A Method of Determining Selectivity Coefficients Based on the Practical Slope of Ion Selective Electrodes

It is a problem to be solved that the experimental selectivity coefficients of ion selective electrodes( ISEs) depend on the activity. This paper studied the new method of determining selectivity coefficients. A mixed ion response equation, which was similar to Nicolsky-Eisenman (N-E) equation recommended by IUPAC, was proposed. The equation includes the practical response slope of ISEs to the primary ion and the interfering ion. The selectivity coefficient was defined by the equation instead of the N-E equation. The experimental part of the method is similar to that based on the N-E equation. The values of selectivity coefficients obtained with this method do not depend on the activity whether the electrodes exhibit the Nernst response or non-Nernst response. The feasibility of the new method is illustrated experimentally.     

Cubic Spline Method for Solving Second-Order Differential Equations Theory and application to the Thomas-Fermi Model for Ions

A cubic spline method is described which can be used to generate solutions to arbitrary second-order differential equation. The principle advantage is that this method provides accurate solutions which are cast in an analytic form, thus allowing the solutions to be applied to other problems. The method is applied to the solution of the Thomas-Fermi equation for singly and doubly ionized ions, and to the solution of the Schrödinger equation for the ground and first excited states of alkali metal atoms.     

从特性粘数和GPC图谱获得数均分子量的新方法

本文提出了从未知K和a的样品的特性粘数和GPC图谱计算数均分子量的新方法。该法对窄分布或宽分布的样品都适用,并用不同分子量和分子量分布的样品进行了验证,和已知K和a的计算值进行比较,得到非常满意的结果。该法是计算M_n的近似解法,但计算简便。     

从特性粘数和GPC图谱获得重均分子量的新方法

本文提出了从未知样品的特性粘数和GPC图谱计算重均分子量的新方法,该方法可用于计算窄分布和宽分布的未知Mark-Houwink 常数的样品的重均分子量。用七个不同分子量和不同分布的实例验证了所提出的方法。结果与已知Mark-Houwink常数用普适校准法得到的结果一致。     

Solving the finite-difference,nonlinear, Poisson–Boltzmann equation under a linear approach

Electrostatic interactions are among the key factors in determining the structure and function of biomolecules. Simulating such interactions involves solving the Poisson equation and the Poisson-Boltzmann (P-B) equation in the molecular interior and exterior region, respectively. The P-B equation is a nonlinear partial differential equation. The central processing unit (CPU) time for solving the full nonlinear P-B equation has been severalfold greater than the equivalent linear case. Here a simple method is proposed to solve the full nonlinear P-B equation under a linear approach, which has been tested both on a spherical case and on small molecules. Results show that our method converges rapidly even under highly charged cases. With this method, the CPU time for solving the full nonlinear P-B equation is somewhat less than the equivalent linear case in our calculations. © 1995 by John Wiley & Sons, Inc.     

Comparison between theoretical values and simulation results of viscosity for the dissipative particle dynamics method

In order to investigate the validity of the dissipative particle dynamics method, which is a mesoscopic simulation technique, we have derived an expression for viscosity from the equation of motion of dissipative particles. In the concrete, we have shown the Fokker-Planck equation in phase space, and macroscopic conservation equations such as the equation of continuity and the equation of momentum conservation. The basic equations of the single-particle and pair distribution functions have been derived using the Fokker-Planck equation. The solutions of these distribution functions have approximately been solved by the perturbation method under the assumption of molecular chaos. The expressions of the viscosity due to momentum and dissipative forces have been obtained using the approximate solutions of the distribution functions. Also, we have conducted nonequilibrium dynamics simulations to investigate the influence of the parameters, which have appeared in defining the equation of motion in the dissipative particle dynamics method. The theoretical values of the viscosity due to dissipative forces in the Hoogerbrugge-Koelman theory are in good agreement with the simulation results obtained by the nonequilibrium dynamics method, except in the range of small number densities. There are restriction conditions for taking appropriate values of the number density, number of particles, time interval, shear rate, etc., to obtain physically reasonable results by means of dissipative particle dynamics simulations.     

The PVT Properties Calculation of Water by BWR Equation

The classical theory of the equation of state first formulated by van der Waals1 has shown that a universal relation among pressure, volume, and temperature exists. In spite of the large number of equations of state that have appeared during the last century, very few people seems to have developed an equation with a reasonable number of empirical constants, and few constants have a clear physical significance. The successful empirical development is the virial type equation of state. It is…     

A new explicit Bessel and Neumann fitted eighth algebraic order method for the numerical solution of the Schrödinger equation

An explicit eighth algebraic order Bessel and Neumann fitted method is developed in this paper for the numerical solution of the Schrödinger equation. The new method has free parameters which are defined in order the method is fitted to spherical Bessel and Neumann functions. A variable-step procedure is obtained based on the newly developed method and the method of Simos [17]. Numerical illustrations based on the numerical solution of the radial Schrödinger equation and of coupled differential equations arising from the Schrödinger equation indicate that this new approach is more efficient than other well known methods.     

奥美拉唑对映体在自制手性柱上吸附等温线的测定

采用吸附-脱附法研究了奥美拉唑对映体在自制的纤维素三苯基氨基甲酸酯涂敷型手性柱上的吸附性能,并用Langmuir方程拟合出吸附平衡方程.当样品浓度降低时,Langmuir方程退化为直线关系,直线斜率即为吸附平衡常数.这一吸附平衡常数与从脉冲色谱流出曲线求得的吸附平衡常数相等,表明吸附-脱附法测得的吸附平衡关系准确,Langmuir方程能描述这一吸附平衡关系.     

Iterative Solution Method for the Linearized Poisson-Boltzmann Equation: Indirect Boundary Integral Equation Approach

An iterative solution scheme is proposed for solving the electrical double-layer interactions governed by the linearized Poisson-Boltzmann equation. The method is based on the indirect integral equation formulation with the double-layer potential kernel of the linearized Poisson-Boltzmann equation. In contrast to the conventional direct integral equation approach that yields Fredholm integral equations of the first kind, the indirect integral equation approach yields well-posed Fredholm integral equations of the second kind. The eigenvalue analysis reveals that the spectral radius of the double-layer integral operator is always less than one. Thus, iterative solution schemes can be successfully implemented for solving the electrical double-layer interactions for very large and complex systems. The utility of the iterative indirect method is demonstrated for several examples which include spherical and spheroidal particles. Copyright 2001 Academic Press.     

Solution of the Schrödinger equation for two different molecular potentials by the Nikiforov-Uvarov method

In this study, the solution of the Schr?dinger equation by a method developed by Nikiforov and Uvarov which is not based on the manipulation of formal power series has been schematically presented. The method gives elegant, easy and exact solutions of the Schr?dinger equation. In order to demonstrate the applications of the method, solutions of the Schr?dinger equation for the well-known pseudo-harmonic oscillator and a new symmetrical potential proposed by the authors are given. The concrete energy spectra and corresponding wave functions are obtained. The superiority and the limitations of the method compared to other methods have also been emphasized. Received: 20 November 1996 / Accepted: 1 October 1997     

Exact solution of the adsorption integral equation for a heterogeneous surface

An exact method of solution of the adsorption integral equation using finite and realistic limits of the heats of adsorption and which is applicable to any analytic site-energy distribution is given. The optical feature is the postulation and use of a set of transformations to reduce the finite-limit problem to one of limits (0, ∞). The transformed integral equation is then inverted using the theory of Stieltjes transforms. The method thus avoids the limitations inherent in earlier work which employed restrictive assumptions about the heats of adsorption.     

Numerical technique and computational procedure for isotachophoresis

This paper presents a new numerical method for computation of solutions of prototypical equations of isotachophoresis. Numerical computation is complicated because the Poisson equation, which relates electrostatic potential to space charge density, contains a small parameter. This parameter is usually assumed to have the value of zero. Under this assumption the Poisson differential equation is replaced by an algebraic equation, which is often called the equation of electroneutrality, because it indeed states that the electrolyte is electrically neutral this assumption were not studied in the past. Here we propose an iterative procedure which allows for computation of solutions without the assumption of electroneutrality. The accuracy is controlled by a number of iterations and is limited by a computer round-off error only. The method is based on our previously published theory of existence and uniqueness of solutions of isotachophoretic equations. Details of the computational algorithm for prototypical equations of isotachophoresis are given. A numerical example and comparison with previously published data are also provided.     

On a nonelementary progress curve equation and its application in enzyme kinetics

The analytical equation describing progress curves of an enzyme catalyzed reaction acting upon the Michaelis-Menten mechanism has been known for the case in which only the free enzyme incurs a loss of its activity, either spontaneously or as a result of an irreversible inhibitor action. The solution of differential equations which defines the rates of enzyme inactivation and substrate utilization is expressed by a nonelementary function in equation of an implicit type that precludes direct calculation of the extent of reaction at any time. Previously, the implicit equations have been rearranged to the alternative formulas and solved by the Newton-Raphson method, but this procedure may fail when used upon the presented equation. For this reason the other root-finding numerical method was applied, and the enzyme kinetic parameters of such numerically solved implicit equation for the reaction mechanism of irreversibly inhibited acetylcholinesterase were fitted to the experimental data by a nonlinear regression computer program.     

A boundary element method for molecular electrostatics with electrolyte effects

A boundary element method is developed to compute the electrostatic potential inside and around molecules in an electrolyte solution. A set of boundary integral equations are derived based on the integral formulations of the Poisson equation and the linearized Poisson-Boltzmann equation. The boundary integral equations are then solved numerically after discretizing the molecular surface into a number of flat triangular elements. The method is applied to a spherical molecule for which analytical solutions are available. Use is made of both constant and linearly varying unknowns over the boundary elements, and the method is tested for various values of parameters such as the dielectric constant of the molecule, ionic strength, and the location of the interior point charge. The use of the boundary integral method incorporating the nonlinear Poisson-Boltzmann equation is also briefly discussed.     

Determining partial differential cross sections for low-energy electron photodetachment involving conical intersections using the solution of a Lippmann-Schwinger equation constructed with standard electronic structure techniques

A method for obtaining partial differential cross sections for low energy electron photodetachment in which the electronic states of the residual molecule are strongly coupled by conical intersections is reported. The method is based on the iterative solution to a Lippmann-Schwinger equation, using a zeroth order Hamiltonian consisting of the bound nonadiabatically coupled residual molecule and a free electron. The solution to the Lippmann-Schwinger equation involves only standard electronic structure techniques and a standard three-dimensional free particle Green's function quadrature for which fast techniques exist. The transition dipole moment for electron photodetachment, is a sum of matrix elements each involving one nonorthogonal orbital obtained from the solution to the Lippmann-Schwinger equation. An expression for the electron photodetachment transition dipole matrix element in terms of Dyson orbitals, which does not make the usual orthogonality assumptions, is derived.     

On a partial differential equation method for determining the free energies and coexisting phase compositions of ternary mixtures from light scattering data

In this paper we present a method for determining the free energies of ternary mixtures from light scattering data. We use an approximation that is appropriate for liquid mixtures, which we formulate as a second-order nonlinear partial differential equation. This partial differential equation (PDE) relates the Hessian of the intensive free energy to the efficiency of light scattering in the forward direction. This basic equation applies in regions of the phase diagram in which the mixtures are thermodynamically stable. In regions in which the mixtures are unstable or metastable, the appropriate PDE is the nonlinear equation for the convex hull. We formulate this equation along with continuity conditions for the transition between the two equations at cloud point loci. We show how to discretize this problem to obtain a finite-difference approximation to it, and we present an iterative method for solving the discretized problem. We present the results of calculations that were done with a computer program that implements our method. These calculations show that our method is capable of reconstructing test free energy functions from simulated light scattering data. If the cloud point loci are known, the method also finds the tie lines and tie triangles that describe thermodynamic equilibrium between two or among three liquid phases. A robust method for solving this PDE problem, such as the one presented here, can be a basis for optical, noninvasive means of characterizing the thermodynamics of multicomponent mixtures.     

On the Use of the Knox Equation. I. the Fit Problem

Abstract

To obtain significant A, B, and C parameters of the Knox equation, which corresponds to the plot of the reduced plate height, h, versus the reduced linear velocity, v, actual results must be fitted correctly. The Knox equation is analyzed, the rôle of each individual parameter is shown. The equation producing the coordinates of the minimum plate height (maximum efficiency) is fully derived from the derivative of the Knox equation. Two tables giving the minimum coordinates for usual A, B, and C ranges are listed. The classical fit method is derived and analyzed. A graphical fit method which uses the remarkable graphical capabilities of modern spreadsheet packages is described. A synthetical set of data was fitted. It is demonstrated that each Knox parameter, obtained through fit procedures, must be given with the fitting confidence interval which is often in the 20% range.     

The application of the brönsted concept to the calculation of pH in systems involving two acid-base couples : Systems involving two independent acid-base couples

The BRÖNSTED concept has been applied to the derivation of a single equation which is valid for the calculation of pH in any system containing two independent acid-base couples in an amphiprotic solvent. Bxamples of the application of this equation to a few representative systems have been given. The advantages of the method of approach which, has been outlined have been emphasized.