Debye-Huckel function

As can be seen in Figure 8.1, the Davies equation does not decrease monotoni-cally with ionic strength, as the Debye-Huckel equation does. Beginning at ionic strengths of about 0.1 molal, it deviates above the Debye-Huckel function and at about 0.5 molal starts to increase in value. The Davies equation is reasonably accurate to an ionic strength of about 0.3 or 0.5 molal. 

EXCESS GIBBS ENERGY AND THE DEBYE-HUCKEL FUNCTION... 

If the coefficients dy vanish, dy = 28y, we recover the exact Debye-Huckel limiting law and its dependence on the power 3/2 of the ionic densities. This non-analytic behavior is the result of the functional integration which introduces a sophisticated coupling between the ideal entropy and the coulomb interaction. In this case the conditions (33) and (34) are verified and the... 

This is the electrostatic energy arising from ions approaching within a of each other. When subtracted from the free energy functional above the corrected Debye Huckel equation becomes... 

FIG. 10 The calculated internal energy of a one-component plasma as a function of coupling strength is compared with corresponding simulation results (open circles) by Brush, Sahlin, and Teller (J. Chem. Phys. 45 2102 (1966). The Debye-Huckel (DH) and hole-corrected Debye-Huckel (DHH) theories were used with results as shown (indicated lines). 

Variable di in Equation 8.2 is the ion size parameter. In practice, this value is determined by fitting the Debye-Huckel equation to experimental data. Variables A and B are functions of temperature, and I is the solution ionic strength. At 25 °C, given I in molal units and taking a, in A, the value of A is 0.5092, and B is 0.3283. 

Fig. 8.2. Values of A, B, and B for the B-dot (modified Debye-Huckel) equation at 0 °C, 25 °C, 60 °C, 100 °C, 150 °C, 200 °C, 250 °C, and 300 °C (squares) and interpolation functions (lines). Values correspond to I taken in molal and a in A. Data from the LLNL database, after Helgeson (1969) and Helgeson and Kirkham (1974).

Coefficients cl-c4 are used to approximate the integral function "J" aphi is the Debye-Huckel constant at 25 C /... 

Friedman (1962) has used the cluster theory of Mayer (1950) to derive equations which give the thermodynamic properties of electrolyte solutions as the sum of convergent series. The first term in these series is identical to and thus confirms the Debye-Huckel limiting law. The second term is an I2.nl term whose coefficient is, like the coefficient in the Debye-Huckel limiting law equation, a function of the charge type of the salt and the properties of the solvent. From this theory, as well as from others referred to above, a higher order limiting law can be written as... 

The statistical thermodynamic approach of Pitzer (14), involving specific interaction terms on the basis of the kinetic core effect, has provided coefficients which are a function of the ionic strength. The coefficients, as the stoichiometric association constants in our ion-pairing model, are obtained empirically in simple solutions and are then used to predict the activity coefficients in complex solutions. The Pitzer approach uses, however, a first term akin to the Debye-Huckel one to represent nonspecific effects at all concentrations. This weakens somewhat its theoretical foundation. 

In the conventional Debye-Huckel treatment the equilibrium radial distribution function for a pair of reactants g(r) is simply equal to exp(-w/RT) with w given by (15)... 

Although more complex pair-correlation functions are available, the Debye-Huckel expression is adequate for our present purpose. It is valid when the work required to bring the reactants... 

The activity a and concentration c are related by a = (c/c ) x y (equation (3.12)), where y is the mean ionic activity coefficient, itself a function of the ionic strength /. Approximate values of y can be calculated for solution-phase analytes by using the Debye-Huckel relationships (equations (3.14) and (3.15)). The change of y with ionic strength can be a major cause of error in electroanalytical measurements, so it is advisable to buffer the ionic strength (preferably at a high value), e.g. with a total ionic strength adjustment buffer (TISAB). 

A modification of GB that includes the effects of dissolved electrolytes in the formalism, i.e., an extension analogous to the Poisson-Boltzmann extension of the Poisson equation, has been proposed by Srinivasan et al. (1999). In this model, the dielectric constant is a function of the interatomic distance and the Debye-Huckel parameter (Eq. (11.7)). 

Let us summarize by modeling the velocity autocorrelation function using Debye-Huckel type interactions between charged point defects in ionic crystals, one can evaluate the frequency-dependent conductivity and give an interpretation of the universal dielectric response. 

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 ... 

Stokes-Robinson Modification of Debye-Huckel Theory Effect of Ion-Solvent Interaction. Debye-Huckel theory explains the activity and activity coefficient data on the basis of ion-ion interaction for dilute solution. According to Eqs. (5.29) and (5.33), the activity coefficient is a decreasing function of concentration. 

In this appendix, we summarize the coefficients needed to calculate the thermodynamic properties for a number of solutes in an electrolyte solution from Pitzer s equations.3 Table A7.1 summarizes the Debye-Huckel parameters for water solutions as a function of temperature. They provide the leading terms for Pitzer s equations, and can also be used to calculate the Debye-Huckel limiting law values from the equations... 

The BMREP and SDM currently use the Davies technique for activity coefficient prediction. The Davies technique is a combination of the extended Debye-Huckel equation (6) and the Davies equation (7). The Davies technique (and hence both equilibrium models) is accurate up to ionic strengths of 0.2 molal and may be used for practical calculations up to ionic strengths of 1 molal (8). Ion-pair equilibria are incorporated for species that associate (e.g., 1-2 and 2-2 electrolytes). The activity coefficients (y ) are calculated as a simple function of ionic strength (I) and are represented as ... 

The left-hand side of this equation can be calculated from measurements of cell voltage as a function of concentration. The second term on the right-hand side becomes zero at infinite dilution. However, because no meaningful measurements can be made at zero concentration of reactants, we must extrapolate the equation to infinite dilution using the known concentration behavior of activity coefficients. In approaching infinite dilution, it is sufficient to use the Debye-Huckel... 

Figure 15. Excess thermodynamic functions for various 1 1 salts in water at 298 Kj mj = molality of salt and the dotted lines indicate the behaviour predicted by the Debye-Huckel limiting law (Fortier et al., 1974).

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

The free space Green function for three spatial dimension problems has the familiar Debye-Huckel form... 

Debye-Huckel-Onsager theory — (- Onsager equation) Plotting the equivalent conductivity Aeq of solutions of strong electrolytes as a function of the square root of concentration (c1/2) gives straight lines according to the - Kohlrausch law... 

It is a function expressing the effect of charge of the ions in a solution. It was introduced by -> Lewis and Randall [iii]. The factor 0.5 was applied for the sake of simplicity since for 1 1 electrolytes I = c (electrolyte). It is an important quantity in all electrostatic theories and calculations (e.g., - Debye-Huckel theory, - Debye-Htickel limiting law, - Debye-Huckel-Onsager theory) used for the estimation of -> activity coefficients, -> dissociation constants, -> solubility products, -> conductivity of -> electrolytes etc., when independently from the nature of ions only their charge is considered which depends on the total amount (concentration) of the ions and their charge number (zj). 

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... 

---