Isotherm fitting

The existence of this situation (for nonporous solids) explains why the ratio test discussed above and exemplified by the data in Table XVII-3 works so well. Essentially, any isotherm fitting data in the multilayer region must contain a parameter that will be found to be proportional to surface area. In fact, this observation explains the success of Ae point B method (as in Fig. XVII-7) and other single-point methods, since for any P/P value in the characteristic isotherm region, the measured n is related to the surface area of the solid by a proportionality constant that is independent of the nature of the solid. 

Typically, sorption isotherms are constructed for a single food ingredient or food system. An alternative approach is to plot the moisture content versus water activity (or relative vapor pressure) values for a variety of as is food ingredients and food systems. The result is a composite food isotherm (Figure 17). The composite isotherm fits the typical shape observed for a sorption isotherm for an individual food system, with a few products falling above or below the isotherm curve (chewing gum, honey, raisins, bread, and colby and cheddar cheeses). Slade and Levine (1991) were the first to construct such a plot using moisture content and aw values from van den... 

Figure 6. Dimensionless interaction parameters as determined from isothermal fits at various temperatures ((- -) KBr-water ( ) NaCl—water)...

The interaction parameters are weak, linear functions of temperature, as shown in Table 5, Table 6 and Figure 6. These tables and figure show the results of isothermal fits for activity coefficient data of aqueous NaCl and KBr at various temperatures. The Pitzer equation parameters are, however, strongly dependent on temperature (Silvester and Pitzer, (23)). 

Figures 7 and 8 illustrate the behavior of the intercepts and slopes from Figure 6 corresponding to the functional forms of Equation 29. The error bars on Figures 7 and 8 represent one standard deviation as determined from isothermal fits. The intercepts have deviations on the order of 0.5% which is consistent with an apparatus analysis. The slopes, however, have much larger uncertainties ranging up to 15%. Increasing the pressure range would greatly reduce this large and important error.

Figure 6. Experimental data for determining slopes ((-------) isotherm fit (----)...

Figure 7. Predicted intercept as a function of temperature ((— —) isotherm fit (---------------------------- -------) surface fit)...

Brunauer, Deming, Deming and Teller, based upon an extensive literature survey, found that all adsorption isotherms fit into one of the five types shown in Fig. 3.1. 

The most notable criticism of the Langmuir adsorption equation concerns the simplifying assumption that the heat of adsorption is independent of surface coverage, which, as discussed in the next section, is not likely to be the case. Nevertheless, many experimental adsorption isotherms fit the Langmuir equation reasonably well. 

Figure 1.7 ITC data at 25 °C for the binding of NBu4+C1 by 1.9 in nitromethane - the top plot represents the raw data with the calorimetric response in ptcal s 1 for each addition of NBu4+C1 while the lower plot is the titration isotherm fitted to a 1 1 model with kcal per mol NBu4+C1 added vs. mole ratio of NBu4 1 Cl to 1.9. (Reproduced with permission from [8] 2006, American Chemical Society).

Isotherm fitting forms. There are two published measurements of the room temperature isotherm for /3-HMX. As part of a series of experiments to determine isotherms for various explosive crystals, Olinger, Roof, and Cady reported in 1978 an x-ray determination of the room temperature lattice parameters of / -HMX in the pressure interval 0 < p < 1A1 Gpa [68]. They fit the isotherm to an equation of state (EOS)... 

In Fig. XVII.l the points labeled by O are those for a Freundlich isotherm, fitted by B 0.665 and 6 — 0.86 to the complex l.angmuir isotherm. Over the range of concentrations (A)/X2.i from 0.5 to 8, the two isotherms have a maximum deviation of 8 per cent and an average deviation of about 2 per cent, showing the closeness of the fit which is attainable between them. The fit could have been shifted to lower pressures by choosing different parameters. 

Figure 21 summarizes experimental isotherm data for several LSC systems 4, 2). In each case, the solid curve is calculated from Eq. (39). It should be noted that values of WS are calculated from the B-solvent molecular weight and dimensions, whereas the Bb values are based on LSC retention data and derived values ofcb, A,Wb, etc. Thus, these calculated curves are not adjusted to the experimental data of Fig. 21—they represent totally independent calculations. This should be contrasted with other attempts at isotherm fitting, where up to three adjustable parameters are invoked to achieve as close a fit as possible. 

Commercially available coal based activated carbons Filtrasorb-400, Norit ROW 0.8 and Norit ROX 0.8 were chosen as the adsorbents. The adsorbates used for determination of the pore network connectivity of the carbons were ethyl propionate, ethyl butyrate, ethyl isovalerate, isobutyl acetate, benzaldehyde, hexyl acetate, methyl salicylate and 2-ethyl hexyl acetate. For the evaluation of the pore network connectivity, we used the fitted maximum capacity obtained fi om isotherm fitting to calculate the accessible pore volume [7]. 

This study firstly aims at understanding adsorption properties of two HSZ towards three VOC (methyl ethyl ketone, toluene, and 1,4-dioxane), through single and binary adsorption equilibrium experiments. Secondly, the Ideal Adsorbed Solution Theory (IAST) established by Myers and Prausnitz [10], is applied to predict adsorption behaviour of binary systems on quasi homogeneous adsorbents, regarding the pure component isotherms fitting models [S]. Finally, extension of adsorbed phase to real behaviour is investigated [4]. 

In all cases the BET isotherm fits satisfactory with correlating index of about... 

The right hand graphs show the composite isotherms (upper panel) and enthalpies (lower panel). The isotherm fits are indistinguishable on this scale. 

Fig. 3.2. Adsorption isotherms of gases on the microporous silica membranes (a) the isotherms (b) the plot of Eq. (3.6). The solid lines in (a) are also the Langmuir isotherm fits.

Figure 14.13 Adsorption-desorption isotherms of naphthalene on (plots a and b correspond to two different samples of C q small aggregates ). Solid diamonds, adsorption data empty diamonds, desorption data solid line, Freundlich isotherms (fitted with the adsorption data). (Reprinted with permission from Ref. [102]. Copyright 2004 American Chemical Society.)...

Will the Temkin isotherm fit the n-hexane data shown in Prob. 8-1 ... 

What kind of isotherm fits these results ... 

For given a set of data, which isotherm equation (or equations) fits best And what is the impact of the quality of fit on predicted performance Unfortunately, neither qnestion cam be answered fully. It is fair to say that the greater the number of parameters in an equation, the more likely it is to fit well and the better it fits, the more valid will be snbseqnent process simulations. That should be balanced against the statistical significance of the parameters. Finally, the isotherm fit that best accommodates heat effects and multicomponent aspects, if any, will be superior. An example that illustrates different degrees of quality of fit of four equations to one set of data is provided in Section 14.5.4. Specialized programs are available that fit equations and plot the results. 

FIGURE 14.14 Adsorption data of water vapor on silica gel at 25°C. Isotherm fits Brunauer-Deming-Dem-ing-Teller Freundlich, Langmuir, and Redhch-Peterson equations. 

In order to test which isotherm fits the data best, linear regression analysis was carried out for the adsorption data. The results are as follows The mass fraction of the solute in the feed can be calculated by assuming the density of the feed is the same as water. 

Figure 9.6b shows a q-p plot of the experimental data and the corresponding predictions of the Langmuir and Freundlich isotherms. It is evident from the plot that in this case, the Langmuir isotherm fits the data significantly better than the Freundlich isotherm. 

Figure 9.6b Langmuir and Freundlich isotherms fitted to data of Example 9.4.

The sorption model was derived from studies to determine the quantity of solvent adsorbed to the surface of silica. Experimental evidence suggests that silica adsorbs water to its surface, some of which can be removed by either heating to 110 °C, or by sequential organic phase extraction. More can be removed at very high temperatures, but this also results in the breakdown of silanol groups (Scott, 1982). Measurement of the adsorption of different concentrations of polar modifiers in an inert solvent allows the adsorption isotherms to be calculated (Fig. 6.2). When the concentration of the modifier is low, the isotherm fits closest to a monolayer function. When the polar modifier is present at a high concentration a bilayer adsorption isotherm function is produced ... 

The first two-box model includes a reversible Freundlich reaction followed by an irreversible first order process. There are four independent parameters the Freundlich parameters, K and n, a reversible rate constant, r, and an irreversible rate constant, k. Initial estimates for the Freundlich parameters were obtained from independent isotherm fits of the data after approximately 2 weeks of adsorption. Fitting the kinetic data by eye provided the initial estimates for r and k. 

Figure 9. Isotherm fits of the surface precipitation model for the sorption of ferrous iron on iron oxides ( ) magnetite, ( ) goethite, ( ) lepidocrocite, (A) hematite (for parameter values see Table I, Fe(II)soi = concentration of dissolved ferrous iron, rpe= concentration of surface-bound ferrous iron per total concentration of iron oxide, pH 7.2,1=20 mM, T=25 C, 25 m L l, teq=15 min) adapted from (7).

Table I. Calculated Parameter Values of Isotherm Fits of the Surface...

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