the same locale thus an electrode or any other indicator system cannot measure a unique h or pc- If tho electrode (or the indicators) reached equilibrium with one of the redox couples, it would indicate the redox intensity of that couple only. A few conditions are necessary to obtain meaningful operational values ... 

The thermodynamic interpretation of first-order phase transitions assumes that the thermodynamic potentials of each of the phases exist on either side of the phase-equilibrium line, and this line is in no way distinguished for the potentials of each of the phases. At the same time the appearance of a growth channel for pre-critical nuclei in the metastable phase makes the analyticity of a thermodynamic potential on the phase-equilibrium line nonobvious. The uncertainty arises because the system is essentially relaxing. Evaluations show that at WJkgT> % the uncertainty is small as compared with the level of thermal fluctuations, which allows one to speak about the uniqueness of extension of the substance properties deep beyond the phase equilibrium line into the metastable region. ... 

First, it has been supposed that the faradaic current instantaneously adjusts to a change in the double-layer potential. This means that all other quantities that affect the current, such as concentration of the reactive species and the coverage of adsorbates, are assumed to vary on a time scale that is much shorter than the time scale on which typical variations of the potential occur. In other words, all other quantities are assumed to adjust immediately to their equilibrium values and can be adiabatically eliminated. Second, chemical instabilities have been excluded. In the presence of chemical instabilities, the current is no longer a unique function of ( l and the state of the system is only defined when taking into account another variable. The absence of chemical instability also implies that a negative differential resistance can only be realized if the current-potential characteristics of the interface exhibit the shape of an N (or multiple Ns) as shown in Fig. 2(a). In contrast, an S-shaped characteristic, being just one example of another characteristic possessing an NDR, would require the existence of a chemical instability. 

The existence of an isosbestic point is not proof of the presence of only two components. There may be a third component with e = 0 at this particular wavelength. The absence of an isosbestic point, however, is definite proof of the presence of a third component, provided the possibility of deviation from Beer s law in the two-component system can be dismissed. For a two-component system, the isosbestic point is a unique wavelength for quantitative determination of the total amount of two absorbing species in mutual equilibrium. 

With the distinction now made between intensive and extensive variables, it is possible to rephrase the requirement for the complete specification of a thermodynamic state in a more coherent manner. The experimental observation is that the specification of two state variables uniquely determines the values of aifother state variables of an equilibrium, single-component, single-phase system. [Remember, however, that to determine the size of the system, that is, its mass or total volume, one must also specify the mass of the system, or"the value of one other extensive parameter (total volume, - total energy, etc.).] The implication of this statement is that for each substance there exist, in principle, equations relating each state variable to two others. For example. 

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