Equilibrium - Necessary Conditions

With aqueous solutions of electrolytes we have two types of equilibrium to consider phase equilibrium and chemical or ionic reaction equilibrium. Phase equilibrium of interest are primarily vapor-liquid and liquid-solid, though vapor-llquid-solid is often of great importance as, for example, in carbonate systems. The necessary condition of phase equilibrium is that the chemical potential of any species i in phase a is equal to the chemical potential of that same species 1 in phase b or 

For chemical or ionic equilibria in a particular phase, the condition of equilibrium is of the same form as the chemical equation. Thus if the reaction at equilibrium is represented by 

The two conditions of equilibrium for phase and chemical equilibrium can be combined to represent the heterogeneous liquid-solid equilibrium of an aqueous solution of a salt B in equilibrium with the solid of salt B B.s B.aq 

The chemical potential of the solid crystal salt B is in phase equilibrium with the dissolved salt B in the liquid or aqueous phase. In aqueous systems we are primarily dealing with salts of strong electrolytes, which in water dissociate completely to the constituent cations and anions of the salt. The chemical potential of the dissolved salt is then given by 

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The sufficient and necessary condition is therefore Cb iCa. As a consequence of imposing the more restrictive condition, which is obviously not correct throughout most of the reaction, it is possible for mathematical inconsistencies to arise in kinetic treatments based on the steady-state approximation. (The condition Cb = 0 is exact only at the moment when Cb passes through an extremum and at equilibrium.)... 

A third approach has been suggested by Jaroudi (Jl), who points out that one necessary condition to prevent reignition of the propellant is to ensure that the gas temperature resulting from thermal equilibrium between the injected fluid and the combustion products is below the propellant autoignition temperature. This approach leads to the conclusion that the ratio of coolant mass flow to propellant mass flow is the critical correlating parameter. 

Synergism can be observed for mixtures of amines with 2,6-bis(l,l-dimethylethyl)phenol but not with monosubstituted phenols [19], There are two reasons for this. First, 2,6-dialkylphenols are characterized by D0 hn—H therefore, the equilibrium of the above reaction is displaced toward the formation of ArO. Second, phenoxyls like these are sterically hindered and, hence, must be less reactive in abstraction reactions. Thus, the necessary conditions for synergism to occur are the following. 

In summary, the necessary condition for temperature jump experiments is that the equilibrium for the chemical system of interest changes with a change in temperature. The advantages of temperature jump experiments are that the perturbation is achieved by a change in a property of the solvent, a fast time resolution can be achieved, as short as picoseconds when using lasers, and a time domain over more than 6 orders of magnitude can be probed with the same technique. The disadvantage of the technique... 

In the experiments on the Jt-A characteristics, it has been usually assumed that thermal equilibrium will be attained easily if the experiment is performed using a slow rate of compression of thin film at the interface. Measurements under thermal equilibrium are, of course, the necessary condition to obtain the physico-chemical properties of the individual "phase" of the lipid ensemble. 

The structural constraints used in the first case study namely, Eqn s 27,28 and 29 are used again. The melting point, boiling point and flash point, are used as constraints for both solvent and anti-solvent. Since the solvent needs to have high solubility for solute and the anti-solvent needs to have low solubility for the solute limits of 17 <8 < 19 and 5 > 30 (Eqn s. 33 and 37) are placed on the solubility parameters of solvent and anti-solvents respectively. Eqn.38 gives the necessary condition for phase stability (Bernard et al., 1967), which needs to be satisfied for the solvent-anti solvent pairs to be miscible with each other. Eqn. 39 gives the solid-liquid equilibrium constraint. 

The necessary condition for equilibrium in the chemical reactions of equation (1) is that... 

Many investigators 35 have reported experimental results on the necessary conditions for the static equilibrium of a sphere. The results of all such studies may be represented by a factor Z which is proportional to the ratio of the forces due to the yield stress xY and those due to gravity. 

The mixed-potential model demonstrated the importance of electrode potential in flotation systems. The mixed potential or rest potential of an electrode provides information to determine the identity of the reactions that take place at the mineral surface and the rates of these processes. One approach is to compare the measured rest potential with equilibrium potential for various processes derived from thermodynamic data. Allison et al. (1971,1972) considered that a necessary condition for the electrochemical formation of dithiolate at the mineral surface is that the measmed mixed potential arising from the reduction of oxygen and the oxidation of this collector at the surface must be anodic to the equilibrium potential for the thio ion/dithiolate couple. They correlated the rest potential of a range of sulphide minerals in different thio-collector solutions with the products extracted from the surface as shown in Table 1.2 and 1.3. It can be seen from these Tables that only those minerals exhibiting rest potential in excess of the thio ion/disulphide couple formed dithiolate as a major reaction product. Those minerals which had a rest potential below this value formed the metal collector compoimds, except covellite on which dixanthogen was formed even though the measured rest potential was below the reversible potential. Allison et al. (1972) attributed the behavior to the decomposition of cupric xanthate. 

The necessary condition for hydrogen storage is that the thermodynamic and kinetic conditions are fulfilled. In that case, a metal exposed to hydrogen gas absorbs hydrogen until equilibrium is established. There are a number of reaction steps that kinetically may hinder a hydrogen-storing system from reaching its... 

Equation (7) gives the necessary condition for equilibrium at 0 K. Notice that it may be written in the form of an equation of state for the crystal at 0 K ... 

A possibility to achieve negative (positive) A5 by its minimization (maximization) even for weak modulation (i.e., for small E) is to adapt the temperature of the bath such that for an undriven system Hamiltonian Hq, no change is observed, A5 = 0, which is a necessary condition for a system-bath equilibrium. This would require nonunitary system modulation, for example, the effect of repeated measurements [92-94],... 

There are at least three possible mechanisms for the spontaneous breakdown of hemiorthoesters, hemiacetals, and related species. Firstly, there may be a rapid and reversible ionization equilibrium followed by hydronium-ion catalysed breakdown of the anion (9) (Gravitz and Jencks, 1974). A necessary condition for this mechanism to be valid is that k2 calculated from kHi0 and Ka should fall below the diffusion controlled limit of c. 10loM 1s 1. The second mechanism (10) is similar to this but involves formation of the anion and hydronium ion in an encounter pair which react to give products faster than the diffuse apart (Capon and Ghosh, 1981). With this mechanism therefore the ionization equilibrium is not established and the rate constant for... 

If a closed system is in equilibrium with reservoirs maintaining constant potentials (e.g., P and T), that system has a free-energy function [e.g., G(P, T)] that is minimized at equilibrium. Therefore, a necessary condition for equilibrium is that any variation in G must be nonnegative (6G)p,t > 0. 

Clearly, the data contain information about both the equilibrium constant and the rate constants for the conformational interconversion. In this instance, the quantitative analysis was based upon the cyclic voltammetric data. The points in Figure 16.3 are the background-corrected experimental data, and the curves were computed by digital simulation with values of the equilibrium and rate constants selected to achieve best agreement with the experimental data. A given set of parameters was found to account for the data at a variety of scan rates, a necessary condition if the kinetic model is to be judged adequate. 

Thus, when studying atmospheric chemistry, it is necessary always to take into account the vertical and horizontal movements in the atmosphere, as well as the conditions controlling those chemical reactions that do not spontaneously lead to photochemical equilibrium. These conditions are applicable not only to ozone in the lower stratosphere, but also to atomic oxygen in the upper mesosphere above 75 km. In fact, equation (4) shows that, with increasing height, the formation of O3 becomes less and less important because of the decrease in the concentration of 02 and N2. Above 60 km the concentration of atomic oxygen exceeds that of ozone, but it is still in photochemical equilibrium up to 70 km. However, at the mesospause (85 km), it is subject to atmospheric movements, and its local concentration depends more on transport than on the rate of production. 

A second common approximation is the steady-state condition. That arises in the example if A is fast compared with in which case [7] remains very small at all times. If [J] is small then d[I /dt is likely to be approximately zero at all times, and this condition is commonly invoked as a mnemonic in deriving the differential rate equations. The necessary condition is actually somewhat weaker (9). For equations 22a and b, the steady-state approximation leads, despite its different origin, to the same simplification in the differential equations as the pre-equilibrium condition, namely, equations 24a and b. 

Irrespective of the experiment to be done, sample preparation contains a number of necessary conditions. First, aggregation must be prevented if one wants to investigate structure and conformation of single molecules. Second, the adsorption process must be reversible, or at least, very slow in order to approach the equilibrium state and allow statistical analysis of the molecular assembly. Third, adhesion of the molecules to the substrate must be strong enough to sustain the mechanical and adhesive interactions with the tip. However, it should be relatively low to prevent the native structure from deformation. 

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