Statistical Rate Theory Approach Research Articles

Atomistic investigation on the kinetic behavior of vapour adsorption and cluster evolution using a statistical rate theory approach.

The kinetic behavior of vapor adsorption on a solid surface in an isobaric-isothermal system is investigated by means of molecular dynamics simulations combined with theoretical studies through a statistical rate theory approach. The molecular insights into the formation and evolution of clusters in the adsorbate are presented. Results show that the argon vapor is adsorbed on the silicon surface as different types of clusters. In the initial stage of adsorption, the empty adsorption sites on the surface decrease, and the adsorbed single-molecule-cluster grows rapidly and dominates the interface. The increasing rate of the adsorbed cluster and the declining rate of the empty adsorption site are dependent on the pressure ratio. For a large pressure ratio, the single-molecule-clusters are aggregated to incubate large clusters, and the fraction of a single-molecule-cluster is decreased with time. When the adsorption isotherm is determined, the chemical potential of the adsorbed cluster is expressed from the zeta isotherm model. Then the adsorption kinetics are analyzed through the statistical rate theory. The molecular exchange rate and the instantaneous driving force are calculated. The higher pressure ratio induces the larger chemical potential difference and accelerates the net adsorption rate. The adsorption kinetics derived from MD simulations are in close agreement with the theoretical analysis of the statistical rate theory.

Adsorption kinetics at the solid/solution interface: Statistical Rate Theory (SRT) at close to equilibrium

In this study, derivation of the modified pseudo first order kinetics model (MPFO) from Statistical Rate Theory (SRT) has been investigated. I have shown that derivation of MPFO model from SRT cannot be true. I have proved that the assumptions that applied for derivation of MPFO from SRT in the previous research are false. These assumption cause that K1s` (adsorption/desorption rate at equilibrium) and KL(Langmuir constant) will become zero simultaneously. In this research work I have shown that K1s` and KL cannot be zero simultaneously. Finally, I have concluded that derivation of MPFO from SRT approach is false.

Vapour adsorption kinetics: statistical rate theory and zeta adsorption isotherm approach.

The equilibrium zeta adsorption isotherm for vapours indicates the amount adsorbed is finite for vapour-phase pressures approaching the saturation value, and is strongly supported by experimental measurements for a number of different vapour-solid surface systems. This isotherm assumes the adsorbate consists of differently sized molecular clusters in local equilibrium rather than the adsorbate being in layers. We use the local-equilibrium approximation and develop a method to determine the expression for chemical potential of the adsorbate in terms of the amount adsorbed, nA(t). This allows us to apply statistical rate theory to calculate nA(t) at five different vapour-phase pressures, xV (≡PV/Psat), in terms of a parameter, re. Statistical rate theory indicates that re describes the dynamics of a given isolated system under equilibrium conditions. We consider two methods for determining the value of re that give the best agreement with the measurements performed at each of five values of xV. In one method, we assume, re, is a function of both temperature, T, and pressure, xV, and determine the five best-fit values of re. In the experiments, xV changes by a factor of more than two, but the standard deviation in the re values is 6%. In the second method, we assume re is a function only of T; and find that the value of re is not changed significantly. In all cases, the calculated nA(t) agree with the measurements.

Solute permeation through lipid membrane: Statistical Rate Theory approach

A new model describing transport across lipid membrane was developed based on the Statistical Rate Theory (SRT) of interfacial transfer. In our calculations we replaced lipid membrane with well-defined octanol membrane. According to SRT the rate of solute transfer across water/octanol interface depends on the ratio of solute concentration in the both phases and the water/octanol partition coefficient. In the case of thick membranes and thick diffuse layers our model predicts almost linear dependence of flux on concentration gradient. However, for thin membranes and thin diffuse layers flux becomes nonlinear function of concentration gradient, especially for more polar solutes and higher gradients.

The interaction effects between adsorbed molecules in the Statistical Rate Theory approach to the kinetics of mixed-gas adsorption

Kinetics of adsorption of single gases and their binary mixture on an energetically homogeneous surface are studied theoretically using the Statistical Rate Theory (SRT) of interfacial transport. Additionally, the influence of intermolecular interactions on the rate of adsorption is modeled using the Regular Adsorbed Solution Theory. The theoretical results are classified and assigned to different types of experimental conditions under which the changes of the adsorbate concentration near the adsorbing surface are considerable or they are negligible. Predictions of the theory are verified using adsorption data measured in real systems.

Statistical Rate Theory Approach to Description of the pH-Dependent Kinetics of Metal Ion Adsorption

The influence of noncontrolled pH conditions on the kinetics of metal ion adsorption at solid/solution interfaces is studied by applying the SRT (statistical rate theory) approach. The quasi-equilibrium assumption inherent in SRT imposes the identical mathematical form of expressions for both the competitive adsorption and ion exchange models. However, they can be differentiated on the basis of a time-dependent profile of pH in the bulk solution. The developed model accounts for both multidentate binding mode and different proton/metal exchange ratios. Its applicability was shown by the quantitative analysis of the kinetic data reported by Choi and Yun.

A Novel Two-Resistance Model for Description of the Adsorption Kinetics onto Porous Particles

A novel two-resistance model for description of the sorption kinetics by porous particles has been proposed. The model takes into account two kinetic steps of different kinds which are involved in the overall sorption process rate: (i) the rate of solute diffusion in pores of the sorbent particles having uniform sizes and characterized by the homogeneous intraparticle diffusion coefficient and (ii) the rate of a direct adsorption/desorption process on the surface, described by applying the statistical rate theory (SRT) approach. Two different kinds of sorbent particles geometry are considered: the spherical and the plane particles, having their dimension characterized by radius and thickness, respectively. The meaning of the parameters which influence the sorption kinetics has been discussed. The results make it possible to judge which conditions have to be fulfilled to consider only one kinetic step and, thus, to simplify the theoretical description of a given system. The conclusion has been drawn that the concave character of kinetic sorption isotherms plotted in the function of the square root of time is of a general nature and is connected with the situation when at least two different processes are involved in controlling the sorption kinetics.

On the applicability of the pseudo-second order equation to represent the kinetics of adsorption at solid/solution interfaces: a theoretical analysis based on the statistical rate theory

It is shown that the empirical pseudo-second order kinetic equation is a very efficient formula to correlate the kinetic data generated by applying theoretical expressions developed from the fundamental SRT (Statistical Rate Theory) approach to the interfacial transport. This is especially true when the most popular linear representation is used in which time/adsorbed amount is plotted vs. time. However, the commonly observed goodness of such linear plots does not necessarily speak for the applicability of the pseudo-second order kinetic equation. A reliable estimation, for instance, of the equilibrium adsorbed amount is possible only when a substantial part of a kinetic isotherms corresponds to the conditions close to equilibrium. Energetic surface heterogeneity increases the goodness of these linear regressions. Then, experimental errors have only little effect on the pseudo-second linear plots.

Description of Desorption Kinetics at the Solid/Solution Interface Based on the Statistical Rate Theory

Desorption is one of the popular methods for the design and regeneration of catalysts. For better understanding and modeling of this process, it is important to have models with theoretical basis. In the present work, the statistical rate theory (SRT) approach was used for the description of desorption kinetics at the solid/solution interface. Based on the SRT approach, two rate equations at initial times of desorption have been derived. A comparison between these two rate equations was done based on numerically generated kinetic data points ( t, q) by the SRT equation. On the basis of experimental data, it has been shown that the kinetics of desorption can be analyzed by the SRT rate equation. Also, the experimental data approve the accuracy of derived rate equations at initial times of desorption.

Statistical Rate Theory Approach to Kinetics of Competitive Adsorption at the Solid/Solution Interface

Removal of pollutants from aqueous solution is an important process, and adsorption is one of the most popular methods for this. Wastewater usually contains several solutes, and therefore the adsorption process in some systems is competitive. The lack of an adequate model for kinetics of competitive adsorption and also difficulties in modeling led only to the reporting of experimental data in most publications. Here we propose the first description of the competitive adsorption kinetics at the solid/solution interface based on the statistical rate theory (SRT) approach. For derivation of rate equations based on the SRT approach, at first we derived the chemical potential of adspecies for a competitive adsorption based on statistical thermodynamics from the partition function of a canonical ensemble. The derived kinetic equations for competitive adsorption are able to describe quantitatively the kinetics of some experimental data. Since the analytical solution of the derived rate equations led to complex expressions, we used stochastic numerical simulation for modeling of experimental data by the derived SRT equations. The present study shows that the stochastic numerical simulation is a powerful technique for modeling adsorption kinetics based on statistical rate theory.

Sticking coefficient and pressure dependence of desorption rate in the statistical rate theory approach to the kinetics of gas adsorption. Carbon monoxide adsorption/desorption rates on the polycrystalline rhodium surface

The statistical rate theory (SRT) fundamental kinetic equation for the Langmuir model of adsorption reveals some features which are difficult to understand. These are the infinitely high adsorption rate for zero coverage limit, infinitely high desorption rate at full saturation of the surface and dependence of the desorption rate on the adsorbate pressure. These features do not allow for a correct formulation of such important physical parameters as sticking and initial sticking probabilities. In this work these peculiar features of the fundamental SRT kinetic equation are discussed and explained. It is shown that the non-physical behavior of the SRT kinetic equation follows from neglecting possible changes of adsorbate concentration near the adsorbing surface. A simple model accounting for the changes of adsorbate concentration close to the adsorbing surface is discussed. As a result, it was shown that application of even such a simple model leads to fully physical behavior of the SRT kinetic equation for the Langmuir model of adsorption. The model is used to build up equations describing the kinetics of adsorption/desorption of carbon monoxide on the energetically heterogeneous rhodium surface. The values of the parameters in these equations were determined from the analysis of equilibrium adsorption isotherms and from the description of experimental conditions. One additional parameter (having well defined limiting values) had to be adjusted in order to satisfactorily reproduce kinetic data. The experimental data used in this work were taken from the article by Yamada and Tamaru (T. Yamada and K. Tamaru, Surf. Sci., 1984, 138, L155).

Kinetics of Isothermal Gas Adsorption on Heterogeneous Solid Surfaces: Equations Based on Generalization of the Statistical Rate Theory of Interfacial Transport

Two generalizations of the statistical rate theory (SRT) approach have been developed for the kinetics of adsorption/desorption on/from heterogeneous surfaces and then compared to each other. The first of them is an improved version of the description based on the condensation approximation (CA) used by us. The other one is an exact generalization for the Langmuir model of adsorption. The latter generalization does not lead to simple analytical expressions for the rate of adsorption/desorption as the CA approach does for typical adsorption energy distributions. The comparison of these two approaches suggests, however, that at small and very high surface coverages the CA approach may not be sufficiently accurate. Very intriguing results are obtained when the developed SRT kinetic equations are applied to describe the kinetics of sorption in carbon molecular sieves. It appears that the fit of experimental data is then equally good as that obtained by assuming that the rate of sorption is controlled by surface diffusion in pores. Also, it is shown that the square-root dependence on time of adsorption at small initial coverages cannot be treated as a definite proof for that sorption proceeds via surface diffusion, as is commonly assumed.

The influence of a small amount of active sites on the adsorption kinetics of nitrogen on ruthenium

Recently, Dahl et al. [S. Dahl, A. Logadottir, R.C. Egeberg, J.H. Larsen, I. Chorkendorff, E. Törnqvist, J.K. Nørskov, Phys. Rev. Lett. 83 (1999) 1814; S. Dahl, E. Törnqvist, I. Chorkendorff, J. Catal. 192 (2000) 381] have proposed very interesting hypothesis that the rate of dissociation and adsorption of nitrogen on Ru(0 0 0 1) facet is totally dominated by the presence of a small amount of step sites on Ru(0 0 0 1) terraces. Following this idea, a kinetic model, based on applying the Statistical Rate Theory approach, was developed in order to explain if such mechanism is able to explain the observed features of the system N 2/Ru(0 0 0 1). As a result, it was stated that the activation barrier for adsorption on the active (step) sites is equal to 36 kJ/mol; in turn, the adsorption energy of nitrogen atoms on the active sites is 43 kJ/mol. It implies that the rate of adsorption via the active sites is much faster than direct adsorption on the three-fold hollow sites; moreover, the occupation of the active sites is always close to zero at the investigated temperatures, so they are not blocked and may act as an indirect channel for adsorption. Thus, the rate of nitrogen adsorption on Ru(0 0 0 1) surface is governed by the rate of diffusion of nitrogen atoms from the active sites into the three-fold hollow sites. The analysis of thermodesorption spectra revealed an important role of repulsive interactions between the N atoms adsorbed on the hollow sites, the associated interaction parameter between nearest neighbors was estimated to be 5 kJ/mol. The presence of small amount of gold on Ru(0 0 0 1), apart of blocking the active sites, seems to remove the repulsion between nitrogen atoms.

A More Accurate Prediction of Liquid Evaporation Flux

In this work, a more accurate prediction of liquid evaporation flux has been achieved. The statistical rate theory approach, which is recently introduced by Ward and Fang and exact estimation of vapor pressure in the layer adjacent to the liquid–vapor interface have been used for prediction of this flux. Firstly, the existence of an equilibrium layer adjacent to the liquid-vapor interface is considered and the vapor pressure in this layer and its thickness calculated. Subsequently, by using the Fick’s second law, an appropriate vapor pressure expression for the pressure of equilibrium layer is derived and by this expression and the statistical rate theory approach, evaporation flux is predicted more accurately than the previous work. Finally, some novel steady state evaporations are simulated and the effects of both liquid and vapor temperature and the effect of the length of the evaporation chamber on the evaporation flux are investigated.

Kinetics of dissociative hydrogen adsorption on the (1 0 0) nickel single crystal face: a statistical rate theory approach

For its importance for many catalytic reactions, the dissociative adsorption of hydrogen on the nickel surfaces has been one of the most extensively investigated subjects during the last few decades. Here the pressure and coverage dependence of that adsorption kinetics is of a particular importance. Meanwhile, although relevant experimental data were measured as long as 30 years ago, no theoretical expressions have been proposed so far which could quantitatively describe the pressure and coverage dependence of that dissociative adsorption kinetics. The attempts based on the classical TAAD approach (theory of activated adsorption/desorption) resulted in a clearly seen failure. Here a new description is proposed based on the SRT (statistical rate theory) approach, linking the features of that dissociative adsorption to the chemical potentials in the gas and adsorbed phases. Based on that approach, theoretical expressions are developed, and next used to analyze the kinetics of hydrogen dissociative adsorption on the single crystal face Ni(1 0 0) surface. By using the same set of adsorption parameters an excellent simultaneous fit is obtained of both kinetic isotherms measured at different pressures, and equilibrium adsorption isotherms measured at different temperatures. These parameters agree well with the related values found for this system in the quantum mechanical calculations.

A Statistical Rate Theory Approach to Kinetics of Dissociative Gas Adsorption on Solids

The reaction rate in some important catalytic systems is governed by kinetics of dissociative adsorption of reactants on a catalyst surface. This, for instance, is the case of ammonia synthesis. Since the beginning of theprevious century, the expressions based on the theory of activated adsorption/desorption (TAAD) approach have commonly been attempted to analyze the pressure and coverage dependence of the observed kinetics. However, neither the simple classical TAAD approach nor its further modifications attempted during the last few decades could explain the kinetics observed in many important systems. In particular, it concerned the kinetics of dissociative adsorption in chemisorption systems. For instance, no successful attempt was published to describe quantitatively the pressure and coverage dependence of the kinetics of the dissociative dinitrogen adsorption on iron surfaces, studied for a large part of the previous century, in relation to ammonia synthesis. Here we propose the first successful description of that dissociative adsorption kinetics based on the new statistical rate theory approach, linking the rate of the dissociative kinetics to the chemical potentials in the bulk and the adsorbed phases. The developed theoretical expressions are able to describe quantitatively the kinetic isotherms of dinitrogen dissociative adsorption, recorded experimentally at the pressures ranging from 10 - 6 to 10 2 Torr and the temperatures ranging from 140 to 485 K. These expressions also explain some unusual features of this dissociative kinetics, such as small negative apparent activation energy for the dissociative adsorption and the pressure invariance of the kinetics at high dinitrogen pressures, when considered as a function of exposure.

Kinetics of gas adsorption on strongly heterogeneous solid surfaces: A statistical rate theory approach

A fundamental theoretical investigation is carried out on the applicability of the Condensation Approximation (CA) method of generalizing the theoretical description of the kinetics of gas adsorption/desorption on/from energetically heterogeneous solid surfaces. Our investigation implies that the commonly observed Elovich’s behavior is a consequence of carrying out kinetic experiments at conditions not far from equilibrium. This striking result seems to suggest that the applicability of that popular approximation deserves further fundamental study.

Effect of pressure on the rate of evaporation from capillaries: statistical rate theory approach

Statistical rate theory (SRT) leads to an expression for the evaporation rate that is in terms of independently determined parameters. A series of experiments have been conducted in which water evaporated from capillaries closed at one end and partially filled to different heights. From the measured evaporation rate, the SRT prediction of the vapor-phase pressure was compared with that determined by an independent technique. When the assumptions used in this technique were valid, the value of the pressure predicted from SRT and from this technique differed by only 0.004 Pa.

Application of the statistical rate theory to the computer simulations of adsorption kinetics

The statistical rate theory (SRT) has frequently been used for interpretation of dynamics of many interfacial processes. In the case of adsorption kinetics it has successfully been applied to well-defined systems as well as to the systems where the energetic heterogeneity of solid surface plays significant role.Because the SRT approach involves the knowledge about the chemical potential of the adsorbed phase, different approximate expressions have been applied for estimating that chemical potential. Today, the most useful tool, for calculation of the chemical potential of the adsorbed phase, are the computer simulations. However, there are no algorithms of computer simulations of adsorption kinetics based on the SRT approach available in the literature. So, the main goal of this publication is design and investigation of such algorithm.The elaborated algorithm gives the results which are in agreement with slightly modified standard SRT equation for adsorption kinetics. This modification has its source in some dynamic limitations which the SRT equation alone does not account for.

Theoretical Description of the Kinetics of Proton Adsorption at the Oxide/Electrolyte Interface Based on the Statistical Rate Theory of Interfacial Transport and the 1pK Model of Surface Charging

A new approach describing ion adsorption kinetics on metal oxide surfaces from electrolyte solution is presented. It was assumed that the diffusion of ions to the oxide surface is fast, and the kinetics of ion adsorption is controlled by the transition of ions from the bulk to the adsorbed state. The transition rate is considered in terms of the statistical rate theory (SRT) of interfacial transport. To simplify the theoretical considerations, the simple Gouy−Chapman model of the electric double layer is applied together with the 1pK protonation model. The above theoretical approach was used to analyze the rate of proton adsorption. Additionally, that kinetic model was applied to study the hysteresis of potentiometric titration curves and the heat of proton adsorption (i.e., difference in the measured quantity at a given pH from acid and base titration). According to the SRT approach, the rate of hydrogen ion adsorption strongly depends on pH, which has a significant influence on numerically simulated titration curves and the heat of proton adsorption.