The Steady-State Concentration Method

Using of approximate methods of chemical kinetics is intended for, first of all, simplifying mathematical models and, respectively, their analysis. The steady-state 

2 Multi-Step Reactions The Methods for Analytical Solving the Direct Problem 

Let us consider a simple example to illustrate the point of the method. Let us have the kinetic scheme of the consecutive reaction with the first step being reversible 

If to assume that the constant the rate of intermediate consumption is 

It is easy to see that the rate of product C formation is then written in the form 

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The partial equilibrium criterion goes back to the steady-state concentration method, which was widely used before the computer age. In the framework of this approach, only species participating in the slowest reaction are considered, while a steady-state equilibrium for all other components and fast reactions is assumed. If two or more species are connected by fast chemical reactions and the rate of their reciprocal conversion in these reactions is much higher than into other compounds, they form a steady-state subsystem. Their ratio in such a subsystem is determined by the rate constants of their reciprocal conversion. In the case of two fast reciprocally transforming compounds, their ratio is determined only by the equilibrium constant and current concentration. The particular values of the rate constants determine only the time of equilibration, and other reactions of these compounds only slightly disturb their ratio. In such a case the use of the partial equilibrium criterion can be fruitful. 

The following kinetic equation, based on the scheme above, can be obtained using the steady-state concentrations method and assuming reaction (VII. 1)... 

Similar solutions (6.28) and (6.29) were reported in [19,20] by using the steady-state concentration method. 

Let us carry out a check of the steady-state principle. For this purpose, let us calculate the time dependence of the end product formation rate from the relationships obtained by accurate solving the direct kinetic problem (see Table 2.1). Next, let us compare the result with the calculations from obtained formula (2.9). The corresponding plots represented in Fig. 2.17 show that the behaviour of the both curves coincide after less than 0.5 s at given values of the rate constants satisfying the condition ki > k. This indicates applicability of the steady-state concentration method to the considered model of the consecutive reaction. 

Fig. 2.17 Checking applicability of the steady-state concentration method or be converted into the end product...

The steady-state concentration method is widely used for an analysis of kinetic mechanisms of complex reactions. In particular, this method plays an important... 

Next, the elements related to the stable reactants should be extracted from the vector of the right parts of the differential equations and the intermediate concentrations, which enter the equations, should be replaced with the obtained expressions. This stage of the s5mibolic evaluation is shown in Fig. 2.20. Thereby, by applying the steady-state concentration method, we obtained the reduced system of the differential equations describing the time-dependent concentration decrease or increase of the stable reactants. 

Due to the rapidity of the spontaneous dismutation reactions, the steady-state concentrations of 02 achieved by chemical or by enzymatic reactions are usually quite low. The physical methods for detecting 02, although direct and unequivocal, are restricted to measurements of steady-state concentrations and are thus often found to lack of sensitivity. For distince, due to the reason mentioned above, when the EPR method was employed for studying the 02 production by xanthine oxidase, it was necessary to use a high concentration of the reactants and to work at elevated pH so... 

Calculations gave for germene Me2Ge = CH2 values of 7.79-7.84 eV,26 slightly depending on the method used. Unfortunately, the experimental determination of IE has not been possible because the steady-state concentration is not sufficient.26... 

The rather time- and cost-expensive preparation of primary brain microvessel endothelial cells, as well as the limited number of experiments which can be performed with intact brain capillaries, has led to an attempt to predict the blood-brain barrier permeability of new chemical entities in silico. Artificial neural networks have been developed to predict the ratios of the steady-state concentrations of drugs in the brain to those of the blood from their structural parameters [117, 118]. A summary of the current efforts is given in Chap. 25. Quantitative structure-property relationship models based on in vivo blood-brain permeation data and systematic variable selection methods led to success rates of prediction of over 80% for barrier permeant and nonper-meant compounds, thus offering a tool for virtual screening of substances of interest [119]. 

Conventional EPR techniques have been successfully used to measure the D and E values of matrix-isolated carbenes in the ground triplet state because the steady-state concentration of triplet species is sufficiently high in the system. The technique cannot be used, however, for excited species having triplet hfetimes of the order of 10-100 ns, since their steady-state concentration is too low. The D parameters are estimated from the external magnetic field effect on the T—T fluorescence decay in a hydrocarbon matrix at low temperamre. The method is based on the effect of the Zeeman mixing on the radiative and nonradiative decay rates of the T -Tq transition in the presence of a weak field. The D values are estimated by fitting the decay curve with that calculated for different D values. The D T ) values estimated for nonplanar DPC (ci symmetry) is 0.20... 

The determinants (given by the method of graphs) which provide the steady-state concentrations of [E] and of the various exzyme-substrate and enzyme-product complexes are... 

If, during the bioconcentration test, the chemical concentrations in the organism and water reach steady-state, the bioconcentration factor can be calculated from the steady-state concentrations in the organism (CB) and the water (Cw) as CB/ Cw. However, when steady-state is not achieved during the test because the test was conducted for an insufficiently long period of time or because exposure concentrations were variable during the test, the derivation of the BCF and the rate constant for chemical uptake and elimination require a more specific method of data analysis. 

The build-up is often very fast, e.g. over in 10 4 s, and not observed experimentally under normal steady state conditions. Special fast reaction techniques are required to study the build-up, and analysis of the data requires non-steady state methods. If build-up continues after the steady state concentrations occur, the rate of reaction continues to increase and explosion can occur. 

As a point of note, the axial dispersion coefficient can also be obtained from the steady-state concentration-versus-distance profile of the tracer. This method, among others, has been used by Watson and McNeese, 09 Shestopalov et al.,97 Kato et al.,49 Imafulcu et al.,47 and Deckwer et al.32 The two-dimensional axial dispersion models for the evaluation of RTD are considered by Schugerl94 and Tanoka and Inoue.103... 

In the present research, we use a dispersion-type model. Techniques for estimating dispersion coefficients are classed as non-steady or steady state. Steady-state methods usually incorporate the continuous addition of a fixed tracer concentration into the liquid stream followed by the measurement of the steady-state concentration profile along the column length. 

The problem of secondary electron transfer normally observed in thermal and electrochemical oxidation can be partially circumvented by using photo- and radiation chemical methods, because in such processes the oxidizing species is generated as a transient and the steady-state concentration of radical intermediates is usually very low (< 10 m). 

The Ce(IV)-As(III) reaction is catalyzed by iodine and thereby furnishes the basis for the determination of traces of iodine. A possible pathway involves the formation of I or I by reaction of I with Ce(IV). The rate of the overall reaction (15-23) is determined by that of (15-27), which is proportional to the steady-state concentration of iodide ion. Because each iodide ion enters the catalytic cycle many times, the method is extremely sensitive. Chloride is beneficial in the iodide-catalyzed reaction, presumably through formation of ICl as an intermediate that inhibits formation of HIO3 as an unreactive product. ... 

The steady-state approximation is a more general method for solving reaction mechanisms. The net rate of formation of any intermediate in the reaction mechanism is set equal to 0. An intermediate is assumed to attain its steady-state concentration instantaneously, decaying slowly as reactants are consumed. An expression is obtained for the steady-state concentration of each intermediate in terms of the rate constants of elementary reactions and the concentrations of reactants and products. The rate law for an elementary step that leads directly to product formation is usually chosen. The concentrations of all intermediates are removed from the chosen rate law, and a final rate law for the formation of product that reflects the concentrations of reactants and products is obtained. 

The determinants of blood-brain barrier penetration are similar to the determinants of membrane permeability. They include lipophilicity (log P), H-bonding capacity, ionization prohle, size, and flexibility. An example of a simple quantitative structure-activity relationship (QSAR) equation to calculate the ratio of the steady state concentration of the drug molecule in the brain and in the blood have been described" (for a comprehensive review of the in silico methods see" ) ... 

The GEM method is based upon the assumption that the size and shape of the testing chamber does not influence the emission. During the testing the formaldehyde concentration in the chamber will rise and stabilize at a steady state concentration. At constant climate the steady-state concentration or emission rate from the test object depends on the relation between the loading factor and the air change rate. Good air circulation in the chamber is also essential ( ). 

Thus, although there is no air exchange between the glass bell and the surroundings, the Bell method can be used to provide data to calculate the steady state concentration in a ventilated system. 

The initial acceleration of enzyme reactions can be observed by a study of the rate of appearance of the final product during the short time interval between mixing of enzyme and substrate and the attainment of the steady-state concentrations of all the intermediate compounds. Apart from the final steady-state velocity, this method can, in principle, give information about the kinetics of two reaction steps. In the first place, the second-order constant ki which characterizes the initial enzyme-substrate combination can be determined when [ S]o, the initial substrate concentration, is sufficiently small to make this step rate-determining during the pre-steady-state period. Kinetic equations for the evaluation of rate constants from pre-steady-state data have recently been derived (4). Under suitable conditions ki can be evaluated from... 

An efficient way to treat such a system is to assemble all coefficients of the different terms of the mass-balance equations in a matrix and to apply methods of matrix algebra to solve the system for steady-state concentrations (level III) or for the concentrations as functions of time (level IV) [19]. We denote the matrix of coefficients (the fate matrix ) by S, the vector of concentrations in all boxes of the model by c, and the vector of all source terms by q. The set of mass-balance equations describing the temporal changes of the concentrations in all boxes then reads c = -S c + q. The steady-state solution is obtained by setting c equal to zero and solving for c. This leads to ss -1. j obtain the steady-state concentrations the emission vector has to be multiplied by the inverse of the matrix S. For the dynamic solutions of the system, the eigenvalues and eigenvectors of S have to be determined. 

The technique of transient isotope-switching can provide fundamental information about the kinetics of catalytic reactions that is difficult or impossible to obtain with other methods. Following a gas-phase isotope switch at steady state, transients of products labeled with the original isotope result from reactions of intermediates that have been left behind on/in the catalyst after the gas-phase switch. The area under the transients provides a direct measure of the steady-state concentrations of intermediates on/in the catalyst at the time of the switch. The temporal shape of the transients provides information on the reaction kinetics of the intermediates. In the present study, we compliment the transient techniques with steady-state measurements to obtain information on the rates of the various... 

Two reactions similar to (6A) have been studied. Wadiida, Nfartinez, and Bayes measured the steady-state concentration of HCO in the presence of O2, using a photo-ionization mass spectrometer (Method Xm), and calculated kso = (3.4 0.7) X 10 dm mol s at 297 K. Reaction (SO) almost certainly IHroceeds via a direct bimolecular st involving virtually no activation enogy. 

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