Isotherm crossing

Fig. 7. Isothermal cross section of the system H20-CH4-CsH8 on a water-free basis at —3° C. The points represent experimental results and the curves have been obtained from a theoretical analysis. The line AB represents the four-phase equilibrium HiHn ice G the gas G consists of almost pure methane, Hj contains only methane. Consequently, the composition of the latter two phases almost coincide in the figure, and the situation around point A has therefore been drawn separately on an enlarged scale.

Fig. 10. Isothermal cross-section of the system HaS-CaHg-HaO on a water-free basis at — 3°C.

Fig. 11. Tentative isothermal cross section of the system HaO-HaS-CH,CHF2 on a water-free basis below 0°C. In the two binary systems H2G-H2S and H20-CHsCHF2 only hydrates of Structure I occur, but in the ternary system mixed hydrates of Structure II have been reported.44...

Figure 6.4 On the left is a phase diagram for carbon dioxide. Broken lines indicate isotherm crossing at either constant pressure or density. On the right is illustrated the change in solubility of naphthalene as a function of temperature and pressure.

Guseva, A.N. and Parnov, E.I. Isothermal cross-sections of the systems cyclanes-water, Vestn. Most Univ., Ser. 11 Khim., 19 77-78, 1964. 

Dlepen and Scheffer ( 6) were the first to show that near either the lower or upper critical end point the solubility of a solid in a supercritical fluid is enhanced and also very sensitive to changes in temperature and pressure our solubility isotherms show this effect for both end points. First, the isotherms cross at about 140 bar so that the solubility at the lowest temperature (50.0°C) is largest at 120 bar. This is a result of approaching the lower critical end point region (which should be close to the critical point of pure C02 as previously mentioned). At temperatures and pressures near this LCEP the solubility enhancement results in lower temperature isotherms having the greater solubilities. The effect of the upper critical end point is also well shown by our data. The 58.5°C isotherm shows a large increase in solubility at about 235 bar the slope of the isotherm is near zero. As Van Welie and Diepen... 

Many trial-and-error experiments can be avoided during the development of a displacement chromatographic separation, when the isotherm of at least the most strongly adsorbed sample component is known. Therefore, as the next step, the adsorption isotherms of the most retained iscmers of chloroaniline and Ibuprofen, the examples discussed above, were determined as shown in Figs. 8 and 9. It can be seen by cxnparing the isotherms of the solute and prospective displacer pairs that indeed p-nitrophenol can be used as a displacer for the separation of the chloroaniline iscmers. The situation is more complicated with Ibuprofen and 4-t-butylcyclchexanol because their isotherms cross each other at 1.5 irM. This indicates that successful separations can be expected only below this concentration level. Other examples of crossing isotherms were also reported (69). 

Stationary phase are fortuitous. In exceptional cases, where the concave adsorption isotherm crosses the convex one, even a reversal of the elution order is obtained and can be used to achieve a higher productivity as has been demonstrated by Roussel et al. [51] for the separation of the enantiomers of 3-(2-propylphenyl)-4-methyl-4-thiazolin-2-one on microcrystalline cellulose triacetate. 

When the solute isotherms cross one another, the situation becomes more complex. It then becomes possible to experience selectivity reversal that is, at one displacer concentration, the solutes elute in the order A first, then B, whereas at another displacer concentration, the order is B first, then A. In a study of this problem, Antia and Horvath showed the existence of the separation gap. This is a region in the isotherm plane, the position of which depends on the ratio of the saturation capacities of the solutes in question. If the operating line is outside the separation gap, displacement occurs in the normal fashion. The elution order of the solutes then depends on the position of the operating line relative to the separation gap. However, if the operating line is within the separation gap, displacement operation does not separate the displaced solutes, but results in the elution of a mixture of the solutes. 

When the p-isotherms cross at a given pressure, (dup/d T)p changes its sign and this property is responsible for the above-mentioned minima. In addition, as the temperature increases, the minima are shifted to higher pressures. At higher temperatures, especially when approaching the critical temperature, the minimum becomes a shallow one. As an example for associated liquids, some selected Cp-isotherms and the pressure effects at different temperatures on the isobaric heat capacity of m-cresol are presented in Figures 2(al) and (a2), respectively. 

The maltitol syrup and carbohydrate mixtures sorption isotherms cross at the two different temperatures in the region between 0.3 and 0.4. Above this region, the water content was smaller at 308 K than at 298 K. A crossing of isotherms at two different temperatures was also observed by A3Tranci et al. (1990) in dried apricot, fig, and raisin. Analysis of variance showed that except for isomalt, the equilibrium water contents of the different products did not differ (p > 0.05) for each product at different temperatures. 

The solubility isotherms cross one another. Namely, at pressures roughly ranging from 40 bar to 170-180 bar (zone B of the plot) a retrograde behavior is seen, where the solubility is decreased by a temperature increase, whereas above the crossover pressure the usual dependence is encountered (zone C of the plot). 

Kal] Kaloev, N.I., Abram yan, A.Kh., Kabanov, S.V., Kulova, L.K., Isothermal Cross Section of the System Fe-Cr-Nb at 973 K , Russ. Metall, 1(1), 202-204 (1988), translated from Izv. Akad. NaukSSSR, Metally, (1), 205-206 (1988) (Phase Diagram, Phase Relations, Experimental, Meehan. Prop.,, 5)... 

One of the earliest studies on the use of BAM examined monolayers of the racemic mixture and of the pure l- and D-enantiomers of N-dodecylgluconamide. This system presented jt-A isotherms at 10 °C that were more condensed for the racemic mixture than for the pure enantiomers, which were equivalent however, at 25 °C, the opposite was true, although the isotherms crossed at a certain point. Significant surface pressure relaxation at fixed area was observed, which differed in extent between the enantiomers and the racemic mixture and was much greater at 25 °C than at 10 °C, as expected on the basis of the short chain length. BAM observation during compression of either of the pure enantiomers at 25 °C showed extensive growth of two-dimensional dendritic domains. Differences could not be found between the dendritic forms of the two enantiomers. The monolayers of the racemic... 

T and K - triple and critical points of component A and B f - eutectic equilibrium (L-G-Sa-Sb). Open circles - nonvariant points in one-component systems. Solid circles - nonvariant points in binary systems (p-T projection) and compositions of fluid (liquid, gas and critical) phases in binary nonvariant equilibria (p-T-x, T-x and p-x schemes). Diamonds - compositions of solid phases in binary nonvariant equilibria. Heavy lines - monovariant curves in one-component systems (A, B) solid lines - compositions of Uquid phases in monovariant equilibria of binary systems dashed lines - compositions of vapor (gas) phases in monovariant equUibria of binary systems dash-dotted lines - critical curves thin lines - isothermal cross-sections of the p-T-x diagram and tie-lines in T-x and p-x projections. For clarity, only tie-lines between fluid phases are shown in monovariant equilibria. 

Figure 9.8 Isothermal cross section of the phase diagram for the ternary system chosen as a model of investigation.

Figure 9.9 Isothermal cross section of the Mo-Co-Fe phase diagram.

Figure 12.5 Effect of temperature on the solubility curves and the co-crystal domain in the ternary phase diagram of (a) a congruently melting cocrystal (c/.. Figure 12.1(a)) and (b) an incongruently melting co-crystal cf, Figure 12.1(b)). Ideal miscibility of all components in the liquid phase is assumed. The eutectic grooves are indicated by continuous lines, the peritectic groove by a dashed line. The two-phase regions in the 75 °C isothermal cross section are shaded.

CHARACTERIZATION OF MICROPOROSTTY AND SURFACE HOMOGENEITY BY THE STUDY OF ARGON AND NITROGEN ISOTHERM CROSSING AND MEASUREMENT OF DIFFERENTIAL ENTHALPIES OF ADSORPTION... 

On the other hand, there is another kind of isotherm crossing found when plotting adsorption isotherms of two different adsorptives at a given temperature as a function of the relative pressure (refs. 8,9). Of course, this crossing point has a different meaning and could be associated to the micropoious texture of a given solid. 

Since the Carnot cycle is reversible, any reversible cycle can be presented as a sequence of elementary (Carnot) cycles. Ten such cycles consisting of isotherms crossed by adiabats are depicted in Figure 3.20. 

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