Solute equilibrium relations

The crystal-solution equilibrium relates to that between a particular form of the solute in solution and the same form in the crystalline phase. The isoelectric (zwitterionic) form usually exists in the crystalline phase over most of the pH range. Such compounds usually exhibit the lowest apparent solubility at their isoelectric pH since at other pH s some fraction of the solute exists in solution in the acid or base form. However, at very high acid or base concentrations, a salt, e.g., H2A" Cr, could be the crystalline form. 

When only the total system composition, pressure, and temperature (or enthalpy) are specified, the problem becomes a flash calculation. This type of problem requires simultaneous solution of the material balance as well as the phase-equilibrium relations. 

An ideal gas obeys Dalton s law that is, the total pressure is the sum of the partial pressures of the components. An ideal solution obeys Raoult s law that is, the partial pressure of the ith component in a solution is equal to the mole fraction of that component in the solution times the vapor pressure of pure component i. Use these relationships to relate the mole fraction of component 1 in the equilibrium vapor to its mole fraction in a two-component solution and relate the result to the ideal case of the copolymer composition equation. 

In Chap. 8 we saw how the equilibrium osmotic pressure of a solution is related to AG for the mixing process whereby the solution is formed. Any difference in the concentration of the solution involves a change in AG j, ... 

Example 2 Calculation of Variance In mixed-hed deionization of a solution of a single salt, there are 8 concentration variables 2 each for cation, anion, hydrogen, and hydroxide. There are 6 connecting relations 2 for ion exchange and 1 for neutralization equilibrium, and 2 ion-exchanger and 1 solution electroneiitrahty relations. The variance is therefore 8 — 6 = 2. 

Pi =f Ci) or Pi = HCi, equilibrium relation at the interface a = interfacial area/iinit volume Zg, Z-L = film thicknesses The steady rates of solute transfer are... 

With a reactive solvent, the mass-transfer coefficient may be enhanced by a factor E so that, for instance. Kg is replaced by EKg. Like specific rates of ordinary chemical reactions, such enhancements must be found experimentally. There are no generalized correlations. Some calculations have been made for idealized situations, such as complete reaction in the liquid film. Tables 23-6 and 23-7 show a few spot data. On that basis, a tower for absorption of SO9 with NaOH is smaller than that with pure water by a factor of roughly 0.317/7.0 = 0.045. Table 23-8 lists the main factors that are needed for mathematical representation of KgO in a typical case of the absorption of CO9 by aqueous mouethauolamiue. Figure 23-27 shows some of the complex behaviors of equilibria and mass-transfer coefficients for the absorption of CO9 in solutions of potassium carbonate. Other than Henry s law, p = HC, which holds for some fairly dilute solutions, there is no general form of equilibrium relation. A typically complex equation is that for CO9 in contact with sodium carbonate solutions (Harte, Baker, and Purcell, Ind. Eng. Chem., 25, 528 [1933]), which is... 

Consider an aqueous caustic soda solution whose molarity mi = 5.0 kmol/m (20 wt.% NaOH). This solution is to be used in >scH(t>ing H2S from a gaseous waste. The operating range of interest is 0.0 < xi kmoUn ) < 5.0. Derive an equilibrium relation for this chemical absorption over the operating range of interest. 

A typical example is as follows. Benzoic acid, C6H5COOH, is a solid substance with only moderate solubility in water. The aqueous solutions conduct electric current and have the other properties of an acid listed in Section 11-2.1. We can describe this behavior with reaction (42) leading to the equilibrium relation (43) ... 

Simultaneous solution of these equilibrium relations (coupled with the conservation equations x+ x-f = 1 and x/ + x/ = 1) gives the coexistence curve for the two-phase system as a function of pressure. 

Identify the equilibrium relations between the solute species (often by setting up an equilibrium table). 

It remains to evaluate the quantity c — Cs. Since an explicit general solution is not to be had, we resort to the consideration of special cases. First, suppose that the external electrolyte concentration Cs is very small compared with the concentration ic /z- of the ge-gen ions belonging to the polymer and occurring in the gel. Then the second term in the left-hand member of Eq. (45) may be neglected in comparison with the first. Furthermore, the very large ionic osmotic pressures developed in such cases will cause V2m to be very small, thus justifying adoption of the dilute solution approximations (see, for example, Eq. 40) for the right-hand member. The equilibrium relation reduces in this case to... 

Fig. 2.7. Relation between the pH and CP concentration of geothermal waters. The solid line indicates the albite-K-feldspar-muscovite-quartz-solution equilibrium at 250°C. For symbols used see caption to Fig. 2.2. (Shikazono, 1978a).

Fig. 2.8. Relation between the Ca and CR concentrations of geothermal waters and inclusion fluids. Solid lines indicate (1) albite-K-feldspar-muscovite-quartz-caleite-solution equilibrium at OHaCOs = 10 (2) albite-K-feldspar-muscovite-quartz-calcite-solution equilibriumn at oh2C03 = 10 (3) anhydrite-solution at SSo (total dissolved sulfate concentration) = 10 and (4) anhydrite-solution equilibrium at SSq = 10. For symbols used see caption to Fig. 2.2 (Shikazono, 1978a).

Thus Y1 is obtained not as the result of the numerical integration of a differential equation, but as the solution of an algebraic equation, which now requires an iterative procedure to determine the equilibrium value, Xj. The solution of algebraic balance equations in combination with an equilibrium relation has again resulted in an implicit algebraic loop. Simplification of such problems, however, is always possible, when Xj is simply related to Yi, as for example... 

We shall be dealing throughout this chapter with many situations in which various atomic solutes in a solid solution can react to form a variety of complexes, which in turn can redissociate into their atomic constituents. Some of these may exist in different charge states, which can interconvert by emission or absorption of electrons or holes. When the various atomic or electronic reactions have come to equilibrium, the concentrations of the various species involved will have to obey certain equilibrium relations. In this section, we shall review these in a language suitable for analysis of the various experiments to be discussed in Section III. 

Potentiometric EDTA titrations are best carried out with a mercury pool electrode (Figure 5.6) or a gold amalgam electrode. When this electrode dips into a solution containing the analyte together with a small amount of added Hg-EDTA complex, three interdependent reactions occur. For example, at pH = 8 the half cell reaction (a) which determines the electrode potential is related to the solution equilibrium by (b) and (c). 

H2S is soluble in such a solution and the equilibrium relation may be taken as 7 = 2X, where 7 is kmol of H2S kmol inert gas and X is kmol of H2S/kmol of solvent. 

A solution of 5 per cent acetaldehyde in toluene is to be extracted with water in a five stage co-current unit. If 25 kg water/100 kg feed is used, what is the mass of acetaldehyde extracted and the final concentration The equilibrium relation is given by ... 

The equilibrium relation is of the form y = mx Making a balance in terms of the solute A gives ... 

It can be assumed for almost all practical cases that equilibrium exists at the interface between the two phases. The concentrations of solute A at the interface, C A G and CAL, are related by the equilibrium relation of Eq. (5) and which, for gases in general and hydrogen in particular, is often described using simplified Henry s law applied at the interface (Eq. (6)). 

What is inaccurate about this solution is that kga depends on the extent of chemical reaction at each position in the tower. Also the equilibrium relation is more complex than linear and depends on the extent of chemical reaction. Use of a mean value of kga between the ends, however, gives at least an order of magnitude value of 2. 

The phenomena of surface precipitation and isomorphic substitutions described above and in Chapters 3.5, 6.5 and 6.6 are hampered because equilibrium is seldom established. The initial surface reaction, e.g., the surface complex formation on the surface of an oxide or carbonate fulfills many criteria of a reversible equilibrium. If we form on the outer layer of the solid phase a coprecipitate (isomorphic substitutions) we may still ideally have a metastable equilibrium. The extent of incipient adsorption, e.g., of HPOjj on FeOOH(s) or of Cd2+ on caicite is certainly dependent on the surface charge of the sorbing solid, and thus on pH of the solution etc. even the kinetics of the reaction will be influenced by the surface charge but the final solid solution, if it were in equilibrium, would not depend on the surface charge and the solution variables which influence the adsorption process i.e., the extent of isomorphic substitution for the ideal solid solution is given by the equilibrium that describes the formation of the solid solution (and not by the rates by which these compositions are formed). Many surface phenomena that are encountered in laboratory studies and in field observations are characterized by partial, or metastable equilibrium or by non-equilibrium relations. Reversibility of the apparent equilibrium or congruence in dissolution or precipitation can often not be assumed. 

The phase-equilibrium relation for volatile electrolytes, such as HC1, has the advantage that the electrolyte in aqueous solution... 

Gas, from a petroleum distillation column, has a concentration of H2S reduced from 0.03 (kmol H2S/kmol of inert hydrocarbon gas) to 1 per cent of this value by scrubbing with a triethanolamine-water solvent in a countercurrent tower, operating at 300 K and atmospheric pressure. The equilibrium relation for the solution may be taken as Ye = 2X. 

Of these possibilities, types (b), (c), and (d) all give rise to systems that may be used, although those of types (b) and (c) are the most promising. With conditions of type (b), the equilibrium relation is conveniently shown by a plot of the concentration of solute in one... 

It is also assumed that the transfer coefficients are independent of concentration. For dilute solutions and where the equilibrium relation is a straight line, a simple expression may be obtained for determining the required height of a column, by the same method as given in Chapter 12. 

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