Solid-liquid equilibria isotherm

Now consider the case depicted in figure 3.20c, an isotherm at the UCEP temperature (see figure 3.19). At the UCEP pressure there is a vapor-liquid critical point in the presence of solid. This requires the solid-liquid equilibrium curve to intersect the liquid-gas envelope precisely at the binary liquid-gas critical point and, hence, exhibit a negative horizontal inflection, i.e., (dPldx)T = 0. Notice that the vapor-liquid envelope has not shrunk to a point, as it did at the naphthalene-ethylene UCEP. The solid curve shown in figure 3.20d is the solubility isotherm obtained if a flow-through apparatus is used and only the solubility in the SCF phase is determined. This solid curve has the characteristics of the 55°C biphenyl-carbon dioxide isotherm shown in figure 3.17. So the 55°C isotherm represents liquid biphenyl solubilities at pressures below 475 bar and solid biphenyl solubilities at pressures above 475 bar. 

Figure 5. Isothermal solid-liquid equilibrium phase diagram for the solvent + lauric acid + myristic acid system, (ideal solution)...

Essentially, extraction of an analyte from one phase into a second phase is dependent upon two main factors solubility and equilibrium. The principle by which solvent extraction is successful is that like dissolves like . To identify which solvent performs best in which system, a number of chemical properties must be considered to determine the efficiency and success of an extraction [77]. Separation of a solute from solid, liquid or gaseous sample by using a suitable solvent is reliant upon the relationship described by Nemst s distribution or partition law. The traditional distribution or partition coefficient is defined as Kn = Cs/C, where Cs is the concentration of the solute in the solid and Ci is the species concentration in the liquid. A small Kd value stands for a more powerful solvent which is more likely to accumulate the target analyte. The shape of the partition isotherm can be used to deduce the behaviour of the solute in the extracting solvent. In theory, partitioning of the analyte between polymer and solvent prevents complete extraction. However, as the quantity of extracting solvent is much larger than that of the polymeric material, and the partition coefficients usually favour the solvent, in practice at equilibrium very low levels in the polymer will result. 

The equilibrium model for the adsorption of polymers at solid-liquid interfaces recently presented by Hogg and Mirville (1) has been evaluated at some length. It has been shown that, for polymers consisting of a reasonably large number of segments, the adsorption isotherms can be closely approximated by an expression of the form ... 

Figure 26 shows the ternary phase diagrams (solubility isotherms) for three types of solid solution. The solubilities of the pure enantiomers are equal to SA, and the solid-liquid equilibria are represented by the curves ArA. The point r represents the equilibrium for the pseudoracemate, R, whose solubility is equal to 2Sd. In Fig. 26a the pseudoracemate has the same solubility as the enantiomers, that is, 2Sd = SA, and the solubility curve AA is a straight line parallel to the base of the triangle. In Figs. 26b and c, the solid solutions including the pseudoracemate are, respectively, more and less soluble than the enantiomers. 

Adsorption isotherms are used to quantitatively describe adsorption at the solid/ liquid interface (Hinz, 2001). They represent the distribution of the solute species between the liquid solvent phase and solid sorbent phase at a constant temperature under equilibrium conditions. While adsorbed amounts as a function of equilibrium solute concentration quantify the process, the shape of the isotherm can provide qualitative information on the nature of solute-surface interactions. Giles et al. (1974) distinguished four types of isotherms high affinity (H), Langmuir (L), constant partition (C), and sigmoidal-shaped (S) they are represented schematically in Figure 3.3. 

Table 3.24 shows the computed data for k, for both solid/liquid ratios and the mean values if we consider the hypothesis of a linear equilibrium isotherm. 

Material balance calculations on separation processes follow the same procedures used in Chapters 4 and 5. If the product streams leaving a unit include two phases in equilibrium, an equilibrium relationship for each species distributed between the phases should be counted in the degree-of-freedom analysis and included in the calculations. If a species is distributed between gas and liquid phases (as in distillation, absorption, and condensation), use tabulated vapor-liquid equilibrium data, Raoult s law, or Henry s law. If a solid solute is in equilibrium with a liquid solution, use tabulated solubility data. If a solute is distributed between two immiscible liquid phases, use a tabulated distribution coefficient or equilibrium data. If an adsorbate is distributed between a solid surface and a gas phase, use an adsorption isotherm. 

Some essential discoveries concerning the organization of the adsorbed layer derive from the various spectroscopic measurements [38-46]. Here considerable experimental evidence is consistent with the postulate that ionic surfactants form localized aggregates on the solid surface. Microscopic properties like polarity and viscosity as well as aggregation number of such adsorbate microstructures for different regions in the adsorption isotherm of the sodium dedecyl sulfate/water/alumina system were determined by fluorescence decay (FDS) and electron spin resonance (ESR) spectroscopic methods. Two types of molecular probes incorporated in the solid-liquid interface under in situ equilibrium conditions... 

Before the solubility isotherms of low-melting solid solutes can be predicted, it is necessary to ascertain whether there is a solid-liquid transition along the solubility isotherm due to the melting point depression. For RESS, one needs to study the S-V equilibrium and for PGSS it is required to study the L-V equilibrium at a temperature above the melting point. 

Shown in figure 3.18c is a solubility isotherm at a temperature, Tb, that is less than the previous temperature, Tj, but still higher than the UCEP temperature. The solubility behavior at T is similar to the behavior in figure 3.18b. But at T, the three-phase SLV line is intersected at a higher pressure, closer to the UCEP pressure. Hence, the vapor-liquid envelope has diminished in size and the solid-gas equilibrium curve is shifted toward higher solvent concentrations. As a result, the solid-gas curve is now much closer to the vapor branch of the vapor-liquid envelope. 

For the adsorption from a solution into a solid, the Langmuir equation is often used to describe the equilibrium isotherms. But the parameters in the Langmuir equation should not be considered as having the same physical meaning as those used for the vapor adsorption. Nevertheless, the parameters n and b in Equation (10a) can be used for comparison as n may indicate the "saturated" adsorption capacity and b is related to the initial slope of the isotherm. The liquid adsorption equilibrium results given in Table II show that the saturation adsorption capacity for the five alcohols decrease in the following order (based on the adsorption quantity in mmol/g) ... 

The essential features of vapor-liquid equilibrium (VLE) behavior are demonstrated by the simplest case isothermal VLE of a binary system at a temperature below the critical temperatures ofboth pure components. Forthis case ( subcritkaT VLE), each pure component has a well-defined vapor-liquid saturation pressure ff, and VLE Is possible for the foil range of liquid and vapor compositions xt and y,. Figure 1.5-1 ffiustrates several types of behavior shown by such systems. In each case, (he upper solid curve ( bubble curve ) represents states of saturated liquid (he lower solid curve ( dew curve ) represents states of saturated vtqtor. 

Sorption isotherms of the wet solid are, from the point of view of model structure, equilibrium relationships, and are a property of the solid-liquid-gas system. For the most common air-water system, sorption isotherms are, however, traditionally considered as a solid property. Two forms of sorption isotherm equations exist—explicit and implicit ... 

P8.14 The vapor pressures of liquid anthracene and phenanthrene can be described by the Antoine equation using the Antoine parameters given below. Calculate the vapor-liquid-solid equilibrium (VLSE) along the solid-liquid saturation curve assuming ideal mixture behavior in the liquid phase. Melting points and heats of fusion of both components are given in example P8.1 above. Compare the vapor-liquid separation factors to those of an isothermal VLE data set at 220 "C (calculate assuming ideal liquid mixture behavior). 

Since concentrations xj, X2, X3 represent equilibrium values (i.e. concentrations in the bulk phase after adsorption takes place) it is impossible to prepare the original samples of ternary solutions in such a way that the X2/X3 ratio stays constant without prior knowledge of the adsorption isotherm. This is the reason that adsorption isotherms seem to depend on the solid/liquid ratio in the system. An increase in the amount of the solid phase increases the total amount of surfactants adsorbed, which results in a change of X2/X3 ratio and a shift of the experimental point on the adsorption isotherm surface. Obviously, this effect is more pronounced in systems with large differences in individual surfactant adsorption characteristics. 

Evaluation of Sorbent Capacity. The quantity of pollutant that a sorbent can adsorb is defined by the equilibrium isotherm. This is a mathematical relationship between the liquid phase concentration of the spill chemical, Q, and the solid phase concentration of the spill chemical on the sorbent, when the contacting system has reached equilibrium. Figure 15.10 shows this relationship for cadmium salt in solution that has been spilled. 

In order to understand this isotherm, it is worthwhile to apply Gibbs phase rule (93) which, at constant temperature and pressure, states that the number of degrees of freedom (F) in a system at equilibrium is equal to the number of components (C) in the system minus the number of phases (solid, liquid, gas) present in the system, or F — C — P. Crisp (94) has derived a two-dimensional phase rule to apply to a single plane surface containing q surface phases. The rule predicts that the number of degrees of Freedom (F) will he F = C — Ph — q — ), where C is the total number of components in the system, F is the number of bulk phases, and q is the number of surface phases. In the case of deoxycholic acid spread on aqueous substrate the number of components (C) can be considered to be two, the water of the aqueous phase and the deoxycholic acid. The number of bulk phases, that is the substrate, can be 1 or 2 and the number of surface phases can be 1 or 2. When the area per molecule is very large, for instance 10,000 A- molecule (right side of Fig. 11), the surface pressure is very low (>0.1 dynes/cm) but... 

At very low pressures and temperatures close to 0 K, a fourth state of matter has been discovered. This is the Bose-Einstein condensate. Eric A. Cornell obtained the Nobel prize in 2001 for his work on the Bose-Einstein condensate and fourth state of matter. The critical point of the pure substance can be seen in Figure 2.1. Beyond this point, the liquid and gas are indistinguishable from each other and exist as a fluid. Are there similar critical points at the end of the fusion curve and sublimation curves The PVT and other phenomena at very low pressures and temperatures are subjects of exploratory research. The critical point may also be viewed as the highest pressure and temperature at which a pure chemical species is observed to exist in vapor/liquid equilibrium [2]. The vertical line in Figure 2.1 is an isotherm, and a horizontal line in Figure 2.1 is an isobar. The solid lines in Figure 2.1 indicate a... 

---