Phase Equilibria Involving Solutions

The chemical potentials in Equation 5.51 can be expressed in terms of the standard state chemical potentials and the activities via Equation 5.2, but then the standard state chemical 

As previously discussed, a prime designates the after-mixing values. 

For each component, we can make a substitution for the activities in Equation 5.52. First, assuming the unmixed components are in their standard states, their activities are identically 1, and so In a = 0. Second, using Equation 5.35, the activities of the mixed components can be expressed in terms of the mole fraction and the component s activity coefficient a i = Xiji. This simplification of Equation 5.52 leads to 

Notice that if the solutions are ideal, the activity coefficients are 1.0 and the preceding equations simplify considerably. The volume change and the enthalpy change are zero in the limit of ideal behavior. 

Progressive effect on phase equilibria of a solvent from increasing the amount of solute. The heavy lines are those of the pure solvent. As the concentration of the solute increases, phase equilibria curves for the liquid-vapor interface and the solid-vapor interface are displaced further and further from those of the pure solvent. This leads to elevation of the boiling point and lowering of the freezing point. 

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There is one other three-phase equilibrium involving clathrates which is of considerable practical importance, namely that between a solution of Q, the clathrate, and gaseous A. For this equilibrium the previous formulas and many of the following conclusions also hold when replacing fiQa by fiQL, the chemical potential of Q in the liquid phase. But a complication then arises since yqL and the difference are not only... 

Electrode processes are a class of heterogeneous chemical reaction that involves the transfer of charge across the interface between a solid and an adjacent solution phase, either in equilibrium or under partial or total kinetic control. A simple type of electrode reaction involves electron transfer between an inert metal electrode and an ion or molecule in solution. Oxidation of an electroactive species corresponds to the transfer of electrons from the solution phase to the electrode (anodic), whereas electron transfer in the opposite direction results in the reduction of the species (cathodic). Electron transfer is only possible when the electroactive material is within molecular distances of the electrode surface thus for a simple electrode reaction involving solution species of the fonn... 

The separation of components by liquid-liquid extraction depends primarily on the thermodynamic equilibrium partition of those components between the two liquid phases. Knowledge of these partition relationships is essential for selecting the ratio or extraction solvent to feed that enters an extraction process and for evaluating the mass-transfer rates or theoretical stage efficiencies achieved in process equipment. Since two liquid phases that are immiscible are used, the thermodynamic equilibrium involves considerable evaluation of nonideal solutions. In the simplest case a feed solvent F contains a solute that is to be transferred into an extraction solvent S. 

If we look at the mechanistic and crystallographic aspects of the operation of polycomponent electrodes, we see that the incorporation of electroactive species such as lithium into a crystalline electrode can occur in two basic ways. In the examples discussed above, and in which complete equilibrium is assumed, the introduction of the guest species can either involve a simple change in the composition of an existing phase by solid solution, or it can result in the formation of new phases with different crystal structures from that of the initial host material. When the identity and/or amounts of phases present in the electrode change, the process is described as a reconstitution reaction. That is, the microstructure is reconstituted. 

In our discussion of (vapor + liquid) phase equilibria to date, we have limited our description to near-ideal mixtures. As we saw in Chapter 6, positive and negative deviations from ideal solution behavior are common. Extreme deviations result in azeotropy, and sometimes to (liquid -I- liquid) phase equilibrium. A variety of critical loci can occur involving a combination of (vapor + liquid) and (liquid -I- liquid) phase equilibria, but we will limit further discussion in this chapter to an introduction to (liquid + liquid) phase equilibria and reserve more detailed discussion of what we designate as (fluid + fluid) equilibria to advanced texts. 

Other measurements of AfG involve measuring AG for equilibrium processes, such as the measurement of equilibrium constants, reversible voltages of electrochemical cells, and phase equilibrium measurements. These methods especially come into play in the measurement of Afand AfG for ions in solution, which are processes that we will now consider. 

In order to begin this presentation in a logical manner, we review in the next few paragraphs some of the general features of polymer solution phase equilibrium thermodynamics. Figure 1 shows perhaps the simplest liquid/liquid phase equilibrium situation which can occur in a solvent(l)/polymer(2) phase equilibrium. In Figure 1, we have assumed for simplicity that the polymer involved is monodisperse. We will discuss later the consequences of polymer polydispersity. 

This illustrates the statement made earlier that the most convenient choice of standard state may depend on the problem. For gas-phase problems involving A, it is convenient to choose the standard state for A as an ideal gas at 1 atm pressure. But, where the vapor of A is in equilibrium with a solution, it is sometimes convenient to choose the standard state as the pure liquid. Since /a is the same for the pure liquid and the vapor in equilibrium... 

The purpose of this chapter is to outline the simplest methods of arriving at a description of the distribution of species in mixtures of liquids, gases and solids. Homogeneous equilibrium deals with single phase systems, such as electrolyte solutions (e.g., seawater) or gas mixtures (e.g., a volcanic gas). Heterogeneous equilibrium involves coexisting gaseous, liquid and solid phases. 

As already stated, one of the important pieces of data for biotransformation processes is knowledge of phase equilibrium and the activity of solutes involved. Hence, assuming that gas and liquid phases are at thermodynamic equilibrium, we can write... 

The calculation of the equilibrium conversion of heterogeneous reactions is in most cases much more complicated then in the case of homogeneous reactions, because the calculations involve in general the solution of the conditions for chemical equilibrium and the conditions for phase equilibrium. In the following a relatively simple example is given. 

In applying equation 33, Cpsl (the constant-pressure molar heat capacity of the stoichiometric liquid) is usually extrapolated from high-temperature measurements or assumed to be equal to Cpij of the compound, whereas the activity product, afXTjafXT), is estimated by interjection of a solution model with the parameters estimated from phase-equilibrium data involving the liquid phase (e.g., solid-liquid or vapor-liquid equilibrium systems). To relate equation 33 to an available data base, the activity product is expressed... 

A general formulation of the problem of solid-liquid phase equilibrium in quaternary systems was presented and required the evaluation of two thermodynamic quantities, By and Ty. Four methods for calculating Gy from experimental data were suggested. With these methods, reliable values of Gy for most compound semiconductors could be determined. The term Ty involves the deviation of the liquid solution from ideal behavior relative to that in the solid. This term is less important than the individual activity coefficients because of a partial cancellation of the composition and temperature dependence of the individual activity coefficients. The thermodynamic data base available for liquid mixtures is far more extensive than that for solid solutions. Future work aimed at measurement of solid-mixture properties would be helpful in identifying miscibility limits and their relation to LPE as a problem of constrained equilibrium. 

Chapters 17 and 18 use thermodynamics to describe solutions, with nonelectrolyte solutions described in Chapter 17 and electrolyte solutions described in Chapter 18. Chapter 17 focuses on the excess thermodynamic properties, with the properties of the ideal and regular solution compared with the real solution. Deviations from ideal solution behavior are correlated with the type of interactions in the liquid mixture, and extensions are made to systems with (liquid + liquid) phase equilibrium, and (fluid -I- fluid) phase equilibrium when the mixture involves supercritical fluids. 

When a solute is distributed between two immiscible liquids, different species, formed from the solute, may exist in the two liquids. Thus, when an organic liquid such as benzene or carbon tetrachloride and water are used as the two liquids, a weak acid may dimerize in the organic phase and partially ionize in the aqueous phase. The condition of equilibrium is the equality of the chemical potential of the monomeric, nonionized species in the two phases. If the dimerization is complete, the condition of equilibrium involves half of the chemical potential of the dimer in the organic phase. 

Equilibrium between solution and adsorbed or sorbed phases is a condition commonly used to evaluate adsorption or sorption processes in soils or soil-clay minerals. As previously stated, equilibrium is defined as the point at which the rate of the forward reaction equals the rate of the reverse reaction. Two major techniques commonly used to model soil adsorption or sorption equilibrium processes are (1) the Freundlich approach and (2) the Langmuir approach. Both involve adsorption or sorption isotherms. A sorption isotherm describes the relationship between the dissolved concentration of a given chemical species (adsorbate) in units of micrograms per liter (pg L 1), milligrams per liter (mg L-1), microequivalents per liter (pequiv L-1), or millimoles per liter (mmol L-1), and the sorbed quantity of the same species by the solid phase (adsorbent) in units of adsorbate per unit mass of adsorbent (solid) (e.g., pg kg-1, mg kg-1, peq kg-1, or mmol kg 1) at equilibrium under constant pressure and temperature. Sorption isotherms have been classified into four types, depending on their general shape (Fig. 4.13) ... 

The discussion in Sections 5.3.1 and 5.3.2 was based on the electrochemical cell shown in Figure 5.1, which included a platinum and a hydrogen electrode. This electrode pair gave us an understanding of pe and Eh, but one should keep in mind that the electrode potential in cell I is independent of the SHE and platinum electrode. The redox potential is determined only by the redox couple under consideration (e.g., Fe3+, Fe2+) when both members are present in solution or in contact with solution. In natural soil-water systems, more often than not, several redox pairs maybe present (e.g., Fe3+, Fe2+ SO2-, H2S N03, N02). In such situations, each pair will produce its own pe or Eh, but the three values may or may not be the same. When a chemical equilibrium between the three redox pairs is met, the three pe or Eh values will be the same (Drever, 1982). In nature, redox couples may also involve solution and solid phases, for example, Fe2+, Fe(II)(OH)3 Mn2+, Mn(III)(OOH) Fe(II)CO , Fe(II)(OH)3 their role in determining redox potential is demonstrated below. 

IR-spectra of pseudostrychnine in the solid phase and in solution indicate that it exists almost entirely in the carbinolamine form CXCVI. Two other forms might be expected to exist in equilibrium with CXCVI in solution the immonium form CXCVII, produced by proton-catalyzed loss of the hydroxyl group, is totally excluded on steric grounds the keto-amine form CXVIII, on the other hand, almost certainly exists in equilibrium with CXCVI in solution, but in concentrations which escape detection by physical methods. Its presence is deduced from some of the chemical reactions of pseudostrychnine. The easy formation of O-alkyl ethers (125) by interaction with methanol or ethanol even at room temperature almost certainly involves nucleophilic attack of the alcohol molecule on the carbonyl carbon in CXCVIII or in the O-protonated form CXCIX (R = H) to yield a hemiacetal as a first step likewise, the formation of 16-cyanostrychnine (XXV CN instead of OH) must involve addition of cyanide ion to CXCVIII, and the reaction with... 

Since the type of solutions encountered in extractive distillation involve mixtures of polar compounds or polar with nonpolar ones, the solutions are usually nonideal, and predicting the phase equilibrium from pure component data only is practically impossible. Theoretical and experimental studies through the years, however, have established certain trends which are used to search for and screen potential solvents. 

Subsurface solute transport is affected by hydrodynamic dispersion and by chemical reactions with soil and rocks. The effects of hydrodynamic dispersion have been extensively studied 2y 3, ). Chemical reactions involving the solid phase affect subsurface solute transport in a way that depends on the reaction rates relative to the water flux. If the reaction rate is fast and the flow rate slow, then the local equilibrium assumption may be applicable. If the reaction rate is slow and the flux relatively high, then reaction kinetics controls the chemistry and one cannot assume local equilibrium. Theoretical treatments for transport of many kinds of reactive solutes are available for both situations (5-10). 

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