Robinson-Stokes equation

Of course, (4.9) cannot be used at higher ionic strengths. A more precise procedure is then required, such as the Robinson-Stokes equation for the mean molal activity coefficient [11] (see also [86a], chapter 1)... 

Solvation Effects. Many previous accounts of the activity coefficients have considered the connections between the solvation of ions and deviations from the DH limiting-laws in a semi-empirical manner, e.g., the Robinson and Stokes equation (3). In the interpretation of results according to our model, the parameter a also relates to the physical reality of a solvated ion, and the effects of polarization on the interionic forces are closely related to the nature of this entity from an electrostatic viewpoint. Without recourse to specific numerical results, we briefly illustrate the usefulness of the model by defining a polarizable cosphere (or primary solvation shell) as that small region within which the solvent responds to the ionic field in nonlinear manner the solvent outside responds linearly through mild Born-type interactions, described adequately with the use of the dielectric constant of the pure solvent. (Our comments here refer largely to activity coefficients in aqueous solution, and we assume complete dissociation of the solute. The polarizability of cations in some solvents, e.g., DMF and acetonitrile, follows a different sequence, and there is probably some ion-association.)... 

Equations 14 and 15 differ from the work by Robinson, Stokes, and Marsh... 

Another equation which has received use is the Wishaw-Stokes equation, which is discussed in Robinson and Stokes [32], This equation is based on the extended Onsager equation and is corrected for viscosity in a Walden rule sort of term ... 

Observed molar conductivities were analyzed by assuming the ion association (ion-pair formation) between the complex ions and the counter ions in the same manner as described previously. The closest distances of approach of ions (a) in the Robinson-Stokes conductivity equation and in the Debye-HUckel equation were taken as 6.8 and 7.3 A for chlorides and perchlorates of the tris(phen) complexes 6.6 and 7.1 A for those of the tris(bpy) complex, respectively, using the effective ionic radii of the complex ions, shown in T le 1, and those of Cl (1.81 A) and C104 (2.30A). The values of ref were estimated from the ionic partial molar volumes (f i°°) by use of Glueckauf equation. > ... 

When Ka <0.5 the treatments of conductance which have been described break down. Conductance of concentrated electrolyte solutions in non-aqueous solvents has been reported to follow the Wishaw and Stokes equation,which is essentially the Robinson and Stokes equation 5.2.35 with the conductances being corrected for the increase of viscosity of the solution. Association and conductance in concentrated solutions in various solvents have also been considered by several in-vestigators. " ... 

Inaccuracies arise at modest electrolyte concentrations because the concentrations of ions predicted within an ionic atmosphere are imrealistic if a finite ionic radius is not considered. The assumption of a finite ionic radius leads to the extended Debye-Hiickel limiting law. This is still inaccurate for concentrated ionic solutions because the depletion of solvent molecules due to solvation shells makes the assumption of zero ion-solvent interaction highly inaccurate. Only empirical formulas such as the Robinson-Stokes or Pitzer equations are able to address this issue. Most well-supported electrolyte solutions have behaviours in this latter regime. 

Rg. 2.7 Experimental activity coefficients ( ) and their ionic strength dependence shown schematically. DHL = Oebye-Huckel limiting law EOHL = Extended Debye-HQckel law RS s Robinson and Stokes equation. 

The effects of hydration-ion-solvent interactions-on activity coefficients can be quantified by means of the Robinson and Stokes Equation ... 

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

The derivative equations for osmotic and activity coefficients, which are presented below, were applied to the experimental data for wide variety of pure aqueous electrolytes at 25°C by Pitzer and Mayorga (23) and to mixtures by Pitzer and Kim (11). Later work (24-28) considered special groups of solutes and cases where an association equilibrium was present (H PO and SO ). While there was no attempt in these papers to include all solutes for which experimental data exist, nearly 300 pure electrolytes and 70 mixed systems were considered and the resulting parameters reported. This represents the most extensive survey of aqueous electrolyte thermodynamics, although it was not as thorough in some respects as the earlier evaluation of Robinson and Stokes (3). In some cases where data from several sources are of comparable accuracy, a new critical evaluation was made, but in other cases the tables of Robinson and Stokes were accepted. 

Generally, agreement has been found between our correlations and those of Pitzer, and others (1972, 1973, 1974, 1975, 1976) and Rard, and others (1976, 1977). Many of our correlations agree fairly well with Robinson and Stokes, (1965) and Harned and Owen, (1958) but in most cases a much larger data base and more recent measurements have been incorporated into the evaluations. It has been observed that agreement with Pitzer s equations is found below moderate concentrations (several molal), but often deviate at higher concentrations where the Pitzer equations do not contain enough parameters to account for the behavior of the activity (or osmotic) coefficient. 

An extended form of the Debye-Huckel equation is the hydration one of Robinson and Stokes (11). It contains two adjustable parameters, ap and h, where h is rilated to the hydration number. It can be fitted to y for several electrolytes for concentrations in excess of 1 m. Their equation has the valuable feature of describing not only the salting-in but also the salting-out part of the y+ versus m curve. It should be noted, however, that the... 

In Equation 4.21, the activity of pure water (a) is unity and the activity of the water with the inhibitor (a ) is the product of the water concentration (xw) and the activity coefficient (xw). The water concentration is known and the activity coefficient is easily obtained from colligative properties for the inhibitor, such as the freezing point depression. For instance the activity of water in aqueous sodium chloride solutions may be obtained from Robinson and Stokes (1959, p. 476) or from any of several handbooks of chemistry and physics. 

Liu and Ruckenstein [17] presented a semiempirical equation to estimate diffusivities under supercritical conditions that is based on the Stokes-Einstein relation and the long-range correlation, respectively. The parameter 20 was estimated from the Peng-Robinson equation of state. In addition, f = 2.72 — 0.3445 TcB/TcA for most solutes, but for C5 through C14 linear alkanes,/ = 3.046 — 0.786 l cBl l cA. In both cases Ta is the species critical temperature. They compared their estimates to 33 pairs, with a total of598 data points, and achieved lower deviations (5.7 percent) than the Sun-Chen correlation (13.3 percent) and the Catchpole-King equation (11.0 percent). 

Fig. 3.39. Comparison of experimental activity coefficients with those predicted by the one-parameter equation as shown. (Reprinted from R. H. Stokes and R. A. Robinson, J. Am. Chem. Soc. 70 1870, 1948.)...

A similar result is obtained in the Stokes and Robinson application ofthe idea, i.e., that there is a removal of effective free solvent into hydration shells around ions (Section 2.4.1). Both ideas are similar to the effect of the y - h term in van der Waals s equation of state for gases. If the a/V attraction term is neglected, P = kT/(V- b). As Vis reduced to be comparable in value to b, P (which is analogous to the ionic activity) inaeases above that for the simple PV = kl equation. 

In most of the modem versions of the Debye-Hiickel theory of 1923, it is still assumed that the dielectric constant to be used is that of water. The dielectric constant of solutions decreases linearly with an increase in the concentration of the electrolyte. Using data in the chapter, calculate the mean activity coefficient for NaCl from 0.1 M to 2 M solutions, using the full equation with correction for the space taken up by the ions and the water removed by hydration. Compare the new calculation with those of Stokes and Robinson. Discuss the change in a you had to assume. 

Equation 8 can be used to calculate excess free energies for single solute solutions (n = 1) using tabulated values such as those appearing in Robinson and Stokes (13). 

In saline soils and soils contaminated with geothermal brines, the ionic strengths of the soil solution may exceed 0.5 M. This fact poses the necessity of using equations which have been developed to describe the activity coefficients of ions in concentrated, multicomponent electrolyte solutions. As part of a study on the chemistry of ore-forming fluids, Helgeson (50) has proposed that the true individual ion activity coefficients for ions present in small concentrations in multicomponent electrolyte solution having sodium chloride as the dominant component be approximated by a modified form of the Stokes-Robinson equation. The equation proposed is ... 

Stokes and Robinson achieved remarkable success with a one-parameter equation, the single parameter being a hydration number. For a salt of the type AB or AB2, they wrote the equation... 

Liu and Ruckenstein [17] presented a semiempirical equation to estimate diffusivities under supercritical conditions that is based on the Stokes-Einstein relation and the long-range correlation, respectively. The parameter 20 was estimated from the Peng-Robinson equation of state. In addition, f = 2.72 — 0.3445 for most... 

Using an argument based on the Gibbs-Duhem equation, Stokes and Robinson (1) derived the following equation for a solution of a single aqueous electrolyte (A), presumed to be fully dissociated ... 

Stokes and Robinson (1) fit their equation to data reported for 35 pure aqueous electrolytes of the 1 1 and 2 1 types and obtained good results to fairly high concentrations. For example, for NaCl, the reported average difference in In y was only 0.002 for over a concentration range of 0.1-5.0 m (1 = 0.1-5.0 m) for CaCl2, 0.001 from 0.01-1.4 m (I = 0.06 to 8.4 m). These statistics can not be directly compared with those reported for fitting with Pitzer s equations (e.g., ref. 9) as the... 

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