Solutions and Condensed Phases

Up to this point the equilibrium constants have been expressed in terms of partial pressures. However, for real gases the fugacities of the species should be used. If the pressures are low enough, the pressures themselves can be used, because at low pressures the pressure is approximately equal to the fugacity But many chemical reactions involve phases other than the gas phase. Solids, liquids, and dissolved solutes also participate in chemical reactions. How are they represented in equilibrium constants  

We answer this by defining activity a, of a material in terms of its standard chemical potential yu,° and its chemical potential yu, under nonstandard pressures  

Comparison of this equation with equation 4.59 shows that for a real gas, activity is defined in terms of the fugacity as 

Reaction quotients (and equilibrium constants) are more formally written in terms of activities, rather than pressures  

Unless otherwise noted, all art on this page is Cengage Learning 2014. 

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Usually, in plots such as those in figure 8.20, for the sake of simplicity, the boundaries between solutes and condensed phases are not drawn for the various molal concentrations in solution, as seen for cerium, but for a selected bulk molal amount, which in geochemistry is normally 10 . This choice is dictated by the fact that, at this bulk molal concentration, the solid phase can be considered effectively inert from the point of view of reactivity (cf Garrels and Christ, 1965). In the various heterogeneous equilibria involving aqueous solutions and condensed or gaseous phases, it is nevertheless opportune always to specify the molal concentrations to which the various boundaries refer. 

Table 8.17 Main predominance limits of aqueous complexes and saturation limits between solutes and condensed phases in iron-bearing aqueous solutions (see figure 8.22). Standard state Gibbs free energies of formation of species are listed in table 8.18. (c) = crystalline ... 

The inability to adequately understand complex systems — such as solutions and condensed phases in general — on the basis of molecular theory has made it necessary to develop alternative methods. Basically... 

Zaharchenko et al. [110] have investigated the laser-induced luminescence of CdSe/ZnS nanoparticles in solution and condensed phase. The luminescence properties of monodispersed CdSe/ZnS nanoparticles and Rhodamine 6G in solution and condensed phase are compared upon laser excitation in the visible spectral range for a wide range of the radiation power density and the PL quantum yield of nanoparticles is determined. It is demonstrated that, in the condensed phase, the luminescence quantum yield of nanoparticles is higher than that of the dye by two orders of magnitude. 

The synthesis of highly substituted rigid tricyclic nitrogen heterocycles via a tandem four-component condensation (the Ugi reaction)/intramolecular Diels-Alder reaction was investigated in both solution and solid phase [24]. The Ugi reaction in MeOH (Scheme 4.2) involves the condensation of furylaldehydes 17, benzylamine 18, benzyl isocyanide 19 and maleic or fumaric acid derivatives 20, and provides the triene 21 which immediately undergoes an intramolecular Diels-Alder reaction, affording the cycloadduct 22 in a diastereoisomeric mixture with high yield. 

The thermodynamic development above has been strictly limited to the case of ideal gases and mixtures of ideal gases. As pressure increases, corrections for vapor nonideality become increasingly important. They cannot be neglected at elevated pressures (particularly in the critical region). Similar corrections are necessary in the condensed phase for solutions which show marked departures from Raoult s or Henry s laws which are the common ideal reference solutions of choice. For nonideal solutions, in both gas and condensed phases, there is no longer any direct... 

As we did for equilibria between solute species, we can also define the boundaries between solute species and condensed phases. Assuming the condensed forms to be pure phases (i.e., assuming unitary activity), in the presence of metallic cerium we have... 

In contrast to solid state crystallization, crystallization from vapor, solution, and melt phases, which correspond to ambient phases having random structures, may be further classified into condensed and dilute phases. Vapor and solution phases are dilute phases, in which the condensation process of mass transfer plays an essential role in crystal growth. In the condensed melt phase, however, heat transfer plays the essential role. In addition to heat and mass transfer, an additional factor, solute-solvent interaction, should be taken into account. 

The basic criteria to consider in crystal growth from vapor, solution, and melt phases are therefore whether the phase is condensed or dilute, and whether the phase involves a solute-solvent interaction or not. 

Once A" is determined by the extrapolation procedure of Sec. 10.11 or from a table of electrode potentials (which implies that someone else has done the extrapolation), measurement of as a function of concentration allows determination of Qa as a function of concentration of the cell electrolyte from Eq. (47). Often, the electrolyte consists of a single solute, whose activity can then be determined from Qa. For example, for the cell in Fig. 5, assuming the activities of the H2 gas and condensed phases are unity,... 

Solution of condensed-phase heat transfer equation is needed to analyze structural response to fires and simulate flame spread on solid surfaces. The solution of this conjugate heat transfer problem simulate is typical for fires, but rarely found in commercial CFD packages. Over the years, different techniques have been developed to tackle this problem. Since solid-phase heat transfer... 

Developments in experimental and computational science have shed light on phenomena in bioenvironments and condensed phases that pose significant challenges for theoretical models of solvation [27]. Tapia [22] raises the important distinction between solvation theory and solvent effects theory. Solvation theory is concerned with direct evaluation of solvation free energies this is extensively covered by recent reviews [16,17]. Solvent-effect theory concerns changes induced by the medium onto electronic structure and molecular properties of the solute. Solvent-effect theory is concerned with molecular properties of the solvated molecule relative to the properties in vacuo as such it focuses on chemical features suitable for studying systems at the microscopic level [23]. Extensive reviews of different computational methods are given in a book by Warshel [24]. 

Liquid-vapor phase diagrams, and boiling-point diagrams in particular, are of importance in connection with distillation, which usually has as its object the partial or complete separation of a liquid solution into its components. Distillation consists basically of boiUng the solution and condensing the vapor into a separate receiver. A simple one-plate distillation of a binary system having no maximum or minimum in its boiling-point curve can be understood by reference to Fig. 3. Let the mole fraction of B in the initial solution be represented by... 

The vapor pressure of pure chloroform at 40°C is 366 torr, and that of pure carbon tetrachloride is 143 torr. If a solution with mole fraction 0.180 in chloroform is allowed to evaporate, the vapor phase is separated from the solution and condensed, and the resulting solution is allowed to evaporate, what would be the mole fraction of the vapor phase after the second evaporation ... 

With adiabatic combustion, departure from a complete control of m by the gas-phase reaction can occur only if the derivation of equation (5-75) becomes invalid. There are two ways in which this can happen essentially, the value of m calculated on the basis of gas-phase control may become either too low or too high to be consistent with all aspects of the problem. If the gas-phase reaction is the only rate process—for example, if the condensed phase is inert and maintains interfacial equilibrium—then m may become arbitrarily small without encountering an inconsistency. However, if a finite-rate process occurs at the interface or in the condensed phase, then a difficulty arises if the value of m calculated with gas-phase control is decreased below a critical value. To see this, consider equation (6) or equation (29). As the value of m obtained from the gas-phase analysis decreases (for example, as a consequence of a decreased reaction rate in the gas), the interface temperature 7], calculated from equation (6) or equation (29), also decreases. According to equation (37), this decreases t. Eventually, at a sufficiently low value of m, the calculated value of T- corresponds to Tj- = 0, As this condition is approached, the gas-phase solution approaches one in which dT/dx = 0 at x = 0, and the reaction zone moves to an infinite distance from the interface. The interface thus becomes adiabatic, and the gas-phase processes are separated from the interface and condensed-phase processes. 

Acid salts of imidazole- and benzimidazole-related compounds have been evaluated as alternative promoters to the various activators developed for the condensation of a nucleoside phosphoramidite and a nucleoside. The acid/azole complexes were developed to circumvent some of the disadvantages most commonly encountered in both solution- and solid-phases. Azolium promoters were shown to achieve high yielding coupling reactions even with nucleosides of low reactivity. Hayakawa has also reviewed and broadened the recent phosphoramidite methodologies by describing the versatility of allyl and allyloxycar-... 

Let us now consider the systems which have one degree of freedom (so-caUed monovariant systems). According to the phase rule, these must consist of three phases, and may contain gas, solution, and sohd phase, or gas and two sohd phases. Systems composed of sohd phases alone, or of sohd and liquid phases alone, are caUed condensed systems. Condensed systems See Abegg, Handb. d. anorg. Chemie, vol. ii. p. 95. 

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