Guggenheim equation modified

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

Various attempts have been made to modify the Langmuir model. One of the best known is that of Fowler and Guggenheim (1939), which allowed for adsorbate-adsorbate interactions in a localized monolayer on a uniform surface. However, on an empirical basis the Fowler-Guggenheim equation turns out to be no more successful than the original Langmuir isotherm. The highly complex problem of localized adsorption on heterogeneous surfaces has been discussed by Rudzinski and Everett (1992). 

Saturated liquid density (5) the data were fitted to the modified Guggenheim equation as proposed by Haynes and Hiza (8). 

Guggenheim Equations 1 and 2 are in corresponding-states form that is, the reduced-form equations pr(Tr) explicitly contain no material constants and are formally applicable to all substances (in practice, to substances having small nearly spherical molecules, with modest accuracy). The desirable corresponding-states formulation will be retained in the equation presented here in a modified form, by including (as is now usual) the Pitzer-Curl acentric factor w as an explicit material constant ... 

Mh parameter in modified Guggenheim equation for density at temperatnres >71,... 

D. The Uniform and Consistent interpretation of the Modified Fowler-Guggenheim Equation... 

FIG. 7 The modified Fowler-Guggenheim equation can describe four types of isotherm (Types I, III, and y and condensation). 

B. The Modified Fowler-Guggenheim Equation Applied to Heterogeneous Surfaces (FT Equation)... 

Step 11. Calculate the least squares fit of the selected high tenqjerature values to the modified Guggenheim equation using weighting factors Wj. 

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

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.)...

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. 

These Nernstian equations should be modified to indicate the effect of the electric field superimposed on the chemical potential of the surface hydrogen/metal hydride. One possible way to treat this is with the thermodynamic concept of electrochemical potential defined by Guggenheim. The electrochemical potential is defined as ... 

Table 4.2 summarizes the isotherm equations discnssed so far pins some others, added for comparison. The first four have been discussed earlier in the text. The Guggenheim-Anderson-de Boer isotherm (GAB) is a multilayer sorption equation, a modified form of the BET equation that has found wide application in food science and technology (Timmermann 2003 and references therein). Basically, it does not consider the layers beyond the first as the liquid adsorbate but as an intermediate... 

At first the BET equation was derived from the kinetic considerations analogous to those proposed by Langmuir while deriving the monomolecular adsorption isotherm. First, statistical thermodynamic derivation was carried out by Cassie [123]. Lately, a slightly modified derivation has been proposed by HiU [124-126], Fowler and Guggenheim [127]. 

The above equation is modified when interaction between adsorbing molecules are taken into account. Fowler and Guggenheim (1939) gave... 

Measurements of the pH in aqneous-organic solvent mixtures and in nonaqueous mixtures pose similar problems, but the need for such data is extensive, for instance for mobile phases used in chromatography, for electrochemistry, and for corrosion studies. When in cell (8.3) the solvent S is not pure water but its mixture with another solvent or a non-aqueous solvent altogether. Equation 8.4 still holds, but Ihe activity coefficient of the chloride ion must be changed, using the modified Bates-Guggenheim... 

It has been proven that the consistent Langmuir equation has the function i/ miC ) defined by Eq. (109) therefore, the consistent (modified) Fowler-Guggenheim (mFG) equation should have the following ftmetion ... 

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