On the solution of diffusion controlled adsorption kinetics for any adsorption isotherms

Summary The problem of diffusion controlled adsorption kinetics is solved by application of an implicite difference method. Thereby all forms of adsorption isotherms can be used in the calculations. Computation examples are given concerning Langmuir, Freundlich, Volmer, Frumkin, and van der Waals isotherms.

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Zusammenfassung Es wird das Problem der diffusions-kontrollierten Adsorptionskinetik durch Anwendung eines impliziten Differenzenverfahrens gelöst. Dabei können bei der Berechnung alle Formen von Adsorptionsisothermen zugrunde gelegt werden. Rechenbeispiele für die Langmuir-, Freundlich-, Volmer-, Frumkin- und van-der-Waals-Isotherme werden angegeben.

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Nomenclature c concentration - c _o bulk concentration - D diffusion coefficient - x distance from the surface - t time - surface concentration - ₀ equilibrium surface concentration - saturation surface concentration - k _o/c _o - K ₁,K ₂ parameters of Henry and Freundlich isotherm with the dimension of a length - n parameter of Freundlich isotherm - a ₁,a ₂ interaction parameters - B ₁,B ₂,B ₃,B ₄ parameters of different isotherms with the dimension of a concentration - R gas law constant - T absolute temperature dimensionless parameters C c/c _o - / _o - X x/k - X step size ofX - Dt/k ² - step size of - _e time at - 0.95 Zentralinstitut für Organische Chemie,Bereich Grenzflhenaktive Stoffe, der Akademie der Wissenschaften der DDR, Berlin-AdlershofWith 6 figures and 1 table     

Neural network modelling of adsorption isotherms

This paper examines the possibility to use a single neural network to model and predict a wide array of standard adsorption isotherm behaviour. Series of isotherm data were generated from the four most common isotherm equations (Langmuir, Freundlich, Sips and Toth) and the data were fitted with a unique neural network structure. Results showed that a single neural network with a hidden layer having three neurons, including the bias neuron, was able to represent very accurately the adsorption isotherm data in all cases. Similarly, a neural network with four hidden neurons, including the bias, was able to predict very accurately the temperature dependency of adsorption data.     

吸附等温线的插值方法的研究

关于用函数形式确切地描述吸附等温线已有不少研究.沈中和等认为,难于找到适合各种形状吸附等温线的理想函数形式,建议采用线性插值.但线性插值显然会使计算结果带来额外偏差.Buchner等和Zingales等曾提出用函数平滑或拟合整个吸附等温线,但此法受所选函数形式的限制并存在拟合误差,无法如实描绘某些吸附等     

Pattern of adsorption isotherms in Ono-Kondo coordinates

The Ono-Kondo lattice density functional theory is used to analyze adsorbate-adsorbate interactions for supercritical systems. In prior work, this approach has been used to study intermolecular interactions in subcritical adsorbed phases, and this has included the study of adsorbate-adsorbate repulsions in the regime of adsorption compression. In this paper, we present the general pattern of adsorption isotherms in Ono-Kondo coordinates; this has not been done in the past. For this purpose, experimental isotherms for adsorption of supercritical fluids (including nitrogen, methane, and carbon dioxide) are plotted in Ono-Kondo coordinates. In addition, we performed Grand Canonical Monte Carlo simulations of adsorption for Lennard-Jones molecules and plotted isotherms in Ono-Kondo coordinates. Our results indicate a pattern of isotherms with regimes of adsorbate-adsorbate attractions at low surface coverage and adsorbate-adsorbate repulsions at high surface coverage. When the generalized Ono-Kondo model is used over a wide range of pressures - from low pressures of the Henry's law regime to supercritical pressures - the slope of the isotherm varies from positive at low pressures to negative at high pressures. The linear sections of these graphs show when the adsorbate-adsorbate interaction energies are approximately constant. When these linear sections have negative slopes, it indicates that the system is in a state of adsorption compression.     

Differential heats of adsorption and adsorption isotherms of alcohols on silicalite

Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 9, pp. 2117–2120, September, 1989.     

Analytical solutions for diffusion-controlled adsorption kinetics with non-linear adsorption isotherms

A regular perturbation series method provides a solution to the diffusion equation when the boundary condition is a non-linear adsorption isotherm. For adsorption at the interface the Freundlich and Langmuir isotherms yield power series in the square root of time. Convergence of the power series solutions is improved by applying the Shanks transformation. The solutions are compared to limiting cases and to published numerical solutions. The results are most accurate for small time where the numerical finite difference method is least reliable.     

Enthalpy of adsorption and adsorption isotherms of polyacrylamide on sea sand

The enthalpy of dilution of polyacrylamide 3530s (HPAM) solution, the enthalpy of adsorption of the polymer on sea sand, and the adsorption isotherms, were determined at 298. 15 K. The effect of NaCl on these properties were also determined. From the experimental results and certain assumptions, it was deduced that, for an adsorbedHPAM molecule, a large number of segments are in contact with the surface of the solid, some of them forming hydrogen-bonds, most of them belonging to Van der Waals interactions, which are much weaker than hydrogenbonds, and theHPAM molecules adsorbed on the surface of the solid are spread out.     

Calculation of immersion enthalpy data from adsorption isotherms

The thermodynamic equations for the calculation of binary and ternary immersion data in excess formalism are presented. Immersion enthalpies and entropies of the n-hexane/n-octane, n-octane/n-tetradecane and n-hexane/n-tetradecane binary mixtures as well as the n-hexane/n-octane/n-tetradecane ternary mixture on activated carbon are calculated from the temperature dependence of adsorption isotherms. In order to evaluate the quality of the calculations, the calculated immersion enthalpies of the binary mixtures on activated carbon are compared with those that were measured calorimetrically. It is shown that phenomenological thermodynamics can be used successfully to predict calorimetric data on the basis of adsorption excess isotherms.     

Intersection of isotherms for phosphate adsorption on hematite

Adsorption isotherms for phosphate on hematite were prepared at pH 3.39, 4.16, 5.10, 5.63, and 6.71 in this study. It was found that the adsorption isotherms at pH 5.63 and 6.71 intersected those at pH 4.16 and 5.10. Using surface complexation theory, this study demonstrates that the intersection of adsorption isotherms results from (1) phosphate being adsorbed mainly as protonated complexes at pH 4.16 and 5.10 but as nonprotonated complexes at pH 5.63 and 6.71; (2) the electric potential (psi) at the surface of hematite changing with pH at a rate less than 29.5 mV per pH unit (-d psi /dpH approximately equal 8.9 mV/pH). Fundamentally, however, it seems that the dominance of an imperfect (001) crystal face in the hematite sample is responsible for a low value of -d psi/dpH and the intersection of adsorption isotherms. The adsorption behavior may be regarded as characteristic behavior of protonation of adsorbed phosphate on an oxide with a small value of -d psi / dpH.     

Predicting adsorption isotherms of asphaltenes in porous materials

In this paper we present a molecular thermodynamics approach for the modeling of adsorption isotherms of asphaltenes adsorbed on Berea sandstone, Bedford limestone and dolomite rock, using a model for bulk asphaltenes precipitation and a quasi-two-dimensional approach for confined fluids [E. Buenrostro-González, C. Lira-Galeana, A. Gil-Villegas, J. Wu, AIChE J., 50 (2004) 2552–2570; A. Martínez, M. Castro, C. McCabe A. Gil-Villegas, J. Chem. Phys. 126 (2007) 074707, respectively], both based on the Statistical Associating Fluid Theory for Potentials of Variable Range [A. Gil-Villegas, A. Galindo, P.J. Whitehead, S.J. Mills, G. Jackson, A.N. Burgess, J. Chem. Phys. 106 (1997) 4168–4186]. The theory is applied to model adsorption isotherms from experimental data of asphaltenes extracted from a dead sample of heavy crude oil from a Mexican reservoir. The theoretical results give the right Langmuir Type II adsorption isotherms observed experimentally. The model requires the determination of ten molecular parameters related to the size of the particles and the square-well potentials used to describe the particle–surface and particle–particle interactions at the bulk and adsorbed phases. Nine parameters are taken from previous published results about the behavior of asphaltenes in bulk phases and the adsorption of several molecular fluids onto activated carbon and graphite surfaces. The remaining parameter, the energy strength of the particle–surface interaction, is adjusted to reproduce the experimental data, obtaining values that are consistent with Molecular Mechanics calculations for asphaltenes adsorbed on different surfaces and solutions. Although the agreement between theory and experiments shows some deviations at low bulk concentrations, the model reproduces adsorption data at high concentrations where other semi-empirical approaches fail.     

Carbon dioxide adsorption by synthetic mordenite isotherms and differential heat of adsorption

Carbon dioxide adsorption isotherms by synthetic mordenite are determined over the ranges 10^?5 atm<P<50 atm and ?77°C<T<+160°C, differential heats of adsorption calorimetrically measured over 10^?5atm<P<l atm and ?77°C< T< + 120°C.Differential heat curves show two adsorption ranges with different energetic characteristics. At the transition a very marked maximum appears.The integral molar entropy and specific heat of adsorbed phase are calculated.Two methods for adsorbed phase density determination are proposed:(i) The first, straight from maximal adsorbed amount at 50 atm. where adsorption isotherms level out.(ii) The second, based on adsorption potential theory with an elementary graphical determination.When the adsorbed phase density is so determined, Dubinin's equation can be successfully applied.     

Substantiation of the method of an “ideal adsorption solution” for calculating the adsorption of binary vapor mixtures from the individual isotherms

Conclusions Substantiation was offered for the method of an ideal adsorption solution, which was proposed by Myers and Prausnitz to calculate the adsorption equilibrium of binary vapor mixtures with a solid adsorbent from the individual isotherms, for the particular case of adsorption systems for which is fulfilled the condition of an affinity of the adsorption isotherms of the vapor mixtures at a constant composition of the adsorption solution, and the condition of the additivity of the affinity coefficient of such a solution.Translated from Izvestiya Akademii Nauk SSSR, Seriya, Khimicheskaya, No. 1, pp. 171–173, January, 1972.     

Calculation of the adsorption isotherm from solution on a nonporous adsorbent according to the adsorption isotherms of the components from the gas phase

Conclusions A solution of the problem of calculating the isotherm of excess adsorption from binary liquid solutions on a nonporous (or wide-pored) adsorbent according to the individual adsorption isotherms of the components from the gas phase and the phase diagram of the volume solution is given.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 1, pp. 3–6, January, 1973.     

Enantioseparation on molecularly imprinted monoliths--preparation and adsorption isotherms

In this paper we report the synthesis of monolithic molecularly imprinted polymers (monolithic MIPs) for the chromatographic separation of Z-phenylalanine by “in situ” polymerisation. Four different functional monomers were used for the preparation and the columns were tested with a standard mobile phase of 1% acetic acid in acetonitrile. 4-Vinylpyridine has shown the best separation performance with a resolution factor R>1 and was chosen for a further improvement of the imprinted stationary phase, which leads to a resolution factor R=2.06. All monoliths investigated have surface areas of around 400 m² g⁻¹. The columns containing 4-vinylpyridine as functional monomer were characterised by frontal analysis and peak fitting method. These methods allow the determination of the adsorption isotherm which quantify the separation potential of the imprinted monoliths.