Precision measurements of the colligative properties of solutions of strong electrolytes and of sea and brackish water: Theoretical calculation of the osmotic coefficient

Abstract

The osmotic coefficients of sea and brackish water are determined by means of cryoscopic measurements. From these data the “best values” of the individual ionic radii and the dependence of the dielectric constant on concentration are calculated using a two-parametric minimum-square-method of best fitting by applying the Debye-Hückel theory according to Bonino, with the effect of the “higher terms” of Gronwall and La Mer. The method is valid for recalculating experimental cryoscopies with an error of a few thousandths of a degree over the whole concentration interval of sea and brackish water, which represents its dilutions. These techniques may also be applied to the prediction of other thermodynamic properties of sea water, and above all, of brackish water, which today are assuming great importance in less expensive desalting methods such as reverse osmosis and electrodialysis. In order o make theoretical calculations on the thermodynamic properties of sea and brackish water, it is necessary to refer to the theory of concentrated solutions of strong electrolytes. In this field there have been a great number of studies, some quite recent, but it cannot be said that the problem has been definitely resolved, even though the most modern theoretical formulations, based on Mayer's theory of “ionic clusters” 1 and on the “hyperreticulated chains” equation 2, may hopefully lead to an interpretation more closely bound to the microscopic properties of the ions and of the solvent. In any case, every theory regarding solutions of strong electrolytes amplifies the original formulation of Debye and Hückel ( 3) and in fact is compared to it as a first test of validity for dilute solutions of salts of the NaCl or KCl type. In this theory, an “ionic radius” averaged from all ions present, is introduced for concentrations which are not greatly diluted, and at higher concentrations it becomes necessary to consider even the short range forces and introduce a relative parameter. The first treatment in this sense was carried out by Hückel ( 4) on the basis of dielectric saturation, conglobating such interactions toward the lower powers of l/r. There have also been attempts ( 5) to conglobate toward infinite powers, that is, toward rigid exclusion forces, but only the cluster theory has met the problem satisfactorily. At any rate, even the original theory of Hückel succeeds well in rendering experimental results up to fairly high concentrations for solutions of simple electrolytes, so long as a more ample meaning than the initial one is given to the variation of the dielectric constant. Bonino ( 6) has interpreted this variation as the “efficacious dielectric constant”, finding an exponential law D = D oe -μ (1) with D o as the dielectric constant of the solvent and μ proportional to an appropriate power of the concentration (5/4 in the applications made): in this way absurd negative values of D are avoided and trends closer to the experimental ones are obtained for macroscopic dielectric constants ( 7). An attempt was also made to extend the theory to individual ionic radii or averaged on the dimensions of the second ion only, which approaches the first. In fact, a definitive theory should contemplate the two-by-two minimum distances between all the ion couples for these “maximum approach radii”. In any case, the introduction of Bonino's individual ionic radii also allows better separation of the characteristic behaviours of single ions in different solutions, than the mere average radius, and above all aids in the search for correlations with other physical quantities. In this way linear relations with the crystallographic-radius/polarizability ratio were found both for cations ( 8) and for anions ( 9). This formulation has made possible the determination of these ion radii from experimental data by means of bi-parametric optimization with electronic computers. This therefore requires the most precise and consistent experiments possible for the whole concentration field and on complete series of electrolytes. For this purpose, high precision cryoscopies represent the best experiments, but another series of apparatus for ebullioscopy ( 10), vapour tension and osmotic pressure in order to obtain precise thermodynamic data comparable at different temperatures are also being set up. From the cryoscopies of alkali chlorides, ionic radii and variation constants of D were found ( 11) for wide concentration intervals (the smallest from 0.02 to 1 mol/l) with average disparities between those calculated and those found of one thousandth of a degree, starting from the hypothesis of the equality of the ionic radii of the chlorine anion and the potassium cation, which was discussed in the paper mentioned. This work is being continued and extended to other halides, bivalent ions and different solvents. The theoretical extension to the cluster theory is also being planned.