Equilibrium solubility, phase

The design of crystallizers is based on knowledge of phase equilibria, solubilities, rates and amounts of nuclei generation, and rates of crystal growth. Each system is... 

The use of models and particularly those of a sophisticated nature is, however, seriously restricted by the limitations of the parameters involved in the model equations. It is actually the determination of certain parameters which becomes the crucial point in designing. The major uncertainties originate from two sources. Firstly, the process data, i.e. estimation of phase equilibria (solubilities), diffusivities and especially kinetic rate data,involves inaccuracies. The second major source of large uncertainties is the reliability of the nonadjustable hydrodynamic quantities. 

The design of crystallizers is based on knowledge of phase equilibria, solubilities, rates and amounts of nuclei generation, and rates of crystal growth. Each system is unique in most of these respects and not often predictable. The kind of information needed for design of a continuous crystallizer is indicated by the data supplied for Example 16.1 and as listed in greater detail below. 

We have used this chapter to illustrate how thermodynamics can contribute to the analysis and design of selected engineering processes. The applications considered here included calculations for phase equilibria, solubilities, heat effects in steady-flow processes, and the response of certain variables to changes of state. 

For every extraction process (including SFE), the knowledge of many thermodynamic and transport properties of the solvent phase, the solutes, and all mixtures involved is of primary importance. These include information such as pVTx data, phase equilibria, solubilities, viscosities, diffusion coefficients, and surface tensions. The same holds for chromatographic applications. In SFC, however, the emphasis is on very dilute solutions. 

Chapter 1 is devoted to introductory concepts and de nitions, while Chapter 2 deals with physical and molecular aspects of polymers, that is, those relating to molecular shape and size, distinctive characteristics, conformational and con gurational behavior, structural features, morphology, thermal transitional phenomena, and relaxation properties. Chapter 3 discusses polymer solution behavior, the emphasis being on thermodynamics, phase equilibria, solubility, swelling, frictional properties, and viscosity. Molecnlar weight determination, which is one of the rst steps of polymer characterization and a centrally important topic of polymer science, mostly involves... 

We describe simple models of complex s> stems, including polymers, colloids, surfaces, and catalysts. Chapters 25 and 2G focus on cooperativity phase equilibria, solubilities, critical phenomena, and conformational transi-... 

Phase Equilibria, Solubility Product, Solubility, Complex Species in Solution, Competing Phases... 

Crystal Formation There are obviously two steps involved in the preparation of ciystal matter from a solution. The ciystals must first Form and then grow. The formation of a new sohd phase either on an inert particle in the solution or in the solution itself is called nucle-ation. The increase in size of this nucleus with a layer-by-layer addition of solute is called growth. Both nucleation and ciystal growth have supersaturation as a common driving force. Unless a solution is supersaturated, ciystals can neither form nor grow. Supersaturation refers to the quantity of solute present in solution compared with the quantity which would be present if the solution were kept for a veiy long period of time with solid phase in contac t with the solution. The latter value is the equilibrium solubility at the temperature and pressure under consideration. The supersaturation coefficient can be expressed... 

The alloy aluminium-4 wt% copper forms the basis of the 2000 series (Duralumin, or Dural for short). It melts at about 650°C. At 500°C, solid A1 dissolves as much as 4 wt% of Cu completely. At 20°C its equilibrium solubility is only 0.1 wt% Cu. If the material is slowly cooled from 500°C to 20°C, 4 wt% - 0.1 wt% = 3.9 wt% copper separates out from the aluminium as large lumps of a new phase not pure copper, but of the compound CuAlj. If, instead, the material is quenched (cooled very rapidly, often by dropping it into cold water) from 500°C to 20°C, there is not time for the dissolved copper atoms to move together, by diffusion, to form CuAlj, and the alloy remains a solid solution. 

Now interpret phase X as pure solute then Cs and co become the equilibrium solubilities of the solute in solvents S and 0, respectively, and we can apply Eq. (8-58). Again the concentrations should be in the dilute range, but nonideality is not a great problem for nonelectrolytes. For volatile solutes vapor pressure measurements are suitable for this type of determination, and for electrolytes electrode potentials can be used. 

Thus we see that the equilibrium solubility of a gas again involves a balance between randomness and energy as it does for a solid, but the effects are opposite. For a gas, the tendency toward maximum randomness favors the gas phase, opposing dissolving. The tendency toward minimum energy favors the liquid state, hence favors dissolving. 

Let us first consider the three-phase equilibrium ( -clathrate-gas, for which the values of P and x = 3/( +3) were determined at 25°C. When the temperature is raised the argon content in the clathrate diminishes according to Eq. 27, while the pressure can be calculated from Eq. 38 by taking yA values following from Eq. 27 and the same force constants as used in the calculation of Table III. It is seen that the experimental results at 60°C and 120°C fall on the line so calculated. At a certain temperature and pressure, solid Qa will also be able to coexist with a solution of argon in liquid hydroquinone at this point (R) the three-phase line -clathrate-gas is intersected by the three-phase line -liquid-gas. At the quadruple point R solid a-hydroquinone (Qa), a hydroquinone-rich liquid (L), the clathrate (C), and a gas phase are in equilibrium the composition of the latter lies outside the part of the F-x projection drawn in Fig. 3. The slope of the three-phase line AR must be very steep, because of the low solubility of argon in liquid hydroquinone. 

In processing, it is frequently necessary to separate a mixture into its components and, in a physical process, differences in a particular property are exploited as the basis for the separation process. Thus, fractional distillation depends on differences in volatility. gas absorption on differences in solubility of the gases in a selective absorbent and, similarly, liquid-liquid extraction is based on on the selectivity of an immiscible liquid solvent for one of the constituents. The rate at which the process takes place is dependent both on the driving force (concentration difference) and on the mass transfer resistance. In most of these applications, mass transfer takes place across a phase boundary where the concentrations on either side of the interface are related by the phase equilibrium relationship. Where a chemical reaction takes place during the course of the mass transfer process, the overall transfer rate depends on both the chemical kinetics of the reaction and on the mass transfer resistance, and it is important to understand the relative significance of these two factors in any practical application. 

Englezos, P. and S. Hull, Phase Equilibrium Data on Carbon Dioxide Hydrate in the Presence of Electrolytes, Water Soluble Polymers and Montmorillonite , CanJ. Chem. Eng, 72, 887-893 (1994). 

Coprecipitation is a partitioning process whereby toxic heavy metals precipitate from the aqueous phase even if the equilibrium solubility has not been exceeded. This process occurs when heavy metals are incorporated into the structure of silicon, aluminum, and iron oxides when these latter compounds precipitate out of solution. Iron hydroxide collects more toxic heavy metals (chromium, nickel, arsenic, selenium, cadmium, and thorium) during precipitation than aluminum hydroxide.38 Coprecipitation is considered to effectively remove trace amounts of lead and chromium from solution in injected wastes at New Johnsonville, Tennessee.39 Coprecipitation with carbonate minerals may be an important mechanism for dealing with cobalt, lead, zinc, and cadmium. 

When determining the solubility and dissolution rate of amorphous or partially crystalline solids, the metastability of these phases with respect to the highly crystalline solid must be considered. While the low diffusivity of the molecules in the solid state can kinetically stabilize these metastable forms, contact with the solution, for example during measurements of solubility and dissolution rate, or with the vapor, if the solid has an appreciable vapor pressure, may provide a mechanism for mass transfer and crystallization. Less crystalline material dissolves or sublimes whereas more crystalline material crystallizes out. The equilibrium solubility measured will therefore approach that of the highly crystalline solid. The initial dissolution rate of the metastable form tends to reflect its higher... 

It is important to ascertain whether the solid phase of the solute changes during equilibration to produce a different polymorph or solvate, by analyzing the solid phase (using either chemical or thermal analysis, or x-ray diffraction). If a solid-solid phase transition occurs during equilibration, the measured equilibrium solubility will be that of the new solid phase of the solute. Methods of circumventing this problem have been proposed and evaluated [26]. 

Phase solubility analysis is a technique to determine the purity of a substance based on a careful study of its solubility behavior [38,39]. The method has its theoretical basis in the phase mle, developed by Gibbs, in which the equilibrium existing in a system is defined by the relation between the number of coexisting phases and components. The equilibrium solubility of a material in a particular solvent, although a function of temperature and pressure, is nevertheless an intrinsic property of that material. Any deviation from the solubility exhibited by a pure sample arises from the presence of impurities and/or crystal defects, and so accurate solubility measurements can be used to deduce the purity of the sample. 

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. 

It should be noted that constant temperature and pressure are assumed, and that A must exist in exactly the same form in both phases. Equilibrium is established when the chemical potentials (free energies) of the solute in the two phases are equal and is usually achieved within a few minutes by vigorous shaking. The value of KD is a reflection of the relative solubilities of the solute in the two phases. 

It is recommended that concentration measurements for this type of modeling work are based on analytical standards of mole or mass fraction, to avoid the conversion error caused by density effects. The excess solid phase should always be characterized by a suitable analytical technique, before and after the equilibrium solubility measurements, to confirm that the polymorphic form is unchanged. It should be noted that the crystal shape (habit) does not always change significantly between different polymorphic forms, and visual assessments can be misleading. 

Sol id Sol utions. The aqueous concentrations of trace elements in natural waters are frequently much lower than would be expected on the basis of equilibrium solubility calculations or of supply to the water from various sources. It is often assumed that adsorption of the element on mineral surfaces is the cause for the depleted aqueous concentration of the trace element (97). However, Sposito (Chapter 11) shows that the methods commonly used to distinguish between solubility or adsorption controls are conceptually flawed. One of the important problems illustrated in Chapter 11 is the evaluation of the state of saturation of natural waters with respect to solid phases. Generally, the conclusion that a trace element is undersaturated is based on a comparison of ion activity products with known pure solid phases that contain the trace element. If a solid phase is pure, then its activity is equal to one by thermodynamic convention. However, when a trace cation is coprecipitated with another cation, the activity of the solid phase end member containing the trace cation in the coprecipitate wil 1 be less than one. If the aqueous phase is at equil ibrium with the coprecipitate, then the ion activity product wi 1 1 be 1 ess than the sol ubi 1 ity constant of the pure sol id phase containing the trace element. This condition could then lead to the conclusion that a natural water was undersaturated with respect to the pure solid phase and that the aqueous concentration of the trace cation was controlled by adsorption on mineral surfaces. While this might be true, Sposito points out that the ion activity product comparison with the solubility product does not provide any conclusive evidence as to whether an adsorption or coprecipitation process controls the aqueous concentration. 

Phase Equilibrium Calculations by Equation of State for Aqueous Systems with Low Mutual Solubility... 

Therefore, the physical meaning of the solubility curve of a surfactant is different from that of ordinary substances. Above the critical micelle concentration the thermodynamic functions, for example, the partial molar free energy, the activity, the enthalpy, remain more or less constant. For that reason, micelle formation can be considered as the formation of a new phase. Therefore, the Krafft Point depends on a complicated three phase equilibrium. 

On the other hand, micelle formation has sometimes been considered to be a phase separation of the surfactant-rich phase from the dilute aqueous solution of surfactant. The micellar phase and the monomer in solution are regarded to be in phase equilibrium and cmc can be considered to be the solubility of the surfactant. When the activity coefficient of the monomer is assumed to be unity, the free energy of micelle formation, Ag, is calculated by an equation... 

Equilibrium Solubility. The solubility of the reacting components will limit their movement from phase to phase. This factor will certainly influence the form of the rate equation since it will determine whether the reaction takes place in one or both phases. 

The equilibrium solubility of an Fe oxide can be approached from two directions -precipitation and dissolution. The first method involves precipitating the oxide from a supersaturated solution of ions with stepwise or continuous addition of base und using potentiometric measurements to monitor pH and calculate Fej- in equilibrium with the solid phase until no further systematic change is detected. Alternatively the oxide is allowed to dissolve in an undersaturated solution, with simultaneous measurement of pH and Fejuntil equilibrium is reached. It is essential that neither a phase transformation nor recrystallization (formation of larger crystals) occurs during the experiment this may happen with ferrihydrite which transforms (at room temperature) to a more condensed, less soluble phase. A discussion of the details of these methods is given by Feitknecht and Schindler (1963) and by Schindler (1963). 

Information about the kinetics of dissolution reactions is provided by Delmon (1969) and by Brown et al. (1980). Dissolution may be either diffusion (i.e., transport) or surface controlled. If diffusion controlled, i. e., if the concentration of dissolved species immediately adjacent to the surface corresponds to the equilibrium solubility (Ce) of the solid phase, the concentration, c, of the dissolved species is diffusion controlled and increases with the square root of time, t, i. e.. 

Solubilities and phase distributions of species between phases require careful application of principles of thermocfynamic phase equilibrium. 

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