Analytical solutions for the rate equations of irreversible two-step consecutive processes with second order later steps

Analytical solutions are presented for four consecutive two-step kinetic schemes, which involve a first reaction with zeroth, first, second or mixed second order dependence and a second step, which is second order with respect to the intermediate formed in the first step. Two of the analytical solutions found use elementary functions not very commonly encountered in chemistry, as the rate equations are shown to be related to the Legendre or modified Bessel differential equations. The solutions are analyzed not only as a function of time, but by plotting two concentrations as a function of each other as well. The dependence of the kinetic traces on the parameter values is also investigated. In all cases, two scaling parameters are identified. Three of the four cases are characterized by a single shape parameter, which is basically the ratio of the rate constant scaled with a suitable concentration unit if necessary. The mixed second order–second order scheme has an additional shape parameter, which is the ratio of the initial concentrations of the two reactants.     

Analytical solutions for the rate equations of irreversible two-step consecutive processes with mixed second order later steps

A general strategy was developed to solve the ordinary differential equations defined by two-step chemical processes with a mixed second order later process for all possible cases of parameter values (initial concentrations and rate constant values). As the earlier process, first order, second order, mixed second order and zeroth order cases were considered. For the scheme with a mixed second order first step, several different variations were considered. The analytical solutions contain moderately advanced, but still elementary functions such as the error function, the incomplete gamma function, the hypergeometric function or the Legendre functions. When coupling between the two steps or some reversibility is present in the system, no analytical solutions are found.     

The general theory of irreversible processes in solutions of macromolecules

The general theory of irreversible processes in solutions of macromolecules, previously formulated by the author, is reviewed. The theory is based upon the Oseen method for determining the perturbation in the hydrodynamic flow pattern produced by the frictional forces exerted by the macromolecule on the solvent, and on a generalized theory of Brownian motion in molecular configuration space. Applications of theory to viscoelastic behavior, flow birefringence, and the Kerr effect, and to dielectric dispersion are presented in outline.     

Photosynthetic models with maximum entropy production in irreversible charge transfer steps

Steady-state bacterial photosynthesis is modelled as cyclic chemical reaction and is examined with respect to overall efficiency, power transfer efficiency, and entropy production. A nonlinear flux–force relationship is assumed. The simplest two-state kinetic model bears complete analogy with the performance of an ideal (zero ohmic resistance of the P–N junction) solar cell. In both cases power transfer to external load is much higher than the 50% allowed by the impedance matching theorem for the linear flux–force relationship. When maximum entropy production is required in the transition with a load, one obtains high optimal photochemical yield of 97% and power transfer efficiency of 91%. In more complex photosynthetic models, entropy production is maximized in all irreversible electron/proton (non-slip) transitions in an iterative procedure. The resulting steady-state is stable with respect to an extremely wide range of initial values for forward rate constants. Optimal proton current increases proportionally to light intensity and decreases with an increase in the proton-motive force (the backpressure effect). Optimal affinity transfer efficiency is very high and nearly perfectly constant for different light absorption rates and for different electrochemical proton gradients. Optimal overall efficiency (of solar into proton-motive power) ranges from 10% (bacteriorhodopsin) to 19% (chlorophyll-based bacterial photosynthesis). Optimal time constants in a photocycle span a wide range from nanoseconds to milliseconds, just as corresponding experimental constants do. We conclude that photosynthetic proton pumps operate close to the maximum entropy production mode, connecting biological to thermodynamic evolution in a coupled self-amplifying process.     

Exact solutions for interacting finite potential wells

Exact wave functions and eigenenergy relations are obtained for a biwell potential as a function of interwell displacement and single well parameters. The proliferation of energies is shown explicitly as the wells are brought into close proximity. The model is compared with the opaque division-wall model. These results find important application in laser and nuclear physics.     

A model for codependent reversible/irreversible growth processes

A model for codependent growth that combines reversible and irreversible bond formation is developed. The system is composed of two processes: A reversible process which is fast but does not lead to a stable growth by itself, while the irreversible process is stable but is too slow to occur by itself. Therefore, neither the reversible nor the irreversible growth processes will occur separately, but their combination is shown to yield a new type of stable, codependent growth. Using kinetic Monte Carlo techniques we simulate and analyze the general properties of this codependent growth. We discuss the general conditions for such growth and its applications to self-organization processes.     

Exact solutions for kinetic models of macromolecular dynamics

Dynamic biological processes such as enzyme catalysis, molecular motor translocation, and protein and nucleic acid conformational dynamics are inherently stochastic processes. However, when such processes are studied on a nonsynchronized ensemble, the inherent fluctuations are lost, and only the average rate of the process can be measured. With the recent development of methods of single-molecule manipulation and detection, it is now possible to follow the progress of an individual molecule, measuring not just the average rate but the fluctuations in this rate as well. These fluctuations can provide a great deal of detail about the underlying kinetic cycle that governs the dynamical behavior of the system. However, extracting this information from experiments requires the ability to calculate the general properties of arbitrarily complex theoretical kinetic schemes. We present here a general technique that determines the exact analytical solution for the mean velocity and for measures of the fluctuations. We adopt a formalism based on the master equation and show how the probability density for the position of a molecular motor at a given time can be solved exactly in Fourier-Laplace space. With this analytic solution, we can then calculate the mean velocity and fluctuation-related parameters, such as the randomness parameter (a dimensionless ratio of the diffusion constant and the velocity) and the dwell time distributions, which fully characterize the fluctuations of the system, both commonly used kinetic parameters in single-molecule measurements. Furthermore, we show that this formalism allows calculation of these parameters for a much wider class of general kinetic models than demonstrated with previous methods.     

Phenomenological coefficients in the equations of thermodynamics of irreversible processes for anisotropic media

The phenomenological coefficients of the phenomenological equations (laws) obtained for anisotropic media, in particular, interphase separation regions, were studied. Generally, these coefficients are tensors of rank from 0 to 4. An attempt was made to find a general procedure for the determination of nonzero tensor components when cross phenomenological coefficient are present in phenomenological equations including generalized thermodynamic forces with various tensor ranks.     

Exact solutions of lattice polymer models

We consider directed path models of a selection of polymer and vesicle problems. Each model is used to illustrate an important method of solving lattice path enumeration problems. In particular, the Temperley method is used for the polymer collapse problem. The ZL method is used to solve the semi-continuous vesicle model. The Constant Term method is used to solve a set of partial difference equations for the polymer adsorption problem. The Kernel method is used to solve the functional equation that arises in the polymer force problem. Finally, the Transfer Matrix method is used to solve a problem in colloid dispersions. All these methods are combinatorially similar as they all construct equations by considering the action of adding an additional column to the set of objects.     

Exact solutions and coherent states for a generalized potential

Exact solutions of the vibrational Schrödinger equation for a generalized potential energy function \(\hbox {V(R)}=\hbox {C}_{0}(\mathrm{{R}-\mathrm {R}}_{\mathrm{e}})^{2}/[\hbox {aR}\,+\,(\mathrm{{b}-\mathrm {a}})\hbox {R}_{\mathrm{e}}]^{2}\) are obtained. It includes those of Dunham, Ogilvie and Simons–Parr–Finlan potentials as special cases corresponding to b \(=\) 1, a \(=\) 0, 1/2, 1, respectively. The analytical wave functions derived are useful to test the quality of numerical methods or to perform perturbative or variational calculations for the problems that cannot be solved exactly. Coherent states for generalized potential, which minimize the position–momentum uncertainty relation are also constructed.     

Experimental evidence for excess entropy discontinuities in glass-forming solutions

Glass transition temperatures T(g) are investigated in aqueous binary and multi-component solutions consisting of citric acid, calcium nitrate (Ca(NO(3))(2)), malonic acid, raffinose, and ammonium bisulfate (NH(4)HSO(4)) using a differential scanning calorimeter. Based on measured glass transition temperatures of binary aqueous mixtures and fitted binary coefficients, the T(g) of multi-component systems can be predicted using mixing rules. However, the experimentally observed T(g) in multi-component solutions show considerable deviations from two theoretical approaches considered. The deviations from these predictions are explained in terms of the molar excess mixing entropy difference between the supercooled liquid and glassy state at T(g). The multi-component mixtures involve contributions to these excess mixing entropies that the mixing rules do not take into account.     

The solution of the rate equations for Berry processes

The symmetry of the matrix describing the rate equations for Berry intramolecular rearrangements is studied. It is shown that this matrix is invariant with respect to any element of a cyclic group of order ten which has been defined previously [9]. This property is used to obtain the relaxation times, the chemical normal modes and the solution of the rate equations corresponding to Berry processes. Other possible applications are announced.

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Zusammenfassung Die Symmetrie der Matrix für die Kinetik der intramolekularen Berry-Umlagerung wird untersucht. Sie ist invariant gegenüber jedem Element der früher behandelten cyclischen Gruppe 10. Ordnung [9]. Damit lassen sich Relaxationszeiten, chemische Normalmode und die Lösung der zugehörigen kinetischen Gleichungen ermitteln. Auf weitere Anwendungsmöglichkeiten wird außerdem hingewiesen.

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Résumé On étudie la symétrie de la matrice qui décrit les équations de vitesse pour les réarrangements intramoléculaires de Berry. On montre que cette matrice est invariante pour tout élément d'un groupe cyclique d'ordre dix qui a été défini précédemment [9]. Cette propriété est utilisée pour obtenir les temps de relaxation, les modes normaux chimiques et la solution des équations de vitesse correspondant aux processus de Berry. La possibilité d'autres applications est annoncée.

     

Measurement of rate processes of free radicals in homogeneous and micellar solutions by cidnp

CIDNP is used to study rate processes of free radicals in both homogeneous and micellar solution. An estimate of the lifetime of the phenyl-acetyl radical at ambient temperature (τ__co?10^?7 sec) produced during photolysis of dibenzyl ketone is made based on quantitative CIDNP measurements and computer simulations. Observation of CIDNP in micellar solution is shown to be consistent with an isotropic medium which restricts diffusion on a short time scale, allowing for an increased tendency toward cage reaction. In the case of t-butyl/pivaloyl radical pairs, escape of the radical fragments from the micelle is shown to be competitive with decarbonylation of the pivaloyl radical Likewise, CIDNP is consistent with product yield results which show the enhanced tendency of triplet born benzyl radical pairs to undergo cage reaction when they are sequestered in a micelle.     

Exact solutions of the rigid rotor in the electric field

We first present a new constraint condition on the confluent Heun function H_C(α, β, γ, δ, η;z) (β, γ ≥ 0, z ∈ [0, 1]) and then illustrate how to solve the rigid rotor in the electric field. We find its exact solutions unsolved previously through solving the Wronskian determinant. The present results compared with those by the perturbation methods are found to have a big difference for a large parameter a. We also present 2D and 3D probability density distributions by choosing different angular momentum quantum numbers l. We observe that the original eigenvalues with degeneracy (2 l + 1) are split into the (l + 1) state with approximate eigenvalues l(l + 1) for small a but large l.     

Exact solutions for spherically confined hydrogen‐like atoms

We present exact analytical solutions for the much‐studied problem of a hydrogen‐like atom confined in a spherical box of radius R. These solutions, which are obtained for all states and all R, are expressed directly in terms of the Kummer M‐functions whose analytical and numerical properties are well known, and may be calculated using standard computing packages. The solutions are illustrated by precise calculations that yield accurate energies E for any given radius R, or for R when E is known. In the special case where E = 0, it is shown that the solution may be expressed in terms of Bessel functions. Finally, the physical assumptions made in applying this model to describe atomic confinement are discussed critically. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

A system-independent correlation between the enthalpy and the entropy of dilution for polymer solutions

The second osmotic virial coefficient (A₂) and its entropic and enthalpic parts (A_2,s and A_2,H) have been determined, by means of light-scattering measurements, for solutions of polystyrene, polymethylmethacrylate and cellulose nitrate of different molecular weights in 19 solvents. A distinct qualitative correlation exists between A₂ and A_2,H and between A_2,s and A_2,H. The elimination of the “geometric” parameters of the polymer, by dividing these coefficients by suitably chosen reduction parameters, shows that the reduced coefficients obtained A₂⁰ and A_2,s⁰ are predominantly functions of the reduced enthalpy coefficient A_2,H⁰.     

Classical theory of collisional entropy production

A classical dynamical theory of elementary collision processes is formulated in analogy to the quantum theory of the dynamical scattering matrix, which can be defined for a pure quantum stationary scattering state. The elements of this matrix are probability amplitudes for transitions between internal states defined for given values of a reaction coordinate. The squared magnitudes of these amplitudes, modeled in the proposed classical theory, define normalized internal state population distributions suitable for information theoretical analysis. Statistical entropy and surprisal are defined as dynamical functions of a reaction coordinate. This formalism differs fundamentally from concepts based on the classical Liouville equation.