Uniqueness of equilibrium

Since the publication of some prolegomena to the rational analysis of systems of chemical reaction [1] other cognate work has come to light and some earlier statements have been made more precise and comprehensive. I would like first to advert to an earlier work previously overlooked and to mention some recent publications that partially fill some of the undeveloped areas noticed before. Secondly, I wish to extend the theorem on the uniqueness of equilibrium to a more general case and to establish the conditions for the consistency of the kinetic and equilibrium expressions ( 2, 3). Thirdly, the conception of a reaction mechanism is to be reformulated in a more general way and the metrical connection between the kinetics of the mechanism and those of the ostensible reactions clarified. The notation of the earlier paper ([1], hereinafter referred to as P) will be followed and augmented where necessary. In particular the reader is reminded that the range of each affix is carefully specified and the summation convention of tensor analysis is employed. 

In a recent paper Shapiro Shapley [4] have considered the problem of the uniqueness of equilibrium of systems of reactions in several phases in great detail. The computation of equilibrium compositions by direct minimization of the Gibbs free energy function has proved a valuable tool in the discussion of very complex systems and it is important to show that this minimum is unique and achieved under the same conditions that satisfy the mass action laws. This is what Shapiro Shapley have done Sellers has suggested some improvements [5]. 

Method 3. Univalent mapping argument. Another method for demonstrating uniqueness of equilibrium is based on verifying that the best response mapping is one-to-one that is, if f x) is 2i RT map-... 

Rosen, J.B. 1965. Existence and uniqueness of equilibrium points for concave N-person games. Econometrica, Vol.33, 520-533. 

For high temperatures, the spin-glass system behaves essentially the way conventional Ising-spin systems behave namely, a variety of different configurations are accessible, each with some finite probability. It is only at low enough tempera tures that a unique spin-glass phase - characterized chiefly by the appearance of a continuum of equilibrium states - first appears. 

If one of the compounds of interest has a unique spectroscopic feature, or can be synthesized with a fluorescent or radioactive label, then a variety of equilibrium... 

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 ... 

For a binary mixture under constant pressure conditions the vapour-liquid equilibrium curve for either component is unique so that, if the concentration of either component is known in the liquid phase, the compositions of the liquid and of the vapour are fixed. It is on the basis of this single equilibrium curve that the McCabe-Thiele method was developed for the rapid determination of the number of theoretical plates required for a given separation. With a ternary system the conditions of equilibrium are more complex, for at constant pressure the mole fraction of two of the components in the liquid phase must be given before the composition of the vapour in equilibrium can be determined, even for an ideal system. Thus, the mole fraction yA in the vapour depends not only on X/ in the liquid, but also on the relative proportions of the other two components. 

We should be clear as to what a catalyst can and cannot do in a reaction. Most important, no catalyst can alter the equilibrium composition in a reactor because that would violate the Second Law of Thermodynamics, which says that equilibrium in a reaction is uniquely defined for any system However, a catalyst can increase the rate of a reaction or increase the rate of one reaction more than another. One can never use a catalyst to take a reaction from one side of equilibrium to another. The goal in reaction engineering is typically to find a catalyst that will accelerate the rate of a desired reaction so that, for the residence time allowed in the reactor, this reaction approaches equihbrium while other undesired reactions do not. Attempts to violate the laws of thermodynamics always lead to failure, but maity engineeis still try. 

Some materials might produce a unique failure surface providing measurements could be conducted under first stretch conditions in a state of equilibrium. Tschoegl (110), at this writing, is attempting to produce experimental surfaces by subjecting swollen rubbers to various multiaxial stress states. The swollen condition permits failure measurements at much reduced stress levels, and the time dependence of the material is essentially eliminated. Studies of this type will be extremely useful in establishing the foundations for extended efforts into failure of composite materials. 

A plot of equilibrium r/e as a function of the external parameter a is schematically presented in Fig. 2.3.5a. The plot in Fig. 2.3.5a is markedly different from that in Fig. 2.3.4a by its lack of bifurcation. (Uniqueness of the appropriate solutions of the Poisson-Boltzmann equation for any values of a is proved in [18].) In the (F, ri) or (F, aeS) plane this corresponds to the existence of solutions of the Poisson-Boltzmann equations with finite F (bounded norm of the appropriate solution with a subtracted singular part due to the effective line charge) only for aeS < with adetermined by Conjecture 2.1. This is schematically illustrated in Fig. 2.3.5b. Note that F as a function of creS is constructed in a single counterion case by solving (2.3.3a) with a = of J e rdr and with the boundary conditions tp(a) = -aeS lna, = 0, and by going to the limit a- 0. 

The metric geometry of equilibrium thermodynamics provides an unusual prototype in the rich spectrum of possibilities of differential geometry. Just as Einstein s general relativistic theory of gravitation enriched the classical Riemann theory of curved spaces, so does its thermodynamic manifestation suggest further extensions of powerful Riemannian concepts. Theorems and tools of the differential geometer may be sharpened or extended by application to the unique Riemannian features of equilibrium chemical and phase thermodynamics. 

Another unique feature of ionized NIPA gel has been found recently both of the equilibrium swelling ratio a and the first-order transition temperature T0 depend strongly on the shape of samples [31]. The measurement of equilibrium a has been made on ionized NIPA gel rods of various diameters, and also on plates and cubes. The gel contained 680 mM NIPA, 20 mM acrylic acid (AA), and 8.6 mM BIS. All samples were prepared from the same pregel solution at the same time so as to guarantee that the composition and the structure of all samples were the same. 

A more complete model of equilibrium has been studied by J. H. Hancock T. S. Motzkin [25], who prove some valuable existence and uniqueness theorems. 

As noticed in a review of P [2]t, an important early reference had been overlooked, namely the two notes of Jouguet [3], Observations sur les principes et les theoremes generaux de la statique chimique (pp. 61-180) and Sur les lois de la dynamique chimique relatives aux sens des reactions irreversibles (pp. 181-194). In these papers (of which I was culpably unaware when writing P) Jouguet carefully establishes the notion of independence of reactions and the invariants of a system of reactions and considers the ramifications of the phase rule. In particular he is quite clear as to the way in which stoicheiometry lays a foundation for thermodynamics and in going on to discuss the nature of equilibrium he shows that its uniqueness is a consequence of its stability. Thus much of the content of 2-4 of P is to be found in Jouguet. His second note, however, is concerned with the direction... 

Intuitively, the uniqueness of the chemical equilibrium state of a mixture of reacting gases is more or less obvious. However, it may be of some interest to rigorously prove that the system of equations of the law of mass action (LMA), together with the imposed conditions of conservation of matter for given T and v or T and p, has one and only one real-valued and positive solution. 

The principle of detailed equilibrium accounts for the specific features of closed systems. For kinetic equations derived in terms of the law of mass/ surface action, it can be proved that (1) in such systems a positive equilibrium point is unique and stable [22-25] and (2) a non-steady-state behaviour of the closed system near this positive point of equilibrium is very simple. In this case even damped oscillations cannot take place, i.e. the positive point is a stable node [11, 26-28]. 

It is this property that was used by Zel dovich [22] to prove the uniqueness of the equilibrium specified as a point of the free energy minimum. 

The fluid phase that fills the voids between particles can be multiphase, such as oil-and-water or water-and-air. Molecules at the interface between the two fluids experience asymmetric time-average van der Waals forces. This results in a curved interface that tends to decrease in surface area of the interface. The pressure difference between the two fluids A/j = v, — 11,2 depends on the curvature of the interface characterized by radii r and r-2, and the surface tension, If (Table 2). In fluid-air interfaces, the vapor pressure is affected by the curvature of the air-water interface as expressed in Kelvin s equation. Curvature affects solubility in liquid-liquid interfaces. Unique force equilibrium conditions also develop near the tripartite point where the interface between the two fluids approaches the solid surface of a particle. The resulting contact angle 0 captures this interaction. 

Components are species whose concentrations can be varied independently (in a mathematical sense) and whose combination in reactions can produce all other species in an aqueous system. The number of components is unique to a given system, but not their identity, which is chosen for convenience in developing a thermodynamic (or kinetics) description. For an excellent discussion of equilibrium speciation calculations in terms of components, see Chap, 3 in F. M. M. Morel, Principles of Aquatic Chemistry, Wiley, New York, 1983. 

---