Adsorption isotherms inverse method

While DFT allows us to calculate values for q(p, e), it of course provides no analytic form for the function, and in general the form of f(e) is also unknown. However, by using carefully designed numerical methods, model isotherms calculated by MNLDFT can be used in carrying out the inversion of the discrete form of the integral equation of adsorption. In this way one can determine the effective adsorptive potential distribution of the adsorbent from the experimental adsorption isotherm. The method used can be expressed by... 

Using this methodology via measurement of adsorption isotherms, Guiochon and coworkers investigated site-selectively the thermodynamics of TFAE [51] and 3CPP [54] on a tBuCQD-CSP under NP conditions using the pulse method [51], the inverse method with the equilibrium-dispersive model [51, 54], and frontal analysis [54]. 

Among the different chromatographic approaches available for measnring adsorption isotherms, three fnndamental techniqnes, the FA, the perturbation on the platean (PP), and, finally, the inverse method (IM), will be discnssed briefly. 

Because of the importance of these characterization methods, an experimental test of the thermodynamic consistency of the models was felt to be desirable. One such test is to determine how well the models and inversion methods used predict the temperature dependence of the isotherms as represented by the isosteric heat of adsorption. 

Whereas conventional chromatographic methods manipulate the surface energetics of sorbents to separate fluid mixtures, inverse gas chromatography uses known properties of fluids to characterize surface properties of solids. Specifically, Lewis acids and bases are used, in this study, as probes to deduce the nature and extent of solid/gas attraction from the shape of chromatograms, which are transformed adsorption isotherms. IGC can determine the specific surface (m2/g) of the substrate, whether the surface is acidic, basic, amphoteric, or neutral, and whether the surface is homogeneous or heterogeneous. 

Figure 4.33 Top. Simultaneous fitting to a competitive bi-Langmuir model of the chromatograms obtained with a large (50.7 mg) and a moderate (10.14 mg) sample of the racemic mixture of 1-indanol. Bottom. Comparison of the FA adsorption data points (s)rmboIs) and the best competitive bi-Langmuir isotherms obtained by the inverse method (lines) for the racemic mixture. Reproduced with permission from A. Felinger, D. Zhou, G. Guiochon, f. Chromatogr. A, 1005 (2003) 35 (Figures 7 and 8).

For experimental verification of these models, Foplewska et al. [33] used binary mixtures of methanol-water and acetonitrile-water as the mobile phases and measured the adsorption equilibrium isotherms of cyclopentanone on two similar adsorbents having different degrees of sruface heterogeneity, a Cis non-endcapped and a Cig endcapped silica. Ehie to its structure, cyclopentanone exhibits affinity for adsorption on the bonded alkyl chains and for the polar, im-covered silica sruface of the adsorbent. Overloaded elution bands of cyclopentanone in piue water were recorded (Figrue 15.3) and the isotherms were derived using an inverse method (see Chapter 3). Five independent parameters (the excess coefficients and the eqiulibrirun constants for partition-adsorption and for... 

Two methods can be used for the assessment of the 7s of divided solids contact-angle measurements and adsorption processes. The drawbacks of the contact-angle measurements are associated with surface roughness of the samples. As for the adsorption process, determination of the components of the surface free energy of the solid is based on interpretation of adsorption isotherms, either complete (calculation from spreading pressures) or only from the first linear part of the isotherm. In this respect, inverse gas chromatography (IGC), which appears to be the technique of choice (17), was extensively used in this study. 

Cornel, J., Tarafder, A., Katsuo, S., and Mazzotti, M. (2010) The direct inverse method a novel approach to estimate adsorption isotherm parameters. J. Chromatogr. A, 1217, 1934-1941. 

When the system equations are linear (that is when the adsorption isotherm is linear or the system is perturbed incrementally), we can apply the method of Laplace transform to solve the set of equations and obtain the inverse by either the method of residues or a numerical inversion scheme. For the two types of input, the impulse and the square input, the inversion is not necessary if we are interested in using the response to extract the system parameters. If this is our goal which is the case for the diffusion cell method, then the method of moment can be useful for this purpose. 

In the first situation, a carrier fluid (which is usually an inert fluid but this is not necessary) is passed through the column, and once this is stabilised a tracer is injected into the column with a concentration of Co(t) at the inlet. The concentration is chosen such that the adsorption isotherm of this tracer towards the solid packing is linear. This results in a set of linear equations which permit the use of Laplace transform to obtain solution analytically. Knowing the solution in the Laplace domain, the solution in real time can be in principle obtained by some inversion procedure whether it be analytically or numerically. However, the moment method illustrated in Chapter 13 can be utilised to obtain moments from the Laplace solutions directly without the tedious process of inversion. 

Inverse gas chromatography provides accurate thermodynamic data for solid surfaces [20]. It can be used to determine the DN and AN values of oxides and to establish adsorption isotherms [20, 21]. In this method, the solid is placed in a gas chromatographic column as the stationary phase, and probe gases are introduced into an inert carrier gas. The measured quantity is the retention time, which is related to the adsorption interaction energy and the equilibrium between adsorbed and non-adsorbed gas molecules. 

The adsorption of gas onto a solid surface can also be used to estimate surface energy. Both inverse gas chromatography (IGC) and isotherm measurement using the BET method [19] have been used. Further discussion and detailed references are given by Lucic et al. [20] who compare the application of IGC, BET and contact angle methods for characterising the surface energies of stearate-coated calcium carbonate fillers. 

Already with Jantti s 3PM a timesaving of the stepwise measurement of nitrogen isotherms up to 70 percent was possible 2PM is even faster. For both methods we show that the gain of the shortening of the measurement time is accompanied by a drawback, i.e. the increase of measurement inaccuracy, which is shown to be inversely proportional to the square of the measurement time. In case the molecular adsorption mechanism is not a given factor before the start of the measurement, the application of the time saving measurement will involve extra uncertainties. 

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