Existence of the equilibrium

It has not been found possible to reconcile all these observations with a simple kinetic scheme. A major difficulty is that whilst the stoichiometric concentrations of nitric acid and of acetic acid can be varied independently, the actual concentrations of these species cannot, because of the existence of the equilibrium ... 

The following remarks can be made about Huang s mechanism as on any mechanism involving hydrogen-bonded dimers such as in 19) where the existence of the equilibrium... 

The existence of the equilibrium point and the conditions for the eigenvalues of the linearized model are not difficult to find, nevertheless the hypothesis... 

In water, the equilibrium constant of Eq. (11) is reduced when either the concentration of supporting electrolyte (e.g., NaC104) or the temperature is increased. Existence of the equilibrium is indicated by the absorption coefficients at A.max measured in water for [Ni(L)]2+, which are smaller than those measured in a noncoordinating solvent such as nitromethane or acetone. 

In the molten condition the density of sulphur is dependent only on the temperature, provided that the equilibrium condition between A-sulphur and y-sulphur has been attained at 113° C. the density is 1 811, the value then rising somewhat up to about 100° C.,2 then steadily falling to 1 480 at 446° C. The coefficient of expansion has been measured by several investigators, but on account of the complications introduced by the gradual readjustment of the equilibrium at new temperatures, the results are not altogether concordant, although they supply confirmatory evidence of the existence of the equilibrium. The mean specific heat of mobile sulphur is 0 220 and of viscous sulphur 0-290.4... 

Some workers 26,13 have suggested the existence of the equilibrium represented by equation (31) in which both the forward and reverse steps are catalyzed by Pt. This was first demonstrated by passing H2 through a solution of MV2+ in the presence of colloidal Pt, which led first to the formation of the deep blue MV+ and subsequently to hydrogenation to the piperidine derivative. This led to the postulation of MV+ as an intermediate in the hydrogenation of MV2+. 

Most of the dimer cations and the bonded dimer cations were produced during the pulse irradiation. This can be explained by assuming the existence of the equilibrium between styrene monomers (St) and neutral dimers of styrene (St2) under the experimental conditions [24]. 

The dilution of solutions33 containing equimolar ratios of monomer units of the complex components results in the dissociation of the complexes of PMAA with low-molecular weight PEG. The reduced viscosity of solutions rapidly increases, which indicates the existence of the equilibrium PMAA + PEG complex. In the case of a relatively high-molecular weight PEG, the PMAA macromolecules are firmly connected with PEG and at the dilution of aqueous solution, an increase of the reduced viscosity typical of polyelectrolytes does not occur, i.e. the complex does not dissociate. The absence of temperature dependence of the relative viscosity in the temperature range 15-40 °C is indicative of the stability of this complex (Fig. 4). 

A connection between eqn. (56) and transfer in cationic polymerization has not yet been observed. It is assumed that the great intensity of proton transfer is caused by the existence of the equilibrium (57), comprising a non-classical ion with a loose, more mobile proton... 

Chaotropic Model. If the counteranion concentration is low, some analyte molecules have a disrupted solvation shell, and some do not due to the limited amount of counteranions present at any instant within the mobile phase. If we assume an existence of the equilibrium between solvated and desolvated analyte molecules and counteranions, this mechanism could be described mathematically [151]. 

One major advantage offered by the dppf ligand in Rh-catalyzed olefin hydroformylation is exemplified in its higher linear aldehyde selectivity when present in a dppf Rh ratio of 1.5 or higher [37,242]. This result leads to the proposed key intermediate of a Rh dimer with both chelating and bridging phosphine in the catalytic cycle. It also confirms the significance of the tris (phosphine) moieties at the point when the aldehyde selectivity is determined, i.e., the step in which the hydride is inserted into the M-olefin bond. This involvement of a dinuclear or tris (phosphine) intermediate appears to differ from the intermediate RhH(CO)(PR 3)z (olefin) (which is converted into the square planar Rh(R)(CO)(PR 3)2 by hydride insertion) commonly accepted for hydroformylation catalyzed by monophosphine complexes. P NMR studies also established the existence of the equilibrium in which the disphosphine can be... 

Just as for monomolecular systems, the equilibrium points are structural features that play a central role in the discussion of general complex reaction systems. It is not, however, necessary to introduce them into the system as explicit basic assumptions or to introduce them by means of thermodynamics or statistical mechanics they arise as a consequence of some much more primitive concepts, which are always included in the basic models for closed reaction systems and for many open systems as well. The reader may ask why raise the question as long as the existence of the equilibrium points are assured by some known principles such as those provided by thermodynamics the reason is that a new point of view and an appreciation of the consequences of implicitly and explicitly known basic characteristics often reveal to us the path to a better understanding of nature and to the solution of a particular problem. 

Raman, i.r., and electrical conductance studies73 undertaken on liquid mixtures of C1F3 and HF have established the existence of the equilibrium ... 

Recently, we explored the effects of the additional initiation in detail, and found that the conditions (20) for the existence of the equilibrium (19) must again... 

In the log—log representation of the phase diagram, the dynamic equilibrium relation 19 (/cc[R][Y] = d[I]o) or apt] = 1 appears as a straight line with slope — 1. It starts at (p, rj) = (1/a, 1), and it is practically identical to r] for p > 1/a. The trajectory must closely follow t], and hence, there is a time range where the equilibrium relation is certainly valid. With increasing 1/a, the equilibrium line shifts to the right. It will not be reached at all when 1/a becomes close to b m. Consequently, one condition for the existence of the equilibrium is a2/b 1. Further, because p < 1, one has b > 1, and this implies the third condition a 1 by combination with the first. With the definitions 41, this yields eqs 20. 

The existence of the equilibrium between the free radical and the pinacol salts in the solid phase has been demonstrated by the temperature coefiicient of the paramagnetism of the crystals. From these measurements the heat of dissociation into free radicals in the solid phase was found to be of the order of 0.3 kcal. for the potassium ketyls of benzophenone, phenyl xenyl ketone, and fluorenone. 

A -Benzylepimine 160 reacted with a mixture of ammonium and tetrabutyl-ammonium bromide or iodide to give the respective diequatorial bromo (193) or iodo (194) derivatives as the sole products. This formation is the result of an equilibrium between the starting epimine and the diaxial halo derivative, which is formed initially by aziridine-ring cleavage. This equilibrium could be set up readily because of the instability of the diaxial halo derivatives and their tendency to undergo back-cyclization of the aziridine ring. To prove the existence of the equilibrium, the diaxial bromo derivative 191 was prepared by the action... 

Butenylmagnesium bromide is in mobile equilibrium with a small amount of cyclopropylmethylmagnesium bromide. The existence of the equilibrium has been established by deuterium-labeling techniques. Cyclopropylmethylmagnesium bromide " and cyclopropylmethyllithium have been prepared by working at low... 

Denote the best response function of the retailer by Rr Q) and the best response function of the wholesaler by Ryj A) both defined for zero initial inventory. At this point we have not demonstrated uniqueness or even existence of the equilibrium. It helps, however, to visualize the problem first. We begin by presenting the game graphically (see Figure 14.1, parameters are taken from the example in Section 6 with r = 8). The point A, is a Nash... 

Proof Existence of the equilibrium follows from the concavity of the retailer s and the wholesaler s objective functions (see Moulin 1986 page 114). To show uniqueness it is sufficient to notice that the optimality conditions can be solved sequentially, i.e., condition (14.14) can be solved uniquely for A and then condition (14.13) can be solved uniquely for. ... 

Electrochemical studies (by cyclic voltammetry) on C6oPd(PPh3)2 and on C7oPd(PPh3)2 in acetonitrile in the presence of excess Pd(PPh3)4 revealed the disappearance of some reductive waves characteristic of free C ( = 60, 70), suggesting the existence of the equilibrium shown in Equation (lO). ... 

Several mono- and bis-cyclopentadienyl derivatives of neodymium and ytterbium with o-aldehydophenolato and fiirfuralcoholato chelate ligands have been obtained from the direct reaction of the corresponding alcohol with CpjLn [83]. The products have been characterized by element analysis, IR spectroscopy and mass-spectrometry. Mass-spectroscopy allowed to establish the existence of the equilibrium Cp2LnX CpLnX2 on heating of the complexes. 

Since we can never wait infinitely long time (Xexp cxd) to actually observe a SM S, its existence can never be verified. This is no different from what is customary in equilibrium statistical mechanics, where the existence of the equilibrium state is taken for granted as a postulate. We quote Huang ([41], p. 127) Statistical mechanics, however, does not describe how a system approaches equilibrium, nor does it determine whether a system can ever be found to be in equilibrium. It merely states what the equilibrium situation is for a given system. Ruelle ([42], p. 1) notes that equilibrium states are defined operationally by assuming that the state of an isolated system tends to an equilibrium state as time tends to + oo. Whether a real system actually approaches this state cannot be answered. Therefore, we will... 

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