Diffusion in High Occupancy Zeolites

1987 ◽  
Vol 111 ◽  
Author(s):  
James Wei
Keyword(s):  
Activation Energy ◽  
Random Walk ◽  
Small Molecules ◽  
Irreversible Thermodynamics ◽  
Low Energy ◽  
The Other ◽  
Diffusion Region ◽  
Channel Diameter ◽  
One Dimensional ◽  
Activation Energy Of Diffusion

AbstractDiffusion of small molecules in zeolites most often takes place in the configurational diffusion region, where the molecular diameter is approximately the same or slightly greater than the channel diameter. Since two molecules may not pass each other in a pore, the random walk based diffusion equation does not apply under high occupancy conditions in zeolites with one-dimensional pores.For zeolites with multi-dimensional pores, such as ZSM-5 and A, one sometimes encounters the counter-intuitive result that diffusivity dramatically rises with occupancy. There are two explanations for this behavior: one from irreversible thermodynamics and the Darken equation, which predicts that diffusivity will always rise with occupancy; the other from a Markov model of random activated jumps between low energy positions, such as pore crossings, which predicts that diffusivity will increase with occupancy if the activation energy of diffusion decreases with occupancy– such as due to swelling.

scholarly journals The influence of resistivity gradients on shock conditions for a Petschek reconnection geometry

2016 ◽  
Vol 34 (4) ◽  
pp. 421-425
Author(s):  
Christian Nabert ◽  
Karl-Heinz Glassmeier
Keyword(s):  
Magnetic Field ◽  
Necessary Conditions ◽  
The Other ◽  
Diffusion Region ◽  
Two Dimensional ◽  
Characteristic Velocity ◽  
Magnetohydrodynamic Equations ◽  
One Dimensional ◽  
The Magnetic Field ◽  
Alfven Velocity

Abstract. Shock waves can strongly influence magnetic reconnection as seen by the slow shocks attached to the diffusion region in Petschek reconnection. We derive necessary conditions for such shocks in a nonuniform resistive magnetohydrodynamic plasma and discuss them with respect to the slow shocks in Petschek reconnection. Expressions for the spatial variation of the velocity and the magnetic field are derived by rearranging terms of the resistive magnetohydrodynamic equations without solving them. These expressions contain removable singularities if the flow velocity of the plasma equals a certain characteristic velocity depending on the other flow quantities. Such a singularity can be related to the strong spatial variations across a shock. In contrast to the analysis of Rankine–Hugoniot relations, the investigation of these singularities allows us to take the finite resistivity into account. Starting from considering perpendicular shocks in a simplified one-dimensional geometry to introduce the approach, shock conditions for a more general two-dimensional situation are derived. Then the latter relations are limited to an incompressible plasma to consider the subcritical slow shocks of Petschek reconnection. A gradient of the resistivity significantly modifies the characteristic velocity of wave propagation. The corresponding relations show that a gradient of the resistivity can lower the characteristic Alfvén velocity to an effective Alfvén velocity. This can strongly impact the conditions for shocks in a Petschek reconnection geometry.

Soliton model for collective proton transfer in extended realistic hydrogen-bonded systems

1992 ◽  
Vol 439 (1906) ◽  
pp. 211-226 ◽  
Keyword(s):  
Activation Energy ◽  
Proton Transfer ◽  
Collective Variable ◽  
Continuum Approximation ◽  
Soliton Model ◽  
One Dimensional ◽  
Hydrogen Bonded Systems ◽  
Activation Energy Of Diffusion ◽  
Proton Defect ◽  
Hydrogen Bonded

The discretized version of the collective proton transfer in extended one-dimensional hydrogen-bonded systems with the Lippincott-Schroeder potential is studied. A discrete quasi-kink is compared with its continuum approximation ‘image’. The Peierls-Nabarro activation energy of diffusion of collective proton defect is estimated. The concept of local proton potentials is introduced on the background of the kink-carrier collective variable.

scholarly journals Mathematical approach to human reaction

2020 ◽  
Author(s):  
Takuya Yabu
Keyword(s):  
Random Walk ◽  
Probability Density Function ◽  
Probability Distribution ◽  
Probability Density ◽  
Density Function ◽  
The Other ◽  
Mathematical Approach ◽  
One Dimensional ◽  
Human Reaction ◽  
Fixed Distribution

I thought about how to get the magnitude from the event and the reaction of the other party. Evaluating the values of events and opponents' reactions using a one-dimensional random walk shows that the probability density function of the values of events and opponents' reactions has a fixed probability distribution. Similarly, I have shown that the functions that determine the magnitude of events and reactions are also represented by a fixed distribution. Therefore, I also showed that when individuals gather to form a group, the functions that determine the magnitude of events and reactions as a group are also represented by a fixed distribution. Also, as an application example of this model, I described how to show my reaction and what to do when the magnitude of the event is large.

scholarly journals Moving boundary problems governed by anomalous diffusion

2012 ◽  
Vol 468 (2147) ◽  
pp. 3348-3369 ◽  
Author(s):  
Christopher J. Vogl ◽  
Michael J. Miksis ◽  
Stephen H. Davis
Keyword(s):  
Anomalous Diffusion ◽  
Moving Boundary ◽  
The Other ◽  
Diffusion Region ◽  
Moving Boundary Problems ◽  
Boundary Problems ◽  
One Dimensional ◽  
Moving Interface ◽  
Mean Squared Displacement ◽  
The One

Anomalous diffusion can be characterized by a mean-squared displacement 〈 x 2 ( t )〉 that is proportional to t α where α ≠1. A class of one-dimensional moving boundary problems is investigated that involves one or more regions governed by anomalous diffusion, specifically subdiffusion ( α <1). A novel numerical method is developed to handle the moving interface as well as the singular history kernel of subdiffusion. Two moving boundary problems are solved: the first involves a subdiffusion region to the one side of an interface and a classical diffusion region to the other. The interface will display non-monotone behaviour. The subdiffusion region will always initially advance until a given time, after which it will always recede. The second problem involves subdiffusion regions to both sides of an interface. The interface here also reverses direction after a given time, with the more subdiffusive region initially advancing and then receding.

Understanding Conformational Entropy in Small Molecules

2020 ◽  
Author(s):  
Lucian Chan ◽  
Garrett Morris ◽  
Geoffrey Hutchison
Keyword(s):  
Small Molecules ◽  
Degrees Of Freedom ◽  
Absolute Error ◽  
Low Energy ◽  
Standard Entropy ◽  
Free Energies ◽  
Conformational Entropy ◽  
Wide Range ◽  
High Degree ◽  
Empirical Corrections

The calculation of the entropy of flexible molecules can be challenging, since the number of possible conformers grows exponentially with molecule size and many low-energy conformers may be thermally accessible. Different methods have been proposed to approximate the contribution of conformational entropy to the molecular standard entropy, including performing thermochemistry calculations with all possible stable conformations, and developing empirical corrections from experimental data. We have performed conformer sampling on over 120,000 small molecules generating some 12 million conformers, to develop models to predict conformational entropy across a wide range of molecules. Using insight into the nature of conformational disorder, our cross-validated physically-motivated statistical model can outperform common machine learning and deep learning methods, with a mean absolute error ≈4.8 J/mol•K, or under 0.4 kcal/mol at 300 K. Beyond predicting molecular entropies and free energies, the model implies a high degree of correlation between torsions in most molecules, often as- sumed to be independent. While individual dihedral rotations may have low energetic barriers, the shape and chemical functionality of most molecules necessarily correlate their torsional degrees of freedom, and hence restrict the number of low-energy conformations immensely. Our simple models capture these correlations, and advance our understanding of small molecule conformational entropy.

Electroreduction of Cr(VI) from solutions of bifunctional alcohols

1989 ◽  
Vol 54 (10) ◽  
pp. 2638-2643
Author(s):  
David I. Balanchivadze ◽  
Tamara R. Chelidze ◽  
Jondo J. Japaridze
Keyword(s):  
Activation Energy ◽  
Ethylene Glycol ◽  
Electrode Surface ◽  
Frequency Factor ◽  
Limiting Current ◽  
Factor Log ◽  
Standard Rate ◽  
Standard Rate Constant ◽  
Chromate Ion ◽  
Activation Energy Of Diffusion

The effect of bifunctional alcohols ethylene glycol (EG) and 1,2-propylene glycol (1,2 PG) on the kinetic parameters for the irreversible chromate ion reduction were investigated by polarographic and coulometric methods of analysis. The electroreduction of chromate ion in neutral bifunctional alcohol solutions proceeds according to the scheme: Cr(VI)–Cr(III)–Cr(II) and the values of the standard rate constant k*0 decrease in the order H2O > EG > 1,2 PG. The values of real activation energy, Q, activation energy of diffusion, QD, and frequency factor log A° have been calculated. The obtained values of QD as well as Q proved the diffusion nature of limiting current. The values of the frequency factor log A° decrease in the order H2O > EG > 1,2 PG, which points to a less favourable orientation of the electroactive ions at the electrode surface in glycols.

scholarly journals Assessing a Linear Nanosystem's Limiting Reliability from its Components

2008 ◽  
Vol 45 (03) ◽  
pp. 879-887 ◽  
Author(s):  
Nader Ebrahimi
Keyword(s):  
Present State ◽  
Size Range ◽  
Two Dimensions ◽  
The Other ◽  
One Dimension ◽  
Present Moment ◽  
One Dimensional ◽  
The One ◽  
Fixed Moment ◽  
Nanosized Systems

Nanosystems are devices that are in the size range of a billionth of a meter (1 x 10-9) and therefore are built necessarily from individual atoms. The one-dimensional nanosystems or linear nanosystems cover all the nanosized systems which possess one dimension that exceeds the other two dimensions, i.e. extension over one dimension is predominant over the other two dimensions. Here only two of the dimensions have to be on the nanoscale (less than 100 nanometers). In this paper we consider the structural relationship between a linear nanosystem and its atoms acting as components of the nanosystem. Using such information, we then assess the nanosystem's limiting reliability which is, of course, probabilistic in nature. We consider the linear nanosystem at a fixed moment of time, say the present moment, and we assume that the present state of the linear nanosystem depends only on the present states of its atoms.

Lower Bounds to the Eigenvalues in One-Dimensional Problems by a Shift in the Weight Function

1970 ◽  
Vol 37 (2) ◽  
pp. 267-270 ◽  
Author(s):  
D. Pnueli
Keyword(s):  
Lower Bound ◽  
Weight Function ◽  
Lower Bounds ◽  
Variational Formulation ◽  
Ritz Method ◽  
The Other ◽  
One Dimensional ◽  
Systematic Shift ◽  
The One ◽  
Detailed Computation

A method is presented to obtain both upper and lower bound to eigenvalues when a variational formulation of the problem exists. The method consists of a systematic shift in the weight function. A detailed procedure is offered for one-dimensional problems, which makes improvement of the bounds possible, and which involves the same order of detailed computation as the Rayleigh-Ritz method. The main contribution of this method is that it yields the “other bound;” i.e., the one which cannot be obtained by the Rayleigh-Ritz method.

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