Isotherm deviations from

When one or more of the component elements of sorption is governed by a nonlinear relationship between the solution and the sorbed phase, the composite isotherm deviates from linearity. In these cases, modifications to the Freundhch isotherm have been developed (e.g., Lambert 1967 Weber et al. 1992) to express these conditions. 

In conclusion, the different shapes of isotherms describing equilibrium distributions of a contaminant, between geosorbents and aqueous or gaseous phases, depend on the sorption mechanism involved and the associated sorption energy. At low contaminant concentration, all models reduce to essentially linear correlation. At higher contaminant concentration, when sorption isotherms deviate from linearity, an appropriate isotherm model should be used to describe the retention process. 

The temperature and concentration dependences of the work function, , have been determined from photoemission on Na-Rb, Na-Cs, and K-Cs liquid mixtures. The ease of removing an electron from sodium increases linearly with increasing solute concentration at 25 °C when adsorption processes do not play a role. In the case where surface adsorption of the component with lower (f> occurs, the isotherms deviate from linearity. Where a compound, e.g. Na2Cs, is formed, a minimum exists in the isotherm.55... 

The Hybrid Method of Mass Balance (HMMB) This method is a modification of the MMB method in which, instead of measuring a series of values of C which is cumbersome, these concentrations are estimated through Eqs. 4.85 and 4.86 by using the isotherm parameters, , and determined by the MCV method. This hybrid approach employs MCV only to estimate the mezzanine concentrations, so it is more acciurate than MCV when the actual isotherm deviates from the Langmuir competitive isotherm. 

In the low concentration limit the adsorption isotherm is a linear law as was the partition coefficient, and just as the isotherm deviates from linearity outside of the low concentration limit, so too does the partition relation between the two liquid phases. 

When r = 1, 9 is very small and the adsorption increases linearly with the increase in 0 (Henry s type isotherm). On the other hand, when r = 10 the isotherm deviates from such a straight line and approaches a Langmuirian type with a near-plateau value at high 0 values. However, when r > 20, high affinity isotherms are obtained. In this case, the adsorption rises very steeply at low polymer concentrations and thereafter it reaches a pseudo-plateau region. The adsorption isotherms for chains with r = 100 and above are typical of those obtained experimentally for most polymers that are not too polydisperse (i.e. showing a steep rise followed by a nearly horizontal pseudo-plateau (which only increases by few percent per decade increase in 0. ... 

Type V isotherms deviate from Type IV curves by nearly perpendicular middle portions of the adsorption and the desorption branches often near relative gas pressures p/ps(T) = 0.5, indicating the existence of mesopores in which phase change like pore condensation may occur. 

As was estimated in Ref. [25] only very small isotherm deviations from hnearity of 1-2 K are enough to reach the critical resolved shear stress (CRSS) for dislocation multiplication in GaAs crystals at high temperatures ( 0.5 MPa). In the production of compound semiconductor crystals of diameters 100 and 150 mm it has been not been possible to reduce the thermal inhomogeneities and the related thermal-induced stress to a sufficiently low level at aU process phases in... 

Another distinguishing aspect of copolymer crystallization is that with increasing counit content, the crystallization isotherms deviate from the Avrami relation, given in Equation 11.6 [79], at progressively lower extents of crystallization [69,76]... 

At > 0.85 also capillary condensation in mesopores can take place. As a result, the experimental isotherm deviates from the calculated isotherm of the water vapor absorption. Investigations of isotherms of water vapor sorption by low-porous (5 n2 = 0-4 m /g] and highly porous (5 2 = HO cellophane samples... 

The isotherms related to monomolecular sorption processes were calculated by Eq. (7.38). If < 0.7, then the calculated adsorption or desorption isotherms were equal to the experimental data. However, at

0.7 the experimental isotherms deviate from the calculated values due to the starting of multimolecular sorption processes such as multilayer adsorption and capillary condensation causing the sorption hysteresis. The diameter of mesopores of cellulose samples estimated by Kelvin s equation was in the range from 4 to 12 nm. 

Figure 5 shows the isothermal data of Edwards (1962) for n-hexane and nitroethane. This system also exhibits positive deviations from Raoult s law however, these deviations are much larger than those shown in Figure 4. At 45°C the mixture shown in Figure 5 is only 15° above its critical solution temperature. Again, representation with the UNIQUAC equation is excellent.

We have repeatedly observed that the slowly converging variables in liquid-liquid calculations following the isothermal flash procedure are the mole fractions of the two solvent components in the conjugate liquid phases. In addition, we have found that the mole fractions of these components, as well as those of the other components, follow roughly linear relationships with certain measures of deviation from equilibrium, such as the differences in component activities (or fugacities) in the extract and the raffinate. 

Deviation from the standard isotherm in the high-pressure region offers a means of detecting the occurrence of capillary condensation in the crevices l>etween the particles of a solid and in any mesopores present within the particles themselves. A convenient device for detecting deviations from the standard is the t-plot . In the next section the nature and uses of t-plots will be discussed, together with a,-plots, a later development from them. As will l>e shown, both of these plots may l>e used not only for the detection of capillary condensation in mesopores, but also for showing up the presence of micropores and evaluating their volume. 

Evidence of a different kind is furnished by the fact that the Gurvitsch rule (p. 113) is often obeyed by systems showing Type I isotherms " the amounts of different adsorptives taken up by a given adsorbent, when expressed as a volume of liquid, agree within a few per cent. The order of agreement is illustrated by the typical examples in Table 4.1 for the adsorption of n-alkanes on ammonium phosphomolybdate, and in Table 4.2 which refers to a variety of adsorptives on a silica gel. It must be admitted, however, that there are cases where considerable deviations from the Gurvitsch mle are found, even though the isotherms are of Type 1. Thus, in Table 4.3 the variation in values of the saturation uptake is far outside... 

For banks of in-line tubes,/for isothermal flow is obtained from Fig. 6-43. Average deviation from available data is on the order of 15 percent. For tube spacings greater than 3D(, the charts of Gram, Mackey, and Monroe (Trans. ASME, 80, 25—35 [1958]) can be used. As an approximation, the pressure drop can be taken as 0.32 velocity head (based on V ) per row of tubes (Lapple, et al.. Fluid and Paiiicle Mechanics, University of Delaware, Newark, 1954). 

We developed a sensor for determination of content of phosphorars in metallurgical melts. In quality of ion conductor used orthophosphate of calcium which pressed in tablets 010 mm. Tablets (mass 1-2 g) annealed at a temperature 400°C during 7-10 h. Tablets melts then in a quartz tube and placed the alloy of iron containing 1 mass % P. Control of sensor lead on Fe - P melts. Information on activities (effective concentration) of phosphorars in Fe - P melts was received. It is set that the isotherm of activity of phosphorars shows negative deviations from the Raouls law. Comparison them with reliable literary inforiuation showed that they agree between itself. Thus, reliable data on activities (effective concentration) of phosphorars in metallic melts it is possible to received by created electrochemical sensor for express determination. 

Although a correlation between BET surface areas from 77 K nitrogen isotherms and methane uptake at 298 K and 3.5 MPa has been shown for many carbon adsorbents, [11, 20], deviations from this relationship have been observed [20]. However, as a primary screening process for possible carbonaceous adsorbents for natural gas, this remains a useful relationship. It should be noted that this correlation only seems to be applicable for active carbons. 

In practice, thermal cycling rather than isothermal conditions more frequently occurs, leading to a deviation from steady state thermodynamic conditions and introducing kinetic modifications. Lattice expansion and contraction, the development of stresses and the production of voids at the alloy-oxide interface, as well as temperature-induced compositional changes, can all give rise to further complications. The resulting loss of scale adhesion and spalling may lead to breakaway oxidation " in which linear oxidation replaces parabolic oxidation (see Section 1.10). 

At large distances the curve of Fig. 8b is a plot of — (c2/ r)> where t is the macroscopic dielectric constant of the solvent at the temperature considered. For small values of r the curve deviates from this value but at every point the slope of the curve must represent the mean intensity of the mutual attraction or repulsion at the particular temperature considered. If the curve of Fig. 86 for dissociation in solution is to be useful, every point on this curve must belong to the same temperature T. That is to say, when we consider any change in the distance r between the ions, we are interested in an isothermal change in r. 

Physicochemical properties of molten systems have an applied significance due to their wide use in both technological process planning and in production equipment design. Analysis of various melt properties versus different parameters of the melt enables to infer the interaction mechanism between the initial components, and in some cases, even to estimate the possible composition of the main complex ions formed in the melt [312]. From this point of view, the analysis of isotherms of physicochemical properties versus melt composition and of the magnitude of their deviation from ideal conditions is of most interest. 

Equations 4.55 and 4.57 are the most convenient for the calculation of gas flowrate as a function of Pi and Pi under isothermal conditions. Some additional refinement can be added if a compressibility factor is introduced as defined by the relation Pv -= ZRT/M, for conditions where there are significant deviations from the ideal gas law (equation 2.15). 

There are several reasons for deviations from the LHHW kinetics Surface heterogeneity, surface reconstruction, adsorbate island formation and, most important, lateral coadsorbate interactions.18,19 All these factors lead to significant deviations from the fundamental assumption of the Langmuir isotherm, i.e. constancy of AHa (and AHB) with varying coverage. 

Figure 3. Tg of PVC vs. content of VC (V) data measured by means of deviation from Flory-Huggins isotherm (5) (X) data measured thermomechanically (IS) (O) data obtained from limiting conversion (4).

They assumed that the mobilities of the dissociated species LL and K, wf and respectively, are independent of composition. They have applied this model to (Li, Cs)Cl using three experimental values, which are M[(0), uj 1), and M i( X 2) = 2 (X 2) at the Chemla crossing point. Since the calculated isotherms deviate considerably from the experimental ones, they suppose that the dependence of and K2 on the mixing ratio must be assumed. 

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