Adsorption integral equation via complex approximation with constraints: The Langmuir kernel

The relationship between the measured adsorption isotherm and unknown energy distribution function is described by so‐called adsorption integral equation, a linear Fredholm integral equation of the first kind. We consider the case of the Langmuir kernel when the equation can be reduced to the Stieltjes integral equation. A new method for solving the Stieltjes equation is developed. The method is based on the ideas of complex approximation with constraints. The numerical algorithms constructed on the base of this method allow reduction of the problem under consideration to linear or linear‐quadratic programming problems. The method is compared with the usual regularization methods. The obtained results can be useful for the evaluation of the experimental adsorption energy distribution from experimental data. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 191–200, 2000     

New method for the determination of adsorption space volume for sorbents with an arbitrary porous structure from excess gas adsorption isotherm measurements

A new method for the determination of adsorption space volume has been proposed, which is applicable to adsorbents with an arbitrary porous structure, including nonporous adsorbents with open surfaces. The method is based on the use of an experimental excess adsorption isotherm measured over a wide range of pressures in the equilibrium gaseous phase (as a rule up to 100–150 MPa) and the absolute adsorption isotherm equation with unspecified parameters in the most general form, given by statistical physics. The method has been tested for a number of adsorption systems, and it has been found that the result was always unambiguous, correct, and stable in the sense of input data.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2381–2385, December, 1995.     

What is the correct form of BET isotherm for modeling liquid phase adsorption?

In this work subtleties of application of BET isotherm for liquid phase adsorption is presented. It has been shown that direct use of the classical BET equation (which was developed for gas phase adsorption) to liquid phase adsorption leads to ambiguous and erroneous results. Some cases of misuse of BET equation for liquid phase adsorption have been revisited. By close examination of the development of the classical equation, the causes of misunderstandings were elucidated and the suitable form of the BET equation for liquid phase adsorption was developed. As case studies, the classical form of the BET equation along with the correct form of the equation for liquid phase have been applied for modeling liquid phase adsorption of methyl tert-butyl ether (MTBE) on perfluorooctyl alumina, phenol on activated carbon and pentachlorophenol on carbonized bark. It has been shown that direct application of the classical BET isotherm to liquid phase adsorption results in poor and erroneous estimation of the equation parameters. For example, in aqueous phase adsorption of MTBE on perfluorooctyl alumina, the monolayer adsorption capacity of the adsorbent was calculated as 9.7 mg/g instead of 3.3 mg/g or the saturation concentration of MTBE in water was calculated as 1212 mg/L instead of 42000 mg/L.     

Extension of the Dubinin-Astakhov equation for evaluating the micropore size distribution of a modified carbon molecular sieve

A new method for the characterization of the pore size distribution of microporous solids is applied on data obtained for activated carbon molecular sieve samples. In this method, based on the Dubinin-Astakhov equation, a simple numerical algorithm is used for the reconstruction of the micropore size distribution from the integral equation that represents the experimental nitrogen adsorption isotherm. The results are compared with the ones obtained on the basis of the well-known Horvath-Kawazoe method. The samples used in this study come from a carbon molecular sieve that has been treated with solutions of concentrated HNO3 at various temperatures and with solutions of H2O2 of various concentrations.     

Adsorption energy distribution from the Aranovich-Donohue lattice density functional theory

We propose a new methodology projected for the estimation of the adsorption energy distribution from the monolayer part of a single nitrogen adsorption isotherm determined at 77 K based on the lattice density functional theory (DFT) via the Aranovich-Donohue formalism. At first sight, the presented approach is computationally more difficult than a classical one. However, it is more flexible and comprehensible. Next, we developed a numerical program and used it for the estimation of the adsorption energy distribution from the experimental data on carbon black samples. The main nitrogen molecule-carbon black surface interaction energy can be estimated as approximately 7-8 kJ/mol, but the heterogeneity of the investigated materials differs significantly. Furthermore, we compare the results obtained from the lattice DFT via the Aranovich-Donohue formalism with the solution of the integral equation with the kernel represented by the classical monolayer localized Fowler-Guggenheim isotherm equation. The similarity between these two independent approaches is observed. The proposed methodology can be used for the investigation of the energetic heterogeneity of not only the carbonaceous materials but also the other "flat-surfaced" solids.     

A new integrated method for characterizing surface energy heterogeneity from a single adsorption isotherm

To characterize surface energy heterogeneity of fine particles, this paper presents an integrated strategy from a single adsorption isotherm. By coupling the well-known integral equation method and derivative isotherm summation (DIS) procedure based on a patchwise model, the newly proposed strategy could calculate adsorption energy distributions (AEDs) for different surface patches. Correspondingly, the surface heterogeneity of materials can be described by weighted summation of patch AEDs, that is, the total AED. The validity of this new method is confirmed by both tests of rutile nanoparticles and multiwalled carbon nanotubes (MWNTs). The total AED obtained by the new method agrees well with the result from solving the integral equation directly, and it shows that AED peaks can be assigned to specific energy patches of real surface exactly. Furthermore, a detailed comparison showed that some artificial oscillation in the results can be identified with the new strategy, and the patches with low area and high surface energy could be characterized as well. In conclusion, this strategy constructs a correspondence between derived AEDs and different patches of real surface, so it will be more effective to understand surface heterogeneity by using the adsorption probe method.     

超临界氢在活性炭上的吸附等温线研究

通过７７－２９８Ｋ范围内氢在ＡＸ－２１活性炭上的吸附数据，探讨如何用普通Ｉ－型等温线模型处理超临界条件下的吸附等温线，以获取关于超临界吸附系统的正确信息，结果表明，Ｌａｎｇｍｕｉｒ方程虽然可用来表达实验数据，但不能提供关于该吸附系统的任何可靠信息，Ｖｉｒｉａｌ方程虽不是整组数据的最好模型，但却能够可靠地确定Ｈｅｎｒｙ定律常数，然后可从ｖａｎｔＨｏｆｆ标绘决定等量吸附热，通过将实验数据拟合到Ｄｕｂｉ     

Determination of the entropy of formation of the Pt(111)∣ perchloric acid solution interface. Estimation of the entropy of adsorbed hydrogen and OH species

The entropy of formation of the interface between a Pt(111) electrode and a 0.1-M HClO₄ solution is calculated here for the first time from the temperature dependence of total charge vs potential curves following a thermodynamic analysis based on the electrocapillary equation. From this quantity, the absolute entropies of specifically adsorbed species (hydrogen and OH) can be estimated. The present method is an alternative treatment of data that overcomes some of the limitations involved in the approach that uses a generalized isotherm. However, it requires additional experimental data: the temperature coefficient of the potential of zero total charge of the working electrode and the temperature coefficient of the reference electrode. Comparison of the results obtained by both approaches shows that, for hydrogen adsorption, the agreement is reasonable, but the differences are larger for OH adsorption, thus showing the limitations inherent in the treatment based on the generalized isotherm. Dedicated to Professor Oleg Petrii on the occasion of his 70th birthday on August 24, 2007.     

Computation of dynamic adsorption with adaptive integral,finite difference,and finite element methods

Analysis of diffusion-controlled adsorption and surface tension in one-dimensional planar coordinates with a finite diffusion length and a nonlinear isotherm, such as the Langmuir or Frumkin isotherm, requires numerical solution of the governing equations. This paper presents three numerical methods for solving this problem. First, the often-used integral (I) method with the trapezoidal rule approximation is improved by implementing a technique for error estimation and choosing time-step sizes adaptively. Next, an improved finite difference (FD) method and a new finite element (FE) method are developed. Both methods incorporate (a). an algorithm for generating spatially stretched grids and (b). a predictor-corrector method with adaptive time integration. The analytical solution of the problem for a linear dynamic isotherm (Henry isotherm) is used to validate the numerical solutions. Solutions for the Langmuir and Frumkin isotherms obtained using the I, FD, and FE methods are compared with regard to accuracy and efficiency. The results show that to attain the same accuracy, the FE method is the most efficient of the three methods used.     

Correlation of Adsorption Equilibrium Data for Water Vapor on F-200 Activated Alumina

Single component adsorption equilibrium data for water vapor on commercially available activated alumina F-200 measured in a previous study (Serbezov, 2003) is correlated by two adsorption isotherm equations, both of which are based on the adsorption potential theory. The first equation is the well known Dubinin-Astakhov (D-A) equation. The second equation is obtained from a methodology proposed by Kotoh et al. (1993). It is referred to as a dual mechanism adsorption potential (DMAP) equation because it is a linear combination of two D-A terms with n = 1 where each term accounts for a specific mechanism of water retention. The D-A equation has two fitting parameters; the DMAP equation has three fitting parameters. The DMAP model provides a better fit for the adsorption data than the D-A model, while neither model describes the desorption data well. Analysis of the DMAP equation parameters shows that most of the water is retained by virtue of capillary condensation. In addition to fitting the experimental data, the heat of adsorption was calculated as function of the relative humidity and adsorbent loading. When capillary condensation is present, the heat of adsorption is only slightly higher than the latent heat of vaporization.     

奥美拉唑对映体在自制手性柱上吸附等温线的测定

采用吸附-脱附法研究了奥美拉唑对映体在自制的纤维素三苯基氨基甲酸酯涂敷型手性柱上的吸附性能,并用Langmuir方程拟合出吸附平衡方程.当样品浓度降低时,Langmuir方程退化为直线关系,直线斜率即为吸附平衡常数.这一吸附平衡常数与从脉冲色谱流出曲线求得的吸附平衡常数相等,表明吸附-脱附法测得的吸附平衡关系准确,Langmuir方程能描述这一吸附平衡关系.     

Choice of Model for Estimation of Adsorption Isotherm Parameters in Gradient Elution Preparative Liquid Chromatography

The inverse method is a numerical method for fast estimation of adsorption isotherm parameters directly from a few overloaded elution profiles and it was recently extended to adsorption isotherm acquisition in gradient elution conditions. However, the inverse method in gradient elution is cumbersome due to the complex adsorption isotherm models found in gradient elution. In this case, physicochemically correct adsorption models have very long calculation times. The aim of this study is to investigate the possibility of using a less complex adsorption isotherm model, with fewer adjustable parameters, but with preserved/acceptable predictive abilities. We found that equal or better agreement between experimental and predicted elution profiles could be achieved with less complex models. By being able to select a model with fewer adjustable parameters, the calculation times can be reduced by at least a factor of 10.

     

钙矾石吸附水的模型及等温式

钙矾石３ＣａＯ·Ａｌ２Ｏ３·３ＣａＳＯ４·３２Ｈ２Ｏ是水泥熟料水化的重要产物．它易被空气中的ＣＯ２ 分解,常温下的分解反应为３ＣａＯ·Ａｌ２Ｏ３·３ＣａＳＯ４·３２Ｈ２Ｏ ３ＣＯ２ →３ＣａＣＯ３ Ａｌ２Ｏ３·ｘＨ２Ｏ ３（ＣａＳＯ４·２Ｈ２Ｏ） （２６ -ｘ）Ｈ２Ｏ分解机理［１ ,２］是：Ｈ２Ｏ先吸附在钙矾石表面的活性中心上,吸附态水再吸收ＣＯ２ 成为Ｈ２ＣＯ３ ,然后碳酸与钙矾石反应,实现碳化（即分解）．其分解速率［３］为ｖ＝ｋｐｃｏ２ ｐＨ２ｏ（１）可见水的吸附是钙矾石碳化分解的关键步骤,因此探讨…     

Investigation of Quantitative Relationship between Adsorbed Amount of Solute and Solvent Concentration at Relatively High Solute Concentration by Frontal Analysis in RPLC

IntroductionManyfactors ,includingTraube’srule ,temperature ,solubilityofsolute ,inorganicsalt,thestructureofadsor bentandthenatureandconcentrationofthesolvent,affectsoluteadsorptioninaliquid solidsystem .1Amongthemsolventeffectsareofsignificance .Dispersionsofsilicaarestabilizedbyethanol.Itisreportedthatsolventfilmthick nessisgreatestinmixturesof 5 0percentethanolwithwa ter.2 TherepulsionbetweendispersedparticlesisabovetheeffectsfromthedoublelayerandvanderWaalsforces3andisattributedtostruc…     

Adsorption of inulinases in ion-exchange columns

Theuse of adsorption columns packed with ion-exchange resins for recovering, concentrating and purifying proteins is now widespread. The present work consists of a study on the dyamic behavior ofadsorption columns that uses two kinds of adsorbents: a cationic and an anionic resin. A frontal analysis of the columns was performed with experimental data obtained from Fructozyme, a mixture of inulinase en zymes. The parameters of a Langmuir type of isotherm and adsorption kinetics were obtained from experimental tests in a batch system. A numerical technique based on orthogonal colocation and a fourth-order Runge-Kutta method was coupled with a nonlinear optimization method to predict the coefficients of the rate equations, which are fundamental for scale-up purposes.     

Investigation of Quantitative Relationship between Adsorbed Amount of Solute and Solvent Concentration at Relatively High Solute Con－centration by Frontal Analysis in RPLC

In previous paper a new adsorption isotherm which relates the amount of solute absorbed to the solvent concentration is pro-posed and simplified, and it only can be used at lower solute concentration. In this article the scope of the new adsorption isotherm is extended and the expressions with three and four parameters are obtained. The equations with multi-parameters are valid when the adsorbed mounts are larger and show non-linear logarithmic relationships. Tests with a homologue of are-matic alcohols by frontal analysis in reversed phase liquid chro-matography demonstrate that the experimental results fit those equations well. In addition, the predicted values by the multi-parameters were found to fit the experimental values well also. The parameters have physical meaning only for the two-param-eter equation for the aromatic alcohols.     

Modified Langmuir Isotherm for the Description of Cluster Adsorption on Surface Lyophilic Centers

An adsorption isotherm that generalizes the Langmuir equation for the cluster adsorption of several interacting molecules at one center is proposed. In the case of the adsorption of water molecules at surface hydroxyl groups on modified silica it was shown that the energy and structure characteristics of such a system, determined by the ab initio method, agree with the experimental data and lead to satisfactory agreement with the parameters of the proposed isotherm. __________ Translated from Teoreticheskaya i Eksperimental'naya Khimiya, Vol. 41, No. 5, pp. 283–289, September–October, 2005.     

Measurement and theoretical analysis of the adsorption of supercritical methane on superactivated carbon

Adsorption/desorption isotherms of supercritical methane on superactivated carbon have been measured in the range of 0-10 MPa and 233-333 K (20 K interval). The reversibility of the physical adsorption process is acknowledged. The heat of adsorption of 16.5 kJ/mol is determined from the isotherms, and a new modeling strategy for isotherms with maximum is presented. The model yields fits to the experimental isotherms with precision of ?%, maintaining the constancy of the characteristic energy of adsorption. The exponent of the model equation expresses the pore size distribution feature of the adsorbent. The density of the supercritical adsor-bate is evaluated as a parameter of the model. It is shown that the conventional isotherm theory works too at supercritical condition if the limit state of supercritical adsorption is introduced into isotherm modeling.