Guggenheim Equation Research Articles

The Guggenheim equation for a system of hard spheres and its generalization

In the present work, for the first time, the generalized Guggenheim equation for a system of hard spheres is obtained based on the use of the Euler accelerated convergence method. This equation allows taking into account an arbitrary number of known virial coefficients. For a metastable region, this equation is generalized to the case of taking into account the asymptotic behavior of the free energy at high densities. The resulting expression for the compressibility of the homogeneous phase of a system of hard spheres describes the data of a numerical experiment within their accuracy.

Guggenheim Equation for a System of Hard Spheres and Its Generalization

Guggenheim Equation for a System of Hard Spheres and Its Generalization

Effect of temperature and alkyl chain length on the physicochemical properties of pyridinium bis(trifluoromethylsulfonyl)imide-based ionic liquids

Densities, refractive indices, viscosities, surface tensions and speeds of sounds were measured and discussed for [Cnpy][[NTf2] (n=2,4,6) and [Cnmpy][[NTf2] (n=2,3,4) as a function of temperature and atmospheric pressure. As expected, all properties decrease with increasing temperature, with viscosity being the property most affected by temperature, while surface tension and refractive index being the properties least affected by temperature. Linear correlation was used to calculate the density, speed of sound, surface tension and refractive index as a function of temperature. The viscosity is correlated using Vogel-Flutcher-Tamman (VFT) equation. From the experimental density data, the coefficients of thermal expansion and the isentropic compressibility coefficients were calculated. The Guggenheim equation was used to calculate the critical temperatures. Moreover, the effect of the alkyl chain length of the investigated ionic liquids on the measured thermophysical properties is discussed.

Critical Properties of n-Alkanes Binary Mixtures Based on Equations of State

The aim of this work is the parameters test of the equation of state for hard convex body chain in predicting the critical properties of binary mixtures of n-alkanes. The predicted critical properties agree closely with the experimental results and are better than the predicted values both from the hard convex body equation of state and the Guggenheim equation of state for hard spheres. The agreement between theoretical and experimental results of the critical properties of n-alkanes appears to be improved negligibly, even if the shape of the molecule is considered. However, an improvement in the accuracy of the result is obtained with the equation of state of the hard convex body chain compared to the simple hard convex body equation of state.

The effects of temperature, alkyl chain length, and anion type on thermophysical properties of the imidazolium based amino acid ionic liquids

The viscosity, surface tension, and density as well as molar volume and expansion coefficient of imidazolium based amino acid ionic liquids (AAILs) including alaninate ([ALA]−), glycinate ([GLY]−), glutamate ([GLU]−), serinate ([SER]−), and valinate ([VAL]−) anions were investigated at the extended temperature until 373.15K. Thermophysical as well as electronic properties were taken into account in order to explore the effect of alkyl chain length and anion on these properties. The results indicated that the interaction between ion pairs has a great effect on the properties; the strong interaction caused the higher viscosity and surface tension. The effect of temperature and alkyl chain length as well as anion type on the physicochemical properties was also studied. Raising the temperature resulted in a downward trend on the density, viscosity, and surface tension of all systems. In addition, the hypothetical critical temperatures of the AAILs were estimated making use of the Eötvos and Guggenheim equations. It was shown that the critical temperature decreases with alkyl chain length and its change depends on the ion pair interaction energy density and density of electronic chemical potential. The results of prediction critical temperatures of AAILs using surface tension data revealed that [EMIM][GLY] had the highest critical temperature due to its strongest ion pair interaction and stability and lower alkyl chain length in cation. Moreover, the anion charge density shows a straight correlation with interaction energy density that is in according with the stability of the ion pairs. Investigating viscosity as a transport property demonstrates that the lowest viscosity is observed for AAILs involving [GLY]− anion that is described by high flexibility of the anion. Finally, the change in volume of AAILs is small as the temperature increases confirming AAILs as solvent do not expand much which explains their good solvent characteristics.

On the prediction of critical temperatures of ionic liquids: Model development and evaluation

In this study, the Guggenheim equation was used to estimate the critical temperature (Tc) of 106 ionic liquids using experimental surface tension data as inputs. A group contribution (GC) and a Quantitative Structure-Property Relationship (QSPR) model were also developed to correlate/predict the Tc of ionic liquids. It was shown that a lack of sufficiently large database for Tc leads to the development of models with low prediction capability. The models output as well as the Tc values estimated from the surface tension data were compared with the critical temperatures calculated by the Valderrama et al. method. The results show that the Valderrama et al. method produces different data values compared with the values calculated by the Guggenheim equation. Neither the GC, QSPR, nor Valderrama's method is capable of reliable prediction of the Tc of ionic liquids. Consequently, the calculated Tc values are not reliable enough to be used in the development of corresponding state models for prediction of other thermophysical properties of ionic liquids. Any usage of critical properties of ionic liquids should be done with a serious caution.

Kinetic model of water vapour adsorption by gluten-free starch

Abstract This study evaluated the kinetics of water vapour adsorption on the surface of starch molecules derived from wheat. The aim of the study was to determine an equation that would allow estimation of water content in tested material in any timepoint of the adsorption process aimed at settling a balance with the environment. An adsorption isotherm of water vapour on starch granules was drawn. The parameters of the Guggenheim, Anderson, and De Boer equation were determined by characterizing the tested product and adsorption process. The equation of kinetics of water vapour adsorption on the surface of starch was determined based on the Guggenheim, Anderson, and De Boer model describing the state of equilibrium and on the model of a first-order linear inert element describing the changes in water content over time.

Applicability of different equations in modeling the thermodynamic and transport properties of aqueous and non-aqueous ionic liquid solutions

Applicability of different electrolyte based models has been tested in correlating the osmotic coefficient values of aqueous and nonaqueous binary ionic liquid solutions considering both the complete and partial dissociation of the ionic liquids. In local composition models the sum of a long-range and a short-range term is used. When the partial dissociation of ionic liquid is considered, the Pitzer–Debye–Hückel and Fowler–Guggenheim equations were used to calculate the long-range electrostatic interaction. In the Fowler–Guggenheim equation, for calculating Debye inverse length the necessary degree of association has been determined from osmotic coefficient data by Chemical Model (CM1). The correlation equations for excess molar volume of the NRTL–NRF, mNRTL, NRF–Wilson and NRF–mNRTL models for ionic liquid or electrolyte solutions were also derived. The performance of these equations was compared with the other available models for correlating the excess molar volume of ionic liquid solutions. Taking into account of complete or partial dissociation of ionic liquids, fitting quality of solution viscosities of various ionic liquids were examined using the Eyring’s absolute rate theory and different electrolyte local composition models and Clegg–Pitzer equations.

A Potentiometric Study of the Association of Copper(II) and Sulfate Ions in Aqueous Solution at 25 °C

A study of the association between copper(II) and sulfate ions in aqueous solution has been made using copper ion-selective electrode potentiometry at constant ionic strengths (I) of 0.05, 0.1, 0.25, 0.5, 1.0, 3.0 and 5.0 mol·L−1 in NaClO4 media at 25 °C. Only one complex was detected, corresponding to the equilibrium: \( {\text{Cu}}^{ 2+ } ({\text{aq}}) + {\text{SO}}_{4}^{2 - } ({\text{aq}}) \rightleftarrows {\text{CuSO}}_{4}^{0} ({\text{aq}}). \) No higher order complexes were detected even at sulfate/copper(II) concentration ratios of up to 1,000. The present potentiometric values of log10K1(I) are shown to be consistently higher than those obtained by UV–Vis spectrophotometry because of the failure of the latter technique to detect all of the solvent-separated ion pairs present. Extrapolation of log10K1(I) to infinite dilution using an extended Guggenheim equation yielded a standard state value of log10\( K_{1} \{ {\text{CuSO}}_{4}^{0} ({\text{aq}})\} = 2.32 \pm 0.09 \), which is in excellent agreement with a recent IUPAC-recommended value.

Application of the Eötvos and Guggenheim empirical rules for predicting the density and surface tension of ionic liquids analogues

The recent continuing interest in deep eutectic solvents (DES) as ionic liquids analogues and their successful applications in different areas of separation necessities the existence of reliable physical and thermodynamic properties database. The scarcity of data on the physical properties of such solvents, increases the need for their prediction using reliable methods. In this study, first the critical temperatures of eight DES systems have been calculated based on the Eötvos empirical equation using the experimental data of the density and surface tension at various temperatures, then the density and surface tension values of these systems were predicted from the calculated critical temperatures. For the density prediction the Eötvos and Guggenheim equations were combined to introduce a simple power law equation using the estimated critical temperatures from the Eötvos and the Modified Lydersen–Joback–Reid group contribution methods. Finally, the estimated critical temperatures by these two methods were used in the Guggenheim empirical equation to calculate the surface tension of the DES systems. The prediction quality of the two physical properties under investigation were compared and proper recommendations were postulated.

Kinetics of thermal dehydration of sol-gel derived MgO–ZrO2 composite hydrogel

Abstract The kinetics of thermal dehydration of mixed hydroxide hydrogels in MgO–ZrO2 system was studied as a function of composition by following the isothermal heat treatment route. Kinetic parameters were calculated through the application of the Guggenheim equation. The expulsion of both loosely bound water and constitutional OH groups were not continuous processes but proceeded in steps. The applicability of 1st order reaction kinetics for the major portion of the reaction for all compositions suggests that during the dehydroxylation process the dehydration is essentially controlled by the orientation of H2O molecules and the mutual interaction of hydroxyl groups. Due to a decrease in the concentration of the reacting species the activation energy was always higher at the final stage of dehydration.

Surface tension and refractive index of pure and water-saturated tetradecyltrihexylphosphonium-based ionic liquids

Experimental data on the surface tension and refractive index of tetradecyltrihexylphosphonium-based ionic liquids with bromide, chloride, decanoate, methanesulfonate, dicyanimide, bis(2,4,4-trimethylpentyl)phosphinate and bis(trifluoromethylsulfonyl)imide anions are reported. The data were obtained for pure and water saturated samples at temperatures from 283K to 353K and at atmospheric pressure. The refractive index of the investigated ionic liquids decreases with increasing the water content in the sample. On the other hand, no clearly dependence of the surface tension with the water content up to a weight fraction of 16% was found. The prediction of the refractive index for the studied ionic liquids was also accomplished by a group contribution method and new values for the cation and diverse anions were estimated and proposed. The studied ionic liquids show lower surface tension in comparison with imidazolium-, pyridinium- or pyrrolidinium-based ionic liquids with a similar anion; also they show higher surface entropy than cyclic nitrogen-based fluids which indicates a lower surface organization. The anion dependence of the surface tension and surface entropy for the investigated ionic liquids is weaker than that for short-chain imidazolium-based ionic liquids. Their critical temperatures evaluated from Eötvos and Guggenheim equations are also lower than those of N-heterocyclic ionic fluids.

The unusual importance of activity coefficients for micelle solutions illustrated by an osmometry study of aqueous sodium decanoate and aqueous sodium decanoate + sodium chloride solutions

Freezing-point and vapor-pressure osmometry data are reported for aqueous sodium decanoate (NaD) solutions and aqueous NaD + NaCl solutions. The derived osmotic coefficients are analyzed with a mass-action model based on the micelle formation reaction qNa(+) + nD(-) = (Na(q)D(n))(q-n) and Guggenheim equations for the micelle and ionic activity coefficients. Stoichiometric activity coefficients of the NaD and NaCl components and the equilibrium constant for micelle formation are evaluated. Illustrating the remarkable but not widely appreciated nonideal behavior of ionic surfactant solutions, the micelle activity coefficient drops to astonishingly low values, below 10(-7) (relative to unity for ideal solutions). The activity coefficients of the Na(+) and D(-) ions, raised to large powers of q and n, reduce calculated extents of micelle formation by up to 15 orders of magnitude. Activity coefficients, frequently omitted from the Gibbs equation, are found to increase the calculated surface excess concentration of NaD by up to an order of magnitude. Inflection points in the extent of micelle formation, used to calculate critical micelle concentration (cmc) lowering caused by added salt, provide unexpected thermodynamic evidence for the elusive second cmc.

Test of athermal terms of activity coefficient models by Monte Carlo simulation with hard-core models

The infinite dilution activity coefficients of exactly athermal fluids were calculated by Monte Carlo simulation with hard-core models. The hard-core models used in this work were hard-sphere and hard-spherocylinder models. The Widom test particle method was adopted to calculate the residual chemical potentials of solutes in pure solvent and in pure solute solutions. The infinite dilution activity coefficients of solutes were obtained from the residual chemical potentials of solutes. The infinite dilution activity coefficients calculated by Monte Carlo simulation were compared with those of athermal terms in activity coefficient equations. Staverman–Guggenheim equation overestimates the activity coefficients. The deviations of activity coefficients increase with increasing the hard-core volume of solute. Flory–Huggins equation based on molar volume gives good results for the hard-spherocylinder systems. Elbro-FV equation gives good results for both the hard-sphere and hard-spherocylinder systems.

Adsorption and Competitive Adsorption on Zeolites of Nitrophenol Compounds Present in Wastewater

This work investigates the removal of various nitrophenolic compounds (ortho-nitrophenol (ONP), para-nitrophenol (PNP), meta-nitrophenol (MNP), and 2,4-dinitrophenol (2,4-DNP)) from aqueous solution using hydrophobic FAU zeolites. The adsorption equilibrium of nitrophenols from aqueous solutions by FAU can be well-described using the Fowler−Guggenheim equation. The relative affinity of nitrophenols toward the FAU is dependent on the pH solution and on the pollutant solubility in water. Their sorption capacity is in the following order: ONP > 2.4-DNP > PNP > MNP. In binary mixtures, the most important parameter that governs the adsorption in zeolites seems to be the solubility of pollutants in water. Thus, the less-soluble compound (in this case, ONP) was adsorbed more easily than the other components present in the binary mixture. Finally, hydrophobic FAU zeolite seems to be an efficient adsorbent; it is able to be easily regenerated under air or by solvent leaching, through retention of these initial ads...

Density and Surface Tension Measurements of Imidazolium-, Quaternary Phosphonium-, and Ammonium-Based Room-Temperature Ionic Liquids: Data and Correlations

Presented are experimental densities and surface tensions of imidazolium-, quaternary ammonium-, and phosphonium-based room-temperature ionic liquids (RTILs) in the temperature range of (298 to 350) K. Densities of the RTILs decrease slightly with temperature in the studied range (298 to 333) K. At 298 K, the densities of the phosphonium-based RTILs ranged from (0.88 to 1.05) g·cm-3, while those of the ammonium-based RTILs ranged from (1.08 to 1.37) g·cm-3. The volume expansivities of phosphonium and ammonium RTILs at 1 atm and 298 K are in the range of (5.5 to 6.5)·10-4 K-1. Surface tension and temperature relationships were established using the van der Waals−Guggenheim equation, σ = Es(1 − T/Tc)n, where n ≈ 1, for the RTILs studied. The Macleod−Sugden−Wright (MSW) equation was used to correlate surface tension with the respective molar volume of the various RTILs. The developed equation can accurately correlate surface tension for this grouping of RTILs with a maximum estimated error of 0.15 % within the temperature ranges considered.

Local Structure and Xenon Adsorption Behavior of Metal−Organic Framework System [M2(O2CPh)4(pyz)]n (M = Rh and Cu) As Studied with Use of Single-Crystal X-ray Diffraction, Adsorption Isotherm, and Xenon-129 NMR

The local structure and xenon adsorption behavior of a metal−organic framework system [M(II)2(bza)4(pyz)]n (bza and pyz = benzoate and pyrazine, M = Rh (1a) and Cu (1b)) were investigated by using single-crystal X-ray diffraction, xenon adsorption isotherm, and 129Xe NMR measurements. Single-crystal X-ray diffraction analysis revealed that rare gas atoms were accommodated in a one-dimensional (1D) nanochannel with the dimer structure. Xenon adsorption reached saturation at the xenon uptake of 1.93 Xe per molecular unit in 1a and of 1.85 Xe in 1b, suggesting accommodation of two Xe atoms per host formula unit. Analysis of the xenon adsorption isotherm based on the Fowler−Guggenheim equation gave the xenon−xenon interaction and the isosteric heat of adsorption. The 129Xe NMR spectrum suggested that the environment of the adsorption site for xenon in 1a is very tight and anisotropic. Furthermore, the temperature dependence of the 129Xe chemical shift was explained by using xenon loading, supporting the xenon dimer as the local structure of xenon in 1a. These aspects revealed that cooperative adsorption of xenon to 1a and 1b occurs according to the xenon−xenon interaction. Xenon is stabilized in the nanochannel through formation of a dimer structure.

A comparative study of non-polar solvents effect on dielectric relaxation and dipole moment of binary mixtures of mono alkyl ethers of ethylene glycol and of diethylene glycol with ethyl alcohol

Dielectric relaxation and dipole moment of binary mixtures of homologous series of mono alkyl ethers of ethylene glycol and of diethylene glycol, i.e., mono methyl, mono ethyl and mono butyl ethers of ethylene glycol (ROCH 2CH 2OH) and mono methyl, mono ethyl and mono butyl ethers of diethylene glycol (ROCH 2CH 2OCH 2CH 2OH) with ethyl alcohol (C 2H 5OH) of different concentrations were studied in dilute solutions of benzene, dioxane and carbon tetrachloride at 35 °C. Permittivity ( ε′) and loss ( ε″) at 10.1 GHz, static dielectric constant ε o at 1 MHz and high frequency limiting dielectric constant ε ∞ = n D 2 at optical frequency of these molecules and their binary mixtures at different concentration were measured in dilute solutions of non-polar solvents. The average relaxation time τ o, relaxation times corresponding to overall molecular reorientation τ 1 and group rotations τ 2 were determined using Higasi's single frequency measurement equations for dilute solutions. The evaluated values of relaxation times and free energy of activation Δ F were used to explore the solvent effect on molecular dynamics of these polar binary systems in non-polar solvents. The excess inverse relaxation time and excess free energy of activation were determined to confirm the existence of hydrogen-bonded heterogeneous cooperative domains of the ethers and alcohol molecules at different concentration their binary mixtures in non-polar solvents. The dipole moment of the binary mixtures was evaluated using Higasi's and Guggenheim's equation for dilute solutions. The evaluated values of dipole moments and computed dipole moment values using a simple mixing equation of the polar molecules binary mixture were used to explore the effect of non-polar solvent environment on heterogeneous molecular interactions between ethers and alcohol molecules. The effect of number of carbon atoms in the molecular structure of these homologous series molecules was also considered for the interpretation of various evaluated dielectric parameters.

Activity coefficients of dipotassium phthalate and potassium hydrogen phthalate in water at 298.15 K

Pitzer model parameters and modified Guggenheim equation parameters for dipotassium phthalate (K 2PH) and potassium hydrogen phthalate (KHPh) were determined from cell potential difference (cpd) measurements on galvanic cells without liquid junction. Mean activity coefficients of these salts in water at 298.15 K were calculated by the Pitzer model and modified Guggenheim equations. The Pitzer parameters values obtained for K 2Ph where: β K 2 Ph ( 0 ) + θ Cl,Ph S = 0.25 ± 0.01 mol −1 kg; β K 2 Ph ( 1 ) = 0.86 ± 0.08 mol −1 kg; C K 2 Ph = −0.010 ± 0.002 mol −2 kg 2 and Ψ K,Cl,Ph = 0.014 ± 0.005 mol −2 kg 2. The Pitzer parameters values obtained for KHPh were: β KHPh ( 0 ) + θ Cl,HPh = 0.084 ± 0.013 mol −1 kg, β KHPh ( 1 ) = 0 mol −1 kg. The results were also compared to the values obtained from Pitzer parameters published in literature. The parameter values for the modified Guggenheim equations ( b K,Ph = −0.041 mol −1 kg and b K,HPh = −0.020 mol −1 kg) obtained in this study were used in the mean activity coefficients calculations. The modified Guggenheim equation allows us to obtain similar mean activity coefficient values, as the Pitzer equation, up to 1.5 mol kg −1 ionic strength for K 2Ph and 0.5 mol kg −1 for KHPh.

The new experimental data and a new thermodynamic model based on group contribution for correlation liquid–liquid equilibria in aqueous two-phase systems of PEG and (K 2HPO 4 or Na 2SO 4)

The new experimental data of liquid–liquid equilibria for aqueous two-phase systems PEG–K 2HPO 4–water and PEG–Na 2SO 4–water are presented. The effects of pH and molecular weight of polyethylene glycol were investigated and the tie lines with binodal curves for both systems are shown. A new thermodynamic model based on group contribution has been proposed for studying the phase behavior of aqueous two-phase polymer–salt systems. The assumptions of NRTL-NRF model and the activity coefficient equation of UNIQUAC-NRF model have been used for the groups. In this new model, UNIFAC-NRF, the nonrandom state of groups were selected as a reference state. The binary interaction parameters were adjusted using the data of binary salt–water systems and the ternary systems were correlated with only six binary adjustable parameters. The Debye–Huckel equation based on Fowller–Guggenheim equation was used to calculate the long range electrostatic interaction of the ions. The UNIFAC-NRF model was applied to correlate the experimental data of aqueous two-phase systems: PEG–K 2HPO 4–water and PEG–Na 2SO 4–water for two different molecular weight of PEG at different pH. The results of the new model showed that it can be used to correlate the LLE in aqueous solution of polymer–salt very well.