Debye-Huckel, extended theory

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 parameter67 and the latter by generalizing the Poisson equation to the Poisson-Boltzmann equation.68 The Debye-Huckel modification of the GB model is valid to much higher salt concentrations than the original Debye-Huckel theory because the model includes the finite size of the solute molecules. 

Bromley, L.A. "Approximate Individual Ion Values of 6 (or B) in Extended Debye-Huckel Theory for Uni-univalent Aqueous Solutions At 298.15 K," J.Chem. Thermo., 1972, 4, 669-73. 

Marshall s extensive review (16) concentrates mainly on conductance and solubility studies of simple (non-transition metal) electrolytes and the application of extended Debye-Huckel equations in describing the ionic strength dependence of equilibrium constants. The conductance studies covered conditions to 4 kbar and 800 C while the solubility studies were mostly at SVP up to 350 C. In the latter studies above 300°C deviations from Debye-Huckel behaviour were found. This is not surprising since the Debye-Huckel theory treats the solvent as incompressible and, as seen in Fig. 3, water rapidly becomes more compressible above 300 C. Until a theory which accounts for electrostriction in a compressible fluid becomes available, extrapolation to infinite dilution at temperatures much above 300 C must be considered untrustworthy. Since water becomes infinitely compressible at the critical point, the standard entropy of an ion becomes infinitely negative, so that the concept of a standard ionic free energy becomes meaningless. 

The Ky results of Sweeton, Mesmer and Baes (35) plotted in Fig. 2 were reported in 1974 and although they only extend to 300°C they may well be more accurate above this temperature than the experimental results of Fisher and Barnes(36), since, as mentioned, earlier, the Debye-Huckel theory may not give reliable extrapolations to infinite dilution at temperatures where water is highly compressible. While their work (35) involves extrapolation to infinite dilution as well as to higher temperatures it is very encouraging to note that their ACp at 300°C (-960 J K mol ) is of the magnitude expected on the basis of the NaCl studies referred to in Section 2. The conductance results of Sirota and Shviriaev (37) above 300°C also seem more consistent with the results of Sweeton, Mesmer and Baes (35), than with those of Fisher and Barnes (36). Marshall and Franck s recent representation of data up to 1000°C and 10,000 bars (38) predicts high temperature SVP results somewhat lower than those of Sirota and Shviriaev (37). 

In what way does the Gouy-Chapman theory extend the Debye-Huckel approach ... 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

The Debye-Huckel Theory The Finite-Ion-Size Model. If the approximation of the point charge is removed, the extended form of the Debye-Huckel law is obtained ... 

The activity coefficient yt of an ion depends on the ionic strength (I = ( )Zzfc , where zi is the charge number) according to the Debye-Huckel theory in the limit of low ionic strengths. As discussed in Section 1.2, this equation can be extended... 

The standard solution (which can be purchased from a chemical supply company) is chosen to have a pH close to that of the unknown solution. pHv is calculated (by the company), using extended Debye-Huckel theory, to good accuracy. 

This is not the official definition, which is a practical definition based on the measured potentials of some exactly defined electrochemical cells in which the hydrogen ion activity is calculated by an extended Debye-Huckel theory. 

Electromotive force measurements of the cell Pt, H2 HBr(m), X% alcohol, Y% water AgBr-Ag were made at 25°, 35°, and 45°C in the following solvent systems (1) water, (2) water-ethanol (30%, 60%, 90%, 99% ethanol), (3) anhydrous ethanol, (4) water-tert-butanol (30%, 60%, 91% and 99% tert-butanol), and (5) anhydrous tert-butanol. Calculations of standard cell potential were made using the Debye-Huckel theory as extended by Gronwall, LaMer, and Sandved. Gibbs free energy, enthalpy, entropy changes, and mean ionic activity coefficients were calculated for each solvent mixture and temperature. Relationships of the stand-ard potentials and thermodynamic functons with respect to solvent compositions in the two mixed-solvent systems and the pure solvents were discussed. 

When the proper choice of the ponstants a and b are made, the function (Em° + Eext) should be constant within the limits of the extended Debye-Huckel theory. In calculating Em° the value of the equation for log y (Equation 6) which must be substituted into Equation 4 becomes... 

In conclusion, therefore, it may be said that the treatment of the influence of ion-solvent interactions on ion-ion interactions has extended the range of concentration of an ionic solution which is accessible to theory. Whereas the finite-ion-size version of the Debye-Huckel theory did not permit theory to deal with solutions in a range of concentrations corresponding to those ofreal life, Eq. (3.130) advances theory into the range of practical concentrations. Apart from this numerical agreement with experiment, Eq. (3.130) unites two basic aspects of the situation inside an electrolytic solution, namely, ion-solvent interactions and ion-ion interactions. 

Electrolytes for which the concentration is less than lO Mcan usually be dealt with by the Debye-Huckel limiting law. Utilize the Debye-Huckel theory extended by allowance for ion size and also for removal of some of the active solvent into the ion s primary solvation shell to calculate the activity coefficient of 5 M NaCland 1M LaClj solutions (neglecting ion association or complexing). Take the total hydration number at the 5 M solution as 3 and at the 1 M solution as 5. Take r,- as 320 pm. 

According to the extended Debye-Huckel theory, it is possible to write... 

Note that the standard transformed Gibbs energy of formation of a species depends on the pH, but the standard transformed enthalpy of formation does not. When species have electric charges, their standard thermodynamic properties need to be adjusted for the ionic strength according to the extended Debye-Huckel theory (see Section 1.3) At zero ionic strength,... 

Another well-known example of a medium effect in solution kinetics is the effect of ionic strength on reactions between ions [38, 39]. This is normally treated using the extended Debye-Huckel theory to estimate the activity coefficients in equation (7.10.1). The activity coefficient for species i is given by... 

The distinction between "pure" electrostatic effects and ion association is somewhat arbitrary for example many workers have used an extended Debye-Huckel equation with fixed ion-size parameters to compute the activity coefficients, and assigned all other nonideality to ion-pairing equilibria (3, 13, 14). Recent reviews have surveyed more elaborate solvation-based thermodynamic theories of electrolytes (15, 16). 

The solubilities of 3-2 samarium sulphate hydrates in water and aqueous sulphuric acid have been investigated. The values in water decrease from 0.33 M at 25°C for the octahydrate to 8 x 10 M for lower hydrates at 350°C. The saturation effect of several sets of ionic species on the sulphuric acid solutions were tested by extended Debye-Huckel theory and the 3-2 samarium sulphate was suggested to behave predominantly as a 2-2 sulphate at high temperatures thereby producing Sm2(S04) and sol" ions in solution. Thermodynamic functions for the solubility of samarium sulphate hydrate at 150—250 °C based on its behaviour as a 2-2 salt were presented. 

Normally, the validity of the Debye—HUckel theory extends little further than kR <1. At room temperature, this requires ionic concentrations < 0.1 mol dm for univalent ions in water, 0.03 mol dm for univalent ions in ethanol or <0.01 moldm for univalent ion in ethers. In these cases, ions may be regarded as point particles and the strong repulsive core potential ignored. Furthermore, the time taken for non-reactive ions to diffuse far enough to establish an ionic-atmosphere around an ion, which was suddenly formed in solution containing only univalent ions, is... 

Thus the Debye-Huckel and extended Debye-Htlckel (with a fixed value of the 8 parameter) theories are not very accurate. However, if the 8 parameter is adjusted, much better agreement with experimental data can be achieved. This is left to the student to prove. 

The theory of solutions of flexible uncharged polymers with excluded volume is at present well developed, but the properties of polyelectrolytes and especially polyampholytes have been considered much less from the theoretical point of view. It is well known that polyampholytes exhibit a change in phase from the extended random flight configuration to a condensed microphase. The polyampholyte theory of Edwards et al. [6] considers the isoelectric state of polyampholytes as a microelectrolyte satisfying a Debye-Huckel-type of structure. The criterion of transition from the collapsed conformation to the extended one is described as follows ... 

Friedman and Newton have extended their absolute rate calculations of the [Fe(aq)6] reaction to the case of aq = D2O. With the appropriate allowance for differences in solvent libration and Fe—-O stretch frequency, the rate ratio Ah/ d comes out at 1.7, in comparison with the experimental ratio 2.0. Other effects, not calculated in detail, are expected to bridge the gap. " Activation parameters as functions of ionic strength, for outer-sphere reactions between like-charged ions, have been treated using an extended Debye-Huckel theory. The maximum in or observed for certain reactions in the range / = 0 to... 

In this equation m refers to the aquamolal concentration, the s to the pure solvent effects, a Is the solvent activity and (j) the osmotic coefficient. Data for a number of different electrolyte solutions at one selected temperature are shown In Figure 6. The solutions all show a smaller VPIE than do the pure solvents. In the experiments AlnR Is measured as a function of temperature and concentration, and we have phenomenologically fit the data (58,59) using the extended Debye-Huckel theory. 

We shall just mention here that the simple association theory may be extended by considering the interactions between defects in solution in a medium with dielectric constant a [14]. This is analogous to the Debye-Huckel theory of electrolytic solutions. As a result, the mole fractions of charged point defects of sort i in the mass action laws have to be replaced by their corresponding activities which according to Debye-Huckel are of the form... 

Bl. Bromley, L.A., "Approximate individual ion values of B (or B) in extended Debye-Huckel theory for uni-univalent aqueous solutions at 298.15 K", J. Chem. Thermo., 4, 669 (1972)... 

For ion pairs, Fuoss and Eigen developed an equation to estimate AT, based on extended Debye-Huckel theory and a hard-sphere model for the ions. It is given by... 

Expression (31) for the free energy arising from charge interactions is an exact expression so that, together with Fq as defined previously, it completely defines the total free energy of the system considered here. In the present state of theory an exact evaluation of F according to (31) is out of question however for the same reasons that cannot be calculated immediately. Therefore a more approximate approach is used analogously to the Debye-Huckel treatment of ordinary ionic solutions. This approach, which we shall call the extended Poisson-Boltzmann approach, will be discussed in the next section. 

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