The Debye-Htickel theory

See any standard textbook on physical chemistry for more information on the Debye-Htickel theory and its application to solution equilibrium... 

Kirkwood, J. G. Poirier, J. C., The statistical mechanical basis of the Debye-Htickel theory of strong electrolytes, J. Phys. Chem. 1954, 86, 591-596... 

According to the Debye-Htickel theory, in the limit of the infinitely dilute solution, individual-ion activity coefficients are given by the equation... 

Recall from transition state theory that the rate of a reaction depends on kg (the catalytic rate constant at infinite dilution in the given solvent), the activity of the reactants, and the activity of the activated complex. If one or more of the reactants is a charged species, then the activity coefficient of any ion can be expressed in terms of the Debye-Htickel theory. The latter treats the behavior of dilute solutions of ions in terms of electrical charge, the distance of closest approach of another ion, ionic strength, absolute temperature, as well as other constants that are characteristic of each solvent. If any other factor alters the effect of ionic strength on reaction rates, then one must look beyond Debye-Hiickel theory for an appropriate treatment. 

However, experimentally observed y =f c) functions usually first decrease, pass through a minimum, and then increase at high concentrations. In order to explain the increase of y with concentration, Stokes and Robinson modified the Debye-Htickel theory by introducing the effect of ion-solvent interaction. Thus, the modified theory is based on ion-ion and ion-solvent interactions. The modified theory is in good agreement with experimental results, up to an ionic strength of about 4, as shown in Figure 5.14. 

Reaction (15.37) is usually studied in dilute solution (ionic strength <0.1). If, as in our examples, the ligand is a nonelectrolyte, then it is a reasonable approximation to assume that tl 1. It is also not unreasonable to expect 7 mlj 7m v+ in these dilute solutions, since ions with the same charge behave in a somewhat similar manner, as suggested by the Debye-Htickel theory. Hence, /7 1 and K = Kc. Because we will not be overly concerned with quantitative results of high accuracy in this discussion, we will assume this approximation is sufficient and use K for Kc. It is not absolutely necessary that we do so, however, since corrections can be made for /7. 

Likewise, yAB also cannot be measured experimentally, although, like Qa and Qb, 7a and yB can be measured, and at first sight the conversion given above may seem to give little improvement. However, for ion-ion and ion-molecule reactions, the Debye-Htickel theory, see Equation (7.8), can calculate the activity coefficient for any charged species and convert Equation (7.17) into a useful form. For other reactions the approach is only qualitative, but for them the effects of non-ideality are much smaller. 

In solvents of high dielectric constant such as water, the deviations from ideality caused by ion-ion interactions are reasonably small below concentrations of 0.1 ilf for 1 1 electrolytes and can be treated adequately by means of the Debye-Htickel theory. For polyvalent electrolytes or for higher concentrations of 1 1 electrolytes, or for either in solvents of lower dielectric constant, the situation is less fortunate. The deviations from ideality can become rather large, and there is no adequate theory for either correlating them or predicting them. 

Now, as explained in Section 3.3.2, the principal objective of the Debye-Htickel theory is to calculate the time-averaged spatial distribution of the excess charge density around a reference ion. How is this objective attained ... 

The ion size parameter a has done part of the job of extending the range of concentration in which the Debye-Htickel theory of ionic clouds agrees with experiment. Has it done the whole job One must start looking for discrepancies between theory and fact and for the less satisfactory features of the model. 

This does not mean that the Debye-Htickel theory gives the right answer when there is ion-pair formation. The extent of ion-pair formation decides the value of the concentration to be used in the ionic-cloud model. By removing a fraction 0 of the total number of ions, only a fraction 1 - 0 of the ions remain for the Debye-Hiickel treatment, which interests itself only in the free charges. Thus, the Debye-Htickel expression for the activity coefficient [Eq. (3.120)] is valid for the free ions, with two important modifications (1) Instead of there being a concentration c of ions, there is only (1 - 0)c the remainder Oc is not reckoned with owing to association. (2) The distance of closest approach of free ions is q and not a. These modifications yield... 

Why should one go to all this trouble and do all these integrations if there are other, less complex methods available to theorize about ionic solutions The reason is that the correlation function method is open-ended. The equations by which one goes from the gs to properties are not under suspicion. There are no model assumptions in the experimental determination of the g s. This contrasts with the Debye-Htickel theory (limited by the absence of repulsive forces), with Mayer s theory (no misty closure procedures), and even with MD (with its pair potential used as approximations to reality). The correlation function approach can be also used to test any theory in the future because all theories can be made to give g(r) and thereafter, as shown, the properties of ionic solutions. 

The McMillan-Mayer theory is an alternative to the Debye-Htickel theory. It is called the virial coefficient approach and its equations bear some conceptual resemblance to the virial equation of state for gases. The key contribution in... 

It should be noted that the Debye-Htickel theory yields true, and not stoichiometric, activity coefficients ( 39a), since it is the behavior of the ions only, and not of the whole solute, which is taken into consideration. For strong electrolytes dissociation is virtually complete at all dilutions for which the limiting law may be expected to hold for such solutes, therefore, the distinction between true and stoichiometric activity coefficients may be ignored. 

By means of the Debye-Htickel theory, calculate the activity coefficients of silver iodate in the various potassium nitrate solutions mentioned in Exercise 14, Chapter XVI. Compare the values with those derived from the observed solubilities. 

Lange, Z. pky . Chem., A186,147 (1934) ]. Plot 1 — against Vy and determine the limiting slope compare the value with that given by the Debye-Htickel theory. (The constant A is 0.488 for v/ater at 0 C.)... 

Zi is the algebraic charge on each type, i, of ion in solution, i.e. it includes the sign of the charge. The Debye-Htickel theory (see Sections 10.7, 10.10.1 and 10.10.2) allows a calculation of the activity coefficient, y, for any ion from the known ionic strength. 

The Debye-Htickel theory deals with departures from ideality in electrolyte solutions. The main experimental evidence for this non-ideality is that ... 

The fundamental concepts of the Debye-Htickel theory are also important in the description of the theoretical study of the passage of a current through a solution. Section 10.4 is therefore relevant here. There are other aspects of physics which are pertinent to the experimental study of conductance in solution. These will be discussed below. 

These standard states are used, e.g. in dilute solutions of salt in water, where the Debye-Htickel theory exists for estimation [154] of these Yp- Modern versions of this theory use molalities (number of mols in mass unit, namely 1 kg, of solvent) instead of concentrations with quite analogical, but different in principle, formulation of the standard state at low concentrations the differences, e.g. in are usually negligible. 

The functions f are the electrostatic Debye-Htickel terms and the 5 and C coefficients are electrolyte-specific fitting parameters. Factors involving the charge numbers and stoichiometric coefficients have to be included for electrolyte types other than 1 1. Two universal constants, b =. 2 and a = 2.0 are employed in the full expressions for B and C as well as the solvent- and temperature-dependent A

Debye-Htickel theory. For details the series of papers by Pitzer and coworkers should be consulted, starting with (Pitzer and Mayoraga 1973). 

Macromolecules behave always as real solutions because of the size dependence of the mixing parameters which is approximated by the Flory-Huggins equation, described in Chap. 7. A correction for the mixing of small molecules in the presence of ions is given by the Debye-Htickel theory, not discussed in this book. 

In the broad field of physical chemistry, the Boltzmann distribution law is fundamental to the derivation of the Debye-Htickel theory of electrolyte solutions. In the more narrow arena of interfacial and colloid science, it is applied to the determination of the ionic atmosphere around charged interfaces. In that context, the charge cloud is more commonly referred to as the electrical double layer (EDL). The concept is illustrated schematically (Fig. 5.2) for the situation in which a particle possesses an evenly distributed charge that is just balanced by the total opposite charge, the counterions in the electrical double layer. 

Equation (29.29) was derived by Langelier [3] on the assumption that K and K2 are based on concentrations (moles/liter) rather than on activities. For example, referring to (29.24), if K, is the true activity product, then Ks = Ksjl, where y refers to the mean ion activity coefficient for CaCOs. The activity coefficient was approximated by Langelier using the Debye-Htickel theory, -logy = where p is the ionic strength and z is the valence. Hence,... 

Modified integral equation approaches have yet to enter into industrial process simulations and data descriptions, in contrast to the empirical extensions of the Debye-Htickel theory (see Section III.D.3). 

This approach is satisfactory for mixtures of nonelectrolytes. The situation is a little more complex for mixtures of electrolytes because of the long range of the Coulomb potential. However, each ion in the mixture tends to be surrounded by ions of opposite charge, which causes the potential to decay exponentially with a decay factor, k, the Debye parameter, which is proportional to the square root of the product of the density and T. As a result, an appropriate expansion parameter for electrolytes is k, or which is different from nonelectrolytes, where T is the expansion parameter. These ideas become quantitative in the Debye-Htickel theory. 

The Debye-Htickel theory is a cornerstone of electrolyte theory. It is always used in extrapolating data to infinite dilution, and must be embedded in any generalized treatment of activity coefficients as a function of concentration, as it is in the Pitzer equations. However, at concentrations beyond the validity of the limiting law (Equation 15.26), all attempts at predicting electrolyte behavior at higher concentrations are more or less empirical. 

Soon after the appearance of the Debye-Htickel theory, it was found that the theory did not work well for many electrolytes. In 1926, Bjerrum suggested that electrostatic attraction between pairs of oppositely charged ions resulted in ion-pairs, which would account for the lower measured activity coefficients in these solutions. The problem then, as now, was how best to define and measure the extent of ion-pairing. How close must two ions be to become an ion-pair What is the difference between an ion-pair and a complex Is it really necessary to know this to use thermodynamics Helgeson (1981) notes that The distinction between ion association and short-range ionic interaction is nebulous at best. ... 

Lastly, the uniformity of ionic strength provided to a solution by the presence of ample supporting electrolyte limits effects due to the non-ideality of the solution. According to the Debye-Htickel theory, the presence of electrostatic interactions between ions causes solution non-ideality because these forces are on average stabihsing. Therefore, activity, the quantity appearing in the Nernst equation, differs from concentration by a factor known as the activity coefficient, y, in a maimer which for a dilute (<0.01 M) solution is given by a simplified formula ... 

The electrostatic correlations of the ions in the electrolyte solution contribute to the Helmholtz free energy of the system, depending on the Debye length and ion size. Based on the Debye-Htickel theory for the restricted primitive model (Figure 3.3a), the result (McQuarrie 1976) is... 

The Poisson-Boltzmann formalism is used to compute the electric potentials and charge distributions. In general, this requires numerical work. The Debye-Htickel theory assumes a weak electrical energy compared to the thermal energy allowing closed form analytical formulas for various quantities of interest in electrolyte solutions. 

X is the charge density parameter [cf. (30)]. When the charge density is small (i.e., the distance b between two dissociable groups is large so that X < 1) the polyelectrolyte is completely dissociated. Thus, the first part on the right-hand side of (58) [i.e., pv(l — zci z) ] is the number of moles of dissociated polymer groups times the charge density parameter. Parameter k is the inverse of the radius of the ionic cloud in the aqueous solution, as introduced in the Debye-Htickel theory ... 

Strong and long-range Coulombic forces acting between ions are primarily responsible for the departures from ideality (the activity coefficients are lowered) and dominate all other contributions. The effect has been evaluated in the Debye-Htickel theory and there exist several equations, which are useful in estimating the mean activity coefficient [68, 69]. The latter is related to the ionic strength of the solution ... 

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