Osmotic coefficient and activity coefficient

This property simply considered is the first temperature derivative of the free energy or activity and can be used to obtain osmotic coefficients and activity coefficients by the relationships ... 

The Osmotic Coefficient and Activity Coefficient Equations for calculating the osmotic coefficient have the formd... 

Calculating the osmotic coefficient and activity coefficients of an aqueous solution using the Pitzer approach requires knowing the cation-anion parameters, Bi°J, Bii, and Cca the cation-cation (or anion-anion) parameter, Qcc> (or 9aa>) and the triple particle parameter, important constituents of a solution. If neutral solutes are present at significant concentrations, then the neutral-cation (or neutral-anion) parameter, nc (or Xna), and the triple particle parameter, Cnca, are also needed. Fortunately, there have been many studies using the Pitzer approach in the past 30 years. As a consequence, many of the most important parameters and their temperature dependence have been determined (see, for example, Harvie et al. 1984 Pitzer 1991, 1995 Appendix B). 

While the mathematics of calculating the osmotic coefficient and activity coefficients are complicated (Eqs. 2.39 to 2.69), the great virtue of the Pitzer approach is that it allows one to calculate these quantities at high solute concentrations (/ > 5 m) (Pitzer 1991, 1995 Marion and Farren 1999 Marion 2001, 2002 Marion et al. 2003a,b Marion et al. 2005, 2006). This is particularly important in characterizing the freezing process, which can concentrate solutes rapidly once ice begins to form. 

The relationship between the osmotic coefficient and activity coefficient is given by ... 

In Pitzer s model the Gibbs excess free energy of a mixed electrolyte solution and the derived properties, osmotic and mean activity coefficients, are represented by a virial expansion of terms in concentration. A number of summaries of the model are available (i,4, ). The equations for the osmotic coefficient (( )), and activity coefficients (y) of cation (M), anion (X) and neutral species (N) are given below ... 

Barthel J, Neueder R, Poepke H, Wittmann H (1999) Osmotic coefficients and activity coefficients of nonaqueous electrolyte solutions. Part 2. Lithium perchlorate in the aprotic solvents acetone, acetonitrile, dimethoxyethane, and dimethylcarbonate. J Solution Chem 28 489-503... 

Results for 1-1 Electrolytes. Here we present some of the most recent data obtained for electrolyte solutions within the framework of the MSA. We have shown(5,12) that using the MSA as an empirical theory, osmotic coefficients and activity coefficients of 1-1 electrolytes can be calculated within the experimental accuracy if a density dependent cation radius is used in connection with Pauling crystallographic radii for the anions. Figures 2 and 3 show the kind of... 

Parametrization of the thermodynamic properties of pure electrolytes has been obtained [18] with use of density-dependent average diameter and dielectric parameter. Both are ways of including effects originating from the solvent, which do not exist in the primitive model. Obviously, they are not equivalent and they can be extracted from basic statistical mechanics arguments it has been shown [19] that, for a given repulsive potential, the equivalent hard core diameters are functions of the density and temperature Adelman has formally shown [20] (Friedman extended his work subsequently [21]) that deviations from pairwise additivity in the potential of average force between ions result in a dielectric parameter that is ion concentration dependent. Lastly, there is experimental evidence [22] for being a function of concentration. There are two important thermodynamic quantities that are commonly used to assess departures from ideality of solutions the osmotic coefficient and activity coefficients. The first coefficient refers to the thermodynamic properties of the solvent while the second one refers to the solute, provided that the reference state is the infinitely dilute solution. These quantities are classic and the reader is referred to other books for their definition [1, 4],... 

FIGURE 5 Comparison of experimental (points) and calculated (lines) osmotic (O) and activity (y ) coefficients of Nal in (a) methanol and (b) acetonitrile at 25°C. The experimental data stem from vapor pressure measurements. MSA calculations are executed with one-parameter fits. 

Panagiotopoulos et al. [16] studied only a few ideal LJ mixtures, since their main objective was only to demonstrate the accuracy of the method. Murad et al. [17] have recently studied a wide range of ideal and nonideal LJ mixtures, and compared results obtained for osmotic pressure with the van t Hoff [17a] and other equations. Results for a wide range of other properties such as solvent exchange, chemical potentials and activity coefficients [18] were compared with the van der Waals 1 (vdWl) fluid approximation [19]. The vdWl theory replaces the mixture by one fictitious pure liquid with judiciously chosen potential parameters. It is defined for potentials with only two parameters, see Ref. 19. A summary of their most important conclusions include ... 

The osmotic coefficient is often used as a measure of the activity of the solvent instead of a because a is nearly unity over the concentration range where 7 is changing, and many significant figures are required to show the effect of solute concentration on a. The osmotic coefficient also becomes one at infinite dilution, but deviates more rapidly with concentration of solute than does a. ... 

The osmotic coefficient 4> and activity coefficient are related in a simple manner through the Gibbs-Duhem equation. We can find the relationship by writing this equation in a form that relates a and 2-... 

Rard (1992) reported the results of isopiestic vapor-pressure measurements for the aqueous solution of high-purity NiCl2 solution form 1.4382 to 5.7199 mol/kg at 298.1510.005 K. Based on these measurements he calculated the osmotic coefficient of aqueous NiCb solutions. He also evaluated other data from the literature and finally presented a set of smoothed osmotic coefficient and activity of water data (see Table IV in original reference). 

Estimate Pitzer s electrolyte activity coefficient model by minimizing the objective function given by Equation 15.1 and using the following osmotic coefficient data from Rard (1992) given in Table 15.5. First, use the data for molalities less than 3 mol/kg and then all the data together. Compare your estimated values with those reported by Rard (1992). Use a constant value for in Equation 15.1. 

The activity coefficient of the solvent remains close to unity up to quite high electrolyte concentrations e.g. the activity coefficient for water in an aqueous solution of 2 m KC1 at 25°C equals y0x = 1.004, while the value for potassium chloride in this solution is y tX = 0.614, indicating a quite large deviation from the ideal behaviour. Thus, the activity coefficient of the solvent is not a suitable characteristic of the real behaviour of solutions of electrolytes. If the deviation from ideal behaviour is to be expressed in terms of quantities connected with the solvent, then the osmotic coefficient is employed. The osmotic pressure of the system is denoted as jz and the hypothetical osmotic pressure of a solution with the same composition that would behave ideally as jt. The equations for the osmotic pressures jt and jt are obtained from the equilibrium condition of the pure solvent and of the solution. Under equilibrium conditions the chemical potential of the pure solvent, which is equal to the standard chemical potential at the pressure p, is equal to the chemical potential of the solvent in the solution under the osmotic pressure jt,... 

For a solution of a single electrolyte, the relationship between the mean activity coefficient and the osmotic coefficient is given by the equation... 

The ideality of the solvent in aqueous electrolyte solutions is commonly tabulated in terms of the osmotic coefficient 0 (e.g., Pitzer and Brewer, 1961, p. 321 Denbigh, 1971, p. 288), which assumes a value of unity in an ideal dilute solution under standard conditions. By analogy to a solution of a single salt, the water activity can be determined from the osmotic coefficient and the stoichiometric ionic strength Is according to,... 

Helgeson, H. C., D. H. Kirkham and G. C. Flowers, 1981, Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high temperatures and pressures, IV. Calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to 600 °C and 5 kB. American Journal of Science 281, 1249-1516. 

A wide variety of data for mean ionic activity coefficients, osmotic coefficients, vapor pressure depression, and vapor-liquid equilibrium of binary and ternary electrolyte systems have been correlated successfully by the local composition model. Some results are shown in Table 1 to Table 10 and Figure 3 to Figure 7. In each case, the chemical equilibrium between the species has been ignored. That is, complete dissociation of strong electrolytes has been assumed. This assumption is not required by the local composition model but has been made here in order to simplify the systems treated. 

Pitzer, K. S. and Guillermo Mayorga, "Thermodynamics of Electrolytes. II. Activity and Osmotic Coefficients for Strong Electrolytes with One or Both Ions Univalent," J. Phys. Chem., 1973, 77, 2300. 

As is often the case, after the intense activity of the 1920 s, the investigation of aqueous electrolytes proceeded at a more relaxed pace. But careful and systematic experimental research continued in this area and was summarized by Harned and Owen (2) and by Robinson and Stokes (3) in their excellent monographs. The latter volume contains in the appendix a comprehensive set of tables of the osmotic and activity coefficients of the common inorganic solutes at 25°C and at concentrations up to 6 M in most cases. 

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. 

In addition to the activity and osmotic coefficients at room temperature, the first temperature derivatives and the related enthalpy of dilution data were considered for over 100 electrolytes (26, 29). The data for electrolytes at higher temperatures become progressively more sparse. Quite a few solutes have been measured up to about 50°C (and down to 0°C). Also, over this range, the equations using just first temperature derivatives have some validity for rough estimates in other cases. But the effects of the second derivative (or the heat capacity) on activity coefficients at higher temperatures is very substantial. 

The working equations for osmotic and activity coefficients, derived from equation (3) are given as equations (4) and (5), respectively. The various secondary relationships are defined in several additional equations stated and briefly described thereafter. Additional details and derivations of equations for the entropy, the heat capacity, and other related functions can be found in various published papers (11, 20, 23-29, 32-34). 

Goldberg, R. N. Nuttall, R. L. "Evaluated Activity and Osmotic Coefficients for Aqueous Solutions The Alkaline Earth Metal Halides" J. Phys. Chem. Ref. Data, 1978, 7,... 

Staples, B. R. "Activity and Osmotic Coefficients of Aqueous Sulfuric Acid" J. Phys. Chan. Ref. Data, in press. 

Staples, B. R. Nuttall, R. L. "Computer Programs for the Evaluation of Activity and Osmotic Coefficients" Nat. 

Critical evaluations of activity and osmotic coefficient data were undertaken early in the 1930-1940 period by Harned and Owen (1958) and by Robinson and Stokes, (1965). Wu and Hamer (1968) evaluated activity and osmotic coefficient data for a series of electrolytes but their work on polyvalent electrolytes was not completed. Their work on the 1 1 electrolytes was published in 1972. The evaluation of polyvalent electrolyte data has been continuing in the Electrolyte Data Center at the National Bureau of Standards, and this paper will summarize the methods used in evaluating data for over 100 aqueous polyvalent electrolytes. 

Pitzer et al (1972, 1973, 1974, 1975, 1976) have proposed a set of equations based on the general behavior of classes of electrolytes. Pitzer (1973) writes equations for the excess Gibbs energy, AGex, the osmotic coefficient activity coefficient Y+ for single unassociated electrolytes as... 

Most determinations of activity and osmotic coefficients of an electrolyte solution are based on these experimental techniques ... 

Activity and osmotic coefficient data derived from ten experimental methods have been critically evaluated and correlating equations have been formulated for more than 100 aqueous polyvalent electrolyte systems at 298 K. Evaluations for the major reference solutions KC1 and NaCl (Hamer and Wu, 1972), and CaCl (Staples and Nuttall, 1977) have been published that for (Staples,... 

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. 

In addition, the critical evaluation of enthalpies of dilution and solution, as well as evaluations of heat capacities have been initiated. These evaluations will allow calculations and correlations of activity and osmotic coefficients as a function of temperature and composition. 

The techniques used in the critical evaluation and correlation of thermodynamic properties of aqueous polyvalent electrolytes are described. The Electrolyte Data Center is engaged in the correlation of activity and osmotic coefficients, enthalpies of dilution and solution, heat capacities, and ionic equilibrium constants for aqueous salt solutions. 

Goldberg, R. N., B. R. Staples, R. L. Nuttall and R. Arbuckle, (1977). "A Bibliography of Sources of Experimental Data Leading to Activity or Osmotic Coefficients for Polyvalent Electrolytes in Aqueous Solution", Nat. Bur. Stand. (U.S.) Spec. Publication 485, U.S. Gov t. Printing Office, Washington, D.C. 

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