Extended DH equations

Example 3.2 Calculation of single-ion activity coefficients using the DH and extended DH equations. 

Calculating the single-ion activity coefficient for Na using both the DH and the extended DH equations (use a value of 4.5 for the ion size... 

Clearly, at lower ionic strength, the single-ion activity coefficient is much closer to unity. Also, the DH and the extended DH models give almost exactly the same value. This is because the denominator (l+0.33ai 7) of the extended DH equation approaches a value of unity, i.e. 0.33oi / approaches zero, as the ionic strength decreases towards zero. In other words, the two equations become identical at very low ionic strength. 

The Davies equation (Eq. 4.31) generates the positive change in slope with an add-on term, bl, where b is the same constant for all ions. The denominator of the Davies equation equals I + VT, which is equivalent to assigning a constant a -, value of about 3.0 to all ions in the extended DH equation. These simplifications make the Davies equation less accurate than the extended Debye-Hiickel equation at low ionic strengths, and limit its use to ionic strengths below that of seawater (0.7 mol/kg). 

The TJ equation (Eq. 4.51) improves on shortcomings of the Davies equation by retaining the extended DH equation with its a, parameter and assigning different values to b in the add-on bl term based on the Macinnes convention and on fitting mean salt data for the ions. The TJ equation has been found reliable up to / = 3.5 mol/kg in some cases, but is conservatively limited to / < 2 mol/1. ... 

The denominator of the DH term in the SIT model (Eqs. 4.32 and 4.33) equals 1 + 1.5V/, which is equivalent to assuming a, 4.6 and constant in the extended DH equation. The SIT model accounts for binary interactions of anions and cations with interaction parameters that are specific for individual ions in chloride or other media. Lacking an adjustable ion size parameter, the SIT model is less accurate than the extended DH model at low ionic strengths. It does, however, give accurate results in some solutions up to / = 3.5 mol/kg. 

DH-type, low ionic-strength term. Because the DH-type term lacks an ion size parameter, the Pitzer model is also less accurate than the extended DH equation in dilute solutions. However, a.ssuming the necessary interaction parameters (virial coefficients) have been measured in concentrated salt solutions, the model can accurately model ion activity coefficients and thus mineral solubilities in the most concentrated of brines. 

Originally the extended DH term was similar to that in Equation (3.5), i.e. with an average ion size parameter, a = 3. Scatchard showed that a better fit with experimental data was obtained using an average value of fli = 4.6, i.e. 0.33ai = 1.5, and so the extended DH expression used in the SIT model is often... 

S(f) = limiting theoretical slope of rational activity coefficient in interionic attraction theory a function of y, D, and T as expressed by Equation 5 T = temperature in Kelvin X, Y = function of DH extended theory equation z = valence of an ion... 

The need for an analytical expressions for the equation of state have led to a revival of the macroscopic electrostatic theory due to Debye, Hiickel and Bjerrum. DH theory becomes exact for large particles. In pilot work by Fisher and Levin (FL) [31], DH-Bj theory is extended by considering the interactions of the pairs with the free ions. Weiss and Schroer (WS) [32] have supplemented this theory accounting for dipole-dipole interactions between pairs and the e-dependence of the association constant. 

Besides serving to demonstrate the physical principles involved in applying the PB equation as well as checks for the numerical solution of more complicated geometries, these three systems are also excellent models for systems of extreme biophysical importance, modeling flat-cell membranes, extended polyelectrolytes such as DNA, and spherical micelles. For each geometry, we will solve the PB equation and discuss the resulting potential profile. For those cases in which both nonlinear PB as well as linear DH solutions exist, the similarities and differences (particularly near the interface) are emphasized. 

Having outlined the derivation of the Poisson-Boltzmann equation, we now turn to a discussion of the main assumptions used, under which circumstances they become invalid, and how these problems might be remedied. A comprehensive treatment of the PB equation in which most of the approximations are addressed in a detailed manner was given by Bell and Levine." In their work they derived a modified PB equation which provided corrections for most of the deficiencies in the original theory. Their results have been further extended by Outhwaite and co-workers in an attempt to place the PB equation on par with other more elaborate theories. We discuss here only the generalities of the basic DH assumptions for specific details the reader is referred to the Bell-Levine paper" ° and others cited below. 

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