Guggenheims Equations

A further approach is to consider a more specific form of ion interaction and replace Eq. (5.80) with die Bronsted-Guggenheim equation (Guggenheim and Turgeon 1955)... 

Gibbs s adsorption equation. Guggenheim,2 developing some considerations put forward by Verschaffelt, has worked out the thermodynamics of surfaces, using a conception of the surface layer which is much more easily visualized physically than that used by Gibbs. Gibbs s surface excess is a very difficult quantity to interpret physically, as it is... 

While increasing the relative humidity of the surrounding air, crystalline materials do not absorb significant quantities of water. Only a limited amount of crystal water is embedded in the molecular matrix. Once a substance-specific relative air humidity (or water activity) is reached, the crystals dissolve layer by layer. In contrast, amorphous water-soluble substances absorb significant quantities of water when exposed to an increasing relative humidity of the surrounding air. The amount of absorbed water is a function of the water activity that can be described by various types of isotherm equations (Guggenheim, Anderson and de Boer (GAB), Brunnauer, Emmett and Teller (BET), see Weisser (1986)). 

The Guggenheim and Redlich-Kister equations (Guggenheim, 1937 Redlich and Kister, 19 8a, b King, 1969 pp.325-327) ... 

The values were calculated from the modified Debye-Hiickel equation utilizing the modifications proposed by Robinson and by Guggenheim and Bates ... 

It is, however, possible to calculate the tensile strength of a liquid by extrapolation of an equation of state for the fluid into the metastable region of negative pressure. Burgess and Everett in their comprehensive test of the tensile strength hypothesis, plot the theoretical curves of T /T against zjp, calculated from the equations of state of van der Waals, Guggenheim, and Berthelot (Fig. 3.24) (7], and are the critical temperature and critical... 

Fig. 3.24 Test of the tensile strength hysteresis of hysteresis (Everett and Burgess ). TjT, is plotted against — Tq/Po where is the critical temperature and p.. the critical pressure, of the bulk adsorptive Tq is the tensile strength calculated from the lower closure point of the hysteresis loop. C), benzene O. xenon , 2-2 dimethyl benzene . nitrogen , 2,2,4-trimethylpentane , carbon dioxide 4 n-hexane. The lowest line was calculated from the van der Waals equation, the middle line from the van der Waals equation as modified by Guggenheim, and the upper line from the Berthelot equation. (Courtesy Everett.)...

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... 

Nonlinear least-squares programs have made time lag methods much less important. They are less accurate, for one thing. For another, the linearity of the appropriate plots, although a necessary consequence of first-order kinetics, does not constitute a proof of first-order kinetics. Certain other kinetic equations also lead to linear plots of either function. For example, Problem 2-11 presents data for a product-catalyzed reaction. The data in this case can be plotted linearly according to the Guggenheim equation, although the reaction does not follow first-order kinetics and the plot of In [A] versus time is decidedly nonlinear. 

Data can be fit to this equation by the nonlinear least-squares method. As it turns out, the Guggenheim approach for first-order kinetics is valid, even though the reaction... 

Even with the simpler Frumkin or Fowler-Guggenheim approach (Eqs. 6.50 and 6.52), treating the coadsorption and surface reaction of different adsorbates leads immediately to mathematically intractable expressions and to the introduction of new parameters, whereas equation... 

This isotherm is consistent with the modified electrochemical Langmuir isotherm, the Nemst equation and the potential-work function equivalence. For intermediate 0j and Pj values the isotherm of Eq. (6.58) is well approximated both by the Fowler-Guggenheim and by the Temkin isotherms. 

The Fowler-Guggenheim-Jovanovic model [3] assumes (as it was the earlier case also) the occurrence of intermolecular interactions among the molecules adsorbed as a monolayer but is based on the Jovanovic isotherm. The single-component isotherm is represented by the equation ... 

The most spectacular peak profiles, which suggest self-associative interactions, were obtained for 5-phenyl-1-pentanol on the Whatman No. 1 and No. 3 chromatographic papers (see Figure 2.15 and Figure 2.16). Very similar band profiles can be obtained using the mass-transfer model (Eqnation 2.21), coupled with the Fowler-Guggenheim isotherm of adsorption (Equation 2.4), or with the multilayer isotherm (Equation 2.7). 

Above Th the smoothed values are given by a modification of the Guggenheim equation [67-gug]... 

Step 11. Fit the selected high temperature data with the modified Guggenheim equation using least squares with weighting factors r,. 

Fig. 1.8 Dependence of the mean activity coefficient y tC of NaCl on the square root of molar concentration c at 25°C. Circles are experimental points. Curve 1 was calculated according to the Debye-Hiickel limiting law (1.3.25), curve 2 according to the approximation aB = 1 (Eq. 1.3.32) curve 3 according to the Debye-Hiickel equation (1.3.31), a = 325nm curve 4 according to the Bates-Guggenheim approximation (1.3.33) curve 5 according to the Bates-Guggenheim approximation + linear term 0.1 C curve 6 according to Eq. (1.3.38) for a = 0.4nm, C = 0.055dm5-mor ...

The variable x is usually the mole fraction of the components. The last expression was first introduced by Guggenheim [5]. Equation (3.60) is a particular case of the considerably more general Taylor series representation of Y as shown by Lupis [6]. Let us apply a Taylor series to the activity coefficient of a solute in a dilute binary solution ... 

The Frumkin Equation (also referred to as the Frumkin-Fowler-Guggenheim, FFG, equation) has been specifically developped to take lateral interactions at the surface into account. In the FFG equation, the term 6 / (1-6) in (4.10b) is multiplied by the factor exp(-2 a0) which reflects the extent of lateral interactions... 

As our basis we chose the Bronsted-Guggenheim (6) equation for the mean activity coefficient YR x t ie electrolyte in a solution of several electrolytes having cation R and anion X. The mean activity coefficient Y R>x is given by... 

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