Thermodynamic Properties Of Solutions Research Articles

Experimental measurement – correlation of solubility and dissolution thermodynamics study of itraconazole in pure monosolvents at various temperatures

The information about the solubility and thermodynamic properties of solution is important for pharmaceutically important processes, formulation development, and further theoretical studies. In the present study, the solid–liquid equilibrium (solubility) for itraconazole (ITC) was determined experimentally in 14 monosolvents at temperatures between 293.15 K and 318.15 K under pressure of 0.1 MPa. The mole fraction solubilities were found to increase with increasing temperatures and followed inverse trend with the polarity of selected solvents. Besides, KAT-LSER analysis was performed to study the effect of solvent. The results revealed that the solute–solvent interaction (43.94%) was much higher than that of solvent–solvent interaction (16.59%). Thermodynamic based models like van’t Hoff equation, modified Apelblat equation, Buchowski–Ksiazaczak equation, and polynomial empirical equation were applied to fit and correlate the experimental solubilities. Overall relative average deviation and overall root-mean square deviation () were observed to be minimum with the empirical polynomial equation and attained the values of 0.0033 and 0.0047, respectively. Furthermore, theoretical ideal solubilities, activity coefficients, and thermodynamic properties of dissolution including molar enthalpy, molar entropy, molar Gibbs free energy, and excess enthalpy were estimated. Ideal solubilities were projected considerably higher than experimental solubilities at each studied temperature. Thermodynamic properties of dissolution indicated that the dissolution was not a spontaneous process; observed to be endothermic ( and enthalpy driven Such solid–liquid equilibrium data of ITC will be of immense help in process and formulation development in pharmaceutical sciences.

Ammonium based deep eutectic solvents (DESs) on extraction of benzothiophene from iso-octane: experiment and COSMO-RS model

Densities of pure benzothiophene (BT), iso-octane (ISO), and tetra butyl ammonium bromide decanoic acid {[TBAB][DA]} as deep eutectic solvent and their binary mixtures of [BT] + [ISO], [BT] + {[TBAB][DA]} and {[TBAB][DA]} + [ISO] over the entire mole fractions were measured at different temperatures 303.15 K–353.15 K at 1 atm. From this measured density data, the solution thermodynamic properties such as excess molar volume, partial molar volume, excess partial molar volumes, apparent molar volume, isobaric expansivity, and excess isobaric expansivity were calculated. Further, this calculated excess molar volume data were correlated by Redlich-Kister polynomial fit equation. Subsequently, liquid–liquid extraction (LLE) experiment were conducted for the removal of BT from ISO using {[TBAB + DA]} at 298.15 K and 1 atm. This LLE process was characterized in terms of distribution coefficient and selectivity and thermodynamic consistency test were done by Othmer-Tobias and Hand Equations. Finally, the sigma (σ) profile and sigma (σ) potential of all the studied species were generated and analyzed using COSMO-RS model.

Structure, Molecular Interactions, and Dynamics of Aqueous [BMIM][BF4] Mixtures: A Molecular Dynamics Study.

Molecular dynamics simulations with many-body polarizable force fields were carried out to investigate the thermodynamic, structural, and dynamic properties of aqueous solutions of 1-butyl-3-methylimidazolium tetrafluoroborate ([bmim][BF4]). The radial distribution functions exhibit well-defined features, revealing favored structural correlations between [bmim]+, [BF4]-, and H2O. The addition of water is shown to alter ionic liquid structural organizations by replacing counterions in the coordination shells and disrupt the cation-anion network. At low water concentration, the majority of water molecules are isolated from each other and have lower average dipole moment than that in pure water. With increasing hydration level, while [bmim][BF4] ionic network breaks up and becomes isolated ion pairs or free ions in the dilute limit, water begins to form clusters of increasing sizes and eventually forms a percolating network. As a result, the average water dipole moment increases and approaches its bulk value. Water is also observed to have a substantial influence on the dynamics of ionic liquids. At low water content, the cation and anion have similar diffusion coefficients due to the correlated ionic motion of long-lived ion pairs. As the water concentration increases, both ions exhibit greater mobility and faster rotations from the breakup of ionic network. Consequently, the ionic conductivity of [bmim][BF4] aqueous solutions rises with increasing water composition.

Solubility of sulphadiazine in some {Carbitol® (1) + water (2)} mixtures: determination, correlation, and preferential solvation

ABSTRACT The mole fraction solubility of sulphadiazine (SD) in {Carbitol® (1) + water (2)} binary solvent mixtures at different temperatures from (278.15 to 313.15) K was determined. Solubility values were correlated with the combined nearly ideal binary solvent/Redlich-Kister, modified Wilson and Jouyban-Acree models. Preferential solvation parameters of SD by Carbitol® (δx 1,3) were derived from their thermodynamic solution properties using the inverse Kirkwood-Buff integrals method. δx 1,3 values are negative in water-rich mixtures but positive in Carbitol® mole fractions higher than 0.12. In the former case is conjecturable that hydrophobic hydration around non-polar groups of SD plays a relevant role in drug solvation.

Thermal prehistory, structure and high-temperature thermodynamic properties of Y2O3-CeO2 and Y2O3-ZrO2-CeO2 solid solutions

Ceria-based solid solutions are important materials for high- and medium-temperature electrochemical applications. However, the stabilities of both binary and ternary ceria-based solid solutions are insufficient at elevated temperatures, which limits their application as solid electrolytes or SOFC cathodes. Data on the high-temperature stability of ceria-based ceramics are unavailable in the literature. In the present study, we report a thermodynamic stability investigation of Y2O3-CeO2 and Y2O3-ZrO2-CeO2 solid solutions. The thermal prehistories of binary and ternary systems were investigated using STA, XRD, and ESCA techniques. The vaporization processes were investigated in the temperature range of 1577–2227°С via the Knudsen effusion mass spectrometry technique. Using data on the component activity in solid-phase thermodynamic properties of Y2O3-CeO2 solid solutions, which is represented as the Gibbs energy, the excess Gibbs energy was calculated as a function of the ceria mol. %. It was shown that the reduction of Ce4+ to Ce3+ in Y2O3-CeO2 and Y2O3-ZrO2-CeO2 solid solutions corresponds to less-negative Gibbs energy compared to ZrO2-CeO2 solid solutions.

Black hole solutions in modified gravity induced by quantum metric fluctuations

The inclusion of the quantum fluctuations of the metric in the geometric action is a promising avenue for the understanding of the quantum properties of gravity. In this approach the metric is decomposed in the sum of a classical and of a fluctuating part, of quantum origin, which can be generally expressed in terms of an arbitrary second order tensor, constructed from the metric, and from the thermodynamic quantities describing matter fields. In the present paper we investigate the effects of the quantum fluctuations on the spherically symmetric static black hole solutions of the modified field equations, obtained from a variational principle, by assuming that the quantum correction tensor is given by the coupling of a scalar field to the metric tensor. After reformulating the field equations in a dimensionless form, and by introducing a suitable independent radial coordinate, we obtain their solutions numerically. We detect the formation of a black hole from the presence of a singularity in the metric tensor components. Several models, corresponding to different functional forms of the scalar field potential, are considered. The thermodynamic properties of the black hole solutions (horizon temperature, specific heat, entropy and evaporation time due to Hawking luminosity) are also investigated.

Solubility determination and solution thermodynamics of olmesartan medoxomil in (PEG-400 + water) cosolvent mixtures

The solubility and solution thermodynamic properties of a weakly water-soluble compound olmesartan medoxomil (OLM) in binary ‘polyethylene glycol (PEG-400) + water’ cosolvent compositions were determined. The ‘mole fraction solubility (x e)’ of OLM in binary ‘PEG-400 + water’ cosolvent compositions and pure solvents (PEG-400 and water) was determined at ‘T = 295.15-330.15 K’ and ‘p = 0.1 MPa’. The Hansen solubility parameter (HSP) of OLM, pure PEG-400, pure water, and binary ‘PEG-400 + water’ cosolvent compositions free of OLM were also predicted. The obtained x e values of OLM were correlated using ‘van’t Hoff, modified Apelblat, Yalkowsky-Roseman, Jouyban-Acree and Jouyban-Acree-van’t Hoff’ computational models with the error values of <4.0%. The maximum and minimum x e value of OLM was predicted in neat PEG-400 (1.15 × 10−2 at T = 330.15 K) and neat water (1.90 × 10−7 at T = 295.15 K), respectively. The OLM HSP was predicted to be more close with that of neat PEG-400. The x e value of OLM was found increased significantly with increase in temperature and PEG-400 mass fraction in all ‘PEG-400 + water’ cosolvent compositions including neat PEG-400 and neat water. An ‘apparent thermodynamic analysis’ studies presented an ‘endothermic and entropy-driven dissolution’ of OLM in all ‘PEG-400 + water’ cosolvent compositions including pure PEG-400 and pure water.

Correlation and prediction of thermodynamic properties of dilute solutes in water up to high T and P. III. Polar and aromatic solutes H2S, C6H6, C7H8, CH3SH, SO2, HCN, acetone, NH3, CH3NH2, CH3OH, C2H5OH, phenol

Fugacity coefficients of twelve polar or aromatic species at infinite dilution in water (H2S, benzene C6H6, toluene C7H8, methanethiol CH3SH, SO2, HCN, acetone (CH3)2CO, NH3, methylamine CH3NH2, methanol CH3OH, ethanol C2H5OH and phenol C6H5OH) have been evaluated from ambient conditions (298 K, 0.1 MPa) to high temperatures (up to 2000 K) and water densities (up to 1500 kg m–3). The approach is based on a correlation developed for the function, A12∞=V2∞/κTRT, and includes known constraints: second virial coefficients at low water densities, hard sphere behavior at high water densities, near–critical principles, and corresponding–states relations. A comparison with literature data at subcritical (Henry's and vapor–liquid distribution constants) and supercritical temperatures (fugacity coefficients) is made.

Temperature dependent Broadband dielectric relaxation study of Aqueous Polyvinylpyrrolidone (PVP K-15, K-30 & K-90) using a TDR

ABSTRACT Temperature-dependent dielectric relaxation studies of different viscosity grade Polyvinylpyrrolidones (PVP), PVP K-15, PVP K-30 and PVP K-90 having molecular weight 10,000 gmol−1, 40,000 gmol−1 and 360,000 gmol−1 respectively has been done in the frequency region 10 MHz to 50 GHz using Time domain reflectometry (TDR) technique. The frequency-dependent complex dielectric permittivity spectra have been explored by the Havriliak-Negami expression and is explained by the single relaxation model. The PVP molar concentration comprises the concentration domain of 0.0017 ≤ c/M ≤ 0.0176 for PVP-water system. The values of static dielectric constant (ε0), relaxation time (τ) and the empirical parameter (β) have been determined using non-linear least squares fit method along with the thermodynamic properties (activation enthalpy, entropy and free energy) of aqueous solutions of polymers.

Phase Change Behavior of NaCl-H2O Binary Solution Under the Control of AC Electric Field

Cryopreservation, which refers to preservation of cells or tissues at subzero temperatures, inevitably involves the problem of cryoinjury caused by ice crystals. The application of an external electric field during the freezing process has been shown to be a promising approach to produce miniature ice grains and decrease the fraction of ice crystallization at a slow cooling rate. Thus, the dielectric and thermodynamic properties of NaCl-H2O binary solutions at subzero temperatures were tremendously important for understanding the mechanism of ice formation under the manipulation of an AC electric field in biopreservation. However, there was still a lack of relevant information in the literature. The first objective of this study was to systematically measure the dielectric spectrum of 0.9% NaCl-H2O binary solutions at temperatures ranging from -100°C to 0°C with a cooling/heating rate of 2°C/min. We further measured the thermodynamic properties of a 0.9% NaCl-H2O binary solution while applying a series of electric fields near its dielectric relaxation frequency. The effect of the electric field on the crystal morphology was studied last. Pure water was selected as the control group. The results showed that an AC electric field can alter the thermodynamic process and thus the phase transition and ice crystal structure could be manipulated. It was concluded that the AC electric-assistant preservation method will be a promising technology in cryopreservation.

Thermodynamic first order interaction coefficient between nitrogen and manganese in liquid steel

A simple theory of thermodynamic properties of liquid nitrogen solution in Fe–Mn alloys is proposed. This theory is completely analogous to the theory for liquid nitrogen solution in alloys of the Fe–Cr system proposed previously by the authors in 2019. The theory is based on the lattice model of the considered Fe–Mn solutions. The model assumes a FCC lattice. The atoms of Fe and Mn are in these lattice sites. Nitrogen atoms are located in octahedral interstices. The nitrogen atom interacts only with the metal atoms located in the lattice sites neighboring to it. This interaction is pairwise and it is assumed that the energy of this interaction depends neither on the composition nor on the temperature. Furthermore, the solution in the Fe–Mn system is assumed to be perfect. Within the framework of the proposed theory, a relation was obtained that expresses the value of the Sieverts law constant for N solubility in liquid Mn through the similar constant for the N solubility in liquid Fe and the Wagner N–Mn interaction coefficient in liquid Fe. The values of the Sieverts law constants in this relation are taken directly from the experimental measurements of the solubility of N in liquid Fe and in liquid Mn. In this case, the obtained relation is considered as an equation with respect to the Wagner interaction coefficient $$\varepsilon _{{\text{N}}}^{{{\text{Mn}}}}$$ . The equation’s solution gives the value of Wagner interaction coefficient $$\varepsilon _{{\text{N}}}^{{{\text{Mn}}}}$$ = –5.25 in liquid steel at a temperature of 1873 K. Wagner interaction coefficient $$\varepsilon _{{\text{N}}}^{{{\text{Mn}}}}$$ is related with Langenberg interaction coefficient $$e_{{\text{N}}}^{{{\text{Mn}}}}$$ by the relation deduced by Lupis and Elliott in 1965. The relation includes the atomic masses of Fe and Mn. By substituting the value $$\varepsilon _{{\text{N}}}^{{{\text{Mn}}}}$$ = –5.25 and solving the resulting equation with respect to $$e_{{\text{N}}}^{{{\text{Mn}}}}$$ , we obtain the value $$e_{{\text{N}}}^{{{\text{Mn}}}}$$ = –0.0230. This value corresponds to the experimental data of Beer (1961), which is likely one of the most probable of all experimental values of $$e_{{\text{N}}}^{{{\text{Mn}}}}$$ for liquid steel at 1873 K. Another such value is $$e_{{\text{N}}}^{{{\text{Mn}}}}$$ = 0.0209 obtained by Shin with coworkers in 2011.

Modified embedded-atom method interatomic potentials for Mg–Al–Ca and Mg–Al–Zn ternary systems

Al, Ca, and Zn are representative commercial alloying elements for Mg alloys. To investigate the effects of these elements on the deformation and recrystallization behaviors of Mg alloys, we develop interatomic potentials for the Al–Ca, Al–Zn, Mg–Al–Ca and Mg–Al–Zn systems based on the second nearest-neighbor modified embedded-atom method formalism. The developed potentials describe structural, elastic, and thermodynamic properties of compounds and solutions of associated alloy systems in reasonable agreement with experimental data and higher-level calculations. The applicability of these potentials to the present investigation is confirmed by calculating the generalized stacking fault energy for various slip systems and the segregation energy on twin boundaries of the Mg–Al–Ca and Mg–Al–Zn alloys, accompanied with the thermal expansion coefficient and crystal structure maintenance of stable compounds in those alloys.

Experimental and in-silico studies of a symmetric biomimetic tripod derived from an oxygen-bridged trishydroxyquinolinecyclohexane and its iron and chromium adducts

In this study, complexation behavior, photophysical and solution thermodynamic properties of tripodal chelates of 5,5′,5″-(Cyclohexane-1,3,5-triyltris(oxy)tris(methylene)triquinolin-8-ol (CYTOM5OX) with trivalent Fe and Cr ions are described. The ligand forms protonated complexes [M(H3L)]3+ in lower pH, while at higher pH the neutral complexes result with the overall binding constants (Log β110) 30.18 and 22.02 and pM values of 31.95 and 24.37 for Fe+3 and Cr+3, respectively. The ligand displays high sensitive fluorescence quenching via internal charge transfer (ICT) upon complexation. In-silico studies at DFT level with B3LYP functional and 6-31G* basis set validate the geometry, infrared, absorption, emission spectra and suggest the bonding pattern through NBO and FMOanalysis. The photophysical properties obtained from TD–DFT results showed an excellent correlation with the experimental data.

Kojic acid in fourteen mono-solvents: Solubility data, Hansen solubility parameter and thermodynamic properties

In this research, solubility of kojic acid in fourteen mono-solvents was obtained by laser monitoring approach at various temperatures under 0.1 MPa. The mole solubility of kojic acid exhibited a positive relation with temperature. Moreover, the solubility sequence of kojic acid at 298.15 K is: dimethyl sulfoxide (DMSO) > N-methylpyrrolidone (NMP) > N, N-dimethylacetamide (DMAC) > N, N-dimethylformamide (DMF) > 2-methoxyethanol > 2-ethoxyethanol > acetic acid > methanol > 1,4-dioxane > ethanol > water > n-propanol > acetone > ethyl acetate. The Hansen solubility parameter (HSP) and solvents polarity were applied to explore this solubility behaviour. The experimental solubility data was correlated with five thermodynamic models including two-Suffix Margules, NRTL-SAC, NRTL, modified Apelblat equation and UNIQUAC model, and the calculated results indicated that all of models had great fitting effects with the measured solubility data. Furthermore, the solution thermodynamic properties (ΔdisGo, ΔdisHo, ΔdisSo) of kojic acid in investigated solvents were calculated by van’t Hoff equation. The positive values of ΔdisHo and ΔdisSo indicated that the dissolution process of kojic acid in all the measured solvents was endothermic and entropy-driven.

Relationship of the Permittivity with the Activity Coefficients of Water and Ions in Aqueous Solutions of Lithium and Sodium Sulfates

The activity coefficients in aqueous solutions of lithium and sodium sulfates at 298 K were calculated by the generalized Debye–Huckel theory using the experimental values of the static permittivity of the solutions. The effect of the hydration and ion association on the dielectric and thermodynamic properties of the electrolyte solutions was considered.

Experimental measurement and correlation of solubility of β-carotene in pure and ethanol-modified subcritical water

For the first time, the solubility of β-carotene in pure and ethanol-modified subcritical water (SW) using the static method was determined. The experimental runs were performed at a temperature ranging from 298.15 to 403.15 K and 0–10% (w/w) of ethanol as a modifier at a constant pressure of 5 MPa. Samples were analyzed by UV–vis spectrophotometer. The solubility of β-carotene was found to range from 1.084 × 10 −8 to 227.1 × 10 −8 mol fractions in the subcritical water in above mentioned conditions. The obtained β-carotene solubility data were correlated using the linear model and modified Apelblat model. The obtained results showed the modified Apelblat model was better for estimating the solubility of β-carotene in SW. The values of the root-mean-square deviation (RMSD) between experimental and correlated data were calculated and used as the index of validity and accuracy for the model. Also, thermodynamic properties of the solution such as the Gibbs free energy of solution, enthalpy, and entropy of solution were estimated. • The solubility of β-carotene in pure and ethanol-modified subcritical water using static method was measured for the first time. • Solubility of β-carotene was measured at different temperatures (298.15–403.15 K) and constant pressures (5 MPa). • Solubility data were correlated by semi-empirical models. • Thermodynamic properties of solution were estimated.

Determination and correlation of solubility of 4,4′-difluorobenzophenone in pure and binary mixed solvents and thermodynamic properties of solution

The target of this study was to determine and correlate the solubility of 4,4′-difluorobenzophenone in various solvents from 293.15 K to 333.15 K by gravimetric method at 0.1 MPa, and at the same time got the thermodynamic parameters. In this work, methanol, ethanol, n-propanol, isopropanol, n-butanol, isobutanol, ethyl acetate, methyl acetate, acetone, acetonitrile pure solvents and ethanol + n-propanol mixed solvents were selected as the experimental solvents. The results showed that the solubility of 4,4′-difluorobenzophenone in selected solvents increased with increasing temperature within the investigated temperature range. The effect of solvent properties on 4,4′-difluorobenzophenone solubility behaviors was discussed by solvent polarity, hydrogen bonding donor and acceptor propensity, and cohesive energy density. Furthermore, the dissolution behavior was explained using the Hansen solubility parameter (HSP). In pure solvents, the modified Apelblat equation, van't Hoff equation, λh equation, nonrandom two-liquid (NRTL) equation, Wilson equation and universal quasichemical (UNIQUAC) equation were used to correlate the experimental solubility of 4,4′-difluorobenzophenone. While the solubility in ethanol + n-propanol mixed solvents was correlated by the modified Apelblat equation, van't Hoff equation, λh equation, NRTL model, Wilson model, Jouyban-Acree and the modified Jouyban-Acree model. The results indicated that the calculated values of all selected equations were in good agreement with the experimental data. Furthermore, the thermodynamic properties (∆H°sol, ∆S°sol, ∆G°sol, %ξH and %ξTS) were measured by Van't Hoff analysis. In a word, the physicochemical data of 4,4′-difluorobenzophenone obtained in this work will be helpful for the purification, crystallization, recrystallization and designing a new synthetic route of 4,4′-difluorobenzophenone.

Thermodynamic Properties of Li-Sb Liquid Solution by QAM

The optimization of the thermodynamic properties of a liquid Li-Sb system is performed based on all available literature data. The description of the excess Gibbs energy as a function of temperature and composition is presented using a qualitative associate model (QAM), described in our earlier work on the Li-Pb system. The modeled molar mixing enthalpy changes, excess Gibbs energy and the molar excess entropy values are characterized by strong negative deviations from the ideal solution in the entire concentration range. It was found that the best correlation between experimental data and those calculated using QAM is observed if the Li3Sb and LiSb associates are assumed to be in the liquid Li-Sb solution. Calculations of the microscopic function, i.e., the concentration-concentration in the long-wavelength limit, (Scc(0)) excess stability function (ES) and short-range-order parameter (SRO(α)) show the extreme (maximum or minimum) values almost exactly for the concentrations of Li3Sb associate (and intermetallic phase in solid alloys) and LiSb. The quotient of real Scc(0) to this for the ideal solution (Scc(0)id) is characterized by a similar property to the mentioned structure factors.

Black hole and naked singularity geometries supported by three-form fields

We investigate static and spherically symmetric solutions in a gravity theory that extends the standard Hilbert–Einstein action with a Lagrangian constructed from a three-form field A_{alpha beta gamma }, which is related to the field strength and a potential term. The field equations are obtained explicitly for a static and spherically symmetric geometry in vacuum. For a vanishing three-form field potential the gravitational field equations can be solved exactly. For arbitrary potentials numerical approaches are adopted in studying the behavior of the metric functions and of the three-form field. To this effect, the field equations are reformulated in a dimensionless form and are solved numerically by introducing a suitable independent radial coordinate. We detect the formation of a black hole from the presence of a Killing horizon for the timelike Killing vector in the metric tensor components. Several models, corresponding to different functional forms of the three-field potential, namely, the Higgs and exponential type, are considered. In particular, naked singularity solutions are also obtained for the exponential potential case. Finally, the thermodynamic properties of these black hole solutions, such as the horizon temperature, specific heat, entropy and evaporation time due to the Hawking luminosity, are studied in detail.

Thiamethoxam in aqueous co-solvent mixtures of 1,4-dioxane, N,N-dimethylacetamide, dimethyl sulfoxide and acetonitrile: Solubility solute-solvent and solvent-solvent interactions, and preferential solvation analysis

The determination of equilibrium solubilities of thiamethoxam in solutions of 1,4-dioxane (1) + water (2), N,N-dimethylacetamide (DMAC, 1) + water (2), dimethyl sulfoxide (1) + water (2) and acetonitrile (1) + water (2) was carried out through the shake-flask method from 278.15 to 323.15 K under 101.2 kPa. The relative contributions of solvent–solvent and solute–solvent interactions on the drug solubility variation were studied via the analysis of linear solvation energy relationships. The significant contributions to the solubility variation were the cavity term and dipolarity-polarizability term in aqueous solution of 1,4-dioxane; and cavity term in aqueous solutions of DMAC/DMSO/acetonitrile. The inverse Kirkwood–Buff integrals were employed to investigate the preferential solvation in light of several solution thermodynamic properties. The preferential solvation parameters for 1,4-dioxane, DMAC and acetonitrile presented positive values in the three aqueous solutions within intermediate and 1,4-dioxane/DMAC/acetonitrile-rich compositions, demonstrating that the drug thiamethoxam was solvated preferentially by 1,4-dioxane/DMAC/acetonitrile. It is conjectured that in these regions thiamethoxam serves as a Lewis acid with the 1,4-dioxane and DMAC molecules; while for the (acetonitrile + water) mixture, the preferential solvation could be attributed to polarization effects. Moreover, the solubility data in mole fraction was mathematically described by the van’t Hoff–Jouyban-Acree model and Jouyban-Acree model attaining the average relative deviations no more than 6.67%.