The Osmotic Coefficient and Activity Coefficient

In addition to knowing the TP dependence of equilibrium constants (Eqs. 2.25 and 2.28), we must also know the T-P dependence of solute activity coefficients and the osmotic coefficient of the solution. A theoretical model, such as Pitzer s approach, is necessary for this purpose because activity coefficients and the osmotic coefficient must be defined at finite concentrations and not simply for the infinitely dilute state, which suffices for equilibrium constants (Eqs. 2.25 and 2.28). 

The chemical equilibrium model, FREZCHEM, requires calculation of solute activity coefficients (7) and the osmotic coefficient ((f)) in concentrated solutions (Chap. 3). In this work, the Pitzer approach is used to calculate these quantities. 

The excess Gibbs energy (Gex) is defined as the difference between the actual Gibbs energy (G) and that of an ideal solution (Gld) 

The three terms in these equations reading from left to right are related to 7, a , and to of Eq. 2.13, respectively. The activity coefficient and the osmotic coefficient measure the degree to which solute concentrations and the activity of water (aw) depart from ideal solutions, respectively. For ideal solutions, a = to and 7 = 1.0 (Eq. 2.13) or Gex = 0 (Eq. 2.32). Similarly, aw = 1.0 for an ideal solution. In the real world, solutions are rarely ideal, except in the infinitely dilute case we therefore need a model for calculating and (f [= f(aw)]. An early model based on statistical mechanics was developed by Debye and Hiickel (1923). Their equations are 

The osmotic coefficient ( / ), in turn, is related to the through the relation 

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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 ... 

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],... 

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). 

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,... 

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... 

This expression is analogous to Eiq. (2.3), in that (1 — (p) expresses the contribution of the solvent and In y+ that of the electrolyte to the excess Gibbs energy of the solution. The calculation of the mean ionic activity coefficient of an electrolyte in solution is required for its activity and the effects of the latter in solvent extraction systems to be estimated. The osmotic coefficient or the activity of the water is also an important quantity related to the ability of the solution to dissolve other electrolytes and nonelectrolytes. 

The activity of water is obtained by inserting Eq. (6.12) into Eq. (6.11). It should be mentioned that in mixed electrolytes with several components at high concentrations, it is necessary to use Pitzer s equation to calculate the activity of water. On the other hand, uhjO is near constant (and = 1) in most experimental studies of equilibria in dilute aqueous solutions, where an ionic medium is used in large excess with respect to the reactants. The ionic medium electrolyte thus determines the osmotic coefficient of the solvent. 

Solutions are usually classified as nonelectrolyte or electrolyte depending upon whether one or more of the components dissociates in the mixture. The two types of solutions are often treated differently. In electrolyte solutions properties like the activity coefficients and the osmotic coefficients are emphasized, with the dilute solution standard state chosen for the solute.c With nonelectrolyte solutions we often choose a Raoult s law standard state for both components, and we are more interested in the changes in the thermodynamic properties with mixing, AmjxZ. In this chapter, we will restrict our discussion to nonelectrolyte mixtures and use the change AmjxZ to help us understand the nature of the interactions that are occurring in the mixture. In the next chapter, we will describe the properties of electrolyte solutions. 

Figure 18.5 gives the osmotic coefficient and the activity for NaCl(aq) as a function of temperature and molality as predicted by the equation of Silvester and Pitzer. Note that both and 7 increase as T changes from 273.15 K to 323.15 K, and then decrease in a regular manner with increasing temperature. At T= 523.15 K, 7 has become a small value at high m. 

Figure 18.5 Graph of (a) the osmotic coefficient and (b) the activity coefficient for NaCl(aqueous) at p = 0.1 MPa as a function of temperature. The curves were obtained by using temperature-dependent coefficients in Pitzer s equations. The dotted line is for r=273.15 K and the dashed line is for T = 298.15 K. The solid lines are for T= 323.15, 373.15, 423.15, 473.15, 523.15, and 573.15 K, with both and 7 decreasing with increasing temperature.

Another function, the osmotic coefficient, has been used in place of the excess chemical potential or the activity coefficient. It is a multiplicative factor rather than additive, and is defined in terms of the chemical potential of the solvent. Two such functions are used, one based on molalities and the other on molarities. The first is defined, except for its absolute value, by... 

Equation 17 relates the excess free energy of a mixed electrolyte solution to the osmotic coefficient of the solution and the activity coefficients of its component ions. 

The Activity Coefficient of Any Component Salt in the Aqueous Electrolyte Mixture. Analogous to the derivation of the osmotic coefficient expression, equation 17 was differentiated with respect to m to give a relationship between and the activity coefficient, and then that relationship was evaluated by substituting in the value of obtained by differentiation of G from equation 14 with respect to m, where m is now the result of a change in m, m, added to the initial m, m as caused by a... 

Using the osmotic coefficient and molality of the isopiestic NaCl solution, the activity coefficient for NaCl is given by using equations 20 and 21 to give ... 

The Osmotic Coefficient.—Instead of calculating activity coefficients from freezing-point and other so-called osmotic measurements, the data may be used directly to test the validity of the Debye-Hiickel treatment. If 6 is the depression of the freezing point of a solution of molality m of an electrolyte which dissociates into v ions, and X is the molal freezing-point depression, viz., 1.858° for water, a quantity , called the osmotic coefficient, may be defined by the expression... 

Pitzer and co-workers (1973, 1974) have proposed a more detailed, but at the same time more complex, approach. Whitfield (1973, 1975) has applied these equations to seawater and has shown that this model gives good agreement with available experimental data for the osmotic coefficient and for the mean ion activity coefficient of the major electrolyte components. The results obtained yield numerical results similar to the predictions of the ion association model (see Table A6.2). 

The lysozyme solubilities in aqueous solutions of sodium chloride are predicted for pH=4.5 and pH=6.5. In these predictions only the values of the preferential binding parameter were used and no additional (or adjustable) parameters were involved. The results are presented in Figs. 1 and 2 and the experimental preferential binding parameters used are listed in Table 2. The solubilities at pH=6.5 were predicted from the preferential binding parameter determined at pH=7.0 because the values for pH=6.5 were not available. The concentration dependence of the water activity in solutions of sodium chloride was obtained from Eq. (18) using an accurate semiempirical equation for the osmotic coefficient [37]. 

Instead of characterizing deviations from ideality of the solvent by its activity coefficient y, it is often advantageous to introduce the osmotic coefficient ( ) of Bjerrum and Guggenheim, and to write the chemical potential in the form... 

From (27.76) we can easily derive the mean ionic activity coefficient, and from (27.66) the osmotic coefficient in a very dilute solution. 

The 9mOsmol/kg added to the above equation represents the contribution of other osmoticaUy active substances in plasma, such as K", Ca " ", and proteins, and 1.86 is two times the osmotic coefficient of Na, reflecting the contributions of both Na and CT. The reference interval for plasma osmolality is 275 to 300mOsmol/kg. Comparison of measured osmolality with calculated osmolality can help identify the presence of an osmolal gap, which can be important in determining the presence of exogenous osmotic substances. Comparison of calculated and measured osmolalities can also confirm or rule out suspected pseudohyponatremia caused by the previously discussed electrolyte exclusion effect. 

Although, in principle, equation (39.49) provides a method for determining activity coefficients, the details require consideration. The osmotic coefficients, in the first place, are determined from vapor pressure measurements. The activity ai of the solvent in a given solution is equal to/i// , by equation (31.5), or, approximately, to pi/p d. equation (38.1)], where pi is the vapor pressure of the solvent over the solution and is that of the pure solvent at the same temperature. Hence, by equation (39.46),... 

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