Curve-fitting methods, kinetic

Miscellaneous Methods At the beginning of this section we noted that kinetic methods are susceptible to significant errors when experimental variables affecting the reaction s rate are difficult to control. Many variables, such as temperature, can be controlled with proper instrumentation. Other variables, such as interferents in the sample matrix, are more difficult to control and may lead to significant errors. Although not discussed in this text, direct-computation and curve-fitting methods have been developed that compensate for these sources of error. ... 

Since the kinetics of homogeneous and heterogeneous reactions are fundamentally different, Arnold et al. (157) have shown that the nonisothermal TG curve provides insufficient information for the purpose of reaction kinetic calculations. Mathematical considerations prove also that the parameters of the Arrhenius model cannot be calculated correctly from the TG curve by curve-fitting methods and that there is no unique correlation between the estimated parameters and the measured curves. Also, the correlation between A and D described as a compensation effect is certainly a mathematical... 

The kinetics data analysis carried out using the curve fitting method were second-order reactions for chemical oxidation and carbon adsorption processes cept in the following cases K hiO and Salicylic acid 100 mg/l (zero-order reaction) BCMnO and Salicylic acid 500 mg/l (first-order reacfion), Norit PAC 20B and Resorcinol 500 mg/l (first-order reaction), while the reaction orders determined by tune analysis method confirmed the results of the curve fitting analysis in some cases, in other cases, the reaction orders were greater than 2. 

The kinetics data analysis carried out using the curve fitting method bad reaction rates ranging fiom 0.000005 mg liter min (KMn04 and Vanillin 500 mg/l) and 0.063 mg liter sec (Norit PAC 20B and Vanillin 100 mg/l), while the reaction rates determined by tune analysis had values ranging between 2.7542 X 10 (KMh04 and Vanillin 100 mg/l) and 0.06703 (Norit PAC 20B and Salicylic acid 300 mg/I), in units mg liter t ... 

For these methods to be applicable in practice, the rate-constant ratio, kpjk should generally be at least 3—4. Flowever, novel mathematical approaches such as the multipoint curve-fitting method, which takes full advantage of the information provided by the entire kinetic curve (e.g., the Kalman filter), are applicable to ratios of only 1.5-2. 

Determination of a singie species in a mixture Simuitaneous kinetic-based determinations Classical differential kinetic methods Logarithmic-extrapolation method Proportional-equation method Multipoint methods Curve-fitting methods Kalman filter algorithm Artificial neural networks Multivariate calibration methods... 

There are in fact two slightly different types of non-steady state technique. In the first an instantaneous perturbation of the electrode potential, or current, is applied, and the system is monitored as it relaxes towards its new steady state chronoamperometry and chronopotentiometry are typical examples of such techniques. In the second class of experiment a periodically varying perturbation of current or potential is applied to the system, and its response is measured as a function of the frequency of the perturbation cyclic and a.c. voltammetry are examples of this type of approach. In both cases the rate of mass transport varies with the time (or frequency), and by obtaining data over a wide range of these variables and by using curve fitting procedures, kinetic parameters are obtainable. Pulse techniques will be discussed later in this chapter, whilst sweep methods are described in Chapter 6 and a.c. methods in Chapter 8. 

Kinetic methods can be classified according to how the measurement is made. Differential methods compute the rate of reaction and relate it to the analyte concentration. Rates are determined from the slope of the absorbance versus time curve. Integral methods use an integrated form of the rate equation and determine the concentration of analyte from the absorbance changes that occur over various time intervals. Curve-fitting methods fit a mathematical model to the absorbance versus time curve and compute the parameters of the model, including the analyte concentration. The most sophisticated of these methods use the parameters of the model to estimate the value of the equilibrium or steady-slate response. These methods can provide error compensation because the equilibrium position is... 

The method is quite effective, but is not widely used now because of the ubiquity of digital computers. Zuman and Patel - 36. show circuit designs for some kinetic schemes. Williams and Bruice made good use of the analog computer in their study of the reduction of pyruvate by 1,5-dihydroflavin. In this simulation eight rate constants were evaluated variations in these parameters of 5% yielded discemibly poorer curve fits. 

There have been few discussions of the specific problems inherent in the application of methods of curve matching to solid state reactions. It is probable that a degree of subjectivity frequently enters many decisions concerning identification of a best fit . It is not known, for example, (i) the accuracy with which data must be measured to enable a clear distinction to be made between obedience to alternative rate equations, (ii) the range of a within which results provide the most sensitive tests of possible equations, (iii) the form of test, i.e. f(a)—time, reduced time, etc. plots, which is most appropriate for confirmation of probable kinetic obediences and (iv) the minimum time intervals at which measurements must be made for use in kinetic analyses, the number of (a, t) values required. It is also important to know the influence of experimental errors in oto, t0, particle size distributions, temperature variations, etc., on kinetic analyses and distinguishability. A critical survey of quantitative aspects of curve fitting, concerned particularly with the reactions of solids, has not yet been provided [490]. 

Armentrout et al.69 have developed a method with which the kinetic shift can be taken into account in the curve fitting procedure. The fitting procedure, which corrects for the kinetic shift, includes only the fraction of the precursor ions which... 

Several theoretical models were constructed to describe the chromatographic process in the frontal 116.191 and the zonal elution mode 20. The conventional method of obtaining the kinetic parameters consists in fitting the model to the experimental breakthrough curves. Another method based on the split-peak effect is a direct measurement of the apparent association rate constant (7,211. Because of the slow adsorption process, a fraction of the solute injected as a pulse into the immunochromatographic column is eluted as a nonretained peak. This behavior is observed at high flow rates, with very short or low-capacity columns 121—251. 

Fitzpatrick CP, Pardue HL. Simultaneous determinations of liver- and bone-type alkaline phosphatase by curve-fitting of inhibition kinetic data I. Development and evaluation of an absorbance-based method. Clin Chem 1992 38 238-46. 

The Hill coefficient n obtained from the curve fit of the Cm profile of Class I channels and pores (Fig. 11.7a) corresponds to the number of monomers in the active supramolecule (if self-assembly indeed occurs from an excess monomer in solution. With self-assembly from excess dimer, the number of monomers per active supramolecule is 2n, and so on). The compatibility with the Hill equation further demonstrates the presence of excess monomer besides a small population of active supramolecule. The presence of excess monomer, in turn, reveals that the self-assembly of the channel or pore is an endergonic process. Structural studies of unstable n > 1 supramolecules at concentrations near the EC o by conventional methods are therefore meaningless. For example, NMR or IR measurements will report on the inactive monomers, whereas the unstable active structure of Class I channels and pores is invisible (see Section 11.4 for methods to selectively detect and study minority populations of active supramolecules). In BLMs, the thermodynamic instability of Class I channels and pores is expressed in low open probabilities Po (Fig. 11.4). The n > 1 of Class I channels and pores is unrelated to the kinetic stability expressed in short lifetimes for labile Class lA and long lifetimes for inert Class IB supramolecules. 

Thus, unlike some kinetic models with many parameters that must be determined implicitly by curve fitting, the present equivalent circuit model offers a stringent test of the validity of our electrochemical analysis. The previously published (34, 36) derivation of the equivalent circuit implies that the equivalent circuit is intended for a photocurrent that arises from a single relaxation process that is, either first-order or pseudo-first-order processes. Thus, a composite photosignal that consists of both B1 and B2 is not expected to agree with the equivalent circuit. Therefore, a prerequisite to test the validity of the equivalent circuit analysis is to devise a successful method to separate the two components, for example, by elimination of the B2 component completely but leaving the B1 component intact. 

Rate constants are determined by variation of experimental conditions and chemical composition of the reaction mixture. Data are measured by application of a variety of modem analytical methods. Modem numerical approaches of curve fitting and/or solution of differential equations are applied. Results and consequences influence chemical reaction engineering as well as production costs. Many books cover these formal thermal kinetics in detail, but most are restricted to simple mechanisms. In contrast, analogous treatments of photochemical reactions are restricted to publications of special reactions and examinations. Therefore this book aims to supply an overall treatment of formal photokinetics beyond the scope of normal books on kinetics. 

Once the selection of a possible kinetic model and suitable reactor model are complete (equation (8-1)), a non-linear, least square method can be adopted to determine the kinetic and adsorption parameters. This can be achieved by minimizing an objective function representing the sum of the differences between the model concentration estimates and the measured experimental concentrations. This non-linear, least square fit can be performed using the curve fit functions available in Matlab, as recommended by Ibrahim (2(X)1). 

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