On the application of mean value theorem to reaction kinetics over inhomogeneous surfaces

Monolithic materials have been widely used for over 20 years as catalyst supports in the abatement of emissions from mobile and stationary sources. They have been most successfully used for applications where high volumetric flow rates and potential for thermal shock are required. Their use is expected to further increase as new applications are found. This paper describes three new catalytic technologies which require a monolithic support; 1) the diesel oxidation catalyst commercialized in 1994 and 2) NO_x abatement with hydrocarbons in lean environments and 3) catalytically supported thermal combustion both of which are currently under development.     

Central limit theorem for chemical kinetics in complex systems

The prevalence of apparently first-order kinetics of reactant disappearance in complex systems with many possible reaction pathways is usually attributed to the dominance of a single rate limiting step. Here, we investigate another possible explanation: that apparently first-order kinetics might arise because the aggregate behavior of many processes, with varying order of reaction and rate constant, approaches a central limit that is indistinguishable from first-order behavior. This hypothesis was investigated by simulating systems of increasing complexity and deriving relationships between the apparent reaction order of such systems and various measures of their complexity. Transformation of a chemical species by parallel irreversible reactions that are zero-, first-, or second-order is found to converge to a central limit as the number of parallel reactions becomes large. When all three reaction orders are represented, on average, in equal proportions, this central limit is experimentally indistinguishable from first-order. A measure of apparent reaction order was used to investigate the nature of the convergence both stochastically and by deriving theoretical limits. The range of systems that exhibit a central limit that is approximately first-order is found to be broad. First-order like behavior is also found to be favored when the distribution of material among the parallel processes (due to differences in rate constants for the individual reactions) is more complex. Our results show that a first-order central limit exists for the kinetics of chemical systems and that the variable controlling the convergence is the physical complexity of reaction systems.     

On a general theorem concerning some absolute summability methods

In this note a general theorem covering several absolute summability methods e.g, |N,p_n|_k|R,p_n|_k, is proved.     

On Koopmans' theorem in density functional theory

This paper clarifies why long-range corrected (LC) density functional theory gives orbital energies quantitatively. First, the highest occupied molecular orbital and the lowest unoccupied molecular orbital energies of typical molecules are compared with the minus vertical ionization potentials (IPs) and electron affinities (EAs), respectively. Consequently, only LC exchange functionals are found to give the orbital energies close to the minus IPs and EAs, while other functionals considerably underestimate them. The reproducibility of orbital energies is hardly affected by the difference in the short-range part of LC functionals. Fractional occupation calculations are then carried out to clarify the reason for the accurate orbital energies of LC functionals. As a result, only LC functionals are found to keep the orbital energies almost constant for fractional occupied orbitals. The direct orbital energy dependence on the fractional occupation is expressed by the exchange self-interaction (SI) energy through the potential derivative of the exchange functional plus the Coulomb SI energy. On the basis of this, the exchange SI energies through the potential derivatives are compared with the minus Coulomb SI energy. Consequently, these are revealed to be cancelled out only by LC functionals except for H, He, and Ne atoms.     

On Tanaka-Fillmore's kinetics swelling of gels

Tanaka and Fillmore treated the swelling of a gel as a process where a crosslinked polymer network having been initially under uniform stress is expanded by osmotic pressure, sucking up the surrounding fluid medium. We point out that their physical reasoning is unnatural and leads to an unacceptable conclusion; we propose a more sound approach to the same problem. Our treatment assumes that the gel network is extended not by the osmotic pressure of the gel, but rather by the swelling pressure which is generated by the excess fluid penetrating in against the real nature of a polymer network that tends to shrink. The diffusion equation of the fluid, hence, plays a dominant role and gives the distribution of fluid concentration in contrast to Tanaka-Fillmore's scheme. The expression for the distribution of local strain in a spherical gel is deduced from the relation of mechanical balance between two forces, the one is due to the elasticity of the network and the other due to the gradient in the chemical potential of the fluid. The results obtained have forms analytically similar to Tanaka-Fillmore's, but are differ in the physical meanings.     

On the Wiener—Khintchine theorem for a random time series of finite duration

The work analyzes the existing ambiguity in the ways of application of theoretical relationships derived for infinite numeric series to real experiments of finite duration. A new calculation technique based on constructing a two-dimensional discrete time grid is used as a research tool. The described technique allows simplifying drastically without loss of stringency the derivation of the key relationships of the theory of spectral analysis of random time series of finite duration and formulating the main correlation identity lying in the basis of the Wiener—Khintchine theorem. It is shown that only a biased estimate of the correlation function, as opposed to an unbiased one, satisfies the main correlation identity and therefore it is the only basis for the possibility of correct application of the Wiener—Khintchine theorem to the real data of finite duration. It is also shown that the appearing undesirable increase in the number of spectral lines in the case of using biased correlation estimates is manifested only inside the subsystem of odd spectrum harmonics and the transition to the even harmonic subsystem eliminates this effect completely. In the case of the general theory of spectra, it is of importance that all the even harmonics coincide with the result of the spectrum direct calculation using the Schuster method, which proves the theoretical equivalence of the direct and indirect methods of the energy spectrum calculation.     

On the hypervirial theorem and the scaling problem

It is shown that wave-functions obtained by a limited number of off-diagonal hypervirial relations are often out of scale. Optimum scaling of these functions so as to satisfy the virial theorem gives highly improved wave-functions and expectation values. The harmonic oscillator problem is treated as an example.     

On the use of Ehrenfest''s theorem in molecular scattering

The effective Ehrenfest hamiltonian for systems where the wavefunction of the quantum part depends parametrically upon “classical” coordinates and momenta is presented and a number of examples are discussed.     

On the kinetics of spinodal decomposition: a numerical solution of a dissipative wave equation

A numerical solution of a recently derived dissipative wave equation governing the kinetics of Spinodal decomposition of a Lennard-Jones fluid is presented. In addition, the results are compared with those of Cahn'S and Abraham's generalized diffusion theories for the case of the early stages of the coarsening process.     

On a nonelementary progress curve equation and its application in enzyme kinetics

The analytical equation describing progress curves of an enzyme catalyzed reaction acting upon the Michaelis-Menten mechanism has been known for the case in which only the free enzyme incurs a loss of its activity, either spontaneously or as a result of an irreversible inhibitor action. The solution of differential equations which defines the rates of enzyme inactivation and substrate utilization is expressed by a nonelementary function in equation of an implicit type that precludes direct calculation of the extent of reaction at any time. Previously, the implicit equations have been rearranged to the alternative formulas and solved by the Newton-Raphson method, but this procedure may fail when used upon the presented equation. For this reason the other root-finding numerical method was applied, and the enzyme kinetic parameters of such numerically solved implicit equation for the reaction mechanism of irreversibly inhibited acetylcholinesterase were fitted to the experimental data by a nonlinear regression computer program.     

On a unified treatment of kinetics and diffusion,and a connection to a nonlocal boltzmann equation

Fick's law of diffusion has been generalized to include kinetic processes, the transport term of the Boltzmann equation, and nonlocal interaction processes. It is shown that the collision interaction term can be obtained by the introduction of a quantum stochastic potential equation. Some approximations of a nonlocal Boltzmann equation can be solved exactly. The solutions can be applied to problems of molecular pattern in biology.     

On the kinetics of polymer degradation in solution

Poly(2,2,2-trichloroethyl methacrylate), PTCMA, was irradiated in dilute dichloromethane solution with 16 MeV electron pulses of^{6 0}Co γ-rays. Both in the presence and absence of O₂, G(S)=9–10 was found at [PTCMA]<9 g/dm³. Indirect and direct radiation effects contributed to main-chain scission at a ratio of 7∶3. In contrast to earlier findings with dioxane solutions, at concentrations of up to 20 g/dm³ indications of crosslinking were not obtained. With CH₂Cl₂ solutions two modes of light scattering intensity (LSI) decrease were detected indicating the existence of two different macroradicals, one generated by the direct and the other by the indirect action of radiation. These radicals decayed with rate constants of 1.0·10³ s^?1 and 1.4·10¹ s^?1 and also reacted with C₂H₅SH with rate constants of 8·10⁴M^?1·s^?1 and 1·10³M^?1·s^?1, respectively. In O₂-saturated solutions both radicals formed peroxyl radicals, PO₂, wich combined generating products whose decomposition involved main-chain scission. The combination of PO₂-radicals was rate-determining in the consecutive series of reactions as inferred from the 2nd order decrease of the LSI.     

On the formation kinetics of two-dimensional cytidine films

The kinetics of phase transitions of cytidine adsorbed on mercury are studied by chronoamperometry and capacitance measurements. Cytidine forms highly ordered two-dimensional adlayers in a broad range of pH. In acid solvent, only one kind of condensed layer is formed. In the alkaline solution, cytidine forms two different two-dimensional (2D) adlayers. The minimum capacitance value in adlayer II at pH 5 is 7.0 microF cm(-2) and, at pH 8.3, it is 5.1 microF cm(-2); in adlayer III, the minimum capacitance is 10.6 microF cm(-2). The formation of a physisorbed film of cytidine molecules adsorbed at the mercury surface proceeds by complex mechanisms. From j-t transients, it can be seen that the phase transformations from dilute adlayer Ia to condensed physisorbed film II is accompanied by the reorientation of cytidine molecules at the mercury surface (inverted current transient). The interfacial transformations of the cytidine film yield a sigmoidal C-t transient. This experimentally measured C-t transient were analysed by Avrami theorem. The rate of the transformations from dilute adlayer Ia to condensed film II of cytidine at pH 5 depends strongly on temperature but is only slightly affected by temperature at pH 8.3. The effect of pH and ionic composition of the supporting electrolyte on the rate of transformation of cytidine films was studied as well.     

On the breakdown of the Koopmans' theorem for nitrogen

The nitrogen molecule is one of the simplest and best known examples where Koopmans' theorem not only fails in predicting accurate values for the vertical ionization potentials but even yields the wrong sequence of the two lowest ionizations. In this paper a perturbation treatment we had presented recently is applied to this problem. It turns out that it is necessary to use a very elaborate form of perturbation theory in the case of N₂.     

On an inconsistency connected with koopman's theorem

On the basis of Koopmans' theorem two methodically different ways to estimate the effect of a spin-orbit perturbation on the ionization potentials of I₂ have been proposed. In the present paper a model system is constructed to show the equivalence of both methods.     

Folding kinetics of a polymer

We present the results of computer simulations giving a kinetic insight into the liquid-to-solid transition of a homopolymer chain with short-range interactions. By calculating the absolute rates in each direction of the transition, using molecular dynamics employing the forward flux sampling scheme, we provide the phase diagram based on purely kinetic data, and compare it with the results from Monte Carlo simulations. Additionally, we present and discuss a remarkably simple and general relation between the polymer topology and the folding pathway, and show that the eigenvalue spectrum of a matrix defined by non-bonded contacts (the Laplacian matrix) provides an insight into the nonequilibrium ensembles of these trajectories. In particular, the Laplacian matrix seems to identify a large fraction of configurations on the folding pathway at the free energy maximum that have a very low probability of reaching the crystallized state. This implies that the eigenvalues of this matrix may be suitable additional reaction coordinates to describe the folding transition of chain molecules.     

On the microphase separation kinetics of symmetric diblock copolymers

Time-resolved small-angle x-ray scattering studies were performed on symmetric diblock copolymers of polystyrene and poly(methyl methacrylate), P(S-b-MMA). Freeze-dried powders of P(S-b-MMA) having a molecular weight of 8.4×10⁴ were rapidly heated to temperatures above the glass transition temperature to initiate the microphase separation. The microphase separation process was found to consist of a rapid, local microphase separation followed by a long-term coarsening process. The period characterizing the lamellar microphase separated structure was found to increase initiallv and then saturate at longer times. These results are discussed in light of recent theoretical developments.In celebration of his 65th birthday, this article is dedicated to Prof. E. W. Fischer whose methodic and thorough approach to research has been and continues to be a model for us to follow. Es freut mich, daß ich mit Herrn Fischer gearbeitet habe. Ich habe vieles von ihm gelernt. Ich hoffe, daß auch ich so fleißig sein werde, wenn ich so jung bin wie er.     

On non-isothermal kinetics of two Cu-based bulk metallic glasses

In this paper, two Cu-based bulk metallic glasses, Cu₅₅Zr₃₇Ti₈ and Cu₆₁Zr₃₄Ti₅, have been evaluated in thermodynamics and kinetics. The activation energies with the constant values were generalized by different theoretical models. The E _x of Cu₅₅Zr₃₇Ti₈ and Cu₆₁Zr₃₄Ti₅ are 319 ± 12 and 359 ± 12 kJ mol^?1, respectively, implying that the as-cast alloys have a good stability in thermodynamics. On the other hand, variable activation energies were also determined using Kissinger–Akahira–Sunose method, Ozawa–Flynn–Wall method, and Friedman’s method. The results showed that the Ea(x) at the beginning of the crystallization are higher than that at the end of the crystallization in the first exothermic peak. By introducing the local Avrami exponent, n(x), the growth and nucleation mechanisms were discussed. Furthermore, the effects of different activation energies on local Avrami exponent were also given a discussion.