Bjerrum method equation

An initial step in data analysis is to develop an equation that represents the experimental data reasonably. Although previous sections dealt with this issue, the approach assumed that certain species are formed. Two alternatives to this procedure are discussed here, both yielding the approximate stoichiometry of the complexes formed in the system. The most elementary is referred to as Job s method, while the ligand number method, developed by J. Bjerrum, is slightly more advanced. 

Gruen and Marcelja considered that the electric and polarization fields are not proportional in the vicinity of a surface and that while the electric field has the ion concentrations as its source, the source of the polarization field is provided by the Bjerrum defects. The coupled equations for the electric and polarization fields were derived through a variational method. Attard et al.14 contested the Gruen—Marcelja model because, to obtain an exponential decay of the repulsion, the nonlocal dielectric function was assumed to have a simple monotonic dependence upon the wavelength (eq 33 in ref 13). This was found to be inconsistent with the exact expression for multipolar models.14 In addition, the characteristic decay length for polarization (denoted in eq 18, ref 13) is inversely proportional to the square of the (unknown) concentration of Bjerrum defects in ice. While at large concentrations of Bjerrum defects the disordered ice becomes similar to water and the traditional Poisson—... 

With the potentiometric approach, determination of intrinsic solubility is based upon the measurement of the pH shift caused by compound precipitation during acid-base titration of ionizable compounds. Two commercial potentiometric methods currently available are pSol [30, 39] and Cheqsol [40-42], In the pSol method developed by Avdeef, a minimum of three titrations in the direction of dissolution are performed. Normal pH versus volume titration plots are reexpressed as Bjerrum plots, that is, average number of bound protons versus pH. The Bjermm plots enable the shift in compound pKa to be more readily observed and are used to determine intrinsic solubility (S0) via Equation 2.5 ... 

Solutions to the Poisson—Boltzmann equation in which the exponential charge distribution around a solute ion is not linearized [15] have shown additional terms, some of which are positive in value, not present in the linear Poisson—Boltzmann equation [28, 29]. From the form of Eq. (62) one can see that whenever the work, q yfy - yfy), of creating the electrostatic screening potential around an ion becomes positive, values in excess of unity are possible for the activity coefficient. Other methods that have been developed to extend the applicable concentration range of the Debye—Hiickel theory include mathematical modifications of the Debye—Hiickel equation [15, 26, 28, 29] and treating solution complexities such as (1) ionic association as proposed by Bjerrum [15,25], and(2) quadrupole and second-order dipole effects estimated by Onsager and Samaras [30], etc. 

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