The evaluation of rate constants in rate laws containing two terms

A simple computerized method using APL is described which gives the rate constants for systems obeying the rate law –d[A]/dt = k₁[A][B] + k₂[A] ^m [B]^l from sets of concentrations and times. Applications are discussed.     

Corresponding-states laws for protein solutions

The solvent around protein molecules in solutions is structured and this structuring introduces a repulsion in the intermolecular interaction potential at intermediate separations. We use Monte Carlo simulations with isotropic, pair-additive systems interacting with such potentials. We test if the liquid-liquid and liquid-solid phase lines in model protein solutions can be predicted from universal curves and a pair of experimentally determined parameters, as done for atomic and colloid materials using several laws of corresponding states. As predictors, we test three properties at the critical point for liquid-liquid separation: temperature, as in the original van der Waals law, the second virial coefficient, and a modified second virial coefficient, all paired with the critical volume fraction. We find that the van der Waals law is best obeyed and appears more general than its original formulation: A single universal curve describes all tested nonconformal isotropic pair-additive systems. Published experimental data for the liquid-liquid equilibrium for several proteins at various conditions follow a single van der Waals curve. For the solid-liquid equilibrium, we find that no single system property serves as its predictor. We go beyond corresponding-states correlations and put forth semiempirical laws, which allow prediction of the critical temperature and volume fraction solely based on the range of attraction of the intermolecular interaction potential.     

Algorithm for error-compensated kinetic determinations without prior knowledge of reaction order or rate constant

The development and evaluation of a new algorithm for error-compensating predictive kinetic determinations are described. With the new algorithm it is possible to fit kinetic data without prior knowledge of the rate constant, reaction order or initial and final values of the detector signal. A curve-fitting method is used to obtain values of these parameters that give the best fit of the model to the data. Although intended to be used primarily to compute signal changes between t=0 and ∞, the method can also be used to determine reaction orders. Although the algorithm fails for reaction orders of zero and unity, it works well for orders between these values and orders greater than unity. Simulated data are used to evaluate the effects of reaction order, signal noise and data range on computed values of signal change and, to a lesser extent, reaction order.     

Estimation of second order rate constants using chemometric methods with kinetic constraints

Several methods are described for determining rate constants for second order reactions of the form U + V --> W using chemometrics and hard modelling to analyse UV absorption spectroscopic data, where all species absorb with comparable concentrations and extinctions. An interesting feature of this type of reaction is that the number of steps in the reaction is less than the number of absorbing species, resulting in a rank-deficient response matrix. This can cause problems when using some of the methods described in the literature. The approaches discussed in the paper depend, in part, on what knowledge is available about the system, including the spectra of the reactants and product, the initial concentrations and the exact kinetics. Sometimes some of this information may not be available or may be hard to estimate. Five groups of methods are discussed, namely use of multiple linear regression to obtain concentration profiles and fit kinetics information, rank augmentation using multiple batch runs, difference spectra based approaches, mixed spectral approaches which treat the reaction as two independent pseudospecies, and principal components regression. Two datasets are simulated, one where the spectra are quite different and the other where the spectrum of one reactant and the product share a high degree of overlap. Three sources of error are considered, namely sampling error, instrumental noise and errors in initial concentrations. The relative merits of each method are discussed.     

Analytical solutions for the rate equations of irreversible two-step consecutive processes with second order later steps

Analytical solutions are presented for four consecutive two-step kinetic schemes, which involve a first reaction with zeroth, first, second or mixed second order dependence and a second step, which is second order with respect to the intermediate formed in the first step. Two of the analytical solutions found use elementary functions not very commonly encountered in chemistry, as the rate equations are shown to be related to the Legendre or modified Bessel differential equations. The solutions are analyzed not only as a function of time, but by plotting two concentrations as a function of each other as well. The dependence of the kinetic traces on the parameter values is also investigated. In all cases, two scaling parameters are identified. Three of the four cases are characterized by a single shape parameter, which is basically the ratio of the rate constant scaled with a suitable concentration unit if necessary. The mixed second order–second order scheme has an additional shape parameter, which is the ratio of the initial concentrations of the two reactants.     

Application of chemometrics methods with kinetic constraints for estimation of rate constants of second order consecutive reactions.

To determine the rate constants for the second order consecutive reactions of the form U + V -(k1)--> W -(k2)--> P, a number of chemometrics and hard modeling-based methods are described. The absorption spectroscopic data from the reaction were utilized for performing the analysis. Concentrations and extinctions of components were comparable, and all of them were absorbing species. The number of steps in the reaction was less than the number of absorbing species, which resulted in a rank-deficient response matrix. This can cause difficulties for some of the methods described in the literature. The standard MATLAB functions were used for determining the solutions of the differential equations as well as for finding the optimal rate constants to describe the kinetic profiles. The available knowledge about the system determines the approaches described in this paper. The knowledge includes the spectra of reactants and products, the initial concentrations, and the exact kinetics. Some of this information is sometimes not available or is hard to estimate. Multiple linear regression for fitting the kinetic parameters to the obtained concentration profiles, rank augmentation using multiple batch runs, a mixed spectral approach which treats the reaction using a pseudo species concept, and principal components regression are the four groups of methods discussed in this study. In one of the simulated datasets the spectra are quite different, and in the other one the spectra of one reactant and of the product share a high degree of overlap. Instrumental noise, sampling error are the sources of error considered. Our aim was the investigation of the relative merits of each method.     

A priori rate constants for kinetic modeling

The advent of computer-aided methods for constructing detailed kinetic models of multicomponent reacting systems provides fresh motivation for the development of efficient and accurate methods for estimating rate constants. There is now the real likelihood that a priori rate estimates, formerly of primarily academic interest, could directly impact major public policy and business decisions. This opportunity brings many challenges. The process of building a computer model for a real-world system can require hundreds of thousands of rate estimates, making most existing rate calculation techniques impractical. Also, the demands for tight error bars on model predictions used to make major decisions often imply levels of accuracy unachievable with existing rate calculation techniques. Past and recent progress towards developing fast and accurate rate estimates is selectively reviewed, and our methodology is outlined. New rate estimates for several types of reactions involving O and HO₂ are presented. Several technical issues requiring further work by the theoretical chemistry community are highlighted. Electronic supplementary material to this paper can be obtained by using the Springer Link server located at http://dx.doi.org/10.1007/s00214-002-0368-4. Received: 6 February 2002 / Accepted: 2 June 2002 / Published online: 2 October 2002 Acknowledgements. This work was partially supported by the National Computational Science Alliance under Grant CTS010006 N and utilized the Origin 2000 High-performance Computing and UniTree Mass Storage systems. We are grateful for financial support from the EPA Center for Airborne Organics, the NSF CAREER program, Alstom Power, Dow Chemical, and the Office of Basic Energy Science, U.S. Department of Energy through grant DE-FG02-98ER14914. The authors acknowledge the contribution of Hans-Heinrich Carstensen in the initial stages of this work. Correspondence to: W.H. Green Jr. e-mail: whgreen@mit.edu Electronic supplementary material to this paper can be obtained by using the Springer LINK server located at http://dx.doi.org/10.1007/s00214-002-0368-4     

Analytical solutions for the rate equations of irreversible two-step consecutive processes with mixed second order later steps

A general strategy was developed to solve the ordinary differential equations defined by two-step chemical processes with a mixed second order later process for all possible cases of parameter values (initial concentrations and rate constant values). As the earlier process, first order, second order, mixed second order and zeroth order cases were considered. For the scheme with a mixed second order first step, several different variations were considered. The analytical solutions contain moderately advanced, but still elementary functions such as the error function, the incomplete gamma function, the hypergeometric function or the Legendre functions. When coupling between the two steps or some reversibility is present in the system, no analytical solutions are found.     

Quantum calculations of the third order hyperpolarizabilities for inert gases

For inert gases third order hyperpolarizabilities are calculated with a method based on variation-perturbation theory.Curves showing the variation of those coefficients with the pulsations of the incident perturbing wave are given.     

Exact solutions for kinetic models of macromolecular dynamics

Dynamic biological processes such as enzyme catalysis, molecular motor translocation, and protein and nucleic acid conformational dynamics are inherently stochastic processes. However, when such processes are studied on a nonsynchronized ensemble, the inherent fluctuations are lost, and only the average rate of the process can be measured. With the recent development of methods of single-molecule manipulation and detection, it is now possible to follow the progress of an individual molecule, measuring not just the average rate but the fluctuations in this rate as well. These fluctuations can provide a great deal of detail about the underlying kinetic cycle that governs the dynamical behavior of the system. However, extracting this information from experiments requires the ability to calculate the general properties of arbitrarily complex theoretical kinetic schemes. We present here a general technique that determines the exact analytical solution for the mean velocity and for measures of the fluctuations. We adopt a formalism based on the master equation and show how the probability density for the position of a molecular motor at a given time can be solved exactly in Fourier-Laplace space. With this analytic solution, we can then calculate the mean velocity and fluctuation-related parameters, such as the randomness parameter (a dimensionless ratio of the diffusion constant and the velocity) and the dwell time distributions, which fully characterize the fluctuations of the system, both commonly used kinetic parameters in single-molecule measurements. Furthermore, we show that this formalism allows calculation of these parameters for a much wider class of general kinetic models than demonstrated with previous methods.     

等浓度三级反应的热动力学研究

根据热动力学基础理论,本文首次建立和讨论了等浓度三级反应的无量纲参数法.测定了20℃时环氧氯丙烷与氢溴酸反应的速率常数,实验的重现性很好,计算此反应的速率常数在误差范围内与文献值相符合.证明了等浓度三级反应无纲参数法的正确性,拓宽了无量纲参数法的应用范围.     

Changes in reaction rate for different kinetic models

Conclusions The material presented indicates the expediency of the preliminary analysis of kinetic data based not only on the reaction rate as function of the degree of conversion, proposed earlier [1], but also based on other parameters of the process. The analysis of the relationships between reaction rate and temperature, dilution (or total pressure), and composition of the initial mixture leads to curves, from the specific form and direction of which it is possible to derive the actual form of the kinetic equation and the ratio of its constants. This treatment is complementary to the analysis of the form of the kinetic equation and refers to the initial level [1] of the kinetic relationship at a composition of the reaction mixture or a degree of conversion kept constant while changing each of the other parameters. Based on the relationship between the level of the kinetic function and its form, the analysis provides the information which reduces the possible ambiguity in the interpretation of kinetic data.Evidently this analysis does not encompass all cases of multidirectional reactions, described sometimes by complex and laborious kinetic equations. It can be carried out, however, in each particular case. Of particular interest is the case where the denominator of the total kinetic equation (2) is a product of polynomials with the corresponding exponents, due to inhomogeneity of the catalyst surface [9]. An approximative analysis of the kinetic equation can be obtained with a denominator approximated with a polynomial carrying a certain exponentl.The examples cited indicate that such a preliminary analysis of kinetic functions derived directly from experimental data are useful and necessary for the design of kinetic models, by offering a less ambiguous interpretation.Translated from Teoreticheskaya i éksperimental'naya Khimiya, Vol. 12, No. 6, pp. 789–795, November–December, 1976.     

Decoupling of the Dirac equation correct to the third order for the magnetic perturbation

A two-component relativistic theory accurately decoupling the positive and negative states of the Dirac Hamiltonian that includes magnetic perturbations is derived. The derived theory eliminates all of the odd terms originating from the nuclear attraction potential V and the first-order odd terms originating from the magnetic vector potential A, which connect the positive states to the negative states. The electronic energy obtained by the decoupling is correct to the third order with respect to A due to the (2n+1) rule. The decoupling is exact for the magnetic shielding calculation. However, the calculation of the diamagnetic property requires both the positive and negative states of the unperturbed (A=0) Hamiltonian. The derived theory is applied to the relativistic calculation of nuclear magnetic shielding tensors of HX (X=F,Cl,Br,I) systems at the Hartree-Fock level. The results indicate that such a substantially exact decoupling calculation well reproduces the four-component Dirac-Hartree-Fock results.     

A general computer approach for calculating rate constants from near-equilibrium kinetic studies

A general computer approach for estimating rate constants from relaxation times is described. The programme CORNEK is essentially a least-squares refinement programme applied to non-linear systems. It uses directly the differential forms of the first derivatives of mass-balance and rate equations, thus avoiding the time-consuming derivations of near-equilibrium rate equations. The programme has been tested for binary systems such as Cu-histamine, Cu-serine, and the ternary system Cu-histamine-serine.     

Evaporation rate in containers used for storing radioactive tracer solutions

In radiochemical analysis, the storage of a tracer solution is an important issue to bear in mind. The evaporation of the tracer solution depends on the type of container used for storing. In this paper the evaporation rate in four kinds of containers, i.e., flame-sealed glass ampoule, sealed glass flask, flame-sealed polyethylene ampoule and screw glass vial was studied. It is concluded that the evaporation rate depends on the system of closing.