Harned rule

These relationships are termed the Harned rules and have been verified experimentally up to high overall molality values (e.g. for a mixture of HC1 and KC1 up to 2 mol-kg-1). If this linear relationship between the logarithm of the activity coefficient of one electrolyte and the molality of the second electrolyte in a mixture with constant overall molality is not fulfilled, then a further term is added, including the square of the appropriate molality ... 

Systematizing these results, we see that both in Fig. 72 and in Fig. 73, if we follow tho succession of curves from top to bottom, we go from ions of dissimilar character to ions of similar character in Fig. 73 we go down to Li+ and (Oil)", both strong order-producing ions, while in Fig. 72 we go down to Cs+ and Br", both strong order-destroying ions. If the same rule—from dissimilar character downward to similar character— is to be applied to the rubidium and cesium halides, the order I, Br, Cl, F, will clearly have to be reversed, in order that Rbl and Csl shall be the lowest in each case. It has been known for several years that such an inversion exists. Table 40, compiled by Robinson and Harned, shows the order observed in the whole set of iodides, bromides, and chlorides. It will be seen that, for RbCl, RbBr, and Rbl, and likewise for CsCl, CsBr, and Rbl, the observed order is opposite to that found for the other alkali halides. Hitherto this inversion has been regarded as mysterious. But it falls in line with the facts depicted in Fig. 72,... 

Bronsted-Guggenheim approach to multicomponent equations are only exact if pR,x is constant, the basic equations are reasonable exact if 3r,x is a slowly varying function of concentration and Harned s Rule (7) for mixed electrolytes through an empirical correlation does show such of an effect. 

A similar condition applies to log y+q. One can derive Harned s rule from this expression although it can also be done from the ion pairing model (2). The equations work well for I up to u.l m and permit the calculation of y+b an< Y+c a mixture of the two electrolytes from data obtained in single electrolyte solutions. The method suffers, however, because the B coefficients are not allowed to vary with the Ionic strength. 

Activity coefficients in concentrated solutions are often described using Harned s rule (l ). This rule states that for a ternary solution at constant total molality the logarithm of the activity coefficient of each electrolyte is proportional to the molality of the other electrolyte. The expressions for the activity coefficients are written ... 

For dilute solutions, Equations 4 and 5 reduce to the Bronsted-Guggenheim equations, and the parameters a23 and cu2 can be expressed in terms of the interaction parameters of tne Bronsted-Guggenheim theory. For concentrated solutions, Harned s rule is a simple empirical extension of the Brb nsted-Guggenheim theory. Thus, 1t 1s surprising how well the rule describes activity coefficients 1n highly concentrated solutions. 

The parameter cu3 changes significantly with temperature and molality, however becomes independent of molality at high molalities. On the other hand, a3 varies in a complicated manner with temperature and molality. The results of Figure 2 show that the parameters of Harned s rule cannot be reliably... 

Figure 2. Interaction parameters for Harned s Rule as a function of temperature...

The success of Harned s rule for ternary solutions is largely fortuitous, and the rule has no theoretical basis to expect that it would be useful for solutions containing more than two electrolytes. Furthermore, for high concentrations of several electrolytes, activity coefficients such as Y3(g are hypothetical. There are, unfortunately, few experimental data available to test Harned s rule for concentrated solutions of three or more electrolytes. 

The Bronsted-Guggenheim equations provide a highly satisfactory description of the activity coefficients in dilute solutions however, their empirical extension to concentrated solutions (Harned s rule) introduces several serious problems. 

Equations 11 and 12 were fit to the experimental activity coefficients of HC1 and NaCl as described by Harned s rule. 

Figure 2 was used to calculate the parameters for Harned s rule from 10 to 40°C and for total molalities 0.2 to approximately 6. 

For the ternary solution, the Gibbs-Duhem equation can be easily integrated to calculate the activity coefficient of water when the expressions for the activity coefficients of the electrolytes are written at constant molality. For Harned s rule, integration of the Gibbs-Duhem equation gives the activity of water as ... 

It is important to be able to estimate the values of y for aqueous electrolytes in their mixtnres, say B and C. An expression that has been found to be valid under wide conditions is Harned s rule ... 

As far as mixed strong electrolyte solutions are concerned, Harned s rule [38] holds. At constant ionic strength, the activity coefficient of one electrolyte (A) in the mixture is a function of the fractional ionic strength (y I) of the other electrolyte (B) ... 

The standard deviations of the fit shown in Table I, as well as the smooth plot of an as a function of m in Figure 1, were used as criteria for testing the validity of Harned s Rule. Hamed s Rule is a good description of the data for m = 0.05, 0.1, 0.25, 0.5, and perhaps (as a borderline case, based on the value of ai2) at m = 1.0 mol kg-1. The nonlinear form of Equation 6 was used at m = 1.0 and 1.5 mol kg"1, where the greatest deviations from linearity were thought to occur. 

This simple relationship was derived before as equation 5.24, and was first used by Bauman and Eichorn in 1947 to predict selectivity sequences for simple monovalent cations from mean ionic activity coefficient data for pure aqueous electrolyte solutions containing a common anion. The inaccessibility of resin phase activity coefficients to direct measurement always remains a problem with thermodynamic equilibrium treatments. Therefore Glueckauf and others developed weight swelling and isopiestic water vapour sorption techniques to determine osmotic coefficients of pure salt forms of a resin, from which the mean ionic activity coefficients of mixed resinates could be computed using a modified form of Harned s Rule. Such studies predicted selectivity coefficient values which were in fair agreement with experiment and also demonstrated the fixed ion of the resin to be osmotically inactive. 

Mean sodium chloride activity coefficients have been determined [41] with a sodium ion-selective electrode and silver-silver chloride reference electrode system in mixed sodium chloride-calcium chloride solutions within the range of sodium chloride and calcium chloride levels (0.05-0.5 mol dm ) encountered in extracellular fluids. These show that at constant ionic strength, log /Naci varies linearly with the ionic strength of calcium chloride in the mixture in accordance with Harned s rule [45] ... 

However, since this work by Robinson and Harned, no other convincing paper appeared that could prove or — on the contrary — discard this assumption. An alternative, and as we will see, more realistic idea was forwarded by Lyklema in 2003. He applied the simple rule of thumb like seeks like inferred from specific ion effects at solid surfaces to qualitatively explain why the mean activity coefficients of Csl are lower than those of Lil at concentrations of 1M. This is a simple consequence of the more sophisticated model of matching water affinities proposed by Collins, as will be discussed in subsequent chapters. 

---