Debye-Huckel equation limiting form

DERIVATION OF THE LIMITING FORM FOR THE DEBYE-HUCKEL EQUATION... 

The experimental methods of these authors are described in Chapter 22. For very dilute solutions the second term in the denominator of equation (25) becomes negligible and the Debye-Huckel equation approaches its limiting form,... 

Derivation of the Limiting Form for the Debye-Huckel Equation... 

Limiting and extended forms of the Debye-Huckel equation... 

Figure 7.8 Comparison of experimental ln7 for 1 1, 2 1, and 2 2 electrolytes. The symbols indicate the experimental results, with representing HC1 (z+ = 1, z = — 1) representing SrC ( + = 2, r = — 1) and A representing ZnS04 (z+ = 2, z = -2). The lines are the Debye-Huckel predictions, with the solid line giving the prediction for (z+ = 1, z = -1) the dashed line for (z+ = 2, r = -1) and the dashed-dotted line for (z+= 2, z =-2). In (a), In 7- calculated from the limiting law [equation (7.45)] is shown graphed against I 2. In (b). In 7- calculated from the extended form [equation (7.43)] is shown graphed against 7m2.

In single electrolyte solutions the Debye-Huckel limiting equation works up to 10-3 m. Its well known extended form used at m > 10"3 is... 

The various forms of equation (40.15), referred to as the Debye-HUckel limiting law, express the variation of the mean ionic activity coefficient of a solute with the ionic strength of the medium. It is called the limiting law because the approximations and assumptions made in its derivation are strictly applicable only at infinite dilution. The Debye-Hfickel equation thus represents the behavior to which a solution of an electrolyte should approach as its concentration is diminished. 

It would seem that substitution of E and Q values would allow the computation of the standard redox potential for the couple, However, a problem arises because the calculation of Q requires not only knowledge of the concentrations of the species involved in the cell reaction but also of their activity coefficients. These coefficients are not usually available, so the calculation cannot be directly completed. However, at very low concentrations, the Debye-HUcke limiting law for the coefficients holds. The procedure then is to. substitute the Debye-Hiickel law for the activity coefficients into the specific form of the Nemst equation for the cell under investigation and carefully examine the equation to determine what kind of plot to make of the E[ b ) data so that extrapolation of the plot to zero concentration, where the Debye-Huckel law is valid, gives a plot intercept that equals See Section 7.8 for the details of this procedure and an example for which the relevant graph involves a plot of + (2RT/F) In b against... 

Mitchell et al. (1977) showed that the zeroth- and second-moment conditions follow from the asymptotic form of the direct-correlation function [Eq. (33)]. The MS, HNC, RHNC, and Debye Huckel limiting law approximations satisfy both these conditions the Percus-Yevick equation does not obey the second-moment condition and is less useful for electrolytes than it is for uncharged systems. 

It is possible to understand why solutions of electrolytes do not behave in an ideal manner in terms of both the coulombic attraction on ions which serves to constrain their movement and the thermal agitation which counteracts this restraint. Debye and Huckel developed a theory in which electrostatic forces shaping the behavior of the ions in solution as well as their finite radii formed a basis from which expressions for the activity coefficient of an ion could be derived. One of the simpler usable equations they developed, referred to as the Extended Limiting Law, gives the activity coefficient, y, of an ion i, having a charge Zj in a solution of ionic strength I. 

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