Osmotic coefficient of an electrolyte

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

Formalism According to Pitzer. The most common method for the evaluation of the activity and osmotic coefficients of an electrolyte in a binary mixture of strong electrolytes with a common ion is by Scatchard s Equations (23), the McKay-Perring treatment (24), Mayers Equations... 

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... 

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 immediate importance of the practical osmotic coefficient of the solvent lies in its relationship to the mean activity coefficient of an electrolyte. From equation (39.37),... 

In principle the activity coefficients yb of solute substances B in a solution can be directly determined from the results of measurements at ven temperature of the pressure and the compositions of the liquid (or solid) solution and of the coexisting gas phase. In practice, this method fails unless the solutes have volatilities comparable with that of the solvent. The method therefore usually fails for electrolyte solutions, for which measurements of ye in practice, much more important than for nonelectrolyte solutions. Three practical methods are available. If the osmotic coefficient of the solvent has been measured over a sufficient range of molalities, the activity coefficients /b can be calculated the method is outlined below under the sub-heading Solvent. The ratio yj/ys of the activity coefficients of a solute B in two solutions, each saturated with respect to solid B in the same solvent but with different molalities of other solutes, is equal to the ratio m lm of the molalities (solubilities expressed as molalities) of B in the saturated solutions. If a justifiable extrapolation to Ssms 0 can be made, then the separate ys s can be found. The method is especially useful when B is a sparingly soluble salt and the solubility is measured in the presence of varying molalities of other more soluble salts. Finally, the activity coefficient of an electrolyte can sometimes be obtained from e.m.f. measurements on galvanic cells. The measurement of activity coefficients and analysis of the results both for solutions of a single electrolyte and for solutions of two or more electrolytes will be dealt with in a subsequent volume. Unfortunately, few activity coefficients have been measured in the usually multi-solute solutions relevant to chemical reactions in solution. 

Surface and size parameters are either available in the literature or are calculated following the proposals by Bondi [118]. The degree of counterimi dissociation in infinite dilution is estimated from experimental data for the limiting osmotic coefficient of an aqueous solution of the polyion. Following the ideas outlined in the description of the Pessoa and Maurer model above, one finds when the repeating unit is a 1 1 electrolyte ... 

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.

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

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. 

For any imaginary ideal solution of an electrolyte, at any given T and P, in which all activity and osmotic coefficients are unity, we can write for the chemical potential of a solute s. 

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... 

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. 

In the foregoing method the mean ionic activity coefficient of the solute has been calculated from actual vapor pressure data. If the osmotic coefficients for a reference substance are known over a range of concentrations, the activity coefficients of another electrolyte can be derived from isopiestic measurements, without actually determining the vapor pressures. If w, and p refer to an experimental electrolyte and wr, 0r and vr apply to a reference electrolyte which is isopiestic (isotonic) with the former, then by equation (39.46)... 

As already pointed out the apparent molecular weights of dissolved substances—and consequently the thermodynamic degree of ionisation or activity coefficient a in the case of an electrolyte—as determined by freezing point data are necessarily those which would be obtained from direct measurements of osmotic pressure or from emf measurements, since these different modes of measurement are related thermodynamically It will be recalled that the activity coefficient a for an ion is less than the y value over a wide range of concentration... 

If data are lacking on the system of interest, it is often a fair approximation (especially at low and moderate concentrations) to use a model-substance approach. The behavior of an electrolyte is assumed to be similar to that of a known electrolyte of the same charge type. For example, NaCl is a model substance for 1 1 salts. This approach is particularly useful in estimating the temperature dependence of activity and osmotic coefficients when these coefficients are known only at 25°C, the model-substance approach may be used to estimate the effect of temperature. 

It is known from physical chemistry that the equilibrium vapour pressure is smaller over solutions than over pure water. In the case of ideal solutions this vapour pressure decrease is proportional to x0, the mole fraction of the solvent (Raoult s law). If the solution is real, the interaction of solvent and solute molecules cannot be neglected. For this reason a correction factor has to be applied to calculate the vapour pressure lowering. This correction factor is the so-called osmotic coefficient of water (g ). We also have to take into account that the soluble substance dissociates into ions, forming an electrolyte. 

Values of osmotic coefficients for single electrolytes have been compiled by various authors, e.g., Robinson and Stokes [59ROB/STO]. The activity of water can also be calculated from the known activity coefficients of the dissolved species. In the presence of an ionic medium N, X, of a concentration much larger than those of the reacting ions, the osmotic coefficient can be calculated according to Eq.(B.l 1) (c/ Eqs. (23-39), (23-40) and (A4-2) in [61LEW/RAN]). 

Theoretical discussions of the surface tension increments try to explain the trends noted for the various ions. Aveyard and Saleem (1976) related the 577 of an electrolyte solution to the product of its molality and its osmotic coefficient (p ... 

Pitzer and co-workers have developed an ion interaction model and published a series of papers (Pitzer, 1973a-b, 1974a-b, 1975, 1977, 1995, 2000 Pabalan Pitzer, 1987) which gave a set of expressions for osmotic coefficients of the solution and mean activity coefficient of electrolytes in the solution. Expressions of the chemical equilibrium model for conventional single ion activity coefficients derived are more convenient to use in solubility calculations (Harvie Weare, 1980 Harvie et al.l984 Felmy Weare, 1986 Donad Kean, 1985). 

In a solution of an electrolyte the activity coefi cientf of the eolverU can be determined by measiirement of its partial pressure, or from the freezing-point depression or the osmotic pressure. The relevant equations are the same as those developed in 9 6. Provided that values of the activity or osmotic coefficient have been determined over a wide range of concentrations, including some solutions which are very dilute, it is possible to calculate the activity coefficient of the solute in some particular solution by application of the Gibbs-Duhem equation. This procedure, as applied to solutions of nonelectrolytes, was described in 9 7 and 9 8. 

The osmotic coefficient of a binary solution of an electrolyte is defined by... 

The osmotic coefficient of the solvent in an wig molal electrolyte solution is related to the activity of the solvent, (roughly, its vapor pressure in the solution divided by... 

Thus, the activity of water is a function of the osmotic coefficient of the solution, and the summation extends over all ions k with molality wjj- present in solution. In the presence of an ionic medium NX in dominant concentration, the equation can be simplified by neglecting the contributions of all minor species, that is, the reacting ions. For a 1 1 electrolyte of ionic strength, therefore, I ntjqx and Eq. (2.70) becomes... 

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... 

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