Debye-Hiickel expression

At moderate ionic strengths a considerable improvement is effected by subtracting a term bl from the Debye-Hiickel expression b is an adjustable parameter which is 0.2 for water at 25°C. Table 8.4 gives the values of the ionic activity coefficients (for Zi from 1 to 6) with d taken to be 4.6A. 

From the Debye-Hiickel expressions for lny , one can derive equations to calculate other thermodynamic properties. For example L2, the relative partial molar enthalpy,q and V2, the partial molar volume are related to j by the equations... 

In the tables we find i Ag+/Ag = 0.7996 V and 2 Agci/Ag = 0.2223 V. From the above it is clear that primarily the silver-silver chloride electrode functions as a pAg electrode, i.e., it measures oAg+ at an ionic strength above 0.01 (cf., extended Debye-Hiickel expressions) the calculation of [Ag+ ] becomes more difficult, and even more so for [Cl ], where the solubility product value is also involved. 

If limiting forms of the Debye-Hiickel expression for activity coefficients are used, this equation becomes... 

At ionic strengths near 0.01, it is convenient to use the more complete form of the Debye-Hiickel expression ... 

The Debye-Hiickel term, which is the dominant term in the expression for the activity coefficients in dilute solution, accounts for electrostatic, nonspecific long-range interactions. At higher concentrations, short-range, nonelectrostatic interactions have to be taken into account. This is usually done by adding ionic strength dependent terms to the Debye-Hiickel expression. This method was first outlined by Bronsted [5,6], and elaborated... 

The other method uses an extended Debye-Hiickel expression where the activity coefficients of reactants and products depend only on the... 

As a check on the consistency of our mathematics, it is profitable to verify that Equation (63) reduces to Equation (37) in the limit of low potentials. Expanding the exponentials in T and truncating the series so that only one term survives in both the numerator and denominator results in the Debye-Hiickel expression, Equation (37). 

As a reminder that the level of approximation in Equation (21) is the same as that of the Debye-Hiickel limiting law, the following example continues from this last result to the Debye-Hiickel expression for the mean ionic activity coefficient of an electrolyte solution. 

Debye-Hiickel expression for ionic activity coefficients 539... 

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). 

At this stage, the Debye-Hiickel expressions (3.60) for and/ can be introduced... 

The Debye-Hiickel expression as given by equation (3.36) is valid only in dilute solution (/ < 0.02 mol kg ). At higher concentrations a modified expression has been proposed ... 

This increase in y values with ionic strength can be modeled by adding positive terms to some form of the extended Debye-Hiickel expression for log y. Simple models using this approach have been proposed by Hiickel (see Harned and Owen 1958), and more recently by others, including... 

In this case the experimental concentration equilibrium constant should be independent of ionic strength. However, deviations from non-ideality not accounted for by the Debye-Hiickel expression could be taken care of by plotting log lo conc vs. I. This is likely to be necessary at high ionic strengths. 

At the surface of a large ion, the electric field will be considerably smaller than that at the surface of a small ion. Consequently, non-ideality is to be expected to be less pronounced for large ions than for small ions and the denominator in the Debye-Hiickel expression may be regarded as a reflection of this. 

One way of coping with observed deviations from the behaviour predicted by the Debye-Hilckel expression is not to do any further theoretical calculations, but to work at empirical extensions to the Debye-Hiickel expression to take it to higher concentrations. These methods assume that the Debye-Hiickel theory is valid at low concentrations. 

But take note The Debye-Hiickel expression is not that which is used for unassociated electrolytes. There is one very important difference ... 

His treatment extended the range of the theory to ionic strengths of 0.1 molal, and showed quite unambiguously that values predicted by the numerical integration for 1-1 electrolytes fit almost exactly those predicted by the Debye-Hiickel theory. It follows that the Debye-Hiickel expression is an excellent base-line for describing the properties of 1-1 electrolyte solutions. 

The same analysis showed that the predicted y values for 1-2 and 2-1 electrolytes fit the Debye-Hiickel equation within 2%. So again the Debye-Hiickel expression can be taken as a reasonable base-line for these electrolytes. 

However, for higher charge types matters become more difficult. When 2-2 electrolytes were considered, the predicted values and the Debye-Hiickel values for differed by up to 20%. Values for 1-3 and 3-1 electrolytes are expected to lie between these values and those for 1-2 electrolytes. The inevitable conclusion is that the Debye-Hiickel expression cannot be used as a base-line for these high charge types. 

The numerical integration can, therefore, give a possible base-line for the treatment of electrolyte solutions. Provided that association is taken care of by the Bjerrum treatment, the Debye-Hiickel expression is also a good base-line for the behaviour of free unassociated ions in solution, provided that the appropriate value of q appears in the denominator rather than the conventional a. 

This treatment is only slightly more physically realistic than the 1927 equation outlined above. The improvement hes in the use of the non-approximate form of the truncated Debye-Hiickel expressions in the derivation of the electrophoretic effect, i.e. they used ... 

The Debye-Hiickel expression is used as logjo/ Note the distance of closest approach is now R not a. 

The Debye-Hiickel expression for p does not predict, even in its nonlinear form, a saturation effect for very large values oizexpjkTas would be desirable. Our equations show that for zetpjkT co, f and d are of the same order and thus in the limit pjzen+ —— 1. We encounter then the effect of saturation. In working with a linear approximation one cannot expect to find a saturation effect. 

More detailed models for the ionic strength dependence of E° have been developed that include higher order terms in the Debye-Hiickel expression for y and the actual positions of the charges in the proteins (66, 67). Considering the complexity of proteins and the relative simplicity of the models, remarkable agreement has been achieved with the observed ionic strength-dependent effects. 

The kinetics of formation and dissociation of Ni(SCN)2(aq) have been studied in water and in several organic solvents, using the pressure-jump and shock wave relaxation technique at 20°C. The concentration of Ni(SCN)2(aq) ranged between 0.001 and 0.1 M. In water, only the formation of the monothiocyanato complex was observed. No background electrolyte was used, and the activity coefficients were calculated by an extended Debye-Hiickel expression. Although, this activity model is not compatible with the SIT, the ionic strengths were low. Therefore, the reported result was corrected to 25°C and the resulting value was accepted with an increased uncertainty (log, = (1.79 0.10)). 

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