Isotherm determination method

The first step is devoted to the determination of adsorption isotherms. Several adsorption isotherm determination methods have been described [13], and we developed a laboratory apparatus allowing to determine very precise adsorption isotherms in supercritical fluids. 

Finally, note that the adiabatic and isothermal pipe methods produce results that are reasonably close. For most real situations the heat transfer characteristics cannot be easily determined. Thus the adiabatic pipe method is the method of choice it will always produce the larger number for a conservative safety design. 

The dew-point and non-isothermal isopiestic methods. These are methods where a temperature gradient is maintained between a sample held at a high temperature and a pure metal which is held at a lower temperature. In the dew-point method the vapour pressure of the volatile component is determined by direct observation of the temperature at which it condenses. 

Equations 10.47 through 10.49 constitute the basis for the method of isotherm determination by the perturbation method. The retention time of a small injection of a solute in a column equilibrated... 

In a paper concerned primarily with the determination of neutral sugars, Richey and coworkers examined the O-trimethylsilyl derivatives of 2-acetamido-2-deoxy-D-glucose and 2-acetamido-2-deoxy-D-galactose.171 Similar methods for determining these compounds have also been used in studies on human milk,545 yeasts and bacteria,1,2 blood-group oligosaccharides,405 and urine.118 These methods were also included in a model study on the isothermal determination of sugars.256... 

The isopiestic (isothermal distillation) method for the determination of molecular weights is closely related to the vapor pressure depression method.10 A weighed amount of standard is introduced into one leg of an apparatus and a weighed portion of the unknown is placed in the other leg. Solvent is introduced into the apparatus, which is then evacuated and thermostated. The solvent will distill from one solution to the other until the vapor pressures (and therefore mole fractions) of the two have equalized. If the solutions are ideal, or if the deviations from ideality are similar, equilibrium will occur when the mole fraction of the known equals that of the unknown. 

The molar mass of a newly synthesized organic compound was determined by the isothermal distillation method. In this procedure two solutions, each in an open calibrated vial, are placed side by side in a closed chamber. One of the solutions contained 9.3 mg of the new compound, the other 13.2mg of azobenzene (molar mass 182). Both were dissolved in portions of the same solvent. The experiment was untouched for three days, during which solvent distilled from one vial into the other until the same partial pressure of solvent was reached in both vials. After this, there was no further net distillation of solvent. Neither of the solutes distilled at all. The solution containing the new compound occupied 1.72mL and the azobenzene solution occupied 1.02mL. What is the molar mass of the new compound The mass of solvent in solution may be assumed to be proportional to the volume of the solution. 

In Figure 6.22a, the MPSD of the DAY zeolite calculated with the help of the SF method is reported. The DAY zeolite has an FAU framework, with a 12MR three-dimensional channel system with apertures of 7.4 A (see Section 2.5.1) [83], The obtained PSD for the DAY zeolite shows a clear maximum at about 7 A. In Figure 6.22b, the S-F MPSD of the SWCNT is reported [80], The calculated PSD for the SWCNT indicates that this material has a slightly heterogeneous distribution of pores with a clear, and highly populated, maximum at 13.5 A [80], The adsorption study, in both cases, was carried out using N2 adsorption at 77 K isotherms, obtained with a Quantachrome Autosorb-1 equipment for adsorption isotherm determination by the volumetric method [21],... 

In principle, a continuous procedure can be used to construct the isotherm under quasi-equilibrium conditions the pure adsorptive is admitted (or removed) at a slow and constant rate and a volumetric or gravimetric technique used to follow the variation of the amount adsorbed with increase (or decrease) in pressure. A carrier gas technique, making use of conventional gas chromatrographic equipment, may be employed to measure the amount adsorbed provided that the adsorption of the carrier gas is negligible. In all types of measurement involving gas flow it is essential to confirm that the results are not affected by change in flow rate and to check the agreement with representative isotherms determined by a static method. 

Catalyst surface areas prior to reaction were determined by application of the BET method to nitrogen physisorption isotherms determined at 77K. 

In principle, this method of calculating the differential enthalpy of adsorption could be applied to two adsorption isotherms determined at different temperatures, but the following restrictions must be kept in mind ... 

The as-method was used to analyse the nitrogen isotherms determined after the calcination of very small crystals of bayerite and nordstrandite (Aldcroft et al., 1968). Thus, each value of the external area, u(ext), in Table 10.11 was obtained from the... 

Various procedures have been used to evaluate the micropore capacity from the experimental isotherm data (e.g. the Dubinin-Radushkevich plot), but in practice these are all empirical methods. It should be kept in mind that no theoretical significance can be deduced from the fact that a particular equation gives a reasonably good fit over a certain range of an isotherm determined at only one temperature. In our view, a safer approach is to plot the amount adsorbed against standard data determined on a non-porous reference material (i.e. to construct a comparison plot or Os-plot)-... 

The X-ray diffraction (XRD) patterns were collected on a RIGAKU diffractometer, operated at 30kV and 20 mA and using CuK radiation (A,=0.15405nm). The specific surface areas of the catalysts were tested on a home-made analyzer, following the BET method from air isotherms determined at liquid nitrogen temperature. The structure properties of the catalysts were characterized by IR spectroscopy using a PE-783 IR spectrometer. The amount of samples were mixed with KBr in form of disks, and the spectra were taken at room temperature. 

The aim of this work is to test and to compare the performances of various nitrogen adsorption-desorption isotherms analysis methods. These models were applied to model samples obtained by mechanically mixing two micro- and mesoporous solids respectively in perfectly known proportions. The relevant morphological characteristics of the porous texture of the mixtures, such as the specific surface and volume, are physically additive. A criterion that allows determining the reliability of the analysis methods tested is thus to check the linearity of the relation between a given parameter and the weight percentage of the pure solids. 

Besides the isothermal kinetic methods mentioned above, by which activation parameters are determined by measuring the rate of dioxetane disappearance at several constant temperatures, a number of nonisothermal techniques have been developed. These include the temperature jump method, in which the kinetic run is initiated at a particular constant initial temperature (r,-), the temperature is suddenly raised or dropped by about 15°C, and is then held constant at the final temperature (7y), under conditions at which dioxetane consumption is negligible. Of course, for such nonisothermal kinetics only the chemiluminescence techniques are sufficiently sensitive to determine the rates. Since the intensities /, at 7 ,- and If at Tf correspond to the instantaneous rates at constant dioxetane concentration, the rate constants A ,- and kf are known directly. From the temperature dependence (Eq. 32), the activation energies are readily calculated. This convenient method has been modified to allow a step-function analysis at various temperatures and a continuous temperature variation.Finally, differential thermal analysis has been employed to assess the activation parameters in contrast to the above nonisothermal kinetic methods, in the latter the dioxetane is completely consumed and, thus, instead of initial rates, one measures total rates. 

Several methods allowing the determination of adsorption isotherms exist. The most economical one is the use of saturated salt solutions, generating known water partial pressures, followed by the measurement of the silica weight uptake. However, the time needed to obtain a suitable adsorption isotherm is excessively long (several weeks), and experimental precautions need to be taken to achieve meaningful results. Specialized equipments, quite expensive ones, have been developed for the water adsorption isotherm determination, based on highly sensitive microbalances. 

Application of the Os Method for Analysing Benzene, Dichloromethane and Methanol Isotherms Determined on Molecular Sieve and Superactivated Carbons... 

Application of the a, method for analysing benzene, dichloromethane and methanol isotherms determined on molecular sieve and superactivated carbons... 

Determined by N2 adsorption isotherms. Determined by gravimetric method. 

A close set of equations was formulated in Ref. 16, related to the capillary pressure isotherms determined by the method of standard porosimetry [60], In the latter procedure, the equilibrium amount of the wetting liquid is measured in the porous sample under study. Simultaneously, the amount of the wetting liquid is measured in the standard specimen with a genuine porous structure, in which the capillary equilibrium is established. The standards are kept in thermodynamic equilibrium with the sample. The comparison of the amount of wetting liquid in the membrane with the pore-radius distribution in the standards, enables one to record (with a minimum of theoretical assumptions), the volume-size and surface-size distribution curves, specific pore-space surface area, and absorption isotherm in the membrane of interest, for various wetting liquids. 

An alternative approach to isotherm determination is offered by the so called Pulse methods There are two types of pulse methods, the elution of a pulse... 

In contrast to the well-developed thermodynamic methods for determining gas/ liquid equilibriums the theoretical determination of adsorption isotherms is not yet feasible. Only approaches to determining multi-component isotherms from experimentally determined single-component isotherms are known. Such approaches are explained in more detail in Section 2.5.2.3. Careful experimental determination of the adsorption isotherm is therefore absolutely necessary. The different approaches for isotherm determination are discussed in Chapter 6.5.7. 

Fig. 6.17 Principle of different static methods for isotherm determination.

The method is advantageously combined with the frontal analysis method, which also requires a concentration plateau and thus shares the disadvantage of high sample consumption if operated in open mode. As indicated in Fig. 6.24, the measurement procedure starts at maximum concentration. This concentration plateau is reduced step-by-step by diluting the solution. To reduce the amount of samples needed for the isotherm determination the experiments can be done in a closed loop arrangement (Fig. 6.17). It is also possible to automate this procedure. 

Most determination methods finally lead to discrete loading versus concentration data that have to be fitted to a continuous isotherm equation. For this purpose it is advised to use a least-squares method to obtain the parameters of the isotherm. Nonlinear optimization algorithms for such problems are implemented in standard spreadsheet programs. To select an isotherm equation and obtain a meaningful fit,... 

For the linear part of the isotherm, the Henry coefficient may be determined separately by pulse experiments (Section 6.5.7.2). If a significant deviation from the value obtained with the isotherm equation is encountered, additional experiments in the low concentration range should be carried out. These can be used to clarify whether the isotherm equation (e.g. adding an extra linear term as in Eq. 2.47) or the method of isotherm determination must be changed. 

Frontal analysis is a very popular method of isotherm determination, and rightly so. It has been apphed to the determination of a great number of equi-Ubrlum isotherms, in many modes of chromatography. Among others, it has been used for the measiu ment of the isotherms of peptides [38] and proteins [8,41,65]. It is the method most often used in our group. 

The advantages and drawbacks of FA relative to those of the other methods of isotherm determination are discussed later, in Section 3.6.4. 

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