Guggenheim theory

For dilute solutions, Equations 4 and 5 reduce to the Bronsted-Guggenheim equations, and the parameters a23 and cu2 can be expressed in terms of the interaction parameters of tne Bronsted-Guggenheim theory. For concentrated solutions, Harned s rule is a simple empirical extension of the Brb nsted-Guggenheim theory. Thus, 1t 1s surprising how well the rule describes activity coefficients 1n highly concentrated solutions. 

The synthesis of 2-deoxy-L-ascorbic acid has been described/ and the dipole moment of L-ascorbic acid in dioxane has been calculated by using Guggenheim Theory . 0-Benzyl ether and -benzylidene acetal derivatives of L-ascorbic acid and D-isoascorbic acid have been found to epimerize partly to the corresponding derivatives of L-isoascorbic acid and D-ascorbic acid, respectively, on treatment with triisobutylaluminium. The synthesis of [l- C]- and [2- C]-L-ascorbic acid by application of the KCN chain-extension to suitable compounds has been outlined. ... 

The Flory theory bears certain similarities to the van der Waals and Guggenheim theories in that it makes use of an explicit equation of state. It can be seen from the results given in Table 3 that, although this theory is quite successful in predicting G , the theoretical values of F and are too low by a factor of more than two. 

A number of analytical off-lattice approaches are based on the extension of various simple one-component analytical models (such as Flory-Huggins theory, Flory s equation of state model, or Guggenheim theory) to multicomponent systems by the use of some ad hoc combining rules [5-8,15,16]. These approaches likewise neglect the influence of chain connectivity and monomer... 

One important direetion of study has been to use empirieal adsorption data, together with the preassumed model for loeal adsorption, and attempt to extraet information about the form of x(e) [13,14]. The ehoiee of the model for loeal adsorption, whieh is an important input here, has been eustomarily treated quite easually, assuming that it has rather limited influenee on the form and properties of the evaluated EADFs. Usually, one of so many existing equations developed for adsorption on uniform surfaees is used as the loeal adsorption isotherm. The most often used forms of 0 p, T,e) are the Langmuir [6] and the Fowler-Guggenheim [15] equations for loealized adsorption. Ross and Olivier [4] extensively used the equation for mobile adsorption, whieh results from the two-dimensional version of the van der Waals theory of fluids. The most radieal solution has been... 

For more than two decades researchers have attempted to overcome the inadequacies of Flory s treatment in order to establish a model that will provide accurate predictions. Most of these research efforts can be grouped into two categories, i.e., attempts at corrections to the enthalpic or noncombinatorial part, and modifications to the entropic or combinatorial part of the Flory-Huggins theory. The more complex relationships derived by Huggins, Guggenheim, Stavermans, and others [53] required so many additional and poorly determined parameters that these approaches lack practical applications. A review of the more serious deficiencies... 

Because much experimental work has been stimulated by the quasi-chemical theory, it is important to gain proper perspective by first describing the features of this theory.12 The term, quasichemical will be used to include the Bragg-Williams approximation as the zeroth-order theory, the Bethe or Guggenheim pair-distribution approximations as the first-order theory, and the subsequent elaborations by Yang,69 Li,28 or McGlashan31 as theories of higher order. 

Guggenheim, E. A., Elements of the Kinetic Theory of Gases, Pergamon Press, New York, 1960. 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

Of course, within the general framework of the Flory-Huggins theory a small difference between yn in a swollen network and in a solution is always to be expected, because the crosslinks have four rather than two polymer segments as first neighbours. An estimate of this effect has been made by Rijke 147), based on Stavekman s extension of Guggenheim s formulation of y to branched chains. The result is... 

Current thermodynamic theories for polymer systems are combinations of the Flory -Huggins, Guggenheim, and Equations-of-State approaches. All of these theories make use of empirical parameters and are based on assumptions about the underlying molecular model. 

The Lennard-Jones-Devonshire theory (as summarized by Fowler and Guggenheim, 1952, pp. 336ff) averaged the pair potentials of Equation 5.24a and b between the solute and each water, for Zi molecules in the surface of the spherical cavity to obtain a cell potential r) of... 

As mentioned earlier, I was not aware of the details of the chemical engineering models and Guggenheim s QUAC approximation, when I developed the COSMO-RS model. From a brief glance at that part of the literature I had got the impression that all these models are basically lattice theories, while I wanted to... 

To obtain the pH, it is necessary to evaluate the activity coefficient of the chloride ion. So the acidity function is determined for at least three different molalities mci of added alkali chloride. In a subsequent step, the value of the acidity function at zero chloride molality, lg(flHyci)°, is determined by linear extrapolation. The activity of chloride is immeasurable. The activity coefficient of the chloride ion at zero chloride molality, yci, is calculated using the Bates-Guggenheim convention (Eq. 5) which is based on the Debye-Hiickel theory. The convention assumes that the product of constant B and ion size parameter a are equal to 1.5 (kg mol1)1/2 in a temperature range 5 to 50 °C and in all selected buffers at low ionic strength (I < 0.1 mol kg-1). 

Element abundance data were useful not only in astrophysics and cosmology but also in the attempts to understand the structure of the atomic nucleus. [74] As mentioned, this line of reasoning was adopted by Harkins as early as 1917, of course based on a highly inadequate picture of the nucleus. It was only after 1932, with the discovery of the neutron as a nuclear component, that it was realized that not only is the atomic mass number related to isotopic abundance, but so are the proton and neutron numbers individually. Cosmochemical data played an important part in the development of the shell model, first proposed by Walter Elsasser and Kurt Guggenheimer in 1933-34 but only turned into a precise quantitative theory in the late 1940s. [75] Guggenheimer, a physical chemist, used isotopic abundance data as evidence of closed nuclear shells with nucleon numbers 50 and 82. 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

The enthalpy of mixing of some mixtures of molten salts with monovalent ions could be calculated in terms of the quasi-chemical theory given by Guggenheim [19] as follows ... 

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