Ionic mean diameter

The derivation of the equations of the Debye-Huckel theory did not require differentiation between a solution of a single electrolyte and an electrolyte mixture provided that the limiting law approximation Eq. (1.3.24), was used, which does not contain any specific ionic parameter. If, however, approximation (1.3.29) is to be used, containing the effective ionic diameter ay it must be recalled that this quantity was introduced as the minimal mean distance of approach of both positive and negative ions to the central ion. Thus, this quantity a is in a certain sense an average of effects of all the ions but, at the same time, a characteristic value for the given central... 

Solubilities are calculated from a variety of sources compiled by the author. The mean ionic diameter is calculated by Eq. (6.13) using molar volumes in CRC Handbook of Chemistry and Physics. The value of a for OCPp is calculated from the crystallographic data on OCP in Brown et al. (1962). 

It will be observed from Table XXXV that at ordinary temperatures the value of B with water as solvent is approximately 0.33 X 10 for most electrolytes the mean ionic diameter a is about 3 to 4 X 10 cm. (see Table XXXVI), and hence aB does not differ greatly from unity. A reasonably satisfactory and simple approximation of equation (61) is therefore... 

TABLE XXXVI. MEAN EFFECTIVE IONIC DIAMETERS... 

In the application of equation (69) an arbitrary value of a is chosen so as to give calculated activity coeflScients which agree with those derived by direct experiment the proper choice of a is made by a process of trial and error until a value is found that is satisfactory over a range of concentrations. There is no doubt that the Gronwall-LaMer-Sandved extension represents an important advance over the simple Debye-Hilckel treatment, for it frequently leads to more reasonable values of the mean ionic diameter. The validity of equation (69) has been tested by a variety of activity measurements and the results have been found satisfactory were it not for the tedious nature of the calculations it would probably be more widely used. 

The values of Q(b) as defined above have been tabulated for various values of b from 1 to 80, and so by means of equation (76) it is possible to estimate the extent of association of a uni-univalent electrolyte consisting of ions of any required mean diameter a, at a concentration c in a medium of dielectric constant D, It will be seen from equation (76) that in general d increases as b increases, i.e., d increases as the mean diameter a of the ions and the dielectric constant of the solvent decrease. The values for the fraction of association of a uni-univalent electrolyte in water at 18 have been calculated by Bjerrum for various concentrations for four assumed ionic diameters the results are recorded in Table XXXVII. The extent of association is seen to increase markedly with decceasing ionic diameter and increasing concentration. The values are... 

According to the simple calculations on page 156 the dielectric constant necessary for the solvent in which a uni-univalent electrol3rte whose mean ionic diameter is 6.4 X 10 cm. should be dissolved in order that there may be no appreciable association is 2.79 X 10- /6.4 X IQ-, i.e., 42. 

Utilize the results of the preceding problem, together vnth the known values of A and B for water, to calculate approximate activity coefficients for uni-uni, uni-bi, and bi-bivalent electrolytes in water and in ethyl alcohol, at ionic strengths 0.1 and 0.01, at 25 . The mean ionic diameter may be taken as 3A in each case. 

Calculate the activity coefficients of the silver iodate in the various solutions plot the values of — log / against Vv to see how far the results agree with the Debye-Huckel limiting law. Determine the mean ionic diameter required to account for the deviations from the law at appreciable concentrations. 

By means of the value of K obtained in the preceding problem, calculate the mean ionic diameter, a, of hydrochloric acid in the given solvent. For this purpose, use equation (79) and the tabulation of Q h) given by Fuoss and Kraus, J, Am, Chem, Soc, 55, 1019 (1933). 

Compare the mean ionic activity coefficient of a 0.1 molal solution of (i) a uni-univalent, (ii) a bi-bivalent, electrolyte in water and methanol as solvents respectively, at 25 C. The mean ionic diameter d may be taken as 3A in each case. 

Utilize the mean ionic activity coefficients for solutions up to 0.05 molal in Table XXXIII to determine by a graphical procedure the mean ionic diameter d of hydrochloric acid. 

Estimation of this quantity requires that only one adjustable parameter be specified, namely the mean ionic diameter a. 

Since simple ions have mean ionic diameters of the order 0.3-0.6 nm, it is evident that deviations from the Debye-Htickel expression due to ion association are likely to occur for charge types of 2-1 or greater. These deviations are then handled by Bjerrum s ion-pair... 

The dimension of the ionic strength in this equation is mol dm . The dimension of d is nm. For the mean ionic activity the following approximation with a mean ionic diameter d+ can be used... 

During the last decade, the statistical thermodynamics of electrolytes have been continuously developed. The MSA theory (Mean Spherical Approximation) can yield analytical expressions for parameters which have a certain physical meaning (e.g., ionic diameter). To maintain the advantages of the NRTL electrolyte model and overcome its difficulties, the MSA theory has been successfully combined with the NRTL equation by Kunz and his co-workers (25). 

The exact calculation equations are given in [25], where it has also been proved that the Gibbs-Duhem equation is fulfilled. As well, NRTL parameters have been fitted up to molalities of 30mol/kg for a number of systems. Together with the ionic diameters, they are listed in [25]. Osmotic and mean ionic activity coefficients could be reproduced in an excellent way for a number of systems. Furthermore, the parameters fitted to binary systems have been successfully applied to ternary systems, that is, one salt in a binary solvent mixture, which always causes problems with the Electrolyte NRTL model [25]. 

It is possible to take into account the short range ion-ion interaction effect on the volumetric properties of electrolytes by resorting to integral equation theories, as the mean spherical approximation (MSA). The MSA model renders an analytical solution (Blum, 1975) for the umestricted primitive model of electrolytes (ions of different sizes immersed in a continuous solvent). Thus, the excess volume can be described in terms of an electrostatic contribution given by the MSA expression (Corti, 1997) and a hard sphere contribution obtained form the excess pressure of a hard sphere mixture (Mansoori et al, 1971). The only parameters of the model are the ionic diameters and numerical densities. 

Let the mean ionic diameter be a and assume that no other ion can come into the space inside the radius ajl, then from the electroneutrality principle we obtain... 

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