Electrolyte osmotic coefficient

Activity coefficients of H4[SiWi204o] obtained in 0.0004 to 0.04 M concentrations in queous solutions have also been reported11S. The results obtained agree with the Debye-Huckel theory for 1—4 electrolytes. Osmotic coefficients of 12-tungsto-silicic acid have also been reported116. ... 

The solutions to this approximation are obtained numerically. Fast Fourier transfonn methods and a refomuilation of the FINC (and other integral equation approximations) in tenns of the screened Coulomb potential by Allnatt [M are especially useful in the numerical solution. Figure A2.3.12 compares the osmotic coefficient of a 1-1 RPM electrolyte at 25°C with each of the available Monte Carlo calculations of Card and Valleau [ ]. 

Figure A2.3.12 The osmotic coefficient of a 1-1 RPM electrolyte compared with the Monte Carlo results of...

Figure A2.3.14 Osmotic coefficients for 1-1, 2-1 and 3-1 RPM electrolytes according to the MS and HNC approximations.

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between ([ly and ((ij is a measure of the accuracy of the approximation. The osmotic coefficients calculated via the energy equation in the MS approximation are comparable in accuracy to the HNC approximation for low valence electrolytes. Figure A2.3.15 shows deviations from the Debye-Htickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

Figure A2.3.16. Theoretical HNC osmotic coefTicients for a range of ion size parameters in the primitive model compared with experimental data for the osmotic coefficients of several 1-1 electrolytes at 25°C. The curves are labelled according to the assumed value of a+- = r+ + r-...

Figure A2.3.17 Theoretical (HNC) calculations of the osmotic coefficients for the square well model of an electrolyte compared with experimental data for aqueous solutions at 25°C. The parameters for this model are a = r (Pauling)+ r (Pauling), d = d = 0 and d as indicated in the figure.

In principle, simulation teclmiques can be used, and Monte Carlo simulations of the primitive model of electrolyte solutions have appeared since the 1960s. Results for the osmotic coefficients are given for comparison in table A2.4.4 together with results from the MSA, PY and HNC approaches. The primitive model is clearly deficient for values of r. close to the closest distance of approach of the ions. Many years ago, Gurney [H] noted that when two ions are close enough together for their solvation sheaths to overlap, some solvent molecules become freed from ionic attraction and are effectively returned to the bulk [12]. 

Equation (7.45) is a limiting law expression for 7 , the activity coefficient of the solute. Debye-Htickel theory can also be used to obtain limiting-law expressions for the activity a of the solvent. This is usually done by expressing a in terms of the practical osmotic coefficient

electrolyte solute, it is defined in a general way as... 

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 osmotic pressure of an electrolyte solution jt can be considered as the ideal osmotic pressure jt decreased by the pressure jrel resulting from electric cohesion between ions. The work connected with a change in the concentration of the solution is n dV = jt dV — jrel dV. The electric part of this work is then JteldV = dWcl, and thus jzc] = (dWei/dV)T,n. The osmotic coefficient 0 is given by the ratio jt/jt, from which it follows that... 

Thermodynamic methods also measure the activity coefficient of the solvent (it should be recalled that the activity coefficient of the solvent is directly related to the osmotic coefficient—Eq. 1.1.19). As the activities of the components of a solution are related by the Gibbs-Duhem equation, the measured activity coefficient of the solvent can readily be used to calculate the activity coefficient of a dissolved electrolyte. 

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. 

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. 

Sodium chloride has been studied much more thoroughly at high temperature than any other electrolyte. The osmotic coefficient measurements of Liu and Lindsay (30) and various types of measurements of Federov and associates (31) are particularly noteworthy. 

Arbuckle, R. "A Bibliography of Sources of Experimental Data Leading to Activity or Osmotic Coefficients for Polyvalent Electrolytes in Aqueous Solution" Nat. Bur. Stand. Special Publ. No. 485, July 1979. 

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,... 

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., (1980a). "Evaluated Activity and Osmotic Coefficients for Aqueous Electrolyte Solutions Thirty-Seven Unibivalent Systems," manuscript in preparation. 

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. 

Wu, Y. C. and W. J. Hamer, (1969). "Osmotic Coefficients and Mean Activity Coefficients of a Series of Uni-Bivalent and Bi-Univalent Electrolytes in Aqueous Solutions at 25°C," National Bureau of Standards Report (NBSIR) 10052, Part XIV, 83 p. 

The osmotic coefficients calculated from Eq. (9) can be brought into good agreement with solution data up to about 1M for aqueous solutions of alkali (26) and alkaline earth halides, (30) tetraalkyl ammonium halides, T3l) mixed electrolytes, where the Harned coefficients are measured, (32) and electrolyte-non electrolyte mixtures, where Setchenow coefficients are measured. 

ACTIVITY, ACTIVITY COEFFICIENTS, AND OSMOTIC COEFFICIENTS OF STRONG ELECTROLYTES... 

It can be observed that g is the ratio between the observed osmotic pressure and the osmotic pressure that would be observed for a completely dissociated electrolyte that follows Henry s law [see Equation (15.47)], hence the name, osmotic coefficient. A similar result can be obtained for the boiling point elevation, the freezing point depression, and the vapor pressure lowering. 

The activity of water is related formally to the molality of the electrolyte by means of the osmotic coefficient, (p, of the solution ... 

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