Determination of equilibrium states in multi-phase systems

Volume corrections in calculation of equilibrium state for multi-phase systems, as an extension from one-phase systems, have been described. The algorithms can handle with one, two, three or more phases of different volumes. Several applications of the algorithms to calculation of equilibrium concentrations of all species in two- and three-phase systems have been presented. Different possibilities of the determination of equilibrium constants from extraction data have been discussed.     

Trace analysis of acids and bases by conductometric titration with multiparametric non-linear regression

A chemometric method for analysis of conductometric titration data was introduced to extend its applicability to lower concentrations and more complex acid-base systems. Auxiliary pH measurements were made during the titration to assist the calculation of the distribution of protonable species on base of known or guessed equilibrium constants. Conductivity values of each ionized or ionizable species possibly present in the sample were introduced in a general equation where the only unknown parameters were the total concentrations of (conjugated) bases and of strong electrolytes not involved in acid-base equilibria. All these concentrations were adjusted by a multiparametric nonlinear regression (NLR) method, based on the Levenberg-Marquardt algorithm. This first conductometric titration method with NLR analysis (CT-NLR) was successfully applied to simulated conductometric titration data and to synthetic samples with multiple components at concentrations as low as those found in rainwater (∼10 μmol L⁻¹). It was possible to resolve and quantify mixtures containing a strong acid, formic acid, acetic acid, ammonium ion, bicarbonate and inert electrolyte with accuracy of 5% or better.     

Necessary and sufficient conditions for the appearance of extrema on concentration distribution curves in complex equilibrium systems

It is shown that species distribution curves with at least one minimum may be found in most equilibrium systems with three or more components. Whether such concentration minima are actually observed then depends on the values of the equilibrium constants and on the total (analytical) concentrations of the components. A general algorithm is given for the necessary and sufficient conditions for the appearance of extrema in multicomponent systems. Three-component systems are studied in more detail and special attention is given to the limiting case of a horizontal inflection, i.e., the point where the concentration minimum just disappears. Two well-studied chemical examples, the Cu(2+)-diethylenetriamine-OH(-) and Hg(2+)-Cl(-)-OH(-) systems are discussed, along with a simple model system showing as many as five extrema on a single distribution curve.     

Spectrophotometric study of complexation equilibria with H-point standard addition and H-point curve isolation methods

The use of H-point curve isolation (HPCIM) and H-point standard addition methods (HPSAM) for spectrophotometric studies of complex formation equilibria are proposed. One step complex formation, two successive stepwise and mononuclear complex formation systems, and competitive complexation systems are studied successfully by the proposed methods. HPCIM is used for extracting the spectrum of complex or sum of complex species and HPSAM is used for calculation of equilibrium concentrations of ligand for each sample. The outputs of these procedures are complete concentration profiles of equilibrium system, spectral profile of intermediate components, and good estimation of conditional formation constants. The reliability of the method is evaluated using model data. Spectrophotometric studies of murexide-calcium, dithizone-nickel, methyl thymol blue (MTB)-copper, and competition of murexide and sulfate ions for complexation with zinc, are used as experimental model systems with different complexation stoichiometries and spectral overlapping of involved components.     

Machine computation of equilibrium concentrations-some practical considerations

The calculation of equilibrium concentrations in solution by means of seven published computer programs is critically evaluated. Six equilibrium systems were chosen to illustrate the capabilities of these programs. It was found, in general, that the choice of algorithm determines the efficiency of program execution. In all cases studied the problems were most rapidly solved by a modified Newton-Raphson approach that employs Choleski factoring.     

Calculation for liquid-liquid equilibria of quaternary alkane-ethyl acetate-methanol-water systems used in counter-current chromatography

The calculation of liquid-liquid equilibrium compositions of solvent systems is very important for the work on counter-current chromatography (CCC), especially the phase composition and volume ratio obtained from liquid-liquid equilibrium calculation. In this work, liquid-liquid equilibria of quaternary Arizona solvent systems, alkane-ethyl acetate-methanol-water, and related ternary systems are correlated and predicted using the non-random two-liquid model (NRTL). Hexane, heptane and isooctane are the used alkanes. The parameters in the model are regressed only with the special systems considered. Detailed comparison with experimental data shows that liquid-liquid equilibria of these systems can be predicted with greatly improved accuracy as compared to the group contribution method (UNIFAC).     

Simulation of the rheological properties of liquid media containing solid anisometric particles

Theoretical simulation of the rheological properties of liquid media containing solid anisometric particles is performed. Procedures for the calculation of the degree of ordering in the nematic systems under equilibrium conditions and their rheological characteristics are proposed. Earlier developed notions of the calculation procedures for the properties of systems considered are analyzed and corrections to the theory are substantiated. Based on developed model notions, equilibrium and transport equations are derived and a program for their numerical solution is proposed. Model calculations of the lines of phase transition and the ordering of systems containing isotropic and nematic phases, as well as anisotropic (in orienting field) and effective dynamic viscosities under the conditions of free flow of such systems, are carried out at various concentrations and geometry of anisometric particles. A comparison of calculated and experimental data for the solutions of polymer-salt compositions with induced chain rigidity demonstrates the adequacy of the model proposed.     

Prediction of vapor-liquid-liquid-hydrate phase equilibrium for multicomponent systems containing tetrahydrofuran

Numbers of hydrate-based new techniques require the algorithm to be able to perform multiphase flash calculation where one phase is a gas hydrate phase.Tetrahydrofuran(THF)is frequently used as a thermodynamic promoter in the development of hydrate-based technique,which can reduce the hydrate formation pressure,especially when methane or hydrogen exists.However,it is a hard work to describe accurately the phase behavior of THF-water system due to their polarities.It has been proved that the water-in-oil(W/O)emulsion can raise the hydrate formation rate and improve the single stage separation efficiency,and furthermore prevent the hydrate from agglomeration to plug the facilities.The goal of this work is to present an extension of our previous work to the prediction of the vapor-liquid-liquid-hydrate equilibrium of such complex systems.These include ternary and quaternary mixtures with W/O emulsion containing THF.The proposed algorithm of four phase equilibrium calculation is very simple due to avoiding the complexity of simultaneous solution of the sophisticated equation group.The calculation results were found to be in satisfactory agreement with the experimental data.     

Dynamical Approach to Multi-Equilibria Problems Considering the Debye–Hückel Theory of Electrolyte Solutions: Concentration Quotients as a Function of Ionic Strength

We recently described a dynamical approach to the equilibrium problem that involves the formulation of the kinetic rate equations for each species. The equilibrium concentrations are determined by evolving the initial concentrations via this dynamical system to their steady state values. This dynamical approach is particularly attractive because it can be extended easily to very large multi-equilibria systems and the effects of ionic strength also are easily included. Here we describe mathematical methods for the determination of steady state concentrations of all species with the consideration of their activities using several approximations of Debye–Hückel theory of electrolyte solutions. We describe the equations for a system that consists of a triprotic acid H₃A and its conjugate bases. With these equations, two types of multi-equilibria systems were studied and compared to experimental data. The first system is exemplified by case studies of solutions of acetate-buffered acetic acid and the second system is exemplified by the hydroxide titration of citric acid. The discussion focuses on the effect of ionic strength on pH and on the amplification of acidity by ionic strength. Ionic strength effects are shown to cause significant deviations from the widely used Henderson–Hasselbalch equation.     

多组分体系单级萃取平衡的Newton-Raphson算法

提出了一种多组分体系串级萃取动态仿真中的单级萃取平衡和物料计算的新同了有机相萃取量,水相洗涤量和混合萃取比等限制条件下两相组成关系的２λ元非线性方程组,构造以水相中某一组分的平衡含量为变量的目标函数,采用Ｎｅｗｔｏｎ－Ｒａｐｈｓｏｎ迭代求解,并以萃取平衡和物料平衡得到该级其它所有组分在两相的含量。     

Automated calculation of complete Pxy and Txy diagrams for binary systems

An algorithm for the calculation of global phase equilibrium diagrams has been recently developed [M. Cismondi, M.L. Michelsen, Global phase equilibrium calculations: critical lines, critical end points and liquid–liquid–vapour equilibrium in binary mixtures, J. Supercrit. Fluids 39 (2007) 287–295]. It integrates the calculation of critical lines, liquid–liquid–vapour (LLV) lines and critical end points, and was implemented in the software program GPEC: global phase equilibrium calculations [M. Cismondi, D.N. Nuñez, M.S. Zabaloy, E.A. Brignole, M.L. Michelsen, J.M. Mollerup, GPEC: a program for global phase equilibrium calculations in binary systems, in: Proceedings of the CD-ROM EQUIFASE 2006, Morelia, Michoacán, Mexico, October 21–25, 2006; www.gpec.plapiqui.edu.ar]. In this work we present the methods and computational strategy for the automated calculation of complete Pxy and Txy diagrams in binary systems. Being constructed from the points given by the global phase diagram at a specified temperature or pressure, their calculation does not require the implementation of stability analysis.     

A simplex search for conditions giving maximal and/or minimal concentrations of species in distribution plots

A new method is proposed for finding the conditions which maximize or minimize the concentration of a particular species in a multicomponent system. A Simplex procedure is combined with an iterative method for solving mass-balance equations from the total concentrations of the constituents of the system, its pH and the pertinent equilibrium constants. The following examples of application are given: (a) the optimum conditions for maximizing the concentration of mixed-ligand species in systems with more than one ligand; (b) the conditions for maximal or minimal formation of monomeric and dimeric species in the Cu(II)-triethanolamine system; (c) a search for local maximum/minimum in species distribution plots.     

On linear programming approach for the calculation of chemical equilibrium in complex thermodynamic systems

The advantages of linear programming approach for the calculation of chemical equilibrium in complex thermodynamic systems are demonstrated. The algorithm presented is used for the development of software package for thermodynamic modeling.     

Random Error Analysis in the Determination of Equilibrium Constants of Very Stable Metal Complexes.

Abstract

The analysis of random errors is discussed for the equilibrium constants of very stable metal complexes, determined through competitive potentiometric methods. Prior to this analysis the calculation of standard deviations both for overall and stepwise equilibrium constants is considered. Clues are given as to how a system can be properly characterized. To illustrate the error analysis, the following systems were selected: Cu(II) and L-2,6-diaminohexanoic acid (L-lysine); Cu(II) and ethylene diaminotetraacetic acid (EDTA); and Cu(II), EDTA and 1,2-dihydroxybenzene (catechol).     

Spectrophotometric study of competitive complexation equilibria involving overlapped spectral responding species: Determination of the stability constant of bismuth-pyrophosphate complex

Spectrophotometric study of competitive complex formation equilibria involving overlapped spectral responding species applying a simple and versatile algorithm was carried out. The algorithm involves multivariable regression for calculation of equilibrium concentrations from multiwavelength data and mass action law for the stability constant calculation. The used regression functions are part of common statistical software. Stability constants and complex stoichiometry of competing equilibria were simultaneously determined. The species concentration profiles at several spectral overlapping and ??-coefficient of competing reaction were obtained. Non-absorbing bismuth ?? pyrophosphate (PPh) system was studied as a competitive reaction of bismuth ?? 4-(2-Pyridylazo) resorcinol (PAR) complex. The formation of Bi-PPh complex with 1:1 stoichiometry was proved in the studied concentration region (C_Bi = 1×10^?5 mol L^?1; C_PPh = 5×10^?6 ? 1×10^?4 mol L^?1). The stability constant of the complex at pH 1 and ?? = 1.0 have been determined: log?? = 4.2±0.2.      

End-evaporation dynamics revisited

We present analytical results on the so-called end-evaporation kinetics in equilibrium polymeric systems following a temperature jump (T jump). A T jump prepares the system with a nonequilibrium length distribution, after which it relaxes back to its equilibrium state. Starting from a master equation, we develop a mean-field analytical theory based on a generating function approach, which allows explicit approximate expressions for the monomer and dimer concentrations to be derived in a discrete setting; the concentrations of the other chains as well as the average chain length were shown to be entirely expressible in terms of the monomer and dimer concentrations. We find that the calculated monomer and dimer concentrations as well as the average chain length are in good agreement with numerical simulation results and do not suffer from some of the defects of earlier continuum theories. Furthermore, the relaxation was shown to take place in three different stages. The first stage comprises the very fast relaxation of the monomers to almost their equilibrium concentration; the other polymer chains have hardly relaxed. During the second stage, which is highly nonlinear, a redistribution of material at practically constant monomer density takes place. Only in the final stage of the relaxation process the chain concentrations approach their true equilibrium values. In this stage there are only very small shifts in the concentrations of chains, which are governed by extremely slow "indirect" monomer-mediated processes.     

Activity behaviour of electrolyte solutions: Evaluation of temperature effects by means of the quasi-random lattice model

The paper presents some applications of the Moggia–Bianco (quasi-random lattice) model for the calculation of mean activity and osmotic coefficients of electrolyte solutions, with emphasis on temperature effects. Aqueous uni-univalent and uni-divalent electrolytes from 0 °C to 100 °C are considered. Results show that the model is in general applicable over wide ranges of concentrations and temperatures, despite its dependence on a unique fitting parameter (which is, sometimes, experimentally known).Some theoretical aspects of the model are investigated. Hydration diameters are evaluated for uni-univalent systems. Concerning uni-divalent systems, charge-asymmetry effects are accounted for more rigorously than in previous works, starting from a better assessment of the relationship between the behaviour of the electrolyte at very low concentrations and the behaviour occurring at high concentrations.     

Calculation of Combined Diffusion Coefficients from the Simplified Theory of Transport Properties

The aim of this study is to check if it is possible to use the combined diffusion coefficients introduced by Murphy at equilibrium in a two-temperature model (electron temperature T_e different from that of heavy species T_h), such as that defined by Devoto and Bonnefoi for transport properties. On the one hand, the two-temperature (2-T) theory of transport properties was established by Devoto and Bonnefoi by separating electrons and heavy species because of their mass difference. Their simplified theories allow the calculation of transport coefficients (except diffusion) out of thermal equilibrium, but it has to be noted that when T_e tends toward T_h, the results are those obtained with an equilibrium calculation. On the other hand, Murphy's combined diffusion coefficients describe the diffusive mixing of two nonreactive ionized gases at equilibrium. First, the exact combined diffusion coefficients of Murphy are calculated for an Ar–N₂ (50 wt.%) mixture at atmospheric pressure. Expressions of combined diffusion coefficients are then obtained by using the simplified theory of Bonnefoi at thermal equilibrium. The results of the calculation of combined diffusion coefficients from the simplified theory of transport properties, assuming Te=Th, are compared with those of Murphy at equilibrium. It is shown that large discrepancies occur as soon as the ionization degree is over 10%. These results prove that the simplified 2-T theory of transport coefficients cannot be used for the treatment of diffusion, probably because the mass flux of electrons is no longer constrained. Thus, a new theory of transport coefficients has to be developed, taking into account the coupling of electrons and heavy species.     

Calculation of chemical equilibrium concentrations of complexing ligands and metals A flexible computer programme taking account of uncertainty in formation constant values

Metal complexes often act as substrates in enzymic reactions. Because their concentrations cannot be measured directly, they must be calculated from the formation constants and the total concentrations of the different metal and ligand species. An iteration algorithm devised by Sayce has been used to formulate a FORTRAN (ICT 1900) programme, which meets the following requirements. (1) The programme may be made to calculate the total concentrations for the parent species if the free concentration of any metal or ligand species is known, or to calculate the free concentrations of all species for which the total concentrations are known. This feature of the programme is useful for calculation of the total amounts which must be put into a solution in order to achieve prescribed free concentrations. (2) To allow for the uncertainty in the formation constants of the complex species the programme permits the input values of the formation constants to be varied automatically and the resulting variation of the calculated concentrations to be evaluated.     

Gibbs energy based procedure for the correlation of type 3 ternary systems including a three-liquid phase region

Equilibrium data for type 3 ternary systems comprising a liquid–liquid–liquid equilibrium region (LLLE tie-triangle) are not frequently correlated, as is evidenced by a lack of studies in the literature. In the present paper, a robust algorithm to calculate or correlate equilibrium data for this type of system is proposed. This algorithm is based on geometric aspects related to the Gibbs energy of mixing (g^M) and possesses some interesting advantages that are discussed in the text. The methodology put forward is applied to correlate the equilibrium data of three type 3 ternary systems. The NRTL model is used to define g^M, and the common tangent plane criterion to define the equilibrium condition for the calculations. The lack in flexibility of the NRTL equation to model these systems is discussed.