Equilibrium systems

Figure A2.1.9. Chemically reacting systems, (a) The entropy. S as a fiinction of the degree of advancement of the reaction at constant U and V. (b) The affinity Aas a fiinction of for the same reacting system. Equilibrium is reached at 0.623 where tiis a maxuniim and A= 0.

A. C. Wittiagham, iLiquid Sodium—Hydrogen System Equilibrium and Kinetic Measurements in t/je 610—667 K Temperature Range, NTIS Accession No. 

Chemical vapor deposition processes are complex. Chemical thermodynamics, mass transfer, reaction kinetics and crystal growth all play important roles. Equilibrium thermodynamic analysis is the first step in understanding any CVD process. Thermodynamic calculations are useful in predicting limiting deposition rates and condensed phases in the systems which can deposit under the limiting equilibrium state. These calculations are made for CVD of titanium - - and tantalum diborides, but in dynamic CVD systems equilibrium is rarely achieved and kinetic factors often govern the deposition rate behavior. 

Figure 29.4 shows an example, the energy diagram of a cell where n-type cadmium sulfide CdS is used as a photoanode, a metal that is corrosion resistant and catalytically active is used as the (dark) cathode, and an alkaline solution with S and S2 ions between which the redox equilibrium S + 2e 2S exists is used as the electrolyte. In this system, equilibrium is practically established, not only at the metal-solution interface but also at the semiconductor-solution interface. Hence, in the dark, the electrochemical potentials of the electrons in all three phases are identical. 

We briefly review processes in which isotopic fractionations may be recorded in isotopically distinct reservoirs that are preserved in nature. These concepts have been extensively covered in the H, C, O, and S isotope literature, and we illustrate several examples for the non-traditional stable isotope systems discussed in this volume. One of the simplest processes that produces isotopically distinct reservoirs would be slow reaction of substance A to B, where A and B remain open to complete isotopic exchange during the process. This is commonly referred to as closed system equilibrium, and the changes in isotopic compositions that occur may be defined by the exact relation ... 

Adsorption of Mo to Mn oxyhydroxides produces an isotopic fractionation that appears to follow that of a closed-system equilibrium model as a function of the fraction of Mo adsorbed (Fig. 8). Barling and Anbar (2004) observed that the 5 TVlo values for aqueous Mo (largely the [MoOJ species) were linearly correlated with the fraction (/) of Mo adsorbed (Fig. 8), following the form of Equation (14) above. The 5 Mo-f relations are best explained by a MOaq-Mn oxyhydroxide fractionation of +1.8%o for Mo/ Mo, and this was confirmed through isotopic analysis of three solution-solid pairs (Fig. 8). The data clearly do not lie... 

Figure 8. Example of apparent closed-system equilibrium fractionation, where Mo in solution is sorbed to Mn oxides (Barling and Anbar 2004). The 6 Mo values of the Mo remaining in solution during sorption follow die linear trends that are consistent widi closed-system equilibrium fractionation where isotopic equilibrium is continuously maintained between Mo in solution and diat sorbed to die Mn oxides. Three aqueous-solid pairs (shown widi tie lines) are consistent with this interpretation. The isotopic data cannot be ejqilained dirough a Rayleigh process, where die product of die reaction (sorbed Mo) is isolated from isotopic exchange widi aqueous Mo.

Vapor and brine from the Brandon vent of the East Pacific Rise have identical Fe isotope compositions, implying that phase separation does not produce an isotopic fractionation (Beard et al. 2003a). The role that sulfide precipitation plays in controlling the Fe isotope composition of the fluid remains unknown. The precision of the two sulfide analyses reported by Sharma et al. (2001) was not sufficient to resolve if sulfide precipitation would produce Fe isotope fractionation in the vent fluid. In a detailed study of sulfldes from the Lucky Strike hydrothermal field from the mid Atlantic Ridge, however, Rouxel et al. (2004) found that sulfldes span a range in 5 Fe values from -2.0 to +0.2%o, and that pyrite/marcasite has lower 5 Fe values ( l%o) as compared to chalcopyrite. The variations in mineralogy and isotope composition are inferred to represent open-system equilibrium fractionation of Fe whereby... 

Figure 8. Results of Mo adsoqjtion experiments of Barling and Anbar (2004). Mo-bearing solutions were exposed to synthetic Mn oxides (5-Mn02) for 2-96 hours at pH 6.5-8.5. Residual Mo in solution ( ) was measured for all experiments. Mo adsorbed to oxide particle surfaces ( ) was either measured or inferred from mass balance. Dissolved Mo was systematically heavier than adsorbed Mo with a fractionation factor of 1.0018 0.0005. The data are consistent with closed system equilibrium, in which isotopes exchange continuously between surface and solution, but incompatible with an irreversible, Rayleigh-type process. Figure modified after Barling and Anbar (2004).

If such a reaction is isolated from its surroundings, so that A and B neither enter nor leave the system, creating a closed system, these reactants will eventually reach equilibrium, i.e. the forward and reverse rates are the same. For a closed system, equilibrium is the only state in which the concentrations of A and B do not vary with time. Since AG = 0, the reaction can do no useful work and has no direction. Hence a closed system is not a useful model of a living cell or, indeed, a living animal. 

This relationship holds for any chemical system which is subject to variations in temperature, pressure, and proportions of its basic components and describes the number of phases P present in terms of the system s degrees of freedom F and the number of component species C. Even though the phase rule is simple in form, it is not limited in its ability to describe very complex systems. Equilibrium effects arising from the presence of surface tension, stress, magnetic fields, etc. can be accounted for by the incorporation of additional degrees of freedom into the phase rule. Such effects, however, will not be considered in this discussion. 

In other systems, equilibrium by this reaction does not prevail, as this redox reaction is... 

For a continuous extractive distillation process to be possible there must be adequate enhancement of the nitric acid-water relative volatility, and a system equilibrium which permits virtually complete separation of nitric acid from magnesium nitrate, the latter taking up the water content of the weak acid feedstock. This requires addition to the weak nitric acid of solutions of magnesium nitrate usually containing 60 wt% or more of Mg(NC>3)2. Under these conditions a nitric acid-water relative volatility of greater than 2.0 is obtained at the low end of the liquid phase concentration at a nitric acid mole fraction below 0.05 (4, 7). 

EQUILIBRIUM. In the elementary sense of the macroscopic (visible lo the naked eycl system, equilibrium is obtained if the system does mu tend to undergo any further change of its own uccord. 

Equilibrium is obtained in a system when p is the same everywhere in the system. Equilibrium between the liquid and the vapor phases implies that p is the same in both phases. It is this fact that allows the measurement of the vapor phase to determine the water activity of the sample. 

The steps for constructing and interpreting an isothermal, isobaric thermodynamic model for a natural water system are quite simple in principle. The components to be incorporated are identified, and the phases to be included are specified. The components and phases selected "model the real system and must be consistent with pertinent thermodynamic restraints—e.g., the Gibbs phase rule and identification of the maximum number of unknown activities with the number of independent relationships which describe the system (equilibrium constant for each reaction, stoichiometric conditions, electroneutrality condition in the solution phase). With the phase-composition requirements identified, and with adequate thermodynamic data (free energies, equilibrium con-... 

We shall attempt the working hypothesis that in the intermediate system equilibrium would require the maximum number (seven) of phases. This is not entirely unreasonable since the composition of sea water seems quite stable. Hence, which are the seven phases, and what evidence exists concerning the stable-phase assemblage ... 

Equilibrium Data. Liquid and Solid Phases, NH3-C02-H20 System. Equilibrium in condensed systems in the 20° to 40° C. temperature range and from 1- to 4.5-atm. absolute pressure are reported by Neumann and Domke (21). Similar data are reported by Terres and Weiser (26) and Guyer and Piechowicz (9) but over a wider temperature range and for less complex systems. The latter authors also report that... 

In every system, equilibrium corresponds to the state of lowest free energy. When two components form a solution, the change of free energy on mixing is... 

Hance (1967) investigated the rate of sorption and desorption of four pesticides (monuron, linuron, atrazine, and chlorpropham) on two soils, a soil organic matter fraction, and bentonite, a 2 1 smectitic clay mineral. An equilibrium in sorption was reached in 24 h for every system except one (Table 6.1). With eight of the 18 systems equilibrium was reached in less than 4 h, and in five cases equilibrium was established in 1 hr. Equilibrium was attained for most of the systems in 4-24 h. Desorption was slower than sorption. In only eight systems was an equilibrium reached in 24 h. Hance... 

Only a few full dynamic solutions for systems with more than two transitions have been derived, and for multicomponent adiabatic systems equilibrium theory offers the only practical approach. 

Equations 29-37 provide a practical scheme to compute orbital reactivity indices, in vicinity of the system equilibrium point in the parametric space of the KS-eigenvalues and occupation numbers. 

Approximate procedures have been evolved which permit one to determine the state of the expansion process for a given system. In fact these procedures permit the performance to be calculated when the chemical rates are finite and thus do not correspond to frozen, essentially zero chemical rate or equilibrium, essentially infinite chemical rate,flow. As one would expect intuitively, the results of these finite rate determinations show that the flow remains nearly in chemical equilibrium at the beginning of the expansion process, and at a given temperature or point in the nozzle the composition becomes frozen and remains so throughout the expansion process. Finite rate performance calculations are very complex and are presently limited to only a few systems due to lack of kinetic data at the temperatures of concern. Thus most performance calculations are made for either or both equilibrium and frozen flow and it is kept in mind that the actual results must lie somewhere between the two. For most systems equilibrium calculations are very satisfactory. 

In many separation processes (chromatography, countercurrent distribution, field-flow fractionation, extraction, etc.), the transport of components, in one dimension at least, occurs almost to the point of reaching equilibrium. Thus equilibrium concentrations often constitute a good approximation to the actual distribution of components found within such systems. Equilibrium concepts are especially crucial in these cases in predicting separation behavior and efficacy. 

Each resolution implies its own case of the intra-system equilibrium. For example, the L-resolution corresponds to a totally constrained ( frozen ) electron distribution, with all local, infinitesimal volume elements considered as being mutually closed, while the g-resolution represents the opposite extreme of a totally relaxed electron distribution, with all local volume elements regarded as mutually open. Similarly, the levels of intermediate resolutions apply to cases of partially constrained equilibrium, in which all fragments defining the partitioning of a molecular system M ... 

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