Debye-Hiickel theory parameter

Debye-Hiickel theory 333-50 in electrochemical cells 481-2, 488 and osmotic coefficient 345-8 parameters 342... 

Equation (7.44) is known as the third approximation of the Debye-Hiickel theory. Numerous attempts have been made to interpret it theoretically, hi these attempts, either individual simplifying assumptions that had been made in deriving the equations are dropped or additional factors are included. The inclusion of ionic solvation proved to be the most important point. In concentrated solutions, solvation leads to binding of a significant fraction of the solvent molecules. Hence, certain parameters may change when solvation is taken into account since solvation diminishes the number of free solvent molecules (not bonded to the ions). The influence of these and some other factors was analyzed in 1948 by Robert A. Robinson and Robert H. Stokes. 

Raji Heyrovska [18] has developed a model based on incomplete dissociation, Bjermm s theory of ion-pair formation, and hydration numbers that she has found fits the data for NaCl solutions from infinite dilution to saturation, as well as several other strong electrolytes. She describes the use of activity coefficients and extensions of the Debye-Hiickel theory as best-fitting parameters rather than as explaining the significance of the observed results. ... 

We shall assume that there are s different species of ions. Let = ezj be the charge on an ion of species j, and let rij be their number per cm3. Then the parameter k of the Debye-Hiickel theory is given by... 

To obtain the pH, it is necessary to evaluate the activity coefficient of the chloride ion. So the acidity function is determined for at least three different molalities mci of added alkali chloride. In a subsequent step, the value of the acidity function at zero chloride molality, lg(flHyci)°, is determined by linear extrapolation. The activity of chloride is immeasurable. The activity coefficient of the chloride ion at zero chloride molality, yci, is calculated using the Bates-Guggenheim convention (Eq. 5) which is based on the Debye-Hiickel theory. The convention assumes that the product of constant B and ion size parameter a are equal to 1.5 (kg mol1)1/2 in a temperature range 5 to 50 °C and in all selected buffers at low ionic strength (I < 0.1 mol kg-1). 

The most obvious drawback of the finite-ion-size version of the Debye-Hiickel theory lies in the fact that a is an adjustable parameter. When parameters that have to be taken from experiment enter a theory, they imply that the physical situation has been incompletely comprehended or is too complex to be mathematically analyzed. In contrast, the constants of the limiting law were calculated without recourse to experiment. 

Evidently there are factors at work in an electrolytic solution that have not yet been reckoned with, and the ion size parameter is being asked to include the effects of all these factors simultaneously, even though these other factors probably have little to do with the size of the ions and may vary with concentration. If this were so, the ion size parameter a, calculated back from experiment, would indeed have to vary with concentration. The problem therefore is What factors, forces, and interactions were neglected in the Debye-Hiickel theory of ionic clouds ... 

The electrostatic methods just discussed suitable for nonelectrolytic solvent. However, both the GB and Poisson approaches may be extended to salt solutions, the former by introducing a Debye-Huckel parameter and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation. The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Hiickel theory because the model includes the finite size of the solute molecules. 

Use the extended Debye-Hiickel theory to estimate the mean ionic activity coefficient for Na2S04 at concentrations of 0.01 and 0.1 M and 25°C assuming an ion size parameter of 400 pm. Also calculate the mean electrolyte activity and the electrolyte activity. 

The diffusion coefficient of Na+ is 1.34 x 10 m s at infinite dilution in water at 25°C. Estimate its value for a Na+ concentration of 0.1 M in a solution of the same ionic strength. Use the extended Debye-Hiickel theory to estimate the concentration dependence of the activity coefficient assuming that the ion size parameter is 400 pm. 

Assume that the coefficient Xe given in equation (6.7.46) can be estimated using the Debye-Hiickel theory with an ion size parameter equal to 0.42 nm. Calculate values for the phenomenological coefficients Zmm> using equations (6.7.48)-(6.7.50). 

The dominant parameter in the Debye-Hiickel theory is the ionic strength I, defined as... 

Debye Length A parameter in the Debye—Hiickel theory of electrolyte solutions, k-1. For aqueous solutions at 25 °C, k = 3.288y7 in reciprocal nanometers, where I is the ionic strength of the solution. The Debye length is also used in the DLVO theory, where it is referred to as the electric double-layer thickness. See also Electric Double-Layer Thickness. 

Thus at a distance 1 Jk the potential has dropped by a factor of (1/e). This distance may be used as a measure of the extension of the double layer and is often loosely called the thickness of the double layer. According to the theoretical equations it has the value /k = [ekT/e l.CizlY and is identical with the parameter introduced in the Debye-Hiickel theory of electrolytes in which /k is identified with the radius of the ionic atmosphere. Of particular importance in colloid science is the fact that the thickness of the... 

The first two terms are derived from Debye-Hiickel theory, and the third and fourth terms express short-range interactions (e.g., ion-molecular interactions). A( ) can be calculated as a function of temperature using the polynomial equation given by Clegg et al. (1994), which is based on the study of Archer and Wang (1990). Pitzer and Mayorga (1973) determined three parameters mx)... 

Nonelectrolyte G mcxlels only account for the short-range interaction among non-charged molecules (—One widely used G model is the Non-Random-Two-Liquid (NRTL) theory developed in 1968. To extend this to electrolyte solutions, it was combined with either the DH or the MSA theory to explicitly account for the Coulomb forces among the ions. Examples for electrolyte models are the electrolyte NRTL (eNRTL) [4] or the Pitzer model [5] which both include the Debye-Hiickel theory. Nasirzadeh et al. [6] used a MSA-NRTL model [7] (combination of NRTL with MSA) as well as an extended Pitzer model of Archer [8] which are excellent models for the description of activity coefficients in electrolyte solutions. Examples for electrolyte G models which were applied to solutions with more than one solvent or more than one solute are a modified Pitzer approach by Ye et al. [9] or the MSA-NRTL by Papaiconomou et al. [7]. However, both groups applied ternary mixture parameters to correlate activity coefficients. Salimi et al. [10] defined concentration-dependent and salt-dependent ion parameters which allows for correlations only but not for predictions or extrapolations. 

The equilibrium pdfs can be computed nowadays very accurately by theories such as HNC [22] and some of its improved versions [23], The MSA [24, 25] is the simplest theory that satisfies all of the above conditions. It is the Debye-Hiickel theory, but solved with the condition (5.11) for all pairs of ions. The final result introduces a new screening parameter F (intead of the Debye screening parameter /c) which is calculated from an algebraic equation [26, 27, 28]. It was found that the MSA is sufficiently accurate, in all of the studied cases. 

The Debye-Hiickel theory has achieved enormous success. It is considered among the greatest discoveries of the 20th century in the realm of physical chemistry. However, it is not fully satisfactory. It leads to difficulties in some cases. For example, the parameter a can be endowed with a value that is not that of a hydrated ion radius. It can sometimes be negative Debye himself said that the theory was awarded more success than it deserved. However, the Debye-Hiickel laws are now irreplaceable. As just one example, they justify extrapolation procedures to obtain thermodynamic equilibrium constants to null ionic strength. 

The distance of closest approach, a, attained from the Debye-Hiickel theory, is regarded as an adjustable parameter in several semiempirical equations for the activity coeiScients, has been estimated for a large number of electrolytes in aqueous solutions using data inRef [4] and Eq. (2. 1),... 

The ionic diameters used in the Debye— Hiickel theory have been described as, ... the effective diameter of the ion in solution. Since no independent method is available for evaluating aj this quantity is an empirical parameter, but the aj s obtained are of a magnitude for ion sizes. [26]. Values for these effective ionic diameters have been experimentally evaluated and can be found in the literature [26, 27]. 

FIGURE 2.6 Parameter space spanned by the surface density of electric charge on the particle (o, in units of e/rmd-) and the contact line radius (Tq = R sin 9y, in i,m) of a system of spherical particles at an air/water interface, assuming the Debye-Hiickel theory for water with a screening length of 1 pm. The solid line is the locus of values of Ep such that above that line the capillary attraction given by Equation 2.32 dominates over the electric repulsion. The dashed line is the locus Ep = 1 (the small-deformation approach corresponds formally to Ep < 1). (Reproduced from Dominguez, A. et al. /. Chem. Phys. 127, 204706, 2007. With permission.)... 

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