Multiple Chemical Reaction Equilibria

This model represents the most frequently used description of chemical reaction equilibrium and should be familiar to most chemical engineering students. However, for multicomponent mixtures in which multiple reactions may take place, this type of non-linear problems may be cumbersome to solve numerically. One important obstacle is that the non-linear equilibrium constant definitions may give rise to multiple solutions, hence we have to identify which of them are the physical solutions. The stoichiometric formulation might thus be inconvenient for mixtures containing just a few species for which only a few reactions are taking place. 

In this chapter, you have derived the equations governing chemical reaction equilibrium and seen how the key parameters can be estimated using thermodynamics. You have solved the resulting problems using Excel, MATLAB, and Aspen Plus. You also learned to solve multiple equations using MATLAB when there are several reactions in equilibrium. 

The main contribution of non-linear chemistry is the pitchfork bifurcation diagram. A stable state becomes uixstable and bifurcates into two new stable branches. We are unable to foresee which one of these states will be chosen by the nature of the physico-chemical reaction. The multiplicity of choices gives its full importance to Ae evolution of the systems. This paper has aimed to show the extent to which the concepts of non-equilibrium and of deterministic chaos sublimate the fundamental physical laws by leading us to the creation of new structures and to auto-organization. Chemistry is no exception to this rule. The final conclusion is given by Jean-Marie Lehn [21], Nobel Prize Winner in... 

Unfortunately, few of the published studies of extraction equilibria heve provided complete quantitative models that are useful for extrapolation of data or for predicting multiple metal distribution equilibria from single metal data. The chemical-reaction equilibrium formulation provides a framework for constructing such models. One of the drawbacks of purely empirical correlations of distribution coefficients is that pH has often been chosen as an independent variable. Such a choice is suggested by the form of Pigs. 8-3-5 and 8.3-8. Although pH is readily measured and contmlled on a laboratory scale, it is really a dependent variable, which is detenmined by mass belances and simultaneous reaction equilibria. An appropriate phare-equilibrium model should be able to predict equilibrium pH, at least within a moderate activity coefficient correction, concurrently with other species concemrations. 

Solve the multiple chemical reaction equilibrium problem in Example 9.19 at 800 K using the following set of independent reactions ... 

The treatment of chemical reaction equilibria outlined above can be generalized to cover the situation where multiple reactions occur simultaneously. In theory one can take all conceivable reactions into account in computing the composition of a gas mixture at equilibrium. However, because of kinetic limitations on the rate of approach to equilibrium of certain reactions, one can treat many systems as if equilibrium is achieved in some reactions, but not in others. In many cases reactions that are thermodynamically possible do not, in fact, occur at appreciable rates. 

One useful trick in solving complex kinetic models is called the steady-state approximation. The differential equations for the chemical reaction networks have to be solved in time to understand the variation of the concentrations of the species with time, which is particularly important if the molecular cloud that you are investigating is beginning to collapse. Multiple, coupled differentials can be solved numerically in a fairly straightforward way limited really only by computer power. However, it is useful to consider a time after the reactions have started at which the concentrations of all of the species have settled down and are no longer changing rapidly. This happy equilibrium state of affairs may never happen during the collapse of the cloud but it is a simple approximation to implement and a place to start the analysis. 

Chemical reactions with autocatalytic or thermal feedback can combine with the diffusive transport of molecules to create a striking set of spatial or temporal patterns. A reactor with permeable wall across which fresh reactants can diffuse in and products diffuse out is an open system and so can support multiple stationary states and sustained oscillations. The diffusion processes mean that the stationary-state concentrations will vary with position in the reactor, giving a profile , which may show distinct banding (Fig. 1.16). Similar patterns are also predicted in some circumstances in closed vessels if stirring ceases. Then the spatial dependence can develop spontaneously from an initially uniform state, but uniformity must always return eventually as the system approaches equilibrium. 

Ung S, Doherty MF. Vapor-liquid equilibrium in systems with multiple chemical reactions. Chem Eng Sci 1995 50 23 18. 

For chemical reactions involving charged species in multiple (but contacting) phases, the condition for equilibrium is, in analogy to Eq. (30) of Chapter 7,... 

To understand the reculiarities of multiple layer formation, it suffices to consider the A-B binary system with three chemical compounds ApBq, ArBs and AiBn on the equilibrium phase diagram (Fig. 3.1). The scheme of analysis of the process of their occurrence at the A-B interface is analogous to that of two compound layers (see Chapter 2). First of all, the equations of partial chemical reactions taking place at phase interfaces must be written. These are as follows. 

S. Ung and M. F. Doherty, Calculation of residue curve maps for mixtures with multiple equilibrium chemical reactions. Ind. Engng. Chem. Res.,... 

Example 4.8 Chemical reactions and reacting flows The extension of the theory of linear nonequilibrium thermodynamics to nonlinear systems can describe systems far from equilibrium, such as open chemical reactions. Some chemical reactions may include multiple stationary states, periodic and nonperiodic oscillations, chemical waves, and spatial patterns. The determination of entropy of stationary states in a continuously stirred tank reactor may provide insight into the thermodynamics of open nonlinear systems and the optimum operating conditions of multiphase combustion. These conditions may be achieved by minimizing entropy production and the lost available work, which may lead to the maximum net energy output per unit mass of the flow at the reactor exit. 

This chapter, after introducing the equilibrium constant, discusses briefly the rate of entropy production in chemical reactions and coupling aspects of multiple reactions. Enzyme kinetics is also summarized. 

The screen for each chemical kinetic calculation simultaneously displays a variety of characterizations in multiple windows and allows analysis of time/temperature-dependent species and reaction information including species concentrations, species steady-state analysis, individual reaction rates, net production/destruction rates, reaction equilibrium analysis and the temperature/time history of the system. The interactive user-sorting of the species and reaction information from the numerical simulations is mouse/cursor driven. An additional feature also allows interactive analysis and identification of dependent and independent species and reaction pathways, on-line reaction network analysis and pathway/flowchart construe-... 

These are systems that exchange both energy and matter with the environment through their boundaries. The simplest chemical reaction engineering example is the continuous stirred tank reactor. These systems do not tend toward their thermodynamic equilibrium, but rather towards a state called stationary non-equilibrium state and is characterized by minimum entropy production. Open systems near equilibrium have unique stationary non-equilibrium state, regardless of the initial conditions. However far from equilibrium these systems may exhibit multiplicity of stationary states and may also exhibit periodic states. 

Nonequilibrium transport of solutes through porous media occurs when ground-water velocities are sufficiently fast to prevent attainment of chemical and physical equilibrium. Chemical reactions in porous media often require days or weeks to reach equilibrium. For example. Fuller and Davis Q) reported that cadmium sorption by a calcareous sand was characterized by multiple reactions, including a recrystallization reaction that continued for a period of days. Sorption of oxyanions by metal oxyhydroxides often occurs at an initially rapid rate the rate then decreases until steady-state is achieved (2-4). Unless ground-water velocity in such a situation is extremely slow, nonequilibrium transport will occur. 

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. 

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