Steady state kinetics differential equations

If one deals with a steady-state kinetic experiment, the set of differential equations... 

The term microkinetic analysis has been applied " to attempts to synthesise information from a variety of sources into a coherent reaction model for the hydrogenation of ethene. The input includes steady-state kinetics (most importantly the temperature-dependence of reaction orders ), isotopic tracing, vibrational spectroscopy and TPD it uses deterministic methods, i.e. the solution of ordinary differential equations, for estimating kinetic parameters. It selects a somewhat eclectic set of elementary reactions, and in particular the model... 

Stucki (1978) mainly deals with the stability of the steady states of kinetic differential equations. He gives an introduction to stability theory that is more detailed than those found in the usual textbooks on ordinary differential equations. He also shows how to apply the different methods to problems of biochemical kinetics. His paper also includes some of Clarke s results up to that time. 

Then the induced kinetic differential equation for the corresponding CSTR cannot admit multiple positive steady states, no matter what the feed composition may be and no matter what (positive) values the residence time and rate constants take. 

Prove (by analytical or numerical methods) that the (logarithm of the) steady state iodide concentration as a function of the (logarithm of) / H- is as shown in Fig. 4.3, if the kinetic differential equations of the iodate-arsenous acid reaction is to be taken... 

The solution of the differential equations given below was developed initially by Gutfreund (1955) and Gutfreund Sturtevant (1956) for the study of the pre-steady-state kinetics of proteolytic enzymes like chymo-trypsin, trypsin and ficin. It was possible to determine the position of the rate limiting step of the sequence of interconversions of enzyme complexes. When enzyme and substrates are mixed in a stopped flow apparatus the following conditions prevail at /=0 the concentrations of enzyme and substrate are Ce(0) and Cs(0), the concentrations of products and of all intermediates are zero and dCp/d/=0. The rate of product formation will increase as the concentrations of the intermediate enzyme complexes rise to their steady state concentration. The principle of this approach can be illustrated with what we have called the minimum mechanism, with only the first step reversible at negligible product concentration ... 

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 ]. 

The mathematical model most widely used for steady-state behavior of a reactor is diffusion theory, a simplification of transport theory which in turn is an adaptation of Boltzmann s kinetic theory of gases. By solving a differential equation, the flux distribution in space and time is found or the conditions on materials and geometry that give a steady-state system are determined. 

This chapter takes up three aspects of kinetics relating to reaction schemes with intermediates. In the first, several schemes for reactions that proceed by two or more steps are presented, with the initial emphasis being on those whose differential rate equations can be solved exactly. This solution gives mathematically rigorous expressions for the concentration-time dependences. The second situation consists of the group referred to before, in which an approximate solution—the steady-state or some other—is valid within acceptable limits. The third and most general situation is the one in which the family of simultaneous differential rate equations for a complex, multistep reaction... 

Spin trap, 102 Statistical kinetics, 76 Steady-state approximation, 77-82 Stiff differential equations, 114 Stoichiometric equations, 12 Stopped-flow method, 253-255 Substrate titration, 140 Success fraction approach, 79 Swain-Scott equation, 230-231... 

Catalytic reactions (as well as the related class of chain reactions described below) are coupled reactions, and their kinetic description requires methods to solve the associated set of differential equations that describe the constituent steps. This stimulated Chapman in 1913 to formulate the steady state approximation which, as we will see, plays a central role in solving kinetic schemes. 

In the gas phase, the reaction of O- with NH3 and hydrocarbons occurs with a collision frequency close to unity.43 Steady-state conditions for both NH3(s) and C5- ) were assumed and the transient electrophilic species O 5- the oxidant, the oxide 02 (a) species poisoning the reaction.44 The estimate of the surface lifetime of the 0 (s) species was 10 8 s under the reaction conditions of 298 K and low pressure ( 10 r Torr). The kinetic model used was subsequently examined more quantitatively by computer modelling the kinetics and solving the relevant differential equations describing the above... 

Some economies are possible if equilibrium is assumed between selected compartments, an equal fugacity being assignable. This is possible if the time for equilibration is short compared to the time constant for the dominant processes of reaction or advection. For example, the rate of chemical uptake by fish from water can often be ignored (and thus need not be measured or known within limits) if the chemical has a life time of hundreds of days since the uptake time is usually only a few days. This is equivalent to the frequently used "steady state" assumption in chemical kinetics in which the differential equation for a short lived intermediate species is set to zero, thus reducing the equation to algebraic form. When the compartment contains a small amount of chemical or adjusts quickly to its environment, it can be treated algebraically. 

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. 

Equation (2.19), which concerns a situation without processes in the biofilm, can be extended to include transformation of a substrate, an electron donor (organic matter) or an electron acceptor, e.g., dissolved oxygen. If the reaction rate is limited by j ust one substrate and under steady state conditions, i.e., a fixed concentration profile, the differential equation for the combined transport and substrate utilization following Monod kinetics is shown in Equation (2.20) and is illustrated in Figure 2.8. Equation (2.20) expresses that under steady state conditions, the molecular diffusion determined by Fick s second law is equal to the bacterial uptake of the substrate. 

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

A steady-state treatment of the kinetic scheme yields the differential equation... 

The occurrence of kinetic instabilities as well as oscillatory and even chaotic temporal behavior of a catalytic reaction under steady-state flow conditions can be traced back to the nonlinear character of the differential equations describing the kinetics coupled to transport processes (diffusion and heat conductance). Studies with single crystal surfaces revealed the formation of a large wealth of concentration patterns of the adsorbates on mesoscopic (say pm) length scales which can be studied experimentally by suitable tools and theoretically within the framework of nonlinear dynamics. [31]... 

This kinetic-diffusion problem in the steady state can be described by the coupled second-order differential equations ... 

Easterby proposed a generalized theory of the transition time for sequential enzyme reactions where the steady-state production of product is preceded by a lag period or transition time during which the intermediates of the sequence are accumulating. He found that if a steady state is eventually reached, the magnitude of this lag may be calculated, even when the differentiation equations describing the process have no analytical solution. The calculation may be made for simple systems in which the enzymes obey Michaehs-Menten kinetics or for more complex pathways in which intermediates act as modifiers of the enzymes. The transition time associated with each intermediate in the sequence is given by the ratio of the appropriate steady-state intermediate concentration to the steady-state flux. The theory is also applicable to the transition between steady states produced by flux changes. Apphcation of the theory to coupled enzyme assays makes it possible to define the minimum requirements for successful operation of a coupled assay. The theory can be extended to deal with sequences in which the enzyme concentration exceeds substrate concentration. 

A mathematical simplification of rate behavior of a multistep chemical process assuming that over a period of time a system displays little or no change in the con-centration(s) of intermediate species (i.e., d[intermedi-ate]/df 0). In enzyme kinetics, the steady-state assumption allows one to write and solve the differential equations defining fhe rafes of inferconversion of various enzyme species. This is especially useful in initial rate studies. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

The quasi steady state approximation is a powerful method of transforming systems of very stiff differential equations into non-stiff problems. It is the most important, although somewhat contradictive technique in chemical kinetics. Before a general discussion we present an example where the approximation certainly applies. 

As is thoroughly discussed in Chap. 2 of this volume, the convective diffusion conditions can be controlled under steady state conditions by use of hydrodynamic electrodes such as the rotating disc electrode (RDE), the wall-jet electrode, etc. In these cases, steady state convective diffusion is attained, becomes independent of time, and solution of the convective-diffusion differential equation for the particular electrochemical problem permits separation of transport and kinetics from the experimental data. 

The subject of kinetics is often subdivided into two parts a) transport, b) reaction. Placing transport in the first place is understandable in view of its simpler concepts. Matter is transported through space without a change in its chemical identity. The formal theory of transport is based on a simple mathematical concept and expressed in the linear flux equations. In its simplest version, a linear partial differential equation (Pick s second law) is obtained for the irreversible process, Under steady state conditions, it is identical to the Laplace equation in potential theory, which encompasses the idea of a field at a given location in space which acts upon matter only locally Le, by its immediate surroundings. This, however, does not mean that the mathematical solutions to the differential equations with any given boundary conditions are simple. On the contrary, analytical solutions are rather the, exception for real systems [J. Crank (1970)]. 

The preceding remarks all apply to a steadily propagating one-dimensional detonation. It is a relatively simple task to solve the algebraic equation for the over-all steady-state motion. The differential equations for the structure may be solved in the steady state, but the task is tedious and, in addition, detailed knowledge of reaction rates needed in the equation is not available. It is not too difficult to solve the time-dependent over-all equation if a burning velocity for the flame as a function of temperature and pressure is assumed (J4). It is not practicable to solve the time-dependent equations which govern the structure of the wave with any certainty because of the lack of kinetic information, in addition to the mathematical difficulty. The acceleration of the slowly moving flame front as it sends forward pressure waves which coalesce into shock waves that eventually are coupled to a zone of reaction to form a detonation wave has been observed experimentally (LI, L2). 

Laboratory studies of the reactions at steady-state conditions have the advantage of the much simpler mathematical analysis of the results compared to nonsteady processes since the problem of deriving kinetic equations corresponding to a given reaction mechanism is reduced to the solution of a set of algebraic equations instead of differential equations in the general case of a nonsteady reaction. 

In Fig. 2.10, the boundary between the enzyme-containing layer and the transducer has been considered as having either a zero or a finite flux of chemical species. In this respect, amperometric enzyme sensors, which have a finite flux boundary, stand apart from other types of chemical enzymatic sensors. Although the enzyme kinetics are described by the same Michaelis-Menten scheme and by the same set of partial differential equations, the boundary and the initial conditions are different if one or more of the participating species can cross the enzyme layer/transducer boundary. Otherwise, the general diffusion-reaction equations apply to every species in the same manner as discussed in Section 2.3.1. Many amperometric enzyme sensors in the past have been built by adding an enzyme layer to a macroelectrode. However, the microelectrode geometry is preferable because such biosensors reach steady-state operation. 

This sort of analysis is very important in the formulation of the steady state approximation, developed to deal with kinetic schemes which are too complex mathematically to give simple explicit solutions by integration. Here the differential rate expression can be integrated. The differential and integrated rate equations are given in equations (3.61)—(3.66). 

---