On finding the rate constants of linear and nonlinear systems

In lowest approximation, a certain chemical reaction is described by a system of first-order linear differential equations with unknown constant coefficients. One can therefore write down an expression for the state of the system at time t, and from this find the endpoint of the reaction in terms of the initial state and the rate constants. The relative values of some rate constants can then be estimated from experimental data. A better approximation in which the differential equations are nonlinear is also considered, and it turns out that because of symmetry in the reaction, the relationship between the final state and the ratios of the rate constants is unchanged. Although the differential equations now appear much less tractable, the problem of relating the rate constants to the endpoint of the reaction can be formulated and solved in terms of probabilities. The results illustrate an important property of reaction schemes in which some of the steps are reversible. More generally, this is a property of differential equations: provided that they continue to satisfy certain linear constraints, the parameters of a linear system of ordinary differential equations can vary without affecting the asymptotic solution.     

Application of a new method to the determination of rate constants: Examination of the effect of noise on the data

A new linear regression method is applied to determine the rate constants from time-dependent profiles. The method is effective for obtaining the rate constants of such consecutive processes as vibrational relaxation. The ambiguity of ordinary nonlinear analysis is eliminated by the linearization of regression equations. Because the new method also determines relative detectivity, the nascent vibrational distributions immediately following reactions or photolysis can be obtained simultaneously. In order to demonstrate the reliability of the new method under practical conditions, several examples of analysis of the data with noise are shown in detail. © 1998 John Wiley & Sons, Inc. Int J Chem Kinet: 30: 47–54, 1998.     

Third-order rate constants of atmospheric importance

Input data and results are presented for the calculation of a number of third-order rate constants of atmospheric interest using Troe′s approximate method. A comparison with experimental data indicates that this approach provides a reliable method for predicting unknown rate constants and estimating temperature dependences. These calculations form the basis of the recommendations of the NASA review panel for third-order rate constants to be used in atmospheric modeling.     

Determination of competitive isotherms of enantiomers by a hybrid inverse method using overloaded band profiles and the periodic state of the simulated moving-bed process

A procedure for determination of adsorption isotherms in simulated moving-bed (SMB) chromatography is presented. The parameters of a prescribed adsorption isotherm model and rate constants are derived using a hybrid inverse method, which incorporates overloaded band profiles of the racemic mixture and breakthrough data from a single frontal experiment. The latter are included to reduce the uncertainty on the estimated saturation capacity, due to the dilution of the chromatograms with respect to the injected concentrations. The adsorption isotherm model is coupled with an axially dispersed flow model with finite mass-transfer rate to describe the experimental band profiles. The numerical constants of the isotherm model are tuned so that the calculated and measured band profiles match as much as possible. The accuracy of the isotherm model is then checked against the cyclic steady state (CSS) of the target SMB process, which is readily and cheaply obtained experimentally on a single-column set-up. This experiment is as expensive and time consuming as just a few breakthrough experiments. If necessary, the isotherm parameters are adjusted by applying the inverse method to the experimental CSS concentration profile. The method is successfully applied to determine the adsorption isotherms of Tr?gers base enantiomers on Chiralpak AD/methanol system. The results indicate that the proposed inverse method offers a reliable and quick approach to determine the competitive adsorption isotherms for a specific SMB separation.     

Adsorption Henry constants calculated from the entire isotherm

A method is introduced to derive adsorption Henry constants from experimental isotherm data regardless of whether the measurements extend into the Henry law region. The method uses the intersection between Henry’s law and the Gaussian isotherm model. In this way, the Henry constant can be calculated from Gaussian model parameters obtained from a fit of the model to the entire isotherm. In addition, the Gaussian model is shown to closely agree with Henry’s law over a wide range. To demonstrate the accuracy of the method, Henry constants are derived from 63 experimental isotherms collected from the literature for supercritical n alkanes on 5A zeolite. These Henry constants compare well to values found in the literature.     

A new algorithm for the determination of protolytic constants from spectrophotometric data in multiwavelength mode: Calculations of acidity constants of 4-(2-pyridylazo)resorcinol (PAR) in mixed nonaqueous-water solvents

A new efficient, simple and versatile algorithm is presented for determination of the protolytic constants from spectrophotmetric data in multiwavelength mode based on the combining of hard and soft modeling. The algorithm was checked by determining the acidity constants of a triprotic acid from theoretical and real absorption-pH data. The real spectral data are obtained from photometric titration of different solutions of 4-(2-pyridylazo)resorcinol (PAR) by a standard base solution under an inert atmosphere. The algorithm starts the minimization process using an user supplied number of components and initial guesses of the unknown parameters and refined in a least squares manner. New algorithm is implemented in the new version of DATAN package (version 3.1). The validity of the obtained results was checked by some well known programs such as DATAN 2.1, SPECFIT/32, SQUAD, a modified version of difference spectra and a A-pH curve method. The comparison of the outputs of the DATAN 3.1 with the other programs reveals that there is a very good agreement between the obtained results and mentioned programs.     

Analytic solutions to sets of first-order rate equations with up to six rate constants using a symbolic computer language SMP and application to biochemical kinetics

A symbolic computer language SMP* is employed to analytically solve sets of first-order linear differential equations which occur in kinetic rate-reaction studies. The solutions studied are fully analytic functions of time and the rate constants. Two typical systems are studied: the first contains four species and four rate constants, corresponding to four parallel and consecutive first-order reactions; and the second contains four species and six rate constants, including two additional reverse reactions. These analytic functions allow insight into the mechanism, analytic expressions for the rate constants, and more rapid and precise solutions for the species concentrations than a completely numerical solution of the differential rate equations themselves. The results of the first system are applied to a recent experimental study of enzyme kinetics in which constituent amino acid residues of an enzyme are photooxidized and the corresponding catalytic activity measured with time. A second application of the SMP gives rise to a rapid semianalytic method for obtaining the values of the four and six exponentially nonlinear rate constants from the experimental data.     

Comments on the determination of absolute rate constants in anionic polymerization

Claims have recently been made that absolute rate constants for chain propagation of the unassociated active centers can be made in systems where a high degree of association is present. Anionic polymerization of styrene in nonpolar solvents with lithium as counterion is a typical case. The conditions required to obtain these constants (and the associated aggregate dissociation constants) are described using data from styrene polymerization with lithium and potassium as counterions and data from o-methoxystyrene polymerization. The conclusion reached must be that the k_p and K_ds values obtained for styrene with counterion lithium cannot be obtained from existing literature data and are simply artifacts of the computer analysis. © 1998 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 36: 1065–1068, 1998     

Adsorption of organics on MSC5A in supercritical CO2, chromatographic measurements & stop & go simulation

Chromatographic measurements were made for the adsorption of benzene, toluene and m-xylene on molecular sieving carbon (MSC) in supercritical fluid CO₂ mixed with organics. Supercritical chromatograph packed with MSC was used to detect pulse responses of organics. Adsorption equilibria and adsorption dynamics parameters for organics were obtained by moment analysis of the response peaks. Dependences of adsorption equilibrium constants, K ^?, and micropore diffusivity,?D, on the amount adsorbed were examined. The dependencies of adsorption equilibrium constants, K ^?, and micropore diffusivity, D, of benzene, toluene and m-xylene, on molarity of benzene with each parameters of temperature or pressure were obtained. It was found that the values of K ^? and D for an organic substance depended on the amount adsorbed of other organics strongly. And stop & go method was used as simulation method of perturbation chromatography for investigating adsorption equilibrium and rate. Numerical solution for multicomponent chromatogram in time domain could be obtained by appropriate model equations with experimental conditions. This simulated chromatogram can be compared with experimental chromatogram to determine the adsorption equilibrium and rate parameters. In addition, molecular simulation of multicomponent adsorption equilibria was performed, and potential parameters were determined by comparing the simulation with experimental results. Simulation soft ware is Cerius2 (Version?4.2) made by MSI. The purpose of performing simulation is to elucidate an adsorption mechanism on the molecule level.     

Investigation of solvolysis kinetics of new synthesized fluocinolone acetonide C-21 esters--an in vitro model for prodrug activation

In this study the solvolysis of newly synthesized fluocinolone acetonide C-21 esters was analysed in comparison with fluocinonide during a 24-hour period of time. The solvolysis was performed in an ethanol-water (90:10 v/v) mixture using the excess of NaHCO?. The solvolytic mixtures of each investigated ester have been assayed by a RP-HPLC method using isocratic elution with methanol-water (75:25 v/v); flow rate 1 mL/min; detection at 238 nm; temperature 25 °C. Solvolytic rate constants were calculated from the obtained data. Geometry optimizations and charges calculations were carried out by Gaussian W03 software. A good correlation (R = 0.9924) was obtained between solvolytic rate constants and the polarity of the C-O2 bond of those esters. The established relation between solvolytic rate constant (K) and lipophilicity (cLogP) with experimental anti-inflammatory activity could be indicative for topical corticosteroid prodrug activation.     

The effective temperature of Peptide ions dissociated by sustained off-resonance irradiation collisional activation in fourier transform mass spectrometry

A method for determining the internal energy of biomolecule ions activated by collisions is demonstrated. The dissociation kinetics of protonated leucine enkephalin and doubly protonated bradykinin were measured using sustained off-resonance irradiation (SORI) collisionally activated dissociation (CAD) in a Fourier transform mass spectrometer. Dissociation rate constants are obtained from these kinetic data. In combination with Arrhenius parameters measured with blackbody infrared radiative dissociation, the "effective" temperatures of these ions are obtained. Effects of excitation voltage and frequency and the ion cell pressure were investigated. With typical SORI-CAD experimental conditions, the effective temperatures of these peptide ions range between 200 and 400 degrees C. Higher temperatures can be easily obtained for ions that require more internal energy to dissociate. The effective temperatures of both protonated leucine enkephalin and doubly protonated bradykinin measured with the same experimental conditions are similar. Effective temperatures for protonated leucine enkephalin can also be obtained from the branching ratio of the b(4) and (M + H - H(2)O)(+) pathways. Values obtained from this method are in good agreement with those obtained from the overall dissociation rate constants. Protonated leucine enkephalin is an excellent "thermometer" ion and should be well suited to establishing effective temperatures of ions activated by other dissociation techniques, such as infrared photodissociation, as well as ionization methods, such as matrix assisted laser desorption/ionization.     

ANN-MATOPT hybrid algorithm: determination of kinetic and non-kinetic parameters in different reaction mechanisms

In this work we have applied the computational methodology based on Artificial Neural Networks (ANN) to the kinetic study of distinct reaction mechanisms to determine different types of parameters. Moreover, the problems of ambiguity or equivalence are analyzed in the set of parameters to determine in different kinetic systems when these parameters are from different natures. The ambiguity in the set of parameters show the possibility of existence of two possible set of parameter values that fit the experimental data. The deterministic analysis is applied to know beforehand if this problem occurs when rate constants of the different stages of the mechanism and the molar absorption coefficients of the species participating in the reaction are obtained together. Through the deterministic analysis we will analyze if a system is identifiable (unique solution or finite number of solutions) or if it is non-identifiable if it possesses infinite solutions. The determination of parameters of different nature can also present problems due to the different magnitude order, so we must analyze in each case the necessity to apply a second method to improve the values obtained through ANN. If necessary, an optimization mathematical method for improving the values of the parameters obtained with ANN will be used. The complete process, ANN and mathematical optimizations constitutes a hybrid algorithm ANN-MATOPT. The procedure will be applied first for the treatment of synthetic data with the purpose of checking the applicability of the method and after, it will be used in the case of experimental kinetic data.

     

Experimental and Theoretical Insight into the Kinetics and Mechanism of the Synthesis Reaction of 2,3‐Dihydro‐2‐phenylquinazolin‐4(1H)‐one Catalyzed in Formic Acid

An efficient and simple method has been developed for the synthesis of 2,3‐dihydro‐2‐phenylquinazolin‐4(1H )‐one catalyzed in formic acid. Also, the synthesis reaction between benzaldehyde and 2‐aminobenzamid was monitored spectrally. On the basis of the kinetic data obtained by the UV–vis spectrophotometry, both the first and second steps of the speculative five steps mechanism were enabled to be a rate‐determining step and also reaction showed second‐order kinetics. Considering information obtained by the stopped‐flow technique indicated that the first step is certainly a fast step. Moreover, the reaction was energetically and thermodynamically evaluated using theoretical methods and results were profoundly compared with the experimental approaches. Herein, theoretical rate constants were obtained using potential energy surfaces and the transition state theory at the B3LYP/6–311+G** level of theory. The Winger method was also applied to describe the tunneling effects. Calculations showed that the second step was the rate‐determining step in accordance with the experimental data. It is also found that the oxidation step was the fastest step in the proposed mechanism. For all five steps, two possibilities were considered for generating the probable product by using the thermodynamic parameters and kinetic data. Thermodynamic parameters also showed an exothermic reaction.     

Estimation of kinetic rate constants by turbidity and nephelometry techniques in a homocoagulation process with different model colloids

 In this work turbidimetric and nephelometric techniques have been used to study the homocoagulation of aqueous dispersions of uniform spherical particles of surfactant-free latexes. Cationic and anionic latexes of similar particle sizes (361 and 370 nm) and different surface charge densities (+16.4 and −3.6 μC/cm²) were used throughout. The kinetic constants which control the aggregation processes when the electrical repulsion disappears were estimated by both techniques at different particle concentration and wavelength in order to establish the experimental conditions which provided reliable and similar values for the coagulation rate constant. Both experimental techniques (turbidity and nephelometry) and two ways of fitting the data have been used with both latexes. For the first method, the initial slope of turbidity or total scattered intensity versus time curves were used to calculate the kinetic constants. In the second method, the whole turbidity or total scattered intensity versus time curves were fitted and the kinetic constants calculated. An unambiguous experimental value for the doublet rate constant in diffusion conditions is obtained by turbidity and nephelometry techniques. By nephelometry both data treatments have permitted a distinction between the doublet rate constant and the global rate constant in diffusion conditions. Received: 2 June 1997 Accepted: 14 August 1997     

Adsorption kinetics of proteins onto solid surfaces in the limit of the interfacial interaction control

We have developed an experimental set-up using radiolabelled proteins for the continuous measurement, as a function of time, of the excess concentration of superficial protein at a solid/liquid interface. The experimental conditions were designed in order to minimize the coupling of the interfacial interaction with bulk diffusion, and therefore to work within the limit of the interfacial interaction control. Chemical kinetics were assumed to follow a Langmuir equation. The absorption (K_a) and desorption (K_d) rate constants have been evaluated in the case of fibrinogen and albumin adsorption, onto glass beads and synthetic hydroxyapatite powder, respectively. In order to determine these parameters, a non-linear differential equation (obtained by combining the Langmuir rate equation and a mass-balance equation) was solved by the fourth-order Runge-Kutta method. The results obtained are consistent with the hypothesis of an adsorption rate controlled by the interfacial interaction.     

Isothermal calorimetric measurement of polymer cure kinetics: The mangelsdorf method of analysis

A useful calorimetric method for determining cure kinetics of polymers is demonstrated. The technique is an isothermal one in which ·Q, the rate of thermal energy release (exothermic power), is monitored as a function of time. The time-dependence of heat released, Q(t), can, in principle, be determined by integration of the power data; however, experimental considerations often make it difficult to evaluate ·Q for short times. Nevertheless, the use of a simple mathematical device advocated by Mangelsdorf makes it possible to obtain both cure rate constants and total heat release, even if initial values of ·Q are unknown. The only assumption of the method is that heat release be an exponential function of time. The method is illustrated by following the cure of a polyol/isocyanate system at several temperatures. The kinetics are shown to obey an Arrhenius relation, with an activation energy of 10.0±0.1 kcal mol⁻¹.     

Determination of the cis-trans isomerization barrier of enalaprilat by dynamic capillary electrophoresis and computer simulation

Dynamic capillary electrophoresis (DCE) and computer simulation of the elution profiles with the stochastic model has been applied to determine the isomerization barriers of the angiotensin converting enzyme inhibitor enalaprilat. The separation of the rotational cis-trans isomeric drug has been performed in an aqueous 20 mM borate buffer at pH 9.3. Interconversion profiles featuring plateau formation and peak broadening were observed. To evaluate the rate constants k(cis-->trans) and k(trans-->cis) of the cis-trans isomerization from the experimental electropherograms obtained by dynamic capillary electrophoresis, elution profiles were analyzed by a simulation with iterative convergence to the experimental data using the ChromWin software which requires the total migration times of the individual isomers t(R), the electroosmotic break-through time t(0), the plateau height h(plateau), the peak widths at half height of the individual isomers w(h), as well as the peak ratio of the isomers as experimental data input. From temperature-dependent measurements between 0 degrees and 15 degrees C the thermodynamic parameters Delta G, Delta H and Delta S, the rate constants k(cis-->trans) and k(trans-->cis) and the kinetic activation parameters Delta G*, Delta H*, and Delta S* of the cis-trans isomerization of enalaprilat were obtained. From the activation parameters the isomerization barriers at 37 degrees C were calculated to be Delta G* (trans-->cis) = 87.2 kJ.mol(-1) and Delta G*(cis-->trans) = 91.9 kJ.mol(-1).     

Numerical modelling and simulation of Laviron treatment for poly-phenothiazine derivative-modified glassy carbon electrodes

Electropolymerization of a new phenothiazine derivative (bis-phenothiazin-3-yl methane, BPhM) on glassy carbon electrodes leads to electroactive and conducting layers of poly-BPhM. Based on the Laviron method, the kinetic parameters (the heterogeneous electron transfer rate constants k _s and transfer coefficients α) were calculated out of the experimental and theoretical data. The theoretical data have been obtained by using the analogical modelling and numerical simulating method. The modified electrodes present high values of k _s(~50s⁻¹) in phosphate buffer solutions of different pH values. A good correlation of mathematical and numerical simulated kinetic parameters has been obtained, with the experimental values and the norm of residuals being very close to zero.     

The extrapolation of experimental equilibrium constant data to zero ionic strength: Critical review and new approach

Different Debye-Hückel expressions for the activity coefficients of species in aqueous solution in the ionic strength range I = 0-3.5m (3M) are used for the extrapolation of equilibrium constants data to I = 0 and the interpolation to unknown I values. This may be accomplished using four or more values of the equilibrium constants that are equally well distributed on the I scale. The interpolated and extrapolated equilibrium constant values obtained are quite satisfactory and within the experimental error of the corresponding equilibrium constants. The values at I 0.1 are very important as they can particularly influence the equilibrium constant value calculated at I = 0 and for which the error can reach 0.1 log unit or more. The values at I 1.5 can also influence the extrapolated value at I = 0 and the interpolated value at a given I when an inadequate extrapolation model is selected. Among the expressions used, only those with two or more unknown parameters are suitable for such calculations.     

Application of different polarimetric methods for the analysis of enzyme-catalyzed enantioselective reactions

Identification of model parameters in kinetic equations requires the determination of reactant concentrations in the course of time. In the case of enzyme catalyzed enantioselective reactions, three different polarimetric methods were used for concentration measurements obtained in both initial rate experiments and under dynamically changing conditions. Two on-line methods for the determination of different numbers of chiral species and a more universal off-line method were applied. For the investigated substrate 5-benzylhydantoin and the enzyme hydantoinase it is shown, that the obtained polarimetric data are most suitable for identification of kinetic parameters. Furthermore, it is demonstrated that on-line data of only one conversion allow for an assessment of an enzyme with regard to its enantioselectivity. Received: 20 October 1998 / Revised: 28 December 1998 / Accepted: 4 January 1999