Debye-Htickel theory

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

For gases, pure solids, pure liquids, and nonionic solutes, activity coefficients are approximately unity under most reasonable experimental conditions. For reactions involving only these species, differences between activity and concentration are negligible. Activity coefficients for ionic solutes, however, depend on the ionic composition of the solution. It is possible, using the extended Debye-Htickel theory, to calculate activity coefficients using equation 6.50... 

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

Equation (7.45) is a limiting law expression for 7 , the activity coefficient of the solute. Debye-Htickel theory can also be used to obtain limiting-law expressions for the activity a of the solvent. This is usually done by expressing a in terms of the practical osmotic coefficient

electrolyte solute, it is defined in a general way as... 

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. 

The suspended particles are small compared with distances Z over which the potential

in potential from mean value, the ion concentrations deviate as in linearized Debye-Htickel theory ... 

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. 

In the primitive Debye-Htickel theory—one that did not allow for the size of ions—the value for the activity coefficient is given by Eq. (3.60). The corresponding equation in the MSA is... 

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... 

Figure 2.13 Calculated concenfratinns of monovalent cations and aninns as a function nf distance from a surface with a charge of —8 /rC/cm (5 x lO charges/m ), yielding a dimensionless surface potential of —5.21 (—133.9 mV). The bulk electrolyte concentration is 0.01 M, T = 298 K, and the solvent dielectric constant is 80, that of water. Tlie Debye-Htickel theory [Eqs. (2-49) and (2-50)] fails close to the surface. The exact result is given by Eqs. (2-46) and (2-41). 4/ is defined as exfs/kBT. (From Russel et al. 1989, with permission of Cambridge University Press.) ...

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.)... 

FIG. 18 Zeta potential = //(/, + cr/2) as a function of Manning parameter for the DNA-like models with 0.5 mol/L added 2 2 salt. The dotted line is the prediction of PB theory, the dash-dotted line is from bulk Debye—Htickel theory, and the dashed line is the result from a hypernetted chain calculation [36]. The solid line is a fit that merely serves to guide the eye. 

Debye-Htickel theory. This was taken over by Hasa-Ilavsky-Dusek (HID) theory [44] in which a modification of the elastic term was made by considering the influence of charges on the deformation of the network. Although HID theory has not yet been applied to account for the phase transition in ionic gels, Ilavsky et al. [45-48] as well as Konak and Bansil [49] performed several experiments to examine this theory in comparison with the swelling and elastic data for PMAAc gels and related copolymer gels. 

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. 

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