Chemical Kinetics with Mathcad and Maple

In this work, the results of kinetic studies of Pd(II) chloride complex ion adsorption process on activated carbon are presented. The experiments were conducted for different temperatures, initial concentrations of Pd(II) complex ions as well as for different amount of activated carbon. These results confirmed that the mechanism of the adsorption process is complex and can be described by two step reaction model. A new form of the adsorption isotherm based on kinetic mechanism is suggested and can describe the observed process. Moreover, it was shown that for specific conditions, the proposed isotherm can be transformed into the Freundlich's isotherm. The activation energies of the subsequent stages of the studied process were determined and are equal to E 1 = -9 ± 9, E 2 = -6 ± 1.4 and E 3 = -85 ± 8.9 J mol -1 , respectively. It is also suggested that the observed positive Gibbs energy change during adsorption can be related only to the first step of this process which does not lead to the final product.

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“…4 soln is concentration of Pd(II) in the solution The general solution to these equations has the form [21]:…”

Section: Resultsmentioning

confidence: 99%

Kinetic modeling of the adsorption process of Pd(II) complex ions onto activated carbon

Wojnicki

Fitzner

2018

Reac Kinet Mech Cat

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“…To fit the experimentally determined T -jump kinetic traces, the following master equation is solved numerically by a fourth-order Runge-Kutta algorithm, 55, 73

\frac{d}{d t} C false(t false) = α^{T} \cdot r false(t false)

where C is the concentration vector, r is the rate vector, and α T is the transpose of the stoichiometric matrix. Specifically, the C vector is a 2 14 × 1 matrix, while the α matrix is 14∙2 14 × 2 14 , where the rows correspond to the elementary reactions in the model and the columns represent the states of the peptide.…”

Section: Resultsmentioning

confidence: 99%

Microscopic nucleation and propagation rates of an alanine-based α-helix

Lin

Gai

2017

Phys. Chem. Chem. Phys.

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An infrared temperature-jump (T-jump) study by Huang et al. (Proc. Natl. Acad. Sci. 2002 99, 2788–2793) showed that the conformational relaxation kinetics of an alanine-based α-helical peptide depend not only on the final temperature (Tf) but also on the initial temperature (Ti) when Tf is fixed. Their finding indicates that the folding free energy landscape of this peptide is non-two-state like, allowing for the population of conformational ensembles with different helical lengths and relaxation times in the temperature range of the experiment. Because α-helix folding involves two fundamental events, nucleation and propagation, the results of Huang et al. thus present a unique opportunity to determine their rate constants – a long-sought goal in the study of the helix-coil transition dynamics. Herein, we capitalize on this notion and develop a coarse-grained kinetic model to globally fit the thermal unfolding curve and T-jump kinetic traces of this peptide. Using this strategy, we are able to explicitly determine the microscopic rate constants of the kinetic steps encountered in the nucleation and propagation processes. Our results reveal that the time taken to form one α-helical turn (i.e., an α-helical segment with one helical hydrogen bond) is about 315 ns, whereas the time taken to elongate this nucleus by one residue (or backbone unit) is 5.9 ns, depending on the position of the residue.

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“…Note that these UQ methods are programmed and executed with inhouse codes using MATLAB. These examples include a batch chemical reactor and a dynamic model describing the live‐cell signaling . As compared to other UQ techniques, such as MC and nonintrusive SC, the results show the advantage of the proposed method in terms of UA accuracy and computational efficiency, especially when the number of uncertainties is moderately large and the system has complex as well as highly nonlinear forms.…”

Section: Numerical Examplesmentioning

confidence: 89%

Probabilistic surrogate models for uncertainty analysis: Dimension reduction‐based polynomial chaos expansion

Son

2019

Numerical Meth Engineering

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Summary This paper presents an approach for efficient uncertainty analysis (UA) using an intrusive generalized polynomial chaos (gPC) expansion. The key step of the gPC‐based uncertainty quantification (UQ) is the stochastic Galerkin (SG) projection, which can convert a stochastic model into a set of coupled deterministic models. The SG projection generally yields a high‐dimensional integration problem with respect to the number of random variables used to describe the parametric uncertainties in a model. However, when the number of uncertainties is large and when the governing equation of the system is highly nonlinear, the SG approach‐based gPC can be challenging to derive explicit expressions for the gPC coefficients because of the low convergence in the SG projection. To tackle this challenge, we propose to use a bivariate dimension reduction method (BiDRM) in this work to approximate a high‐dimensional integral in SG projection with a few one‐ and two‐dimensional integrations. The efficiency of the proposed method is demonstrated with three different examples, including chemical reactions and cell signaling. As compared to other UA methods, such as the Monte Carlo simulations and nonintrusive stochastic collocation (SC), the proposed method shows its superior performance in terms of computational efficiency and UA accuracy.

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Chemical Kinetics with Mathcad and Maple

Cited by 40 publications

References 0 publications

Kinetic modeling of the adsorption process of Pd(II) complex ions onto activated carbon

Kinetic modeling of the adsorption process of Pd(II) complex ions onto activated carbon

Microscopic nucleation and propagation rates of an alanine-based α-helix

Probabilistic surrogate models for uncertainty analysis: Dimension reduction‐based polynomial chaos expansion

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