Guggenheims numerical integration

Guggenheim used the complete Poisson-Boltzmann Equation (10.28), and solved it by numerical integration using a computer. He included all terms in his calculations, and his results allow for all electrostatic interactions including those giving rise to ion pairs. Hence with his work there is no need to refer explicitly to ion association since this is automatically included in his calculation. His work was of considerable significance, and the conclusions drawn were  

The numerical integration can, therefore, give a possible base-line for the treatment of electrolyte solutions. Provided that association is taken care of by the Bjerrum treatment, the Debye-Hiickel expression is also a good base-line for the behaviour of free unassociated ions in solution, provided that the appropriate value of q appears in the denominator rather than the conventional a. 

Guggenheim did, however, point out that there are also inherent problems in this approach because of the distinct physical stams of the i/ s used in the Poisson and Maxwell-Boltzmann equations (see Section 10.6.5). 

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ILLUSTRATION 3.4 USE OF GUGGENHEIM S METHOD AND A NUMERICAL INTEGRAL PROCEDURE TO DETERMINE THE RATE CONSTANT FOR THE HYDRATION OF ISOBUTENE IN HYDROCHLORIC ACID SOLUTION... 

Extension to higher concentrations is required, and one of the main advances here has been Guggenheim s numerical integration by computer, but the problem of the superposition of fields and the different V s still remain. 

There have been attempts to modify Bjerrum s treatment to remove this arbitrariness, but none has been used universally to any great extent in the interpretation of experimental data. Nevertheless, despite this artificiality, the Bjerrum theory coupled with the Debye-Hiickel theory has proved a very useful and relatively successfiil tool in discussing electrolyte solutions. This success is especially noteworthy when the Bjerrum-Debye-Huckel theory is compared with the alternative approach of Guggenheim s numerical integration which gives similar results (see Section 10.13.1). Table 10.2 gives values of K ssoc for various charge types and for various values of a and q. 

ILLUSTRATION 3.4 Use of Guggenheim s Method and a Numerical Integral Procedure to Determine the Rate Constant for the Hydration of Isobutene in Hydrochloric Acid Solution... 

The first such solutions were carried out by Ross and Olivier [1, p. 129 6,7]. Using Gaussian distributions of adsorptive potential of varying width, they computed tables of model isotherms using kernel functions based on the Hill-de Boer equation for a mobile, nonideal two-dimensional gas and on the Fowler-Guggenheim equation [Eq. (14)] for localized adsorption with lateral interaction. The fact that these functions are implicit for quantity adsorbed was no longer a problem since they could be solved iteratively in the numerical integration. 

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