Chemical equilibrium in a single-phase system

Chemical Equilibrium in a Single-Phase System 713 Thus, Eq. 13.1-17 can be rewritten as... 

The criterion for chemical equilibrium in a single-phase, single-reaction system is stated in terms of the minimization of the free energy of the system. In terms of the reaction, then... 

To prove this assertion, it is first useful to consider the mathematical technique of Lagrange multipliers, a method used to find the extreme (maximum or minimum) value of a function subject to constraints. Rather than develop the method in complete generality, we merely introduce it by application to the problem just considered equilibrium in a single-phase, multiple-chemical reaction system. 

We identified the equilibrium state for several chemical reactions occurring in a single-phase system at constant temperature and pressure by finding the state for which G = N Gi was equal to a minimum subject to the stoichiometric constraints... 

We may treat the chemical potential /x of a pure substance in a single phase as a function of the independent variables T and p, and represent the function by a three-dimensional surface. Since the condition for equihbrium between two phases of a pure substance is that both phases have the same T, p, and /x, equilibrium in a two-phase system can exist only along the intersection of the surfaces of the two phases as illustrated in Fig. 8.12. 

Equations 8.7-4 also provide a means of identifying the equilibrium state when chemical reactions occur. To see this, consider first the case of a single chemical reaction occurring in a single phase (both of these restrictions will be removed shortly) in a closed system at constant temperature and pressure.The total Gibbs energy for this system, using the reaction variable notation introduced in Sec. 8.3, is... 

The state of chemical equilibrium for M. independent reactions occurring in a single phase is the state that satisfies the constraints on the system, the set of stoichiometric relations... 

Thus we have normalized the process to the bulk temperature of the gas phase and a single component (1). Importantly, the chemical potentials at this normalized state refer still to an equilibrium state and they will cancel therefore. Note that the chemical potential for a single component system is not dependent on the mol number, or in other words on the size of the phase. 

The phase rule is deduced from rather formal arguments of linear algebra pointed out above. For the individual phases, there are certain relations, i.e., equations among the intensive variables in equilibrium. These include equal temperatures, equal pressures, and equal chemical potentials. Further, there are the variables that describe the system. We chose the intensive variables and subdivide these into the chemical potentials and other intensive variables, like temperature, pressure. For each component there is a chemical potential. So the number of variables is K for the chemical potential and I for the other intensive variables in a single phase. 

Reactive distillation columns incorporate both phase separation and chemical reaction in a single unit. In some systems, they have economic advantages over conventional reac-tor/separation/recycle flowsheets, particularly for reversible reactions in which chemical equilibrium constraints limit conversion in a conventional reactor. Because both reaction and separation occur in a single vessel operating at some pressure, the temperatures of reaction and separation are not independent. Therefore, reactive distillation is limited to systems in which the temperatures conducive for reaction are compatible with temperatures conducive for vapor-liquid separation. 

If a mixture of monomers is not in equilibrium, any polymerization that occurs at constant temperature, T, and constant pressure, P, must lead to a decrease in the total Gibbs free energy of the system. A reaction variable may be defined. The fundamental property relation for a single phase system for the total differential of the Gibbs free energy can be written as a function of temperature, pressure, and chemical potential as follows ... 

Intelligent engineering can drastically improve process selectivity (see Sharma, 1988, 1990) as illustrated in Chapter 4 of this book. A combination of reaction with an appropriate separation operation is the first option if the reaction is limited by chemical equilibrium. In such combinations one product is removed from the reaction zone continuously, allowing for a higher conversion of raw materials. Extractive reactions involve the addition of a second liquid phase, in which the product is better soluble than the reactants, to the reaction zone. Thus, the product is withdrawn from the reactive phase shifting the reaction mixture to product(s). The same principle can be realized if an additive is introduced into the reaction zone that causes precipitation of the desired product. A combination of reaction with distillation in a single column allows the removal of volatile products from the reaction zone that is then realized in the (fractional) distillation zone. Finally, reaction can be combined with filtration. A typical example of the latter system is the application of catalytic membranes. In all these cases, withdrawal of the product shifts the equilibrium mixture to the product. 

A phase boundary for a single-component system shows the conditions at which two phases coexist in equilibrium. Recall the equilibrium condition for the phase equilibrium (eq. 2.2). Letp and Tchange infinitesimally but in a way that leaves the two phases a and /3 in equilibrium. The changes in chemical potential must be identical, and hence... 

In this first example, a single-component system consisting of a liquid and a gas phase is considered. If the surface between the two phases is curved, the equilibrium conditions will depart from the situation for a flat surface used in most equilibrium calculations. At equilibrium the chemical potentials in both phases are equal ... 

For any pure chemical species, there exists a critical temperature (Tc) and pressure (Pc) immediately below which an equilibrium exists between the liquid and vapor phases (1). Above these critical points a two-phase system coalesces into a single phase referred to as a supercritical fluid. Supercritical fluids have received a great deal of attention in a number of important scientific fields. Interest is primarily a result of the ease with which the chemical potential of a supercritical fluid can be varied simply by adjustment of the system pressure. That is, one can cover an enormous range of, for example, diffusivities, viscosities, and dielectric constants while maintaining simultaneously the inherent chemical structure of the solvent (1-6). As a consequence of their unique solvating character, supercritical fluids have been used extensively for extractions, chromatographic separations, chemical reaction processes, and enhanced oil recovery (2-6). 

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