Concave isotherm

The viscosity isotherm of the system with nitrobenzene is according to Usa ov jch [S], the result of the formation of an unstable compound, whose viscosity is low er than that of the more viscous component. According to the views expressed by one of us [3], however, this isotherm is the result of the formation of a compound of low stability, but whose viscosity is not necessarily lower than the viscosity of the more viscous component. The contribution of the compound to the viscosity of the mixtures is in this case also insufficient to produce a viscosity maximum, but is nevertheless apparent in the part- of the viscosity isotherm concave towards the composition axis. 

Fig. 16.18 Optical inhomogeneities of ruby crystals as a function of the growth isotherms (concave, flat, convex) and annealing temperature T. i, is the optical resolution power.

An example from Dubinin s paper illustrates the application of the new equation. For benzene at 293 K, the plot of Equation (4.18) with m = 2 gave a line concave to the log (p°/p) axis. A revised value of m was therefore required, and was obtained as follows. A provisional value of IVq (= 408 mm g ) was estimated from the plateau of the isotherm the relative pressure (p/p°), corresponding to IV/tV = 0-368 was then read off, and a first value of S calculated from... 

I (curve D). Thus the micropores had been able to enhance the adsorbent-adsorbate interaction sufficiently to replace monolayer-multilayer formation by micropore filling and thereby change the isotherm from being convex to being concave to the pressure axis. 

The 2eohte and the resin adsorbents show different adsorption isotherm characteristics, particularly at higher concentration (51). The resin adsorbent isotherm is slightly concave upward, whereas the 2eohte isotherm is linear, or even slightly concave downward. Resins, therefore, have an advantage in a UOP Sarex operation that involves high feed-soHds concentration. 

McCabe-Thiele diagrams for nonlinear and more practical systems with pertinent inequaUty constraints are illustrated in Figures 11 and 12. The convex isotherms are generally observed for 2eohtic adsorbents, particularly in hydrocarbon separation systems, whereas the concave isotherms are observed for ion-exchange resins used in sugar separations. 

Historically, isotherms have been classified as favorable (concave downward) or unfavorable (concave upward). These terms refer to the spreading tendencies of transitions in fixed beds. A favorable isotherm gives a compact transition, whereas an unfavorable isotherm leads to a broad one. 

The specific concave shape of the isotherms of the properties of KC1 -K2TaF7 melts can be explained by the following ligand-replacement interaction ... 

Peaking and Non-isothermal Polymerizations. Biesenberger a (3) have studied the theory of "thermal ignition" applied to chain addition polymerization and worked out computational and experimental cases for batch styrene polymerization with various catalysts. They define thermal ignition as the condition where the reaction temperature increases rapidly with time and the rate of increase in temperature also increases with time (concave upward curve). Their theory, computations, and experiments were for well stirred batch reactors with constant heat transfer coefficients. Their work is of interest for understanding the boundaries of stability for abnormal situations like catalyst mischarge or control malfunctions. In practice, however, the criterion for stability in low conversion... 

So far, it has been assumed that elution is independent of analyte load or the presence of multiple components in a mixture. If this condition holds, then the analyte concentration in the mobile phase is directly proportional to the concentration in the stationary phase, no matter what the concentration is. Experimentally, this could be determined by incubating various concentrations of an analyte with a fixed amount of stationary phase and measuring the amount adsorbed. A plot of the concentration of analyte in the mobile phase on the x-axis vs. that in the stationary phase on the y-axis would be linear, and such a plot is called a "linear isotherm". A convex isotherm implies that tailing would be expected, and a concave isotherm implies that fronting is expected. 

The equilibrium adsorption characteristics of gas or vapor on a solid resemble in many ways the equilibrium solubility of a gas in a liquid. Adsorption equilibrium data are usually portrayed by isotherms lines of constant temperature on a plot of adsorbate equilibrium partial pressure versus adsorbent loading in mass of adsorbate per mass of adsorbent. Isotherms take many shapes, including concave upward and downward, and S-curves. Equilibrium data for a given adsorbate-adsorbent system cannot generally be extrapolated to other systems with any degree of accuracy. 

In some systems, three stages of adsorption may be discerned. In the activated alumina-air-water vapour system at normal temperature, the isotherm is found to be of Type IV. This consists of two regions which are concave to the gas concentration axis separated by a region which is convex. The concave region that occurs at low gas concentrations is usually associated with the formation of a single layer of adsorbate molecules over the... 

The regions of Type IV and Type V isotherms which are concave to the gas-concentration axis at high concentrations correspond to the bulk condensation of adsorbate in the pores of the adsorbent. An equation relating volume condensed to partial pressure of adsorbate and pore size, may be found by assuming transfer of adsorbate to occur in three stages ... 

Equation 17.75 is important as it illustrates, for the equilibrium case, a principle that applies also to the non-equilibrium cases more commonly encountered. The principle concerns the way in which the shape of the adsorption wave changes as it moves along the bed. If an isotherm is concave to the fluid concentration axis it is termed favourable, and points of high concentration in the adsorption wave move more rapidly than points of low concentration. Since it is physically impossible for points of high concentration to overtake points of low concentration, the effect is for the adsorption zone to become narrower as it moves along the bed. It is, therefore, termed self-sharpening. 

As is the case with adsorption isotherms, those curves in Figure 18.3 which are concave to the concentration axis for the mobile phase are termed favourable and lead to self-sharpening ion exchange waves. 

The sorption-desorption isotherm for the O-octylated extract is noticably different. The shape of the sorption curve is concave upward and the hysteresis effect is much less pronounced. Nearly all of the benzene (97% wt) could be desorbed under vacuum at 30" C. Clearly, this material is responding much differently than the untreated and O-methylated extracts. 

When Af= 1, the model yields a concave down isotherm. The magnitndes of and fc for each species determine their relative affinities and therefore their ability to snppress each other s isotherms. 

The shape of the adsorption front, the width of the MTZ, and the profile of the effluent concentration depend on the nature of the adsorption isotherm and the rate of mass transfer. Practical bed depths may be expressed as multiples of MTZ, values of 5-10 multiples being economically feasible. Systems that have linear adsorption isotherms develop constant MTZs whereas MTZs of convex ones (such as Type I of Figure 15.1) become narrower, and those of concave systems become wider as they progress through... 

A ads is given by the adsorption isotherm, which may be linear, convex, or concave (Fig. 8). Kiis is the slope of the line or the tangent to the curve at very low concentrations of solute. When a... 

Type B is very common. It is concave with respect to the abscissa. Most surfaces are heterogeneous. There are adsorption sites, which have a high affinity, and regions, which have a low affinity. The high affinity sites are occupied first, which accounts for the steep increase at low pressure. Another reason is sometimes a lateral repulsion between adsorbed molecules. This type of adsorption isotherm is described by the Freundlich1 adsorption isotherm equation [366] ... 

Type E is common for the adsorption of gases. Usually the first concave part is attributed to the adsorption of a monolayer. For higher pressures more layers adsorb on top of the first one. Eventually, if the pressure reaches the saturation vapor pressure, condensation leads to macroscopically thick layers. It can be described by the BET adsorption isotherm equation Eq. (9.37) (see below). 

The discussion directly following Eq (6) provides a simple, physically reasonable explanation for the preceding observations of marked concentration dependence of Deff(C) at relatively low concentrations. Clearly, at some point, the assumption of concentration independence of Dp and in Eq (6) will fail however, for our work with "conditioned" polymers at CO2 pressures below 300 psi, such effects appear to be negligible. Due to the concave shape of the sorption isotherm, even at a CO2 pressure of 10 atm, there will still be less than one CO2 molecule per twenty PET repeat units at 35°C. Stern (26) has described a generalized form of the dual mode transport model that permits handling situations in which non-constancy of Dp and Dh manifest themselves. It is reasonable to assume that the next generation of gas separation membrane polymers will be even more resistant to plasticization than polysulfone, and cellulose acetate, so the assumption of constancy of these transport parameters will be even more firmly justified. 

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