The infinitely diluted system

As we have mentioned in Chapter 2, the accuracy of the kinetic equations derived using the superposition approximation cannot be checked up in the framework of the same theory. It is the analysis of the limiting case of the infinitely diluted system, no 0, which nevertheless permits us to compare approximate results obtained in the linearized approximation with the exact solution of the two-particle problem (Chapter 3). 

At the same time, the concentrations can be easily expressed through the survival probability, n t) = nou t). We could expect a solution in the form 

As no — 0, the predominant term here is of the order of Therefore, the shortcoming of equation (4.1.23) arises due to the incorrect use of the superposition approximation in a situation of very particular (strongly correlated) particle distribution. Strictly speaking, the correct treatment of the recombination process with arbitrary initial distribution requires the usage of the complete set of correlation functions. At the joint correlation level, such description yields reasonable results only for the particle distribution close to a random equation (4.1.12). For the infinitely diluted system the correlation function (4.1.10) of dissimilar closely spaced particles reveals 

The bottleneck of equations (4.1.19), (4.1.23) and (4.1.28) (derived for the first time by Leibfried [4], see also [6]) has stimulated development of particular interpolating schemes (in terms of Antonov-Romanovskii [7]). This point has been discussed more than once ([7] and references therein, as well as [8-15]). 

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In the formal limit of the infinitely diluted system (n(t) —> 0) the dimensionless interaction potential = j3U(r). (Note that il(r, t) could give... 

INFINITE.LST This file has a list of the infinitely dilute systems in this data base. First line - total number of systems. 

The selectivity of 2 ( 2,1) at these conditions is given by Eq.(3). The quantity ni P) in the above equation is the pure component amount adsorbed for gas 1 at total column pressure P. Experimental measurements are required for 1 (obtained from the infinite dilution system) and data for pure component isotherm (obtained independently using a volumetric technique) to calculate selectivity (LHS of Eq.3). A similar equation can be written for the infinite dilution of gas 1. 

Expressions for the infinitely dilute systems can be derived from eqs 3-6 to obtain... 

Next, we turn to the calculation of the solvation quantities. Since we are interested only in the limit of the infinite dilution system, we expand the equation of state to first order in t to obtain... 

Now we make contact between the macroscopic coefficients ky T,p), defined by Equation 8.39, and the microstructural details of the reference system, that is, the solvation behavior of the species in the infinitely dilute system, by invoking the solvation formalism discussed in Section 8.2 (and references therein) from which we have that... 

If the mutual solubilities of the solvents A and B are small, and the systems are dilute in C, the ratio ni can be estimated from the activity coefficients at infinite dilution. The infinite dilution activity coefficients of many organic systems have been correlated in terms of stmctural contributions (24), a method recommended by others (5). In the more general case of nondilute systems where there is significant mutual solubiUty between the two solvents, regular solution theory must be appHed. Several methods of correlation and prediction have been reviewed (23). The universal quasichemical (UNIQUAC) equation has been recommended (25), which uses binary parameters to predict multicomponent equihbria (see Eengineering, chemical DATA correlation). 

In this study, a thermodynamic framework has been presented for the calculation of vapor-liquid equilibria for binary solvents containing nonvolatile salts. From an appropriate definition of a pseudobinary system, infinite dilution activity coefficients for the salt-containing system may be estimated from a knowledge of vapor pressure lowering, salt-free infinite dilution activity coefficients, and a single system-dependent constant. Parameters for the Wilson equation may be determined from the infinite dilution activity coefficients. 

The usual choice of a reference state other than the pure components is the infinitely dilute solution for which the mole fractions of all solutes are infinitesimally small and the mole fraction of the solvent approaches unity that is, the values of the thermodynamic properties of the system in the reference state are the limiting values as the mole fractions of all the solutes approach zero. However, this is not the only choice, and care must be taken in defining a reference state for multicomponent systems other than binary systems. We use a ternary system for purposes of illustration (Fig. 8.1). If we choose the component A to be the solvent, we may define the reference state to be the infinitely dilute solution of both B and C in A. Such a reference state would be useful for all possible compositions of the ternary systems. In other cases it may be advantageous to take a solution of A and B of fixed... 

These equations are used whenever we need an expression for the chemical potential of a strong electrolyte in solution. We have based the development only on a binary system. The equations are exactly the same when several strong electrolytes are present as solutes. In such cases the chemical potential of a given solute is a function of the molalities of all solutes through the mean activity coefficients. In general the reference state is defined as the solution in which the molality of all solutes is infinitesimally small. In special cases a mixed solvent consisting of the pure solvent and one or more solutes at a fixed molality may be used. The reference state in such cases is the infinitely dilute solution of all solutes except those whose concentrations are kept constant. Again, when two or more substances, pure or mixed, may be considered as solvents, a choice of solvent must be made and clearly stated. 

Special consideration must be given to systems involving liquid solutions of at least one solid component, for which the choice of either the pure solid or pure supercooled liquid as the standard state is not convenient. This case is encountered for all solutions in which the pure solute is not chosen as the reference state. As an example, we consider an aqueous solution of a solid B and choose the reference state to be the infinitely dilute solution. Then a general change of state for the formation of the solution from the components is written as... 

In many systems the pure liquid phase of a component is not attainable under the experimental conditions and thus cannot be used as the reference state for the component. It is then necessary to choose some other state, usually the infinitely dilute solution of the component in the liquid, as a reference state. We choose to illustrate the development under such circumstances by the use of Equations (10.28)—(10.30) under the appropriate experimental conditions. The combination of these equations yields... 

Thus, when A/iE is determined as a function of x3 at a fixed k, AGE can be obtained for the pseudobinary systems by integration. The most convenient reference state of component 1 is the pure component at the chosen temperature and pressure, because the reference state chosen as the infinitely dilute solution of the component in a pseudobinary solvent is different for every ratio of x2/x3. 

Here Ap [p = 1, excess chemical potential of the second component in the binary system composed of the second and third components at the composition equal to q. The reference state of the second component may be either the pure component or the infinitely dilute solution of the second component in the third component. In the limit of p=l, q/p becomes l/(k+ 1), where k = x2/x3 and, with the proper choice of components or... 

Two reference states, one for each phase, must be defined. As we decrease the mole fraction of the third component, we approach the two-liquid-phase binary system composed of the first and second components. We thus define the reference states of the third component as the infinitely dilute solutions of the component in the two liquid phases that are at equilibrium in the 1-2 binary system. Thus, the value of A/i x), x 3] approaches zero as x 3 approaches zero, and x j approaches its value in the 1-2 system, and the value of A/if [x i, x3] also approaches zero, and x] approaches its value in the 1-2 system. In the limit Equation (10.251) becomes... 

The discussion in the previous sections concerning solvated species indicates that a complete knowledge of the chemical reactions that take place in a system is not necessary in order to apply thermodynamics to that system, provided that the assumptions made are applied consistently. The application of thermodynamics to sulfuric acid in aqueous solution affords another illustration of this fact. We choose the reference state of sulfuric acid to be the infinitely dilute solution. However, because we know that sulfuric acid is dissociated in aqueous solution, we must express the chemical potential in terms of the dissociation products rather than the component (Sect. 8.15). Either we can assume that the only solute species present are hydrogen ion and sulfate ion (we choose to designate the acid species as hydrogen rather than hydronium ion), or we can take into account the weak character of the bisulfate ion and assume that the species are hydrogen ion, bisulfate ion, and sulfate ion. With the first assumption, the effect of the weakness of the bisulfate ion is contained in the mean activity coefficient of the sulfuric acid, whereas with the second assumption, the ionization constant of the bisulfate ion is involved indirectly. 

Throughout this discussion we have considered cells in which the electrolytic solution is an aqueous solution. The same methods can be used to define standard half-cell potentials in any solvent system. However, it is important to remember that when the reference state is defined as the infinitely dilute solution of a solute in a particular solvent, the standard state depends upon that solvent. The values so obtained are not interchangeable between the different solvent systems. Only if the standard states could all be defined independently of the solvent would the values be applicable to all solvent systems. 

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