Debye-Hiickel theory activity coefficient

Debye-Hiickel theory The activity coefficient of an electrolyte depends markedly upon concentration. Jn dilute solutions, due to the Coulombic forces of attraction and repulsion, the ions tend to surround themselves with an atmosphere of oppositely charged ions. Debye and Hiickel showed that it was possible to explain the abnormal activity coefficients at least for very dilute solutions of electrolytes. 

Although it is not possible to measure an individual ionic activity coefficient,, it may be estimated from the following equation of the Debye-Hiickel theory ... 

In the foregoing derivations we have assumed that the true pH value would be invariant with temperature, which in fact is incorrect (cf., eqn. 2.58 of the Debye-Hiickel theory of the ion activity coefficient). Therefore, this contribution of the solution to the temperature dependence has still to be taken into account. Doing so by differentiating ET with respect to T at a variable pH we obtain in AE/dT the additional term (2.3026RT/F) dpH/dT, which if P (cf., eqn. 2.98) is neglected and when AE/dT = 0 for the whole system yields... 

Although the theory of solutions has been widely used in formulating problems of defects in solids the problems encountered differ in certain respects. The most obvious point is that defects are restricted to discrete lattice sites, whereas the ions in a solution can occupy any position in the fluid. Sometimes no allowance is made for this fact. For example, it has not been demonstrated that at very low concentrations, in the absence of ion-pair effects, the activity coefficients are identical with those of the Debye-Hiickel theory. It can be plausibly argued51 that at sufficiently low concentrations the effect of discreteness is likely to be negligible, but clearly in developing a theory for any but the lowest concentrations the effect should be investigated. A second point... 

The case of activity coefficients in solutions is easily but tediously implemented since well-constrained expressions exist, like those produced by the Debye-Hiickel theory for dilute solutions or the Pitzer expressions for concentrated solutions (brines). The interested reader may refer to Michard (1989) for a recent and still reasonably simple account. However simple to handle, activity coefficients introduce analytically cumbersome expressions incompatible with the size of a textbook. Real gas theory demands even more complicated developments. 

Edwards et al. (6) made the assumption that was equal to 4>pure a at the same pressure and temperature. Further theyused the virial equation, truncated after the second term to estimate pUre a These assumptions are satisfactory when the total pressure is low or when the mole fraction of the solute in the vapor phase is near unity. For the water, the assumption was made that <(>w, , aw and the exponential term were unity. These assumptions are valid when the solution consists mostly of water and the total pressure is low. The activity coefficient of the electrolyte was calculated using the extended Debye-Hiickel theory ... 

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. ... 

Dubye-Hiickel Theory of Activity Coefficient Point-Charge Model. The Debye-Hiickel theory of ion-ion interactions (Chapter 2) gives the following theoretical... 

Stokes-Robinson Modification of Debye-Hiickel Theory Effect of Ion-Solvent Interaction. Debye-Hiickel theory explains the activity and activity coefficient data on the basis of ion-ion interaction for dilute solution. According to Eqs. (5.29) and... 

Experimenters would do well to avoid any unnecessary changes in the ionic composition of reaction samples within a series of experiments. If possible, chose a standard set of reaction conditions, because one cannot readily correct data from one set of experimental conditions in any reliable manner that reveals the reactivity under a different set of conditions. Maintenance of ionic strength and solvent composition is desirable, and correction to constant ionic strength often effectively minimizes or ehminates electrostatic effects. Even so, remember that Debye-Hiickel theory only applies to reasonably dilute electrolyte solutions. Another important fact is that ion effects and solvent effects on the activity coefficients of polar transition states may be more significant than more modest effects on reactants. 

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. 

The activity coefficient of the anion A in dilute solutions can be calculated from Debye-Hiickel theory as follows ... 

Use these data and the Debye-Hiickel theory of electrolyte nonideality to criticize or defend the following proposition Indifferent electrolytes always inhibit the rates of ion combination reactions because the activity coefficients are fractions. The data for CTABr show an enhancement of rate so this cannot be due to an activity effect. In these data, the k s for pure water and aqueous NaCl are essentially identical, so no activity effects operate in the absence of micelles either. 

If the Debye-Hiickel theory is used for the activity coefficients, Eq. (6.11) can be written as ... 

When the permittivities of solvents S and R are very different, this affects the De-bye-Hiickel activity coefficient of M+ (9) in Chapter 2). If necessary, the effect should be estimated (or eliminated) using the Debye-Hiickel theory. 

The part of the activity coefficients depending on k (Equation 33) can be simplified further if suitable average values are introduced. If the restriction d a is sufficiently well fulfilled, the term in y is small compared with the term in <5. The latter therefore represents the main influence caused by the presence of dipoles since the terms in are from ionic charges. They are identical with terms of corresponding order in the Debye-Hiickel theory. 

The Debye-Hiickel Theory of Activity Coefficient The Point-Charge Model. The... 

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 thermodynamic model of Krissmann [53] was used in the calculations of these experiments, though this was limited by the phase equilibrium (Eq. (3)) and the reaction equilibrium (Eq. (4)). Calculation of the activity coefficients of the H+ ions and HSOj was performed according to the extended Debye-Hiickel theory, using the approximation of Pitzer... 

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. 

As mentioned above, the activity coefficients of diluted solutions could be computed from the Debye-Hiickel theory. Since, however, a detailed explanation is beyond the limit of this work, only an explanation of the principles will be given, as well as the results thereof. 

The quantity given in equation (V-58) and expressing the average distance between the nearest ions in the solution (i. e. equalling the total of radii of both ions with opposite charges and being within the range of 3—5 x 10-8 cm) determines the specific influence of the electrolytes on the activity coefficient. Because this quantity iH not directly measurable, verification of the validity of the Debye - Hiickel theory is carried out in such a manner that a value is substituted for which conforms best with the values of y+c obtained by experiments. 

With increase in salt concentration the approximations involved in the Debye-Hiickel theory become less acceptable. Indeed it is noteworthy that before this theory was published a quasi-lattice theory of salt solutions had been proposed and rejected (Ghosh, 1918). However, as the concentration of salt increases so log7 ,7 being the mean ionic activity coefficient, appears as a linear function of c1/3 (the requirement of a quasi-lattice theory) rather than c1/2, the DHLL prediction (Robinson and Stokes, 1959). Consequently, a quasi-lattice theory of salt solutions has attracted continuing interest (Lietzke et al., 1968 Desnoyers and Conway, 1964 Frank and Thompson, 1959 Bahe, 1972 Bennetto, 1973) and has recently received some experimental support (Neilson et al., 1975). 

Debye-Hiickel limiting law — The equation based on the - Debye-Hiickel theory providing mean activity coefficients / for ions of charge z+ and z- at - ionic strength I in dilute solutions lg/ = ().S09 z+ z /l, when mol and dm3 units are used. 

Debye-Hiickel theory — The interactions between the ions inside an electrolyte solution result in a nonideal behavior as described with the concepts of mixed-phase thermodynamics. Assuming only electrostatic (i.e., coulombic) interactions - Debye and - Hiickel suggested a model describing these interactions resulting in - activity coefficients y suitable for further thermodynamic considerations. Their model is based on several simplifications ... 

The values of /3 for each C, were obtained by iteration from the total thermodynamic equilibrium constants based on conductivity measurements. The activity coefficients f2- and / were evaluated from the extended Debye-Hiickel theory. 

The solubility product is equal to Ca x AI(OH)4 x [OH , where curly brackets denote species activities and [OH ] may be replaced by 2[Ca ] - [A1(0H)4"]. As the concentrations are low, activity coefficients may be calculated from simplified Debye-Hiickel theory. Solubility products may thus be obtained from experimental data (NI8,B118,BI 19.C48) (Table 10.2). The variations in solubility products with temperature may be represented by empirical equations of the form... 

A commonly used approximate form of the Debye-Hiickel theory for the activity coefficients of ionic species at 25°C is ... 

The student should construct a table giving the actual initial concentrations of the reactants IO3, I, and H. The concentrations should be calculated from the actual concentrations of NaAc and HAc in the stock solutions employed, with an activity coefficient calculated by use of the Debye-Hiickel theory for the ionic strength (/ = 0.16) of the reacting mixtures. 

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