Debye-Hiickel applications

It is important to realise that whilst complete dissociation occurs with strong electrolytes in aqueous solution, this does not mean that the effective concentrations of the ions are identical with their molar concentrations in any solution of the electrolyte if this were the case the variation of the osmotic properties of the solution with dilution could not be accounted for. The variation of colligative, e.g. osmotic, properties with dilution is ascribed to changes in the activity of the ions these are dependent upon the electrical forces between the ions. Expressions for the variations of the activity or of related quantities, applicable to dilute solutions, have also been deduced by the Debye-Hiickel theory. Further consideration of the concept of activity follows in Section 2.5. 

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... 

Another arena for the application of stochastic frictional approaches is the influence of ionic atmosphere relaxation on the rates of reactions in electrolyte solutions [19], To gain perspective on this, we first recall the early and often quoted triumph of TST for the prediction of salt effects, in connection with Debye-Hiickel theory, for reaction rates In kTST varies linearly with the square root of the solution ionic strength I, with a sign depending on whether the charge distribution of the transition state is stabilized or destabilized by the ionic atmosphere compared to the reactants. 

Among other applications of electrolyte solution theory to defect problems should be mentioned the application of the Debye-Hiickel activity coefficients by Harvey32 to impurity ionization problems in elemental semiconductors. Recent reviews by Anderson7 and by Lawson45 emphasizing the importance of Debye-Hiickel effects in oxide semiconductors and in doped silver halides, respectively, and the book by Kroger41 contain accounts of other applications to defect problems. However, additional quantum-mechanical problems arise in the treatment of semiconductor systems and we shall not mention them further, although the studies described below are relevant to them in certain aspects. 

The Debye-Hiickel approximation is strictly applicable only in the case of low potentials. Nevertheless, there are several reasons why the significance of Equation (37) should be fully appreciated ... 

Debye-Hiickel equation Gives the activity coefficient, y, as a function of ionic strength, p,. The extended Debye-Hiickel equation, applicable to ionic strengths up to about 0.1 M, is log y = — 0.51z2 VpJ/... 

Both, the Gouy-Chapman and Debye-Hiickel are continuum theories. They treat the solvent as a continuous medium with a certain dielectric constant, but they ignore the molecular nature of the liquid. Also the ions are not treated as individual point charges, but as a continuous charge distribution. For many applications this is sufficient and the predictions of continuum theory agree with experimental results. At the end of this chapter we discuss the limitations and problems of the continuum model. 

Potentiometry has found extensive application over the past half-century as a means to evaluate various thermodynamic parameters. Although this is not the major application of the technique today, it still provides one of the most convenient and reliable approaches to the evaluation of thermodynamic quantities. In particular, the activity coefficients of electroactive species can be evaluated directly through the use of the Nemst equation (for species that give a reversible electrochemical response). Thus, if an electrochemical system is used without a junction potential and with a reference electrode that has a well-established potential, then potentiometric measurement of the constituent species at a known concentration provides a direct measure of its activity. This provides a direct means for evaluation of the activity coefficient (assuming that the standard potential is known accurately for the constituent half-reaction). If the standard half-reaction potential is not available, it must be evaluated under conditions where the activity coefficient can be determined by the Debye-Hiickel equation. 

The expression for the excess Gibbs energy is built up from the usual NRTL equation normalized by infinite dilution activity coefficients, the Pitzer-Debye-Hiickel expression and the Born equation. The first expression is used to represent the local interactions, whereas the second describes the contribution of the long-range ion-ion interactions. The Bom equation accounts for the Gibbs energy of the transfer of ionic species from the infinite dilution state in a mixed-solvent to a similar state in the aqueous phase [38, 39], In order to become applicable to reactive absorption, the Electrolyte NRTL model must be extended to multicomponent systems. The model parameters include pure component dielectric constants of non-aqueous solvents, Born radii of ionic species and NRTL interaction parameters (molecule-molecule, molecule-electrolyte and electrolyte-electrolyte pairs). 

The activity a2 of an electrolyte can be derived from the difference in behavior of real solutions and ideal solutions. For this purpose measurements are made of electromotive forces of cells, depression of freezing points, elevation of boiling points, solubility of electrolytes in mixed solutions and other characteristic properties of solutions. From the value of a2 thus determined the mean activity a+ is calculated using the equation (V-38) whereupon by application of the analytical concentration the activity coefficient is finally determined. The activity coefficients for sufficiently diluted solutions can also be calculated directly on the basis of the Debye-Hiickel theory, which will bo explained later on. 

For soil solutions, the Davies equation, which is a modification of the Debye-Hiickel equation, is commonly used, and is applicable to solutions up to approximately I — 0.7 mol (see also Section 3.2.1) ... 

One could use the Debye-Hiickel ionic-atmosphere model to study how ions of opposite charges attract each other, (a) Derive the radial distribution of cation ( +) and anion (nj concentration, respectively, around a central positive ion in a dilute aqueous solution of 1 1 electrolyte, (b) Plot these distributions and compare this model with Bjerrum s model ofion association. Comment on the applicability of this model in the study of ion association behavior, (c) Using the data in Table 3.2, compute the cation/anion concentrations at Debye-HUckel reciprocal lengths for NaCl concentrations of lO and 10 mol dm", respectively. Explain the applicability of the expressions derived. (Xu)... 

The application of Blum s theory to experiment is unexpectedly impressive it can even represent conductance up to 1 mol dm . Figure 4.96 shows experimental data and both theories—Blum s theory and the Debye-Hiickel-Onsager first approximation. What is so remarkable is that the Blum equations are able to show excellent agreement with experiment without taking into account the solvated state of the ion, as in Lee and Wheaton s model. However, it is noteworthy that Blum stops his comparison with experimental data at 1.0 M. 

Figure 4.101 shows the variation of the equivalent conductivity versus concentration for a number of alkali sulfocyanates in a methanol solvent. The agreement with the theoretical predictions demonstrates the applicability of the Debye-Hiickel-Onsager equation up to at least 2 x 10" mol dm . ... 

The Debye-Hiickel Theory.—The first successful attempt to account for the departure of electrolytes from ideal behavior was made by Milner (1912), but his treatment was very complicated the ideas were essentially the same as those which were developed in a more elegant manner by Debye and Hiickel. The fundamental ideas have already been given on page 81 in connection with the theory of electrolytic conductance, and the application of the Debye-Hiickel theory to the problem of activity coefficients will be considered here. ... 

Quantitative Tests of the Debye-Hiickel Limiting Equation.—Although the Debye-Hiickel equations are generally considered as applying to solutions of strong electrolytes, it is important to emphasize that they are by no means restricted to such solutions they are of general applicability and the only point that must be noted is that in the calculation of the ionic strength the actual ionic concentrations must be employed. 

If the solution is sufficiently dilute for the Debye-Hiickel limiting law to be applicable, it follows from equation (54), assuming the ions M" " and A to be univalent, for simplicity, that... 

The rate laws are complex, both for the forward reactions and, where studied, usually also for the back reactions. Some of the complexity is likely to be due to use of inadequate corrections for the activity coefficients just as some of the conflict is due to use of differing corrections. A more modern examination might clarify these points, since much of the work was published early in the period of application of the Debye-Hiickel theory. Correlation of the forms of the rate... 

Problem 7.3(c) (Worked Example) The electrostatic contribution in Eq. (A7-1) is valid in the limit of a weak electrostatic force, so that the Debye-Hiickel theory applies. Show that this limit is applicable under the conditions described in part (a). 

As has already been stressed, the Debye-Hiickel relation, even, in the form (4.2.3a), is of only limited applicability. There have been many attempts to extend the range over which it remains useful one of the most widely used versions reads... 

Equation 6-33 suggests that extrapolation of equilibrium constants to infinite dilution is done appropriately by plotting log vs Yh- For example. Fig. 6-1 shows plots of pfC a for dissociation of H2PO4, AMP , and ADP , and ATP vs Yh- The variation of pK with yjl at low concentrations (Eq. 6-35) is derived by application of the Debye-Hiickel equation (Eq. 6-33) ... 

A solution to these difficulties is a blend of the chemical picture in which clustered ion configurations are described by the mass action law, while the interactions between the various entities are treated by methods applying the high-temperature approximations of the /-functions, e.g. by the MSA. The Debye-Hiickel (DH) theory [26], although derived from classical electrostatics, is also a high-temperature approximation, whose range of applicability can be extended by supplementing a mass action law for ion pair formation [27],... 

The activity coefficients given by the Debye-Hiickel treatment presumably represent deviations from the dilute solution behavior, i.e., from Henry s law, and are consequently based on the standard state which makes the activity of an ion equal to its mole fraction at infinite dilution ( 37b, III B). In the experimental determination of activity coefficients, however, it is almost invariably the practice to take the activity as equal to the molarity or the molality at infinite dilution. The requisite corrections can be made by means of equation (39.13), but this is unnecessary, for in solutions that are sufficiently dilute for the Debye-Hackel limiting law to be applicable, the difference between the various activity coefficients is negligible. The equations derived above may thus be regarded as being independent of the standard state chosen for the ions, provided only that the activity coefficients are defined as being unity at infinite dilution. 

The particular application of the Debye-Hiickel equation to be described here refers to the determination of the true equilibrium constant K from values of the equilibrium function K at several ionic strengths the necessary data for weak acids and bases can often be obtained from conductance measurements. If the solution of the electrolyte MA is sufficiently dilute for the limiting law to be applicable, it follows from equation (40.12), for the activity coe cient of a single ionic species, that... 

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