Statistical physics modeling of water vapor adsorption isotherm into kernels of dates: Experiments,microscopic interpretation and thermodynamic functions evaluation

In this paper, water sorption isotherms into date kernels give interesting insights about the sorption mechanism. The equilibrium adsorption data expressing the change in moisture content of date kernels were collected at three different temperatures using the static gravimetric technique. The adsorption isotherm profiles demonstrated that this process was performed via an infinite number of layers. A modified form of the Brunauer, Emett and Teller (BET) model is obtained based on the use of the real gas law and statistical physics treatment so the interaction between molecules is considered. This advanced model is used to fit experimental isotherms by numerical simulation. The sorption mechanism is theoretically explained by the parameters that could be related to the water adsorption process. Based on fitting results, we find that the number of molecules per site (parameter n) has a linear tendency with temperature thanks to the thermal agitation effect. A deeper analysis of adsorption energy demonstrates that the water vapors are physisorbed in the date kernels. Through the exploitation of our model, three classic thermodynamic functions are investigated to interpret the macroscopic aspect of the adsorption mechanism.     

Numerical determination of the competitive isotherm of enantiomers

A numerical method was developed and used to determine adsorption isotherms in chromatography. The numerical parameters of an isotherm model were derived from the recorded band profiles of the racemic mixture of the 1-phenyl-1-propanol enantiomers, by means of a nonlinear least-squares method. We used the equilibrium-dispersive model of chromatography with several isotherm models. The numerical constants of the isotherm models were tuned so that the calculated and the measured band profiles match as much as possible. We show that this numerical inverse method can be applied even without the knowledge of the individual band profile of the pure enantiomers. The isotherms determined from the--usually unresolved--overloaded band profiles matched extremely well the isotherms determined by frontal analysis. Several isotherm models were used and tested--such as Langmuir, biLangmuir, Tóth, Langmuir-Freundlich. The best-fit isotherm was selected by means of statistical evaluation of the results.     

Determination of the single component and competitive adsorption isotherms of the 1-indanol enantiomers by the inverse method

The inverse method of isotherm determination consists in calculating the numerical values of the coefficients of an isotherm model that give a set of chromatographic profiles in best possible agreement with the set of experimental profiles available. This method was applied to determine the adsorption isotherms of the 1-indanol enantiomers on a cellulose tribenzoate chiral stationary phase. Both single-component and competitive isotherms were determined by using no more than one or two overloaded band profiles. The isotherms determined from the overloaded band profiles agreed extremely well with the isotherms determined by frontal analysis. Several isotherm models were used and tested. The best-fit isotherm was selected by means of statistical evaluation of the results. The results show that the adsorption is best characterized with a model describing heterogeneous adsorption with bimodal adsorption energy distribution.     

A thermodynamically consistent explicit competitive adsorption isotherm model based on second-order single component behaviour

A competitive adsorption isotherm model is derived for binary mixtures of components characterized by single component isotherms which are second-order truncations of higher order equilibrium models suggested by multi-layer theory and statistical thermodynamics. The competitive isotherms are determined using the ideal adsorbed solution (IAS) theory which, in case of complex single component isotherms, does not generate explicit expressions to calculated equilibrium loadings and causes time consuming iterations in simulations of adsorption processes. The explicit model derived in this work is based on an analysis of the roots of a cubic polynomial resulting from the set of IAS equations. The suggested thermodynamically consistent and widely applicable competitive isotherm model can be recommended as a flexible tool for efficient simulations of fixed-bed adsorber dynamics.     

Global statistical predictor model for characteristic adsorption energy of organic vapors-solid interaction: use in dynamic process simulation

Adsorption of Volatile Organic Compounds (VOCs) is one of the best remediation techniques for controlling industrial air pollution. In this paper, a quantitative predictor model for the characteristic adsorption energy (E) of the Dubinin-Radushkevich (DR) isotherm model has been established with R(2) value of 0.94. A predictor model for characteristic adsorption energy (E) has been established by using Multiple Linear Regression (MLR) analysis in a statistical package MINITAB. The experimental value of characteristic adsorption energy was computed by modeling the isotherm equilibrium data (which contain 120 isotherms involving five VOCs and eight activated carbons at 293, 313, 333, and 353 K) with the Gauss-Newton method in a statistical package R-STAT. The MLR model has been validated with the experimental equilibrium isotherm data points, and it will be implemented in the dynamic adsorption simulation model PROSIM. By implementing this model, it predicts an enormous range of 1200 isotherm equilibrium coefficients of DR model at different temperatures such as 293, 313, 333, and 353K (each isotherm has 10 equilibrium points by changing the concentration) just by a simple MLR characteristic energy model without any experiments.     

Theoretical study of the accuracy of the pulse method,frontal analysis,and frontal analysis by characteristic points for the determination of single component adsorption isotherms

The adsorption isotherms of selected compounds are our main source of information on the mechanisms of adsorption processes. Thus, the selection of the methods used to determine adsorption isotherm data and to evaluate the errors made is critical. Three chromatographic methods were evaluated, frontal analysis (FA), frontal analysis by characteristic point (FACP), and the pulse or perturbation method (PM), and their accuracies were compared. Using the equilibrium-dispersive (ED) model of chromatography, breakthrough curves of single components were generated corresponding to three different adsorption isotherm models: the Langmuir, the bi-Langmuir, and the Moreau isotherms. For each breakthrough curve, the best conventional procedures of each method (FA, FACP, PM) were used to calculate the corresponding data point, using typical values of the parameters of each isotherm model, for four different values of the column efficiency (N=500$N=500$, 1000, 2000, and 10,000). Then, the data points were fitted to each isotherm model and the corresponding isotherm parameters were compared to those of the initial isotherm model. When isotherm data are derived with a chromatographic method, they may suffer from two types of errors: (1) the errors made in deriving the experimental data points from the chromatographic records; (2) the errors made in selecting an incorrect isotherm model and fitting to it the experimental data. Both errors decrease significantly with increasing column efficiency with FA and FACP, but not with PM.     

Thermodynamic analysis of the heterogenous binding sites of molecularly imprinted polymers

The thermodynamic interactions of two polymers, one Fmoc-L-Trp-imprinted (MIP), the other one an unimprinted reference (NIP), with the two Fmoc-tryptophan enantiomers were studied by frontal analysis, which allows accurate measurements of the adsorption isotherms. These isotherms were acquired at temperatures of 40, 50, 60, and 70 degrees C, for sample concentrations ranging between 0.005 and 40 mM. The mobile phase used was acetonitrile with one percent acetic acid as an organic modifier. Within the measured concentration ranges, the tri-Langmuir isotherm model accounts best for the isotherm data of both enantiomers on the MIP, the bi-Langmuir model for the isotherm data of Fmoc-L-Trp on the NIP. These isotherm models were selected using three independent processes: statistical tests on the results from regression of the isotherm data to different isotherm models; calculation of the affinity energy distribution from the raw isotherm data; comparison of the experimental and the calculated band profiles. The isotherm parameters obtained from these best selected isotherm models showed that the enantiomeric selectivity does not change significantly with temperature, while the affinity of the substrates for both the MIP and the NIP decrease considerably with increasing temperatures. These temperature effects on the binding performance of the MIP were clarified by considering the thermodynamic functions (i.e., the standard molar Gibbs free energy, the standard molar entropy of adsorption, and the standard molar enthalpy of adsorption) for each identified type of adsorption sites, derived from the Van't Hoff equation. This showed that the entropy of transfer of Fmoc-L-Trp from the mobile to the MIP stationary phase is the dominant driving force for the selective adsorption of Fmoc-L-Trp onto the enantioselective binding sites. This entropy does not change significantly with increasing temperatures from 40 to 70 degrees C.     

Discrimination between adsorption isotherm models based on nonlinear frequency response results

A number of criteria are established for distinguishing between different adsorption isotherm types. These criteria are defined based on the adsorption isotherm derivatives up to the third order, which, on the other hand, can be estimated from nonlinear frequency response data. The criteria for five favourable (Langmuir, Freundlich, Sips, Toth and Unilan) isotherms and two complex isotherms (BET and quadratic) are presented. These criteria enable unique identification of the underlying adsorption isotherm relation if the values of the local first, second and third order isotherm derivatives at several points are known. The method is applied to experimental data from our previous publications, for one case of a favourable and one case of a complex isotherm.     

Separation mechanism of nortriptyline and amytriptyline in RPLC

The single and the competitive equilibrium isotherms of nortriptyline and amytriptyline were acquired by frontal analysis (FA) on the C18- bonded discovery column, using a 28/72 (v/v) mixture of acetonitrile and water buffered with phosphate (20 mM, pH 2.70). The adsorption energy distributions (AED) of each compound were calculated from the raw adsorption data. Both the fitting of the adsorption data using multi-linear regression analysis and the AEDs are consistent with a trimodal isotherm model. The single-component isotherm data fit well to the tri-Langmuir isotherm model. The extension to a competitive two-component tri-Langmuir isotherm model based on the best parameters of the single-component isotherms does not account well for the breakthrough curves nor for the overloaded band profiles measured for mixtures of nortriptyline and amytriptyline. However, it was possible to derive adjusted parameters of a competitive tri-Langmuir model based on the fitting of the adsorption data obtained for these mixtures. A very good agreement was then found between the calculated and the experimental overloaded band profiles of all the mixtures injected.     

Determination of competitive isotherms of enantiomers by a hybrid inverse method using overloaded band profiles and the periodic state of the simulated moving-bed process

A procedure for determination of adsorption isotherms in simulated moving-bed (SMB) chromatography is presented. The parameters of a prescribed adsorption isotherm model and rate constants are derived using a hybrid inverse method, which incorporates overloaded band profiles of the racemic mixture and breakthrough data from a single frontal experiment. The latter are included to reduce the uncertainty on the estimated saturation capacity, due to the dilution of the chromatograms with respect to the injected concentrations. The adsorption isotherm model is coupled with an axially dispersed flow model with finite mass-transfer rate to describe the experimental band profiles. The numerical constants of the isotherm model are tuned so that the calculated and measured band profiles match as much as possible. The accuracy of the isotherm model is then checked against the cyclic steady state (CSS) of the target SMB process, which is readily and cheaply obtained experimentally on a single-column set-up. This experiment is as expensive and time consuming as just a few breakthrough experiments. If necessary, the isotherm parameters are adjusted by applying the inverse method to the experimental CSS concentration profile. The method is successfully applied to determine the adsorption isotherms of Tr?gers base enantiomers on Chiralpak AD/methanol system. The results indicate that the proposed inverse method offers a reliable and quick approach to determine the competitive adsorption isotherms for a specific SMB separation.     

Competitive effect of glucose–fructose adsorption in a fixed‐bed chromatographic column

A continuous separation system such as a simulated moving‐bed process requires adsorption data with precise equilibrium and kinetic model parameters of a single chromatographic column. The adsorption of glucose and fructose in a fixed‐bed chromatographic column was investigated to determine the competition effect of each component resulting from their initial molar ratios. The model parameters including bed porosity and axial dispersion coefficient were determined using the moment analysis method. The equilibrium isotherm parameters were estimated by conducting experiments at various molar ratios and initial sugar concentrations. The parameters obtained were then used for the simulation of dynamic breakthrough curves of glucose and fructose. The equilibrium isotherms revealed that the linear adsorption pattern provided good prediction for each molar ratio using the Henry equation. In addition, the modified Langmuir model was proposed to account for the competitive adsorption, due to the cooperative competition effect whereby glucose was promoted to the active sites by fructose to a greater degree than vice versa. A good agreement between the experimental and numerical data of the adsorption time profiles was also observed.     

Determination of the thermodynamic contribution to peak asymmetry of basic solutes in reversed-phase liquid chromatography

To this day packing materials manufacturers are still trying to develop reversed-phase stationary phases that have silica more completely reacted with bonding ligands to afford more homogeneous particle surfaces. Incomplete bonding causes inhomogeneous effects that are readily observed when separating basic solutes because of the acidic silanols that are unreacted. However, it is still not understood exactly what types of silanol sites are unreacted or if metal impurities are contributing to the resulting peak asymmetry observed. A method is presented which utilizes (1) the frontal analysis method of chromatography to obtain adsorption/partition isotherms, (2) a heterogeneous Langmuir distribution model for the resulting isotherm, (3) an expectation-maximization numerical procedure to solve the mathematical problem to yield the most probable distribution of adsorption parameters, and (4) the equilibrium-dispersive model of chromatography incorporating the fitted isotherm model to check the validity of the sorption model with experimental observations. Correlation of packing materials characterization parameters with results obtained by this procedure will indicate what type of silanols or other sites are responsible for observed tailing behavior. Developers and manufacturers will then be able to more efficiently target their synthetic designs for "base-deactivated" reversed-phase silicas.     

Application of water-activated carbon isotherm models to water adsorption isotherms of single-walled carbon nanotubes

The objective of this study is to understand the interactions of water with novel nanocarbons by implementing semiempirical models that were developed to interpret adsorption isotherms of water in common carbonaceous adsorbents. Water adsorption isotherms were gravimetrically determined on several single-walled carbon nanotube (SWNT) and activated carbon samples. Each isotherm was fitted to the Dubinin-Serpinsky (DS) equation, the Dubinin-Astakov equation, the cooperative multimolecular sorption theory, and the Do and Do equations. The applicability of these models was evaluated by high correlation coefficients and the significance of fitting parameters, especially those that delineate the concentration of hydrophilic functional groups, micropore volume, and the size of water clusters. Samples were also characterized by spectroscopic and adsorption techniques, and properties complementary to those quantified by the fitting parameters were extracted from the data collected. The comparison of fitting parameters with sample characterization results was used as the methodology for selecting the most informative and the best-fitting model. We conclude that the Do equation, as modified by Marban et al., is the most suitable semiempirical equation for predicting from experimental isotherms alone the size of molecular clusters that facilitate adsorption in SWNTs, deconvoluting the experimental isotherms into two subisotherms: adsorption onto hydrophilic groups and filling of micropores, and quantifying the concentration of hydrophilic functional groups, as well as determining the micropore volume explored by water. With the exception of the DS equation, the application of other water isotherm models to SWNTs is not computationally tractable. The findings from this research should aid studies of water adsorption in SWNTs by molecular simulation, which remains the most popular tool for understanding the microscopic behavior of water in nanocarbons.     

Statistical physics study of the interaction of the 5, 10, 15, 20-tetrakis (4-tolylphenyl) porphyrin (H2TTPP) with magnesium ion: New microscopic interpretations

In this article, Tetraphenylporphyrin molecules were synthesized and used as biosensor for Magnesium ion. The metallization of porphyrin is a fundamental step which needs to be studied and educated. In fact, adsorption isotherms of magnesium nitrate on porphyrin coated onto Quartz crystal are achieved at four temperatures. Then, statistical physics treatment has been applied to deduce new microscopic interpretations for the experimental data. The main contribution of this analytical treatment is to deduct some physicochemical parameters related to the adsorption process which could not be achieved by means of empirical models. A Layer by Layer adsorption model with two energies was found to be the best model to reproduce the experimental isotherms of magnesium ion on porphyrin. In this model, seven parameters affecting the adsorption process were adjusted, namely the number of ions per site, the density of receptor sites, the two energetic parameters, the cohesion pressure, the covolume and the number of layers. The adsorption mechanisms were characterized by an energetic investigation which demonstrated that the magnesium ion was physisorbed onto porphyrin. Via the exploitation of our advanced model, three classical thermodynamic functions have been investigated and interpreted.     

The effects of energy sites on adsorption of Lennard-Jones fluids and phase transition in carbon slit pore of finite length a computer simulation study

A Monte Carlo simulation method is used to study the effects of adsorption strength and topology of sites on adsorption of simple Lennard-Jones fluids in a carbon slit pore of finite length. Argon is used as a model adsorbate, while the adsorbent is modeled as a finite carbon slit pore whose two walls composed of three graphene layers with carbon atoms arranged in a hexagonal pattern. Impurities having well depth of interaction greater than that of carbon atom are assumed to be grafted onto the surface. Different topologies of the impurities; corner, centre, shell and random topologies are studied. Adsorption isotherms of argon at 87.3 K are obtained for pore having widths of 1, 1.5 and 3 nm using a Grand Canonical Monte Carlo simulation (GCMC). These results are compared with isotherms obtained for infinite pores. It is shown that the surface heterogeneity affects significantly the overall adsorption isotherm, particularly the phase transition. Basically it shifts the onset of adsorption to lower pressure and the adsorption isotherms for these four impurity models are generally greater than that for finite pore. The positions of impurities on solid surface also affect the shape of the adsorption isotherm and the phase transition. We have found that the impurities allocated at the centre of pore walls provide the greatest isotherm at low pressures. However when the pressure increases the impurities allocated along the edges of the graphene layers show the most significant effect on the adsorption isotherm. We have investigated the effect of surface heterogeneity on adsorption hysteresis loops of three models of impurity topology, it shows that the adsorption branches of these isotherms are different, while the desorption branches are quite close to each other. This suggests that the desorption branch is either the thermodynamic equilibrium branch or closer to it than the adsorption branch.     

Systematic errors in the measurement of adsorption isotherms by frontal analysis Impact of the choice of column hold-up volume, range and density of the data points

Besides the accuracy and the precision of the measurements of the data points, several important parameters affect the accuracy of the adsorption isotherms that are derived from the data acquired by frontal analysis (FA). The influence of these parameters is discussed. First, the effects of the width of the concentration range within which the adsorption data are measured and of the distribution of the data points in this range are investigated. Systematic elimination of parts of the data points before the calculation of the nonlinear regression of the data to the model illustrates the importance of the numbers of data points (1) within the linear range and (2) at high concentrations. The influence of the inaccuracy of the estimate of the column hold-up volume on each adsorption data point, on the selection of the isotherm model, and on the best estimates of the adsorption isotherm parameters is also stressed. Depending on the method used to measure it, the hold-up time can vary by more than 10%. The high concentration part of the adsorption isotherm is particularly sensitive to errors made on t(0,exp) and as a result, when the isotherm follows bi-Langmuir isotherm behavior, the equilibrium constant of the low-energy sites may change by a factor 2. This study shows that the agreement between calculated and experimental overloaded band profiles is a necessary condition to validate the choice of an adsorption model and the calculation of its numerical parameters but that this condition is not sufficient.     

Retention of ionizable compounds in high-performance liquid chromatography. 14. Acid-base pK values in acetonitrile-water mobile phases

Using competitive frontal analysis, the binary adsorption isotherms of the enantiomers of 1-phenyl-l-propanol were measured on a microbore column packed with a chiral stationary phase based on cellulose tribenzoate. These measurements were carried out using only the racemic mixture. The experimental data were fitted to four different isotherm models: Langmuir, BiLangmuir, Langmuir-Freundlich and Tóth. The BiLangmuir and the Langmuir-Freundlich models accounted best for the competitive adsorption data. An excellent agreement between the experimental and the calculated overloaded band profiles for various samples of racemic mixture was obtained when the equilibrium dispersive model of chromatography was used together with the BiLangmuir competitive isotherm. The isotherm parameters measured under competitive conditions were used to calculate the overloaded band profiles of large samples of the pure S- and R-enantiomers, too. A satisfactory agreement between the experimental and calculated band profiles was observed when using in the computation the corresponding single component BiLangmuir isotherm derived from the binary isotherm previously determined. Thus oniy data derived from the racemic mixture are required for computer optimization of the preparative chromatography separation of the enantiomers.     

Numerical determination of competitive adsorption isotherm of mandelic acid enantiomers on cellulose-based chiral stationary phase

The use of inverse method for the determination of competitive adsorption isotherm of mandelic acid enantiomers on cellulose tris(3,5-diethylphenyl carbamate) stationary phase is proposed in this work. Non-dominated sorting genetic algorithm with jumping genes (NSGA-II-JG) was applied to acquire the isotherm parameters by minimizing the sum of square deviations of the model predictions from the measured elution profiles. Three different competitive isotherm models, i.e., Langmuir, biLangmuir and Tóth, combined with transport-dispersive chromatographic model were used in predicting the elution profiles. Orthogonal collocation on finite element (OCFE) method was applied to obtain the calculated elution profiles. Results indicate that biLangmuir isotherm and Tóth isotherm give remarkably similar equilibrium isotherms within the investigated liquid concentration range. Band profiles calculated from both isotherm models are in good agreement with the experimental data. The validity of the determined parameters was verified by comparing the model predictions with experimental elution profiles at various experimental conditions.     

Adsorption of Brilliant Yellow onto Sepiolite: Evaluation of Thermodynamics and Kinetics and the Application of Nonlinear Isotherm Models

We studied the adsorption of Brilliant Yellow (BY) from aqueous solutions onto sepiolite and determined the adsorption equilibrium isotherms. We applied pseudo-first-order and pseudo-second-order kinetic models; the adsorption of BY onto sepiolite was best described by the pseudo-second-order kinetic model. The experimental data obtained at different temperatures were analyzed using various isotherm models; the Koble–Corrigan isotherm model provided the best fits for the BY adsorption data at all temperatures. The thermodynamic parameters of the BY adsorption onto sepiolite indicated that the adsorption process is spontaneous and exothermic in nature.     

Band splitting in overloaded isocratic elution chromatography II. New competitive adsorption isotherms

The equations of two new binary competitive isotherms models are derived. The first of these models assumes that the isotherms of the two pure, single compounds have distinct monolayer capacities. Its derivation is based on kinetic arguments. The ideal adsorbed solution (IAS) framework was applied to derive the second model that is a thermodynamically consistent competitive isotherm. This second model predicts the competitive adsorption isotherm behavior of a mixture of two compounds that have single-component adsorption behavior following a BET and/or a Langmuir isotherms. Both models apply well to the binary adsorption of ethylbenzoate and 4-tert.-butylphenol on a Kromasil-C18 column (with methanol-water, 62:38, v/v, as the mobile phase). The best single-solute adsorption isotherms of these two compounds are the liquid-solid extended multilayer BET and the Langmuir isotherms, respectively. The kinetic and thermodynamic new competitive models were compared, regarding the accuracy of their prediction of the elution band profiles of mixtures of these two compounds. A better agreement between experimental and calculated profiles was observed with the kinetic model. The IAS model failed because the behavior of the ethylbenzoate/4-tert.-butylphenol adsorbed phase mixture is probably non-ideal. The most striking result is the qualitative prediction by these models of the peak splitting of 4-tert.-butylphenol during its elution in presence of ethylbenzoate.