Solubility phase equilibrium calculations

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

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

Although most of the oils tested in this study show a similar solubility behavior, significant differences can occur, depending on the composition of the oils with respect to their hydrocarbon fraction and the chemical nature and the amount of additives. With the specifications given by the producers like density and viscosity at standard conditions (see Table 1) no correlation could be found to the experimental data. Further information about the composition is hardly available and an exact analysis is not only undesired but also nearly impossible. This lack of information also makes phase equilibrium calculations to be not very useful for the correlation or prediction of these solubility data. In every single case the solubility has to be determined experimentally. 

Related Calculations. This illustration outlines the procedure for obtaining coefficients of a liquid-phase activity-coefficient model from mutual solubility data of partially miscible systems. Use of such models to calculate activity coefficients and to make phase-equilibrium calculations is discussed in Section 3. This leads to estimates of phase compositions in liquid-liquid systems from limited experimental data. At ordinary temperature and pressure, it is simple to obtain experimentally the composition of two coexisting phases, and the technical literature is rich in experimental results for a large variety of binary and ternary systems near 25°C (77°F) and atmospheric pressure. Example 1.21 shows how to apply the same procedure with vapor-liquid equilibrium data. 

Vapor-liquid phase equilibrium calculations have to be conducted for the estimation of solubility in the vapor phase (16,17). Alternatively, a cubic EOS can be applied for the estimation of properties of the liquid phase. The equality of fugacity in the two phases can be written as... 

In this chapter we consider several other types of phase equilibria, mostly involving a fluid and a solid. This includes the solubility of a solid in a liquid, gas, and a supercritical fluid the partitioning of a solid (or a liquid) between two partially soluble liquids the freezing point of a solid from a liquid mixture and the behavior of solid mixtures. Also considered is the environmental problem of how a chemical partitions between different parts of the environment. Although these areas of application appear to be quite different, they are connected by the same starting point as for all phase equilibrium calculations, which is the equality of fugacities of each species in each phase ... 

Dissolved arsenic concentrations can be limited either by the solubility of minerals containing arsenic as a constituent element (or in solid solution) or by sorption of arsenic onto various mineral phases. For both the precipitation-dissolution of arsenic-containing minerals and sorption-desorption of arsenic onto solid phases, equilibrium calculations can indicate the level of control over dissolved arsenic concentrations that can be exerted by these processes. However, neither of these types of reactions is necessarily at equilibrium in natural waters. The kinetics of these reactions can be very sensitive to a variety of environmental parameters and to the level of microbial activity. In particular, a pronounced effect of the prevailing redox conditions is expected because potentially important sorbents (e.g., Fe(III) oxyhydroxides) are unstable under reducing conditions and because of the differing solubilities of As(V) and As(III) solids. 

When we apply thermodynamics to industrial and research problems, we should draw fundamental ideas from Parts 1 and 11, devise an appropriate solution strategy, as in Chapter 10, and combine those with a computational technique, as in Chapter 11. Such a procedme provides values for measurables that can be used to interpret novel phenomena, to design new processes, and to improve existing processes. The procedure is illustrated in this chapter for several well-developed situations. They include conventional phase-equilibrium calculations for vapor-liquid, liquid-liquid, and solid-solid equilibria ( 12.1) solubility calculations for gases in liquids, solids in liquids, and solutes in near-critical solvents ( 12.2) independent variables in steady-flow processes ( 12.3) heat effects for flash separators, absorbers, and chemical rectors ( 12.4) and effects of changes of state on selected properties ( 12.5). 

Solubility calculations are merely phase-equilibrium calculations applied to supercritical gases in liquids, solids in liquids, and solutes in near-critical fluids. The last application has drawn substantial attention, for near-critical extraction processes are being applied, not only in the chemical and energy industries, but also in food processing, purification of biological products, and clean-up of hazardous wastes. 

Some attention must be paid if one of the reactants is in the solid phase. Equilibrium calculations are simplified in this case because the activity of the solid does not appear in the equilibrium constant (recall from eq. [14.24 that activity of the solid is practically 1 unless pressure is high). This creates the interesting situation that a reversible reaction involving a solid may proceed to completion. This is very similar to the familiar dissolution of solids in water and other solvents. If an amount of a soluble solid (e.g., salt or sugar) is mixed with a liquid, some of the solid dissolves until the solubility limit is reached, at which point we have equilibrium between the undissolved and dissolved fractions of the solid. If, however, the amount of solid is below the solubility limit, the entire amount dissolves. This is an example of a reversible reaction that goes to completion. Such process is possible only if it involves a solid, because the concentration of the solid does not appear in the equilibrium constant. With components in any other phase, the mol fraction of a reactant appears in the denominator of the equilibrium constant, and this prevents a reversible reaction from reaching completion unless the equilibrium constant is a very large number. 

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. 

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... 

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). 

Many additional consistency tests can be derived from phase equilibrium constraints. From thermodynamics, the activity coefficient is known to be the fundamental basis of many properties and parameters of engineering interest. Therefore, data for such quantities as Henry s constant, octanol—water partition coefficient, aqueous solubility, and solubility of water in chemicals are related to solution activity coefficients and other properties through fundamental equilibrium relationships (10,23,24). Accurate, consistent data should be expected to satisfy these and other thermodynamic requirements. Furthermore, equilibrium models may permit a missing property value to be calculated from those values that are known (2). 

Aqueous Solubility. Solubility of a chemical in water can be calculated rigorously from equilibrium thermodynamic equations. Because activity coefficient data are often not available from the literature or direct experiments, models such as UNIFAC can be used for structure—activity estimations (24). Phase-equilibrium relationships can then be applied to predict miscibility. Simplified calculations are possible for low miscibility, however, when there is a high degree of miscibility, the phase-equilibrium relationships must be solved rigorously. 

The calculation of two-phase (hydrate and one other fluid phase) equilibrium is discussed in Section 4.5. The question, To what degree should hydrocarbon gas or liquid be dried in order to prevent hydrate formation is addressed through these equilibria. Another question addressed in Section 4.5 is, What mixture solubility in water is needed to form hydrates ... 

The experimental data are correlated with equation of state models. The calculation of binary phase equilibrium data for FAEE is commonly based on the Peng-Robinson-equation-of-state, Yu et al. (1994). Up to now only the solubility of the oil components in the solvent has been subject of various studies. No attention was paid to a correlation of ternary data. The computation of ternary or multicomponent phase equilibrium is the basis to analyse and optimise the separation experiments. 

Some binary phase equilibrium and solubility data of limonene and linalool with supercritical C02 can be found in the current literature [11, 12, 13, 14, 15]. However, the different ranges of pressure and/or temperature of these data cause difficulties and inaccuracies to calculate or predict the related compound selectivities. 

Experimental vapor-liquid-equilibrium data for benzene(l)/n-heptane(2) system at 80°C (176°F) are given in Table 1.8. Calculate the vapor compositions in equilibrium with the corresponding liquid compositions, using the Scatchard-Hildebrand regular-solution model for the liquid-phase activity coefficient, and compare the calculated results with the experimentally determined composition. Ignore the nonideality in the vapor phase. Also calculate the solubility parameters for benzene and n-heptane using heat-of-vaporization data. 

Another application of atomistic simulations is reported by De Pablo, Laso, and Suter. Novel simulations for the calculation of the chemical potential and for the simulation of phase equilibrium in systems of chain molecules are reported. The methods are applied to simulate Henry s constants and solubility of linear alkanes in polyethylene. The results seem to be in good agreement with experiment. At moderate pressures, however, the solubility of an alkane in polyethylene exhibits strong deviations from ideal behavior. Henry s law becomes inapplicable in these cases. Solubility simulations reproduce the experimentally observed saturation of polyethylene by the alkane. For low concentrations of the solute, the simulations reveal the presence of pockets in the polymer in which solubility occurs preferentially. At higher concentrations, the distribution of the solute in the polymer becomes relatively uniform. 

As in a thermodynamic system description used for a normal solubility equilibrium calculation, the system contains a gas phase, if considered relevant for the problem at hand, an aqueous solution phase (external to the fibres), and a number of solid phases, which appear either with fixed stoichiometry or as solid solutions. The fibres are described as a separate aqueous phase. The thermodynamic data and stoichiometry for the solute species inside the fibre phase are identical to those describing the species in the external solution volume, with the exception that the charge of the species in the two aqueous phases must be defined separately. This will ensure that, given valid input values, charge neutrality will apply to both aqueous phases individually in the equilibrium composition calculated by Gibbs energy... 

The resistance to mass transfer according to (1.221) and (1.223) is made up of the individual resistances of the gas and liquid phases. Both equations show how the resistance is distributed among the phases. This can be used to decide whether one of the resistances in comparison to the others can be neglected, so that it is only necessary to investigate mass transfer in one of the phases. Overall mass transfer coefficients can only be developed from the mass transfer coefficients if the phase equilibrium can be described by a linear function of the type shown in eq. (1.217). This is normally only relevant to processes of absorption of gases by liquids, because the solubility of gases in liquids is generally low and can be described by Henry s law (1.217). So called ideal liquid mixtures can also be described by the linear expression, known as Raoult s law. However these seldom appear in practice. As a result of all this, the calculation of overall mass transfer coefficients in mass transfer play a far smaller role than their equivalent overall heat transfer coefficients in the study of heat transfer. 

In pursuing an accurate thermodynamic description of the three-phase, three-component system, the phase equilibrium compositions can be calculated after pressure and temperature have been fixed, since it is known from the Gibbs phase rule that there are only 2 degrees of freedom. There are five unknown compositions, assuming that the solid is crystalline and pure and that its solubility in the vapor/fluid phase is negligible. Two of these unknown mole fractions are eliminated by the constraints that the mole fractions in each phase sum up to unity. To find these three unknown mole fractions, namely, xi, X3, and y2, only three equilibrium relations are required. 

Colloidal dispersions and other related systems are present in many applications, e.g., in paints and coatings and detergents. Here, phase equilibrium and surface phenomena are equally important. A unified representation of such phenomena, e.g., of adhesion phenomena and liquid-liquid equilibria with the same model/concepts is of interest. Thermodynamic models can be used to calculate certain surface properties such as surface tension. hi addition, properties such as the solubility parameters can be equally well employed for bulk and surface thermodynamic properties. ... 

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