Reaction first-order steady-state chemical

For first-order steady-state chemical reaction, the material... 

The diffusion cloud (flame) technique developed by Hartel and Polanyi in the 1930s is one of the early methods of studying rapid bimolecular chemical reactions imder pseudo-first-order, steady-state conditions. This method is the source of most measured rates for reactions of alkali metals with halogenated compounds and still serves as a basis for experimental and theoretical studies. In most applications of the technique, sodium metal is heated in an oven, mixed with an inert carrier gas, and allowed to diffuse into a backgroimd of a reactant gas. In very exothermic reactions the sodium flame is chemiluminescent otherwise the cloud is illuminated with a sodium resonance lamp. The reaction rate can be measured either by determining the distance the sodium diffuses until it all reacts or by spectroscopically measuring the total amount of sodium in the cloud. ... 

General first-order kinetics also play an important role for the so-called local eigenvalue analysis of more complicated reaction mechanisms, which are usually described by nonlinear systems of differential equations. Linearization leads to effective general first-order kinetics whose analysis reveals infomiation on the time scales of chemical reactions, species in steady states (quasi-stationarity), or partial equilibria (quasi-equilibrium) [M, and ]. 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

The transition between surface and diffusion control is easily demonstrated by considering a first-order surface reaction. In the steady state the rate at which reactant A diffuses to the surface equals the rate at which it is consumed by the chemical reaction, so that... 

RRDE is significantly simpler than with conventional cyclic voltammetry data in quiescent solutions [88, 89]. As such, these forced convection systems have been widely used in the study of electrocatalysis in general. Of special interest are situations where the rate determining step is chemical (a) or electrochemical (B) (Scheme 3.7) [60], In particular, for an RDE at steady state, the rate at which the reactant is depleted at the interface must be equal to the rate at which it is replenished from the solution via convective mass transport. For a reaction first order in dioxygen this relationship reads ... 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

It may be noted that, in the absence of a chemical reaction, equation 10.203 reduces to equation 10.146. For a steady-state process dCA/dt = 0, and for a first-order reaction n = l. Thus ... 

At steady-state conditions, the rate of supply of S by diffusion is balanced by the rate of consumption by chemical reaction, where assuming a first-order chemical reaction... 

The mechanisms considered above are all composed of steps in which chemical transformation occurs. In many important industrial reactions, chemical rate processes and physical rate processes occur simultaneously. The most important physical rate processes are concerned with heat and mass transfer. The effects of these processes are discussed in detail elsewhere within this book. However, the occurrence of a diffusion process in a reaction mechanism will be mentioned briefly because it can lead to kinetic complexities, particularly when a two-phase system is involved. Consider a reaction scheme in which a reactant A migrates through a non-reacting fluid to reach the interface between two phases. At the interface, where the concentration of A is Caj, species A is consumed in a first-order chemical rate process. In effect, consecutive rate processes are occurring. If a steady state is achieved, then... 

Modeling in drinking water applications is largely confined to describing chemical processes. The mathematical models used in this area are based on the reaction rate equation to describe the oxidation of the pollutants, combined with material balances on the reaction system to calculate the concentrations of the oxidants as a function of the water matrix. As noted above, the reaction rate equation is usually simplified to pseudo-first order. This is based on the assumption of steady-state concentrations for ozone and the radicals involved in the indirect reaction. 

Abstract—The question of the multiplicity of the steady states of a chemical reactor was one of the concerns in the pioneering work ofBilous and Amundson. Their diagrams showed quite clearly the geometry of the situation, and this kind of analysis sufficed for many years. It remained for Balakotaiah and Luss, using the methods of singularity theory, to give a comprehensive treatment of the question. After a brief survey, we take up the case of consecutive first-order reactions and show that up to seven steady states are possible. 

Another type of stability problem arises in reactors containing reactive solid or catalyst particles. During chemical reaction the particles themselves pass through various states of thermal equilibrium, and regions of instability will exist along the reactor bed. Consider, for example, a first-order catalytic reaction in an adiabatic tubular reactor and further suppose that the reactor operates in a region where there is no diffusion limitation within the particles. The steady state condition for reaction in the particle may then be expressed by equating the rate of chemical reaction to the rate of mass transfer. The rate of chemical reaction per unit reactor volume will be (1 - e)kCAi since the effectiveness factor rj is considered to be unity. From equation 3.66 the rate of mass transfer per unit volume is (1 - e) (Sx/Vp)hD(CAG CAl) so the steady state condition is ... 

To incorporate mixing by the dispersed plug flow mechanism into the model for the bubble column, we can make use of the equations developed in Chapter 2 for dispersed plug flow accompanied by a first-order chemical reaction. In the case of the very fast gas-liquid reaction, the reactant A is transferred and thus removed from the gas phase at a rate which is proportional to the concentration of A in the gas, i.e. as in a homogeneous first-order reaction. Applied to the two-phase bubble column for steady-state conditions, equation 2.38 becomes ... 

We should realize that certain chemical/biochemical problems can have no multiplicities of their steady states over their entire range of parameters. Consider, for example, a simple first-order reaction process A => B with the rate equation... 

When a chemical reaction is fast enough to become complete within the diffusion film, the chemical rate and diffusion rates are coupled differently. Fig. 5.4 shows the basis for derivation of the rate expression for a reaction in a two-phase organic reactant/water system when the reaction is first order in solute with rate constant k, the diffusion coefficient of the organic species in water is D and the saturation solubility of the organic reactant in water is Csat. We consider the system at steady state and take a mass balance across a slab... 

In single step voltammetry, the existence of chemical reactions coupled to the charge transfer can affect the half-wave potential Ey2 and the limiting current l. For an in-depth characterization of these processes, we will study them more extensively under planar diffusion and, then, under spherical diffusion and so their characteristic steady state current potential curves. These are applicable to any electrochemical technique as previously discussed (see Sect. 2.7). In order to distinguish the different behavior of catalytic, CE, and EC mechanisms (the ECE process will be analyzed later), the boundary conditions of the three processes will be given first in a comparative way to facilitate the understanding of their similarities and differences, and then they will be analyzed and solved one by one. The first-order catalytic mechanism will be described first, because its particular reaction scheme makes it easier to study. 

In these electrode processes, the use of macroelectrodes is recommended when the homogeneous kinetics is slow in order to achieve a commitment between the diffusive and chemical rates. When the chemical kinetics is very fast with respect to the mass transport and macroelectrodes are employed, the electrochemical response is insensitive to the homogeneous kinetics of the chemical reactions—except for first-order catalytic reactions and irreversible chemical reactions follow up the electron transfer—because the reaction layer becomes negligible compared with the diffusion layer. Under the above conditions, the equilibria behave as fully labile and it can be supposed that they are maintained at any point in the solution at any time and at any applied potential pulse. This means an independent of time (stationary) response cannot be obtained at planar electrodes except in the case of a first-order catalytic mechanism. Under these conditions, the use of microelectrodes is recommended to determine large rate constants. However, there is a range of microelectrode radii with which a kinetic-dependent stationary response is obtained beyond the upper limit, a transient response is recorded, whereas beyond the lower limit, the steady-state response is insensitive to the chemical kinetics because the kinetic contribution is masked by the diffusion mass transport. In the case of spherical microelectrodes, the lower limit corresponds to the situation where the reaction layer thickness does not exceed 80 % of the diffusion layer thickness. 

The mass balance equations of the traditional multicomponent rate-based model (see, e.g., Refs. 57 and 58) are written separately for each phase. In order to give a common description to all three considered RSPs (where it is possible, of course) we will use the notion of two contacting fluid phases. The first one is always the liquid phase, whereas the second fluid phase represents the gas phase for RA, the vapor phase for RD and the liquid phase for RE. Considering homogeneous chemical reactions taking place in the fluid phases, the steady-state balance equations should include the reaction source terms ... 

Increasing the frequency of collision by compression does not affect the quantity of AB (2 AB has the same volume as A2+ B2) and hence there is no effect on the rate of reaction AB—>(BA). However, the initial quantity of A and B determines the amount of AB and hence the rate of production of (BA). This is the criterion of a first order reaction, that the rate of reaction shall be proportional to the quantity of the initial material and yet the reaction rate is independent of collision frequency. The situation is similar to that depicted in the reservoir analogy of Fig. 9 where a steady state is reached—deactivation by collision described there being replaced in this case by chemical dissociation into the original material. 

Solution of the coupled mass-transport and reaction problem for arbitrary chemical kinetic rate laws is possible only by numerical methods. The problem is greatly simplified by decoupling the time dependence of mass-transport from that of chemical kinetics the mass-transport solutions rapidly relax to a pseudo steady state in view of the small dimensions of the system (19). The gas-phase diffusion problem may be solved parametrically in terms of the net flux into the drop. In the case of first-order or pseudo-first-order chemical kinetics an analytical solution to the problem of coupled aqueous-phase diffusion and reaction is available (19). These solutions, together with the interfacial boundary condition, specify the concentration profile of the reagent gas. In turn the extent of departure of the reaction rate from that corresponding to saturation may be determined. Finally criteria have been developed (17,19) by which it may be ascertained whether or not there is appreciable (e.g., 10%) limitation to the rate of reaction as a consequence of the finite rate of mass transport. These criteria are listed in Table 1. 

Heterogeneously catalyzed reactions are usually studied under steady-state conditions. There are some disadvantages to this method. Kinetic equations found in steady-state experiments may be inappropriate for a quantitative description of the dynamic reactor behavior with a characteristic time of the order of or lower than the chemical response time (l/kA for a first-order reaction). For rapid transient processes the relationship between the concentrations in the fluid and solid phases is different from those in the steady-state, due to the finite rate of the adsorption-desorption processes. A second disadvantage is that these experiments do not provide information on adsorption-desorption processes and on the formation of intermediates on the surface, which is needed for the validation of kinetic models. For complex reaction systems, where a large number of rival reaction models and potential model candidates exist, this give rise to difficulties in model discrimination. 

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