Chemical reaction state variables

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

The general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

For a PVnr system of uniform T and P containing N species and 7T phases at thermodynamic equiUbrium, the intensive state of the system is fully deterrnined by the values of T, P, and the (N — 1) independent mole fractions for each of the equiUbrium phases. The total number of these variables is then 2 + 7t N — 1). The independent equations defining or constraining the equiUbrium state are of three types equations 218 or 219 of phase-equiUbrium, N 7t — 1) in number equation 245 of chemical reaction equiUbrium, r in number and equations of special constraint, s in number. The total number of these equations is A(7t — 1) + r -H 5. The number of equations of reaction equiUbrium r is the number of independent chemical reactions, and may be deterrnined by a systematic procedure (6). Special constraints arise when conditions are imposed, such as forming the system from particular species, which allow one or more additional equations to be written connecting the phase-rule variables (6). 

A knowledge of the concentrations of all reactants and products is necessary for a description of the equilibrium state. However, calculation of the concentrations can be a complex task because many compounds may be Imked by chemical reactions. Changes in a variable such as pH or oxidation potential or light intensity can cause large shifts in the concentrations of these linked species. Aggregate variables may provide a means of simplifying the description of these complex systems. Here we look at two cases that involve acid-base reactions. 

The reaction rates Rt will be functions of the state variables defining the chemical system. While several choices are available, the most common choice of state variables is the set of species mass fractions Yp and the temperature T. In the literature on reacting flows, the set of state variables is referred to as the composition vector [Pg.267]

The simplest case of this parameter estimation problem results if all state variables jfj(t) and their derivatives xs(t) are measured directly. Then the estimation problem involves only r algebraic equations. On the other hand, if the derivatives are not available by direct measurement, we need to use the integrated forms, which again yield a system of algebraic equations. In a study of a chemical reaction, for example, y might be the conversion and the independent variables might be the time of reaction, temperature, and pressure. In addition to quantitative variables we could also include qualitative variables as the type of catalyst. 

In chemical equilibria, the energy relations between the reactants and the products are governed by thermodynamics without concerning the intermediate states or time. In chemical kinetics, the time variable is introduced and rate of change of concentration of reactants or products with respect to time is followed. The chemical kinetics is thus, concerned with the quantitative determination of rate of chemical reactions and of the factors upon which the rates depend. With the knowledge of effect of various factors, such as concentration, pressure, temperature, medium, effect of catalyst etc., on reaction rate, one can consider an interpretation of the empirical laws in terms of reaction mechanism. Let us first define the terms such as rate, rate constant, order, molecularity etc. before going into detail. 

Almost all the crystalline materials discussed earlier involve only one molecular species. The ramifications for chemical reactions are thereby limited to intramolecular and homomolecular intermolecular reactions. Clearly the scope of solid-state chemistry would be vastly increased if it were possible to incorporate any desired foreign molecule into the crystal of a given substance. Unfortunately, the mutual solubilities of most pairs of molecules in the solid are severely limited (6), and few well-defined solid solutions or mixed crystals have been studied. Such one-phase systems are characterized by a variable composition and by a more or less random occupation of the crystallographic sites by the two components, and are generally based on the crystal structure of one component (or of both, if they are isomorphous). 

Gibbs phase rule phys chem A relationship used to determine the number of state variables F, usually chosen from among temperature, pressure, and species composition in each phase, which must be specified to fix the thermodynamic state of a system in equilibrium F = C - P - M+2, where C is the number of chemical species presented at equilibrium, P is the number of phases, and M is the number of independent chemical reactions. Also known as Gibbs rule phase rule. gibz faz, rijl I... 

The chemical reactor is the unif in which chemical reactions occur. Reactors can be operated in batch (no mass flow into or out of the reactor) or flow modes. Flow reactors operate between hmits of completely unmixed contents (the plug-flow tubular reactor or PFTR) and completely mixed contents (the continuous stirred tank reactor or CSTR). A flow reactor may be operated in steady state (no variables vary with time) or transient modes. The properties of continuous flow reactors wiU be the main subject of this course, and an alternate title of this book could be Continuous Chemical Reactors. The next two chapters will deal with the characteristics of these reactors operated isothermaUy. We can categorize chemical reactors as shown in Figure 2-8. 

In further discussion given by Taylor it is shown that for one- dimensional steady plane waves X can coincide with Y, so that the reaction and steady zones also coincide. This does not hold for variable waves or when the motion ceases to be one-dimensional here Xj will, in general, lie within Xs Y, so that the latter part of the reaction takes place outside of the steady zone. Eqs (VI.6,9 10) apply thruout the steady zone and in particular at the section Xl in which chemical equilibrium is attained. Since D and appear only in (VI.9 10), they are set aside, leaving (VI.6) which involves p v, e. If the chem compn wete independent of the state variables, e could be defined immediately as a function... 

The progress of a given reaction depends on the temperature, pressure, flow rates, and residence times. Usually these variables are controlled directly, but since the major feature of a chemical reaction is composition change, the analysis of composition and the resetting of the other variables by its means is an often used means of control. The possible occurrence of multiple steady states and the onset of instabilities also are factors in deciding on the nature and precision of a control system. 

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