Stepwise isotherm

Temperature-risiag elution fractionation (tref) is a technique for obtaining fractions based on short-chain branch content versus molecular weight (96). On account of the more than four days of sample preparation required, stepwise isothermal segregation (97) and solvated thermal analysis fractionation (98) techniques usiag variatioas of differeatial scanning calorimetry (dsc) techniques have been developed. 

Reproducibility of the used heating rate for comparison with other investigations. Selection of small and large heating rates depending on the type of investigation. Equilibrium measurements (e.g. 0.2 °C7min) or stepwise isothermal. Exploratory runs (standard rate 4-6 °C/min). 

Fig. 14. Heating programs linear heating and cooling (A), isothermal (B), stepwise isothermal (C), multistep program (D), cycling (E)...

In such cases where even slow linear heating is too fast, stepwise isothermal heating is recommended (Fig. 14 C). Some apparatus have a built-in device or, alternatively, the realization is very simple made possible with two electrical switch clocks (see Sect. 4.1). 

All samples for the tests discussed here were formed during more or less linear cooling, if, however, a stepwise isothermal cooling process was used and stopped above room temperature, a better crystalline substance was obtained. 

Fig. 66. Stepwise, isothermal vaporization of gold in a thermobalance with Knudsen ceil...

Polley, Schaeffer, and Smith (59) observe stepwise isotherms for argon, nitrogen and oxygen on graphite at -195° C. The steps become more pronounced as the graphiti-zation temperature of the carbon black adsorbate is increased. The steps appear at multiples of 0.85 Vm and are to be expected for homogeneous surfaces. 

Wissing S, Craig DQM, Barker SA, Moore WD. An investigation into the use of stepwise isothermal high sensitivity DSC as a means of detecting drug-excipient incompatibility. Int J Pharm 2000 199 141-150. 

Many adsorption isotherms are borderline cases between two or more of the above types. In addition, there are some isotherms which do not fit into Brunauer s classification at all, the most notable being the stepwise isotherms, an example of which is given in Figure 5.6. Stepwise isotherms are usually associated with adsorption on to uniform solid surfaces, each step corresponding to the formation of a complete monomolecular adsorbed layer (see page 133). 

Figure 5.6 Stepwise isotherm for the adsorption of krypton at 90 K on carbon black (graphitised at 3000 K)166 (By courtesy of the Canadian Journal of Chemistry)...

Equation (2.72) is not easy to use in the general case in which the spreading pressure is unknown. But in the particular case of stepwise isotherms where there are two adsorbed phases in equilibrium with the gaseous adsorptive (i.e. in the case of a univariant adsorption system), Larher (1968, 1970) showed that the isosteric method may be used with the transition pressure p" to give integral molar energies un and entropies sn of the quasi-layer ... 

A systematic study of krypton adsorption on exfoliated graphite was subsequently undertaken by Thorny and co-workers (Thorny and Duval, 1969 Thorny et al., 1972). Their stepwise isotherm, determined at 77.3 K, is shown in Figure 4.1. The layer-by-layer nature of the physisorption process is clearly evident - at least up to four molecular layers. This isotherm shape is remarkably similar to that of the krypton isotherm on graphitized carbon black reported by Amberg et al., (1955). 

Figure 9.9. Stepwise isotherm of CH4 on graphite foam (open circles) and exfoliated graphite (solid circles) at 77.3 K (reproduced courtesy of Bienfait eta ., 1990).

The stepwise isotherms of type VI are only observed under a number of idealized conditions for uniform non-porous surfaces. Ideally, the step-height represents monolayer capacity In the simplest case, it remains constant for two or three layers. Argon or krypton adsorption on graphltized carbon blacks at liquid nitrogen temperature are amongst the best examples. 

Figure 1. Comparison Neretnieks and Stepwise Isotherm Linealization Technique...

Sorption isotherms for lactose are shown in Fig. 4, compressible sugar in Fig. microcrystalline cellulose in Fig. and aspirin in Fig. 7. The isotherm for lactose has been adapted from data in the Handbook of Pharmaceutical Excipientthe stepwise isotherm expected is not evident. The dashed line in Fig. 4 is based on the statement that the monohydrate forms at 70% RH. 

Thorny, A. and Duval, X. (1994). Stepwise isotherms and phase transitions in physisorbed films. Surf. Sci., 299—300, 415-25. 

Stepwise isothermal analysis This has been described (46,47) as a method whereby reaction temperatures and mass loss steps can be accurately characterized by a complementary combination of simultaneous TG and DTA methods. The sample is heated at a slow, constant rate until a reaction is detected and, on reaching a specified (low) rate of mass loss, the temperature is maintained constant. When this rate process has been completed, indicated by diminution to a second specified (relatively much lower) rate, the slow rate of temperature increase is recommenced. Thus, each successive rate process is completed isothermally and both the mass loss (evidence of stoichiometry) and the temperature of that step are determined precisely. 

Temperature may not always be raised in a linear fashion. In the case of CRT A (Controlled Rate Thermal Analysis), the heating rate is varied in such a manner as to produce a constant rate of mass loss. Alternatively a sinusoidal temperature rise is superimposed on the linear rise this is known as Modulated TG and allows the continuous calculation of activation energy and pre-exponential factor during a run. Sometimes a Temperature Jump (or stepwise isothermal) " is used, where temperature is held constant for a time, then jumped rapidly to a higher constant temperature (usually quite close in temperature). All of these procedures are supposed to help in the determination of kinetics of reaction. Another system accelerates the temperature rise when no mass loss is experienced, i.e. between reactions. The rate is slowed to a low value during mass loss. Some manufacturers call this High Resolution TG and an example follows. 

A stepwise isothermal mode where the sample is heated until a reaction is detected and then held constant until that reaction is complete. 

Under certain circumstances it is possible to obtain adsorption isotherms that correspond to a layer-by-layer mechanisms. This is a very well known fact in gas adsorption on homogeneous flat surfaces [151] and is known as layering. At the solid-liquid interface stepwise isotherms are also observed, nevertheless one step and sometimes two are found [114]. Sellami et al. [152] studied the adsorption of 2,5-dimethylpyridine (DMP) on silica from aqueous solutions. The experimental results [153] clearly show the stepwise character of the isotherms, which could either mean multilayer adsorption or layering. In their paper [152] they addressed the question of using the adsorption isotherm to discriminate between... 

If E, > Ej > Ej >. .. > El, this type of isotherm will lead to stepwise isotherm, with the j-th step occurring at the pressure ... 

This type of stepwise isotherm is not observed in practice because discrete values of the adsorption energies are not followed by practical solids. By allowing for the distribution of adsorption energy, which we will show in the next section, Halsey (1948) and others developed a new multilayer isotherm, which is known in the literature as the FHH isotherm. 

It is difficult to conceive a molecular interpretation of the stepwise isotherms, because rf a lack of systematic measurements on the thickness and density of the layers, the number of contact points, the structure of the layers, etc. It is observed, however, that the quantity adsorbed from a mixture is greater than that of the individual components. 

The trans fraction isolated by Ag -TLC can also be used to analyze the trans monoenes other than 18 1. However, this requires a 10-fold concentration of the trans fraction because the other monoenes occur at much lower levels. At the higher sample load the minor trans isomers of 16 1, 20 1,22 1, and 24 1 are resolved using a stepwise isothermal temperature program starting a 120°C (15). The individual isomers of 16 1 and 20 1 (39) in milk fat and industrial fats were analyzed by maintaining the GC temperature at 125°C. The different trans monoene isomers can be quantitated... 

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