Limiting laws, ionic strength

The Debye-Htickel limiting law predicts a square-root dependence on the ionic strength/= MTLcz of the logarithm of the mean activity coefficient (log y ), tire heat of dilution (E /VI) and the excess volume it is considered to be an exact expression for the behaviour of an electrolyte at infinite dilution. Some experimental results for the activity coefficients and heats of dilution are shown in figure A2.3.11 for aqueous solutions of NaCl and ZnSO at 25°C the results are typical of the observations for 1-1 (e.g.NaCl) and 2-2 (e.g. ZnSO ) aqueous electrolyte solutions at this temperature. 

At sufficiently low ionic strengths the activity coefficient of each electrolyte in a mixture is given by the Debye-Hiickel limiting law... 

It can be seen from Figure 7.8(b) that the curved lines predicted by the extended form of the Debye-Hiickel equation follow the experimental results to higher ionic strengths than do the limiting law expressions for the (1 1) and (2 1) electrolytes. However, for the (2 2) electrolyte, the prediction is still not very good even at the lowest measured molality.0... 

Experience shows that solutions of other electrolytes behave in a manner similar to the examples we have used. The conclusion we reach is that the Debye-Hiickel equation, even in the extended form, can be applied only at very low concentrations, especially for multivalent electrolytes. However, the behavior of the Debye-Hiickel equation as we approach the limit of zero ionic strength appears to give the correct limiting law behavior. As we have said earlier, one of the most useful applications of Debye-Hiickel theory is to... 

Intense ion-ion interactions which are characteristic of salt solutions occur in the concentrated aqueous solutions from which AB cements are prepared. As we have seen, in such solutions the simple Debye-Hiickel limiting law that describes the strength goes up so the repulsive force between the ions becomes increasingly important. This is taken account of in the full Debye-Hiickel equation by the inclusion of a parameter related to ionic size and hence distance of closest approach (Marcus, 1988). 

In fact, the symbol Ic should be used, as the molality ionic strength Im can be defined analogously in dilute aqueous solutions, however, values of c and m, and thus also Ic and Im, become identical.) Equation (1.1.21) was later derived theoretically and is called the Debye-Huckel limiting law. It will be discussed in greater detail in Section 1.3.1. 

Alternatively, we can calculate a value of y with the Debye-Htickel laws. There are two such laws the limiting and the simplified laws. Calculations with the limiting law are only valid at very low ionic strengths (i.e. 0 < I < 10-3 mol dm-3), which is very dilute. The limiting law is given by... 

At extremely low ionic strengths, the simplified law becomes the limiting law. This follows since the denominator I + bv7 tends to one as ionic strength I tends to zero, causing the numerator to become one. 

At low ionic strengths, the V7 term in the denominator becomes negligible, so (1 + i>V7) tends to unity, yielding the limiting Debye-Huckel law. 

According to Deybe-Huckel limiting law, the activity coefficient/of an ion is related to the ionic strength as... 

Knowing the ionic strength, we are now in a position to determine the mean ionic activity coefficient y by using the Debye-HCickel laws. There are two such laws, namely the limiting law and the extended law. 

Strategy. We will assume that the only solutes present are ZnS04 and H2SO4, and first calculate the ionic strength of the solution. (For the purposes of this calculation, we can safely assume that the amounts of zinc sulfate are negligible when compared with the amounts of sulfuric acid.) We will then calculate y for the zinc cation from I (and, for simplicity, obtain y from the Debye-Huckel limiting law). 

In solution thermodynamics, the concentration (C) of ions is replaced by their activity, a, where a = Cy and y is the activity coefficient that takes into account nonideal behavior due to ion-solvent and ion-ion interactions. The Debye-Hiickel limiting law predicts the relationship between the ionic strength of a solution and y for an ion of charge Z in dilute solutions ... 

B is a constant that depends on the properties of the solution, for example, on its dielectric constant, and on the temperature. For water at 25°C, B = 0.51 Ll/2 mol 1/2. The Debye-Hiickel limiting law applies only for solutions of low ionic strength, for example, below 0.01 M for 1 1 electrolytes, such as NaN03, and below 0.001 M for electrolytes of higher charge. 

Using the Debye-Hiickel limiting law for the relationship between the activity coefficients y and the ionic strength of the solution, one finds... 

For dilute solutions, the Debeye-Huckel law (log 7 = —0.5zf/°5) indicates that 7 will be a constant for a given ionic strength /. Therefore, the same quantity of inert electrolyte, called the support electrolyte, must be added to the sample and to the series of standards to increase the concentration of external ions and stabilise the ionic strength. This addition of ISAB (Ionic Strength Adjustment Buffer) is intended to limit variation in 7. Under these conditions, the measured difference in potential only depends on the concentration of the ion to be analysed and is given by equation (18.3). 

DEBYE-HOCKEL LIMITING LAW. The departure from ideal behavior in a given solvent is governed by the ionic strength of the medium and the valences of the ions of the electrolyte, but is independent of their chemical nature. For dilute solutions, the logarithm of the mean activity is proportional to the product of the cation valence, anion valence, and square root of ionic strength giving the equation... 

Since the plot is linear, the Debye-Hiickel limiting law for low ionic strengths is adequate. 

The neglect of activity coefficients y in equations (12) and (13)—which meant that they were strictly valid only in the limit of infinitely dilute solutions—is less serious in equation (14). The interionic mean activity coefficient y for the hydrogen ion and the anion will, at low ionic strengths, be a function of the nature of the solvent only to the extent that the coefficient A in the Debye-Hiickel limiting law for activity coefficients depends on the dielectric constant of the solvent. In view of the similar values of the dielectric constants of H20 and D20 (see p. 261), the resultant difference between activity coefficients in H20 and D20 solutions of the same low ionic strength should be small. (If both Aha and Kn are results extrapolated to zero ionic strength, the problem disappears in its entirety.) (See also Section IVC.)... 

Debye-Hiickel limiting law — The equation based on the - Debye-Hiickel theory providing mean activity coefficients / for ions of charge z+ and z- at - ionic strength I in dilute solutions lg/ = ().S09 z+ z /l, when mol and dm3 units are used. 

Fig. 3.54. Deviations from the Debye-Htickel limiting law (DHLL) for and of a 2 2 electrolyte for several theories. The ion-pairing cutoff distance d for the Bjerrum curve is 1.43 nm. / is the ionic strength. (Reprinted from J. C. Rasaiah, J. Chem. Phys. 56 3071, 1972.)...

Qualitative Verification of the Debye-Hiickel Equations.—The general agreement of the limiting law equation (54) with experiment is shown by the empirical conclusion of Lewis and Randall (p. 140) that the activity coefficient of an electrolyte is the same in all solutions of a given ionic strength. Apart from the valence of the ions constituting the particular electrolyte under consideration, the Debye-Hiickel limiting equation contains no reference to the specific properties of the salts that may be present in the solution. It is of interest to record that the... 

The experimentally determined activity coefficients, based on vapor pressure, freezing-point and electromotive force measurements, for a number of typical electrolytes of different valence types in aqueous solution at 25 , are represented in Fig. 49, in which the values of log / are plotted against the square-root of the ionic strength in these cases the solutions contained no other electrolyte than the one under consideration. Since the Debye-Htickel constant A for water at 25 is seen from Table XXXV to be 0.509, the limiting slopes of the plots in Fig. 49 should be equal to —0.509 the results to be expected theoretically, calculated in this manner, are shown by the dotted lines. It is evident that the experimental results approach the values required by the Debye-Hiickel limiting law as infinite dilution is attained. The influence of valence on the dependence of the activity coefficient on concentration is evidently in agreement with theoretical expectation. Another verification of the valence factor in the Debye-Hiickel equation will be given later (p. 177). 

---