Sorption Rate Curves Research Articles

Application of kinetic models to study the sorption rate in the ‘ammonium ions–calcined sorbent’ system

This study examines sorption kinetics for the extraction of ammonium ions from an aqueous solution under static conditions using 5 g of sorbent. The sorbent was obtained by calcining ash and slag waste accumulated in the ash dump following hydraulic ash removal. The initial concentrations of ammonium ions in the model solutions were 5, 20, 50 and 100 mg/dm3. The volume of the model solution was 50 cm3. For each initial concentration, the kinetic curves of the ammonium ion sorption at the sorption duration of 10, 30, 60, 90, 120, 150 and 180 min were obtained. The kinetic curves were processed using the Lagergren (pseudo-first order), Ho and McKay (pseudo-second order), Morris-Weber (diffusion) and Elovich kinetic equations. It was demonstrated that all kinetic equations adequately describe the experimental data. The linear correlation of the equations was given. The Boyd equation was used to determine the limiting stage in the Morris–Weber model. It was established that the external diffusion mechanism is limiting for the sorption of ammonium ions by a calcined sorbent. The Lagergren model, with a coefficient of determination R2 of 0.9801–0.9949, best described the experimental data on sorption kinetics. Sorption rate curves as a function of the adsorption value and sorption time were given. The latter are described by exponential and polynomial dependence according to the Lagergren model, polynomial dependence according to the Ho and McKay model and exponential dependence according to the Morris–Weber and Elovich models. The sorption rate was the highest at the initial stage of sorption for all dependencies, which is consistent with previous studies on the sorption of pollutants from aqueous solutions using various sorbents.

Transport of liquid hydrocarbons in the polyurethane-based membranes

The sorption and diffusion of liquid hydrocarbons have been studied in a series of polyurethanes (PU) based on poly(oxytetramethylene) diol (PTMO) extended with 2,4-tolylene diisocyanate (TDI) and 4,4′-bis (2-hydroxyethoxy) biphenyl (BHBP). The effects of the soft and hard segment length, and the penetrant structure, on the transport behaviour were investigated using immersion sorption method. The sorption rate curves were generally Fickian for all the PU/benzene systems. For n-hexane and cyclohexane as penetrants, various non-Fickian anomalies were observed for PUs with shorter soft segments (PTMO-1000, 650). The values of the diffusional exponent indicated that system was tending towards Fickian on decreasing the hard segment length or increasing the soft segment one, on decreasing the penetrant size or increasing its flexibility, and on increasing temperature. Diffusion coefficients ( D) were determined from the initial slope of the sorption curves by the small-time method for the PU/benzene systems, and for the PTMO-2000-based PUs. The variation in D was found to depend strongly on the length of the PU hard and soft segments, and on the penetrant structure. The PU/penetrant interaction parameter ( χ) was estimated from the results of solvent equilibrium uptake. The values of this parameter indicated that strong interaction existed for the PU/benzene and for all the PTMO-2000-based PU/solvent systems. The variations in sorption capability among PUs in those systems were found to be solely the effect of different ability of the hard domains to restrain membrane swelling. For other systems, no particular difference in normalised sorptivity among structurally different PUs was observed. The activation energy for diffusion, and the thermodynamic parameters were also calculated and discussed.

Sorption and diffusion of water vapor in poly(ethylene terephthalate) film

The sorption and diffusion of water vapor in poly(ethylene terephthalate) (PET) film were measured by applying a thermogravimetric analyzer (TG-DTA), which customarily has been used to detect the weight loss of a sample with the increase of temperature under a given atmosphere. In this case, we detected the weight gain of PET film by sorption of water vapor under a given humidity at a constant temperature. Sorption-rate curves were successfully obtained in spite of the low solubility of PET film and the presence of Fickian-type curves. The solubility was better described according to the dual-mode sorption model. The diffusion coefficients were determined in their initial slopes by the short-time method. We found that the diffusion coefficient depended on vapor pressure.

General radial diffusion model for heterogeneous sorbents

AbstractSorption kinetics modeling can be improved by accounting for sorbent heterogeneity. This paper demonstrates how heterogeneity can be accommodated with a composite sorbent particle whose shape is described by a general radial geometry. Batch experiments involving sorption and desorption of anthracene from paraffin spheres are used to test the model. The performance of the model is also demonstrated with simulations based on Borden soil. Experiments and modeling demonstrated that sorption by a heterogeneous mix of sorbent particle sizes can be predicted better using a general composite radial geometry than as an equivalent sphere geometry. However, predicting the composite geometry (shape factor and radius) from sorption rate data is difficult due to the nonuniqueness of sorption rate curves that evolve from higher shape factor geometries. Use of models such as ours that better describe sorption sites at long diffusion path lengths may improve predictions when slow, long‐term desorption is an important factor.

Concentration dependence of diffusion–adsorption rate in activated carbon

A new method based on transfer rate measurements of a radioactive solute inside activated carbon grains is presented which allows to isolate the relative contributions of the chemical (reversible adsorption) and geometrical (tortuous diffusion paths) effects on the overall effective diffusion coefficient. Transfer rate measurements were performed with two different initial conditions: carbon grains either initially saturated with a solution of the same concentration as the bath ( self-diffusion) or free of solute. In the self-diffusion case the adsorption equilibrium for the solute is undisturbed during the process and the adsorption dynamics for tracer particles becomes linear. These processes can be analyzed in terms of an effective self-diffusion coefficient ( D eff). The concentration dependence of the transfer rate is shown to be controlled by nonlinearities of the adsorption isotherm. We also show that the self-diffusion coefficient ( D s ) is one order of magnitude lower than in free bulk diffusion. Finally, in the second case (grains initially free of solute) we predict satisfactorily sorption rate curves, as well as the dependence of the transfer rate on the initial concentration, using results obtained in the self-diffusion case.

Influence of solvent uptake and swelling by poly(styrene–divinylbenzene) column packings on sample sorption rate and band broadening in reversed-phase liquid chromatography

Porous poly(styrene–divinylbenzene) (PS–DVB) HPLC packings give broad, tailed peaks for some types of solutes. This phenomenon is due to sorption of these solutes into the polymer matrix where they experience very slow, hindered diffusion. It is known that the presence of small amounts of tetrahydrofuran (THF) in the methanol (MeOH)–H 2O mobile phase yields narrower and less tailed peaks for such solutes. In this study, Hamilton PRP-∞, a nonporous PS–DVB polymer, serves as a model for the matrix of porous polymers. The following measurements were made: sorption isotherms for MeOH and THF from aqueous solution; swelling of PRP-∞ as a function of activity of MeOH and THF in aqueous solution; and both sorption isotherms and sorption rate curves for the solute naphthalene from various solvent mixtures. The addition of 2% THF to 70:30 MeOH–H 2O produced an additional 0.4% swelling, an 11% decrease in the sorption capacity and a 90% increase in the diffusion coefficient of naphthalene in PRP-∞. The increase in the diffusion coefficient, which is responsible for the improved elution peak shape of naphthalene, is shown to be caused by the decrease in sorption capacity, rather than by the additional swelling. The sorbed THF serves to fill and “block” the smallest, most highly hindered micropores in the polymer matrix.

Permeation and sorption for oxygen and nitrogen into polyimide membranes

AbstractThe permeation rate curves and sorption rate curves of oxygen and nitrogen below 1.3 atm were measured for seven polyimides of which chemical structures were systematically changed. These rate curves were applied Fickian model curves. The Fick's law was found to hold from the pressure independencies of diffusion coefficients for both the experiments. The solubility was better described according to dual‐mode sorption model rather than Henry‐s law from the sorption experiments. The diffusivities of both the gases were correlated with packing density (reciprocal of specific free volume) of the polymer, except two polyimides. The packing density of these two polymers could not be successfully calculated from Bondi's method. However, all the diffusion coefficients decreased linearly with an increase in the cohesive energy density (CED), which was calculated by the group contribution method of van Krevelen. These results suggest that the gas diffusions in polyimides better correlate with CED than with the packing density. Therefore, the cohesive energy density may be considered as a more reliable indication of the efficient gas separation. © 1995 John Wiley & Sons, Inc.

Calculation of concentration dependent mutual diffusion coefficients for the penetration period of sorption process in polymer film.

A new method of calculating concentration dependent mutual diffusion coefficients was proposed for the penetration period of adsorption and desorption processes in polymer film. The diffusion equation for a slab based on a dissolved solid coordinate with constant surface concentration was solved numerically for the sorption process, assuming a power function for the concentration dependence of the diffusion coefficient. The solvent concentrations corresponding to the apparent diffusion coefficients, which were calculated easily from the experimental sorption rate curves of the penetration period, were correlated with the surface concentrations using the power value of the concentration dependence as the parameter. Using this correlation, the relation between the diffusion coefficient and the concentration was obtained easily by trial calculation.

Sorption and diffusion of gases in a polyimide

AbstractSorption and diffusion of gases (CO2, N2, and He) in a polyimide (PI2080) film were measured by using an apparatus which gives the sorption rate curves from the initial state to the equilibrium state. Nonlinear isotherms observed for CO2 sorption were interpreted successfully in terms of the dual‐mode model for sorption in glassy polymers. Linear isotherms observed for N2 and He seemed to obey Henry's law. Two diffusion coefficients (DI and DE) were obtained using the short‐time method and the long‐time method for a Fickian diffusion model, together with the equilibrium solubility (Ce) from each experiment. The initial sorption rate curves agreed with the calculated curves using DI, however near sorption equilibrium the curves are in accord with the calculated curves using DE. These observations suggest that some relaxation process is superimposed on the diffusion process. The non‐Fickian transport data were correlated successfully with a model that combines time‐dependent diffusion and the Fickian model.

The diffusion of dichloroethane in phase-segregated polyurethanes

The sorption and diffusion of 1,2-dichloroethane (DCE) has been studied in a series of polyurethanes based on diphenylmethane diisocyanate and butanediol with 1000 and 2000 molecular-weight poly(tetramethylene oxide) (PTMO). DCE is a strongly swelling solvent with solubilities as large as 450% in the commercial Estane sample (63% soft segment). The sorption isotherms for the Estane sample, determined on sorption and desorption cycles and for three sample thicknesses, coincide. Hysteresis is seen in the sorption isotherms of other samples with lower soft-segment content, the isotherm on desorption being higher than that on sorption. The sorption rate curves are generally Fickian at low concentrations but become sigmoidal at higher concentrations for sorption and desorption in the Estane and the PTMO-2000 sample with the highest soft-segment content (68%). For other samples the curves are two-stage for sorption but Fickian for desorption. Diffusion coefficients ( D) were calculated by application of the analysis of coupled Fickian diffusion and first-order relaxation due to Joshi and Astarita. The resulting D versus C curves showed a maximum, attributable to the competing contributions of the solvent mobility and the thermodynamic factor. In addition, it was found that a correction to D for heating effects was required. There was evidence of significant solvent-induced structural relaxation in the higher hard-segment samples leading to higher than expected solubilities and diffusion constants. A pronounced thickness dependence of the diffusion constant could not be explained in terms of current diffusion models.

Diffusion and sorption of gases in poly(ethylene terephthalate)

AbstractThe sorption and diffusion of gases in poly(ethylene terephthalate) (PET) films have been measured by using a newly devised apparatus which gives the sorption rate curves from the initial to the equilibrium state. By analysis of sorption data for carbon dioxide in PET film, it was shown that the diffusion rate, as well as the equilibrium solubilities, could be explained by the dual‐mode sorption mechanism proposed by Vieth et al. Constants of their equation were used in the analysis of our experimental results, in which the apparent diffusion coefficients were dependent on pressure. The result showed that the dual‐mode sorption mechanism indeed applies to diffusion. The value of the real diffusivity obtained in amorphous PET was 5.7 × 10−9 (± 8%) cm2/sec in the range of pressure from 0 to 1 atm. It is believed that this apparatus has wide applicability, not only to diffusion of gases in polymers, but also to the measurement of surface properties, and can be used to study the morphology of a variety of polymers.

Diffusion and association of water in some polyalkylmethacrylates. Part 2.—Transient state diffusion

Sorption-rate curves were determined for water in polypropyl acrylate at 322 K and in polybutylmethacrylate at 313 K and analyzed to yield the concentration dependence of the diffusion coefficient. An earlier steady state analysis showed that the diffusion coefficient D decreased as the concentration C of sorbed water increased, consistent with clustering of the water in the polymer matrix. The transient and steady state diffusion coefficients are in good agreement, indicating that the process of clustering is rapid compared to that of diffusion. Procedures for obtaining the concentration dependence of D are examined in relation to systems for which D decreases with C. The method of Lin-Hwang is examined in detail.

Sorption of nitrogen by titanium films evaporated in ultra-high vacuum

The sorption rate of Ti films has been measured for nitrogen as a function of time and pressure at 15° C. The films were deposited on the inside of a copper hemisphere (200 cm2) mounted within an ultra-high vacuum chamber. Iodide titanium was degassed and evaporated at pressures below 10-8 torr. Nitrogen was leaked from a reservoir at a pressure PR into the ultra-high vacuum vessel via a conductance (SL) and the pressure above the Ti film (PG) maintained constant during sorption; the mass sorption rate equals SL(PR - PG). Impurities in the sorbate were kept low by thermally degassing the sorption and reservoir vessels and pumping both chambers during evaporation and sorption. The sorption properties of Ti films 23-100 A thick deposited on smooth and grooved substrates were measured at N2 pressures between 2.5 × 10-7 and 10-8 torr. The sorption capacity rose from 1.5 × 10-5 l. torr cm-2 for a 23 A thick film to 4.5 × 10-5 l. torr cm-2 for a 100 A film; the non-linear dependence of the capacity on film mass indicated sorption occurred on interior crystallite surfaces. The mass sorbed as a function of time followed the relation for diffusion into a finite slab with D [similar, equals] 10-18-4 × 10-17 cm2 sec-1. Treating the initial sorption as that for diffusion into a semi-infinite solid the surface gas concentration C0 was found showing that C0 [is proportional to] PG1/2. Values of the initial sorption rate SI were obtained by extrapolating the sorption rate curves to zero time. These results also showed that SI [is proportional to] PG1/2 with SI = 3.75 l. sec-1 at 10-8 torr. SI is usually assumed independent of pressure for an uncovered sorbent at t = 0. However, the dependence of SI and C0 on PG1/2 indicated that gas molecules dissociated on chemisorption giving an equilibrium surface coverage following the Langmuir hyperbolic law. Nitrogen atoms diffused into the film interior along grain boundaries with the outer surface concentration C0 [is proportional to] P1/2. Extrapolating sorption rate curves to zero time would then show an apparent dependence on pressure.