Guggenheim Equation Research Articles

Activity coefficients of sodium benzoate and potassium benzoate in water at 298.15 K

Industrial additives, like sodium benzoate and potassium benzoate, are used in low concentrations in aqueous solutions. However, in general, the ionic strength of the aqueous media is high. The Pitzer model allows the determination of activity coefficients in aqueous media. In this work, Pitzer parameters for sodium benzoate were recalculated ( β (0)=0.162 mol −1 kg and β (1)=0.38 mol −1 kg) and Pitzer parameters for potassium benzoate were determined ( β (0)=0.157 mol −1 kg and β (1)=0.27 mol −1 kg) from cell potential difference (cpd) measurements on galvanic cells without liquid junction. From these values, activity coefficients of potassium benzoate in water and in aqueous potassium chloride solutions were calculated by the Pitzer model. The activity coefficients were also calculated by modified Guggenheim equations with parameters obtained in this work and the results of the two models were compared.

The effect of various cations and pH on the adsorption of U(VI) on Amberlite IR-118H resin

The effects of various metal cations and pH on the adsorption of uranium(VI) on strongly acidic cation exchanger Amberlite IR-118H (AIR-118H) were studied. The metal cations suppress U(VI) adsorption differently depending on their ionic radii. Adsorption of U(VI) on AIR-118H peaks at pH 3.4, which was attributed to the occurrence of different forms of U(VI) at different pH values. The adsorption data were then processed using the Frumkin–Fowler–Guggenheim equation, and the standard free energy of adsorption was calculated.

Modeling Physical and Thermodynamic Properties via Inverse Normalizing Transformations

The inverse normalizing transformation (INT) represents a generalization of the inverse of the Box−Cox transformation. It is shown that several well-known and widely used property correlation equations, such as the Antoine, the truncated Riedel, the Rackett, and the Guggenheim equations can be derived from the INT. Its use is demonstrated for modeling the temperature dependence of vapor pressure, solid and liquid heat capacity, vapor and liquid viscosity, and surface tension data. It is shown that the INT can represent satisfactorily curves of different shapes and, as such, its use can be beneficial in modeling the temperature dependence of various physical and thermodynamic properties.

A simple model for adsorption kinetics at charged solid–liquid interfaces

The kinetics of adsorption of charged nano particles or molecules to a charged surface are modeled on the basis of a simple model that takes into account, (1) the transport step from bulk solution to the subsurface layer and (2) the attachment–detachment step that is involved in the transfer of the particle from the subsurface to the adsorbed state. The transport step is based on the presence of a diffusion layer. Passing through the electric double layer is made part of the attachment–detachment step. The configuration part of the attachment–detachment step is based on either a kinetic model that leads to the Langmuir equation in the equilibrium situation, or one that takes into account the ‘specific’ lateral interactions too and that leads in the equilibrium state to the Frumkin–Fowler–Guggenheim (FFG) equation. In the FFG model the activation energy due to specific lateral interactions is assumed to be proportional to the equilibrium lateral interaction energy. The effect of the electrostatic interactions and the corresponding activation energy barriers for adsorption and desorption are considered to be an additional part of the attachment–detachment step. The electrostatic potential of the activated state for attachment–detachment is made proportional to the equilibrium surface potential at a given adsorbed amount. The Gouy–Chapman model is used to calculate the (smeared-out) surface potential from the known (smeared-out) overall surface charge density, that is to say, from the known bare surface charge plus the effective charge contribution due to particle adsorption. As a result of this treatment the adsorption kinetics are not only a function of the particle concentration and the surface coverage, but also of the surface charge, the particle charge and the salt concentration. The model is illustrated with some calculated results. The first illustrations are based on the Langmuir model extended with electrostatic interactions and show, for a given particle concentration and transport rate constant, the effects of salt concentration, surface charge and particle charge on both the adsorption and desorption kinetics. The next illustrations are based on the FFG model extended with electrostatics and the effect of the specific lateral interactions on the adsorption kinetics of charged and uncharged particles is shown.

On extension of UNIQUAC-NRF model to study the phase behavior of aqueous two phase polymer–salt systems

The UNIQUAC-NRF model for aqueous two-phase polymer–polymer systems is extended to correlate liquid–liquid equilibria of aqueous polymer–salt two-phase systems. The systems investigated are polyethylene glycol+ammonium sulfate+water and polyethylene glycol+potassium monohydrogen phosphate (K 2HPO 4)+water for five molecular weights of PEG. In this model, the nonrandom state is selected as a reference state. The model has six binary adjustable parameters that were determined by an optimization program. The Debye–Huckel equation based on Fowller–Guggenheim equation is used to calculate the long-range electrostatic interaction of the ions. The results obtained by this model are in good agreement with experimental data.

A Calorimetric Study of the Influence of Temperature on the Self-Association of Amphiphilic Antidepressant Drugs in Aqueous Solution

Relative apparent molar enthalpies have been determined as a function of concentration (0.0001 to 0.2 mol kg−1) by heat conduction calorimetry for aqueous solutions of the structurally related antidepressant drugs imipramine and clomipramine in water over the temperature range 288 to 308 K. Critical concentrations determined from inflections in these plots for both drugs had minimum values at 298 K. The concentration dependence of the relative apparent molar enthalpy could be quantitatively described using a mass action model of association based on the Guggenheim equations for the activity coefficients of mixed electrolytes. Derived values of the monomer–counterion interaction coefficient for imipramine became increasingly negative with an increase in temperature over the temperature range 293 to 303 K, indicative of an increasing degree of premicellar association. In contrast, negative monomer–counterion interaction coefficients were obtained for clomipramine at only 303 and 308 K, suggesting an absence of premicellar association at lower temperatures. Values derived for the molar enthalpy of micellization of both drugs from the mass action model indicate an increasingly exothermic process with increase in temperature; positive values at 288 and 293 K arise from hydrophobic interactions while the negative values at higher temperatures suggest the importance of London dispersion interactions as the major driving force for aggregation.

Closed-loop liquid-liquid equilibria and the global phase behaviour of binary mixtures involving hard-sphere + van der Waals interactions

Calculations of the critical properties of binary mixtures of components of equal size are reported using the Guggenheim equation of state. The calculations are used to determine the global phase diagram of binary mixtures. Type VI phase behaviour is predicted successfully indicating that closed-loop liquid-liquid equilibria can be obtained from hard-sphere + van der Waals interactions. Closed-loop liquid-liquid equilibria occur in the region of the global phase diagram characterized by moderately strong unlike interactions and components with very dissimilar critical temperatures. The Guggenheim equation can predict all experimentally known phase behaviour types. In addition, other hypothetical phase behaviour types are also predicted.

A Model for Dissolution of Cao-Sio2-H2O Gel at Ca/Si<1 by Considering Disordered Structure

ABSTRACTA model for dissolution of CaO-SiO2-H2O (C-S-H) gel is proposed by considering a Margules type of non-ideal mixture of binary solid solutions at Ca/Si (C/S)<1. The Guggenheim and Prausnitz equations are used to represent the activities of model solids as a function of mole fractions. The Gibbs-Duhem equation, together with the activities of model solids, is applied to express the conditional solubility products of model solids as a function of C/S ratio. The determination of Guggenheim’s empirical parameters is performed with the geochemical code PHREEQE on experimental data. The dissolution model is developed by considering the effect of disorder in the structure of C-S-H gel at C/S<1. The solubility results predicted by the proposed model are comparable with experimental data. The study predicts some structural information of C-S-H gel at C/S<1.

Phase behaviour of ternary mixtures: a theoretical investigation of the critical properties of mixtures with equal size components

Conformal solution theory in conjunction with the one-fluid model and the Guggenheim equation of state are used to calculate the critical properties of binary and ternary mixtures characterised by molecules of identical size (identical h conformal parameters) but different energy parameters (different f conformal parameters). When the geometric mean combining rule is used to obtain contributions from unlike interactions, binary mixtures of equal size components generate Type I, II and III phase behaviour. The critical properties of ternary mixtures composed of binary subsystems of different phase behaviour types are examined. Results are presented at different temperatures and pressures covering both vapour–liquid and liquid–liquid equilibria. The results illustrate the diversity of phenomena exhibited by ternary mixtures and the relationship of the behaviour exhibited to the relative strength of dissimilar interactions between different pairs of molecules. A progressive transition between vapour–liquid and liquid–liquid equilibria and higher order critical phenomena are likely to be common features of the phase behaviour of ternary mixtures.

Thermodynamics of magnesium sulfate–polypropylene glycol aqueous two-phase system. Experiment and correlation

Liquid–liquid equilibrium data were obtained for magnesium sulfate–polypropylene glycol (PPG) 425 aqueous two-phase system at 25°C. The improved regular solution theory was used to correlate the equilibrium data. In this theory the Fowler–Guggenheim equation was used to calculate the long-range electrostatic interactions of the ions. The volume and interaction energy parameters of polymer in this model were obtained by using isopiestic data. The applicability of the model was further tested for some other aqueous two-phase systems. These systems are PPG425+NaCl+H 2O, ethanol+Na 2SO 4+H 2O, 1-propanol+MgSO 4+H 2O, 2-propanol+MgSO 4+H 2O and t-butanol+MgSO 4+H 2O.

Calorimetric Study of the Influence of Electrolyte on the Micellization of Phenothiazine Drugs in Aqueous Solution

Apparent molar enthalpies have been determined as a function of concentration by heat conduction calorimetry for aqueous solutions of the phenothiazine drugs chlorpromazine hydrochloride, promethazine hydrochloride, and promazine hydrochloride in the presence of added electrolyte (0.025−0.10 mol dm-3 NaCl). The concentration dependence of the apparent molar enthalpy could be quantitatively described using a mass action model of association based on the Guggenheim equations for the activity coefficients for mixed electrolytes. Derived values of the monomer−counterion interaction coefficient became increasingly negative with increase of salt concentration, suggesting that electrolyte addition promoted association at concentrations below the critical micelle concentration (cmc). Calculations of the fraction of each drug in the form of micelles as a function of concentration further confirmed the tendency for premicellar association. Significant differences in properties were observed between promethazine and the other two drugs. The micellization of this drug became increasingly exothermic with increase of electrolyte concentration, whereas the micellization of both chlorpromazine and promazine became increasingly endothermic. Moreover, the premicellar association of promethazine was more pronounced, the predicted fraction of drug in micellar form at the cmc increasing from 2.6% in water to 65% in 0.1 mol dm-3 NaCl.

Binary mixtures of water + five noble gases: comparison of binodal and critical curves at high pressures

The critical properties and high-pressure binodal curves of binary mixtures of water + helium, neon, argon, krypton and xenon are discussed. The critical properties are compared with calculations using the van der Waals, Guggenheim and Heilig-Franck equations of state and the optimal values of the unlike molecular interaction parameters are determined. The agreement with experiment is very good. These interaction parameters are used in equation of state predictions of the binodal curves at various pressures between 50 and 250 MPa. The predicted binodal curves agree reasonably well with experimental data.

Flory and Guggenheim lattice statistics reinterpreted and extended to include molecular clustering and chemical dissociation

A new method is presented for counting the number of distinguishable arrangements available to a mixture of molecules in a lattice model. Chain molecules are used and these are allowed to undergo molecular clustering and chemical dissociation. The resulting equation for the configurational degeneracy is shown to reduce to the equations of Flory and Guggenheim in the appropriate limits. The Guggenheim equation is shown to be exact in one dimension; extension to higher dimensions requires allowing the lattice coordination number to increase beyond 2. The Flory equation is shown to be the low volume fraction approximation of the Guggenheim equation. The counting method developed can be used to model entropy effects associated with chemical reactions. The equation of state derived using the new configurational degeneracy term is shown to reduce to a number of well known equations. Comparison to van der Waals equation of state shows that the improved ability of the new equation to model observed behavior, results from a redefinition of the van der Waals free volume.

Thermodynamics of adsorption of dodecylpyridinium chloride on Na-kaolinite

A study was undertaken to determine the influence of temperature and electrolyte concentration on the adsorption of dodecylpyridinium chloride (DPC) on Na-kaolinite. Enthalpies were measured using an isothermal microcalorimeter by titration of a kaolinite suspension with a concentrated surfactant solution at 6, 20 and 60°C. Curves of cumulative adsorption enthalpies against amount adsorbed show a break around the transition from monolayer to bilayer coverage. The curves are fairly linear both before and after the break, and result in partial molar adsorption enthalpy values that are independent of surface coverage, confirming that the surface is homogeneous with respect to the DPC adsorption. This is additional evidence that the surfactant is hardly adsorbed on the edges. Adsorption isotherms at different electrolyte concentrations exhibit a common intersection point that coincides with the isoelectric point and correponds to the transition from monolayer to bilayer coverage. Isotherms are independent of temperature up to monolayer coverage, implying a zero isosteric adsorption enthalpy, but upon bilayer formation the adsorption has a maximum at about 23°C, implying a transition from an endothermic to an exothermic process. The calorimetric adsorption enthalpies change sign at T ≈ 24–30°C. In this respect, the adsorption process is very similar to micellization. The electrolyte concentration has a minor effect on the adsorption enthalpies of the first and second layers. There are some qualitative agreements and some quantitative discrepancies between the isosteric heats of adsorption and the directly measured heats. Our experimental data fit well to a bilayer model based on the Frumkin—Fowler—Guggenheim equation.

Novel high pressure critical phase transitions in multicomponent fluid mixtures

New results are presented for the critical equilibria of ternary mixtures. The critical conditions of a ternary mixture have been solved (using the Guggenheim equation) for a wide range of pressures, temperatures and volumes, and over the entire range of compositions. The analysis, which has been applied to a wide variety of ternary mixtures, indicates that novel critical transitions are possible which have no direct parallel in binary mixtures. In particular, a critical transition is commonly observed between two different types of two-phase equilibria. This is in contrast to normal critical equilibria in binary mixtures or pure fluids which involve the transformation of two coexisting phases to form a single homogeneous phase. This result has implications for the interpretation of tricritical and other multiphase transitions which may instead be manifestations of a broader type of critical transition.

Apparent molar enthalpies of amphiphilic drugs in aqueous solution

Apparent molar enthalpies have been determined as a function of concentration by heat conduction calorimetry for aqueous solutions of the amphiphilic drugs chlorpromazine hydrochloride, promethazine hydrochloride, promazine hydrochloride, and imipramine hydrochloride. The concentration dependence of the apparent molar entalpy could be quantitatively described using a mass-action model of association based on the Guggenheim equations for the activity coefficients of mixed electrolytes

Thermodynamics of ammonium sulfate—polyethylene glycol aqueous two-phase systems. Part 2. Correlation and prediction using extended unifac equation

An extended UNIFAC equation is proposed for the correlation and prediction of liquid—liquid equilibrium data for ammonium sulfate—polyethylene glycol (PEG) aqueous two-phase systems. The Fowler—Guggenheim equation is also used to calculate the long-range electrostatic interactions of the ions, taking into account the ion association of NH + 4 and SO 2- 4 in both phases. Three experimental series with the PEG molecular weight of 1000, 1540 and 2000 were used to regress the UNIFAC group interaction parameters needed in the proposed equation, and the series with the PEG molecular weight of 4000 is predicted. The results show that the extended UNIFAC equation has good predictive capability.

Thermodynamics of ammonium sulfate—polyethylene glycol aqueous two-phase systems. Part1. Experiment and correlation using extended uniquac equation

Liquid—liquid equilibria of ammonium sulfate—polyethylene glycol (PEG) aqueous two-phase systems were studied experimentally. Experimental techniques and analytical methods are described. Equilibrium data at 25°C along with a phase diagram are given. The molecular weights of PEG used in the four series of experiments were 1000, 1540, 2000 and 4000. An extended UNIQUAC equation is proposed for the correlation of equilibrium data. The Fowler—Guggenheim equation was used to calculate the long-range electrostatic interaction of the ions. The ionic association of NH + 4 and SO 2- 4 in both phases is taken into account. Calculated concentration values by the extended UNIQUAC equation are in good agreement with experimental ones. However, the equation has poor predictive power because of the irregularity of the interaction parameters caused by different molecular wieghts of PEG.

The molar volumes and the surface tensions of melts in the systems CaF2-MF (M: Li, Na) and CaF2-MF2 (M: Mg, Sr, Ba).

Surface tension and density of molten CaF2-based binary system were measured. From the measurement, the following results were obtained.(1) Molar volumes of the melts in the systems CaF2-NaF, and CaF2-BaF2 are deviated positively from the additive rule, while for CaF2-LiF and CaF2-MgF2 melts negative deviation from the additive rule on volume is found, but CaF2-SrF2 melt behaves ideally on volume.(2) Surface tensions of the melts in the binary systems can be expressed by Guggenheim's equation for regular solution, if a suitable interaction energy ω of each binary system is selected.(3) The interaction energy ω, estimated from surface tension of binary fluoride systems, is related to the excess volume on mixing for CaF2-MxF2 equi-molar composition (x=1 for divalent and x=2 for monovalent fluoride).(4) Liquidus lines for the systems, calculated from the interaction energy ω in combination with the thermochemical data are fairly in good agreement with those reported, but that for CaF2-NaF system is exceptional.(5) Surface tensions of binary fluoride melts are strongly depended on packing density of fluorine ions on the surface.

Deiters' equation of state and critical phenomena

A computer program has been developed to calculate the critical properties of binary mixtures from Deiters' equation of state. The critical properties of over 50 binary mixtures of permanent gases, alkanes, benzene and alkanes, cyclohexane and alkanes, and acetone and hydrocarbons have been calculated. The Guggenheim equation has also been used to enable comparison with a simple “hard sphere + attractive term” equation. Deiters′ equation gives good results for mixtures of moderately different size molecules whereas Guggenheim's equation is superior for dissimilar size molecules. The values of an interaction paramter, ξ, obtained from our analysis are also reported.