Approximations for the temperature integral

The temperature integral cannot be analytically integrated and many simple closed-form expressions have been proposed to use in the integral methods. This paper first reviews two types of simple approximation expressions for temperature integral in literature, i.e. the rational approximations and exponential approximations. Then the relationship of the two types of approximations is revealed by the aid of a new equation concerning the 1^st derivative of the temperature integral. It is found that the exponential approximations are essentially one kind of rational approximations with the form of h(x)=[x/(Ax+k)]. That is, they share the same assumptions that the temperature integral h(x) can be approximated by x/Ax+k). It is also found that only two of the three parameters in the general formula of exponential approximations are needed to be determined and the other one is a constant in theory. Though both types of the approximations have close relationship, the integral methods derived from the exponential approximations are recommended in kinetic analysis.     

Error evaluation of integral methods by consideration on the approximation of temperature integral

Summary In this paper, the integral methods in general use are divided into two types in terms of their different ways to in order to deal with the temperature integral p(x): for Type A the function h(x)=p(x)x²e^x is regarded as constant vs. x, while for Type B h(x) varies vs. x and ln[p(x)] is assumed to have the approximation form of ln[p(x)]=alnx+bx+c (the coefficients a, b, and c are constant). The errors of kinetic parameters calculated by these two types of methods are derived as functions of x and analyzed theoretically. It is found that Type A methods have the common errors of activation energy, while the Coats-Redfern method can lead to more accurate value of frequency factor than others. The accuracy of frequency factor can be further enhanced by adjusting the expression of the Coats-Redfern approximation. Although using quite simple approximation of the temperature integral, the Coats-Redfern method has the best performance among Type A methods, implying that usage of a sophisticated approximation may be unnecessary in kinetic analysis. For Type B, the revised MKN method has a lower error in activation energy and an acceptable error in frequency factor, and thus it can be reliably used. Comparatively, the Doyle method has higher error of activation energy and great error of the frequency factor, and thus it is not recommended to be adopted in kinetic analysis.     

Precision of integral methods for the determination of the kinetic parameters

In this paper, a systematic analysis of the errors involved in the determination of the kinetic parameters (including the activation energy and frequency factor) from five integral methods has been carried out. The integral methods analyzed here are Coats-Redfern, Gorbachev, Wanjun-Yuwen-Hen-Zhiyong-Cunxin, Junmeng-Fusheng-Weiming-Fang, Junmeng-Fang and Junmeng-Fang-Weiming-Fusheng method. The results have shown that the precision of the kinetic parameters calculated by the different integral methods is dependent on u (E/RT), that is, on the activation energy and the average temperature of the process.     

The (n,n)-graphs with the first three extremal Wiener indices

Let G = (V, E) be a simple connected graph with vertex set V and edge set E. The Wiener index W(G) of G is the sum of distances between all pairs of vertices in G, i.e., , where d _G (u, v) is the distance between vertices u and v in G. In this paper, we first give a new formula for calculating the Wiener index of an (n,n)-graph according its structure, and then characterize the (n,n)-graphs with the first three smallest and largest Wiener indices by this formula.     

On the Randić Index of Unicyclic Conjugated Molecules

The Randić index R(G) of a graph G is the sum of the weights of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we first present a sharp lower bound on the Randić index of conjugated unicyclic graphs (unicyclic graphs with perfect matching). Also a sharp lower bound on the Randić index of unicyclic graphs is given in terms of the order and given size of matching.     

Dissociation Constants for Citric Acid in NaCl and KCl Solutions and their Mixtures at 25 °C

The constants for the dissociation of citric acid (H₃C) have been determined from potentiometric titrations in aqueous NaCl and KCl solutions and their mixtures as a function of ionic strength (0.05–4.5 mol-dm^–3) at 25 °C. The stoichiometric dissociation constants (K_i^*)

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.     

Kinetic characterization of the reduction of silica supported cobalt catalysts

Abstact  The reduction process of silica supported cobalt catalyst was studied by thermal analysis technique. The reduction of the catalyst proceeds in two steps:

Hosoya polynomials of TUC4C8(S) nanotubes

The Hosoya polynomial of a chemical graph G is , where d _G (u, v) denotes the distance between vertices u and v. In this paper, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(S) nanotubes. Accordingly, the Wiener index, obtained by Diudea et al. (MATCH Commun. Math. Comput. Chem. 50, 133–144, (2004)), and the hyper-Wiener index are derived. This work is supported by the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (Grant No. LZULL200809).     

Homomorphism polynomials of chemical graphs

We say that a graphG ishomomorphic to a graphH if there is a mappingp from the vertices of G onto the vertices ofH such thatp(u) andp() are adjacent inH wheneveru and are adjacent in G. Thehomomorphism polynomial of a graphG is a polynomial in two variables that counts the number of homomorphisms ofG onto the complete graph of each order. This polynomial can be computed recursively in an analog to the chromatic polynomial. In this paper, we present some results regarding the homomorphism polynomials of the graphs of chemical compounds — in particular, alkane isomers. The coefficients of the homomorphism polynomial can be used to predict the rankings of compounds with respect to several chemical properties. Our results seem to refine those obtained by Randi et al. from path lengths.     

Thermal decomposition kinetics of bis(pyridine)manganese(II) chloride

The thermal stability and the decomposition steps of bis(pyridine)manganese(II) chloride (Mn(py)₂Cl₂) were determined by thermogravimetry and derivative thermogravimetry. The initial compound and the solid compounds resulted from each step of decomposition were characterized by FT-IR spectroscopy and RX diffraction. It was pointed out that at the progressive heating of Mn(py)₂Cl₂, the following decomposition reactions occur: I $$ {\text{Mn}}\left( {\text{py}} \right)_{ 2} {\text{Cl}}_{ 2} \left( {\text{s}} \right) \, \to {\text{ Mn}}\left( {\text{py}} \right){\text{Cl}}_{ 2} \;\left( {\text{s}} \right) \, + {\text{ Py }}\left( {\text{g}} \right) $$ II $$ {\text{Mn}}\left( {\text{py}} \right){\text{Cl}}_{ 2} \left( {\text{s}} \right) \, \to {\text{ Mn}}\left( {\text{py}} \right)_{ 2/ 3} {\text{Cl}}_{ 2} \;\left( {\text{s}} \right) \, + { 1}/ 3 {\text{ Py }}\left( {\text{g}} \right) $$ III $$ {\text{Mn}}\left( {\text{py}} \right)_{ 2/ 3} {\text{Cl}}_{ 2} \left( {\text{s}} \right) \, \to {\text{ MnCl}}_{ 2} \left( {\text{s}} \right) \, + { 2}/ 3 {\text{ Py }}\left( {\text{g}} \right) $$ The dependence of the activation energy of these decompositions steps on the conversion degree, evaluated by isoconversional methods, shows that all decomposition reactions are complex. The mechanism and the corresponding kinetic parameters of reaction (I) were determined by multivariate non-linear regression program and checked for quasi-isothermal data. It was pointed out that the reaction (I) consists of three elementary steps, each step having a specific kinetic triplet.     

Study of the influence of a reactive diluent on the rheological properties of an epoxy-diamine system

The generalized temperature integral I(m, x) appears in non-isothermal kinetic analysis when the frequency factor depends on the temperature. A procedure based on Gaussian quadrature to obtain analytical approximations for the integral I(m, x) was proposed. The results showed good agreement between the obtained approximation values and those obtained by numerical integration. Unless other approximations found in literature, the methodology presented in this paper can be easily generalized in order to obtain approximations with the maximum of accurate.     

On the Randić Index of Quasi-tree Graphs

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertex u and v in G. A graph G is called quasi-tree, if there exists such that G − u is a tree. In the paper, we give sharp lower and upper bounds on the Randić index of quasi-tree graphs. Mei Lu: Partially supported by NSFC (No. 10571105).     

Estimation of the Critical Temperature of Thermal Explosion for the Highly Nitrated Nitrocellulose Using Non-isothermal DSC

Two methods for estimating the critical temperature (T_b) of thermal explosion for the highly nitrated nitrocellulose (HNNC) are derived from the Semenov's thermal explosion theory and two non-isothermal kinetic equations, d/dt=Af()e^–E/RT and d/dt=Af()[1+E/(RT)(1–T_o/T)]e^–E/RT, using reasonable hypotheses. We can easily obtain the values of the thermal decomposition activation energy (E), the onset temperature (T_e) and the initial temperature (T_o) at which DSC curve deviates from the baseline of the non-isothermal DSC curve of HNNC, and then calculate the critical temperature (T_b) of thermal explosion by the two derived formulae. The results obtained with the two methods for HNNC are in agreement to each other.This revised version was published online in November 2005 with corrections to the Cover Date.     

On the ABC index of connected graphs with given degree sequences

The atom-bond connectivity (ABC) index of a graph G is defined to be \(ABC(G)=\sum _{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}\) where d(u) is the degree of a vertex u. The ABC index plays a key role in correlating the physical–chemical properties and the molecular structures of some families of compounds. In this paper, we describe the structural properties of graphs which have the minimum ABC index among all connected graphs with a given degree sequence. Moreover, these results are used to characterize the extremal graphs which have the minimum ABC index among all unicyclic and bicyclic graphs with a given degree sequence.     

PbCu3(OH)(NO3)(SeO3)3·1/2H2O und Pb2Cu3O2(NO3)2(SeO3)2: Synthese und Kristallstrukturuntersuchung

Crystals of PbCu₃(OH)(NO₃)(SeO₃)₃·1/2H₂O [a=7.761(3)Å,b=9.478(4)Å,c=9.514(4)Å, =66.94(2)°, =69.83(2)°, =81.83(2)°, space group P ,Z=2] and Pb₂Cu₃O₂(NO₃)₂(SeO₃)₂ [a=5.884(2)Å,b=12.186(3)Å,c=19.371(4)Å, space group Cmc2₁,Z=4] were synthesized under hydrothermal conditions. Their crystal structures were refined with three-dimensional X-ray data toR _w=0.033 resp. 0.055. In PbCu₃(OH)(NO₃)(SeO₃)₃·1/2H₂O the Cu atoms are [4+1] and [4+2] coordinated and via SeO₃ groups a three-dimensional atomic arrangement is built up. In Pb₂Cu₃O₂(NO₃)₂(SeO₃)₂ there are sheets, which are connected only via Pb-O bonds ranging from 2.98 Å to 3.16 Å.

     

Kinetic analysis of solid‐state reactions: Precision of the activation energy calculated by integral methods

The integral methods are extensively used for the kinetic analysis of solid‐state reactions. As the Arrhenius integral function [p(x)] does not have an exact analytical solution, different approximated equations have been proposed in the literature for performing the kinetic analysis of experimental integral data. Since the first approximation of Van Krevelen, a large number of equations have been proposed with the objective of increasing the precision in the determination of the Arrhenius integral, as checked from the standard deviation of the approximated function with regard to the real exact value of the integral. However, the main application of these equations is the determination of the kinetic parameters, in particular activation energies, and not the computation of the Arrhenius integral. A systematic analysis of the errors involved in the determination of the activation energy from these integral methods is still missing. A comparative study of the precision of the activation energy as a function of x and T computed from the different integral methods has been carried out. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 658–666, 2005     

Systematic determination of the Slater-Condon parameters of atoms and ions with 1s 2 2s m 2p n configurations

Semiempirical values of the Slater parameters F ²(2p,2p) and G ¹(2s, 2p) have been determined for the atoms and ions with the electron configurations 1s ²2s ^m2p ⁿ from the experimental atomic energy levels. Particular attention has been paid to get the values of the parameters to be used for semiempirical calculations on molecular electronic structure. The calculation has also yielded E _av's, the average energies of configurations of these atoms. Evaluation of the semiempirical or effective value of F ⁰ from E of an appropriate electron-transfer reaction, based on the idea presented by Anno [15], is also referred to in the present paper. The semiempirical values of the Slater parameters as well as those of E's and E _av's show almost linear dependence upon atomic number Z through isoelectronic series. From the overall tendency of the correlation lines, it is pointed out that the assignment of atomic energy levels of Na⁵⁺ (1s ²2p ⁴) must be wrong.

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Zusammenfassung Semiempirische Werte der Slater-Parameter F ²(2p,2p) und G ¹(2s, 2p) wurden für die Atome und Ionen mit der Elektronenkonfiguration 1s ²2s ^m2p ⁿ aus den experimentellen atomaren Energietermen bestimmt. Insbesondere wurden die Parameter bestimmt, die für semiempirische Berechnungen der Elektronenstruktur von Molekülen benötigt werden. Die Berechnung ergibt weiterhin Werte von E _av, der Durchschnittsenergie der Konfigurationen der genannten Atome. Die vorliegende Arbeit geht auch auf die Bestimmung des semiempirischen bzw. effektiven Wertes von F ⁰ aus E einer geeigneten Elektronenübertragungsreaktion ein, die auf die Arbeit von Anno [15] zurückgeht. Die semiempirischen Werte der Slater-Parameter sowie der E und E _av zeigen annähernd lineare Abhängigkeit von der Atomnummer Z innerhalb isoelektronischer Reihen. Aus dem Gesamtverhalten der genannten Kurven wird geschlossen, da die Zuordnung der atomaren Energieterme von Na⁵⁺(1s ²2p ⁴) falsch sein dürfte.

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Résumé Détermination à partir des niveaux d'énergie atomiques expérimentaux des valeurs semi-empiriques des paramètres de Slater F ²(2p, 2p) et G ¹(2s, 2p) pour les atomes et les ions ayant les configurations 1s ²2s ^m2p ⁿ. On a fait particulièrement attention d'obtenir les valeurs des paramètres à utiliser pour des calculs moléculaires semi-empiriques. Le calcul a aussi fourni les énergies moyennes des configurations de ces atomes. On évoque aussi le calcul de la valeur semi-empirique ou effective de F ⁰ à partir du E d'une réaction appropriée de transfert électronique, selon une idée de Anno [15]. Les valeurs semiempiriques des paramètres de Slater, ainsi que celles des E et E _moy, montrent une dépendance presque linéaire au nombre atomique Z à travers les séries isoélectroniques. A partir des tendances générales des lignes de corrélation, on remarque que l'attribution des niveaux d'énergie atomiques de Na⁵⁺(1s ²2p ⁴) doit être fausse.

     

Influence of Substance on the Equilibrium Translation Rate for the Isothermal and Isobaric Chemical Reactions of Ideal gases

To increase inert substance i will make the equilibrium translation rate α _j of reactant j decrease if ∑ _i ν _i < 0 or increase if ∑ _i ν _i > 0. When or , to increase non-inert substance i will make α _j increase if i is reactant (i ≠ j) or decrease if i is resultant. When has maximum if i is reactant (i≠ j) or minimum if i is resultant. If i is reactant, (x _r ⁰ is “optimum proportion” of reactant)     

Use of the parametric representation method in revealing the root structure and hopf bifurcation

In practical applications of dynamical systems, it is often necessary to determine the number and the stability of the stationary states. The parameric respresentation method is a useful tool in such problems. Consider the two parameter families of functions:f(x) =u _o +u ₁ x +g(x), whereu _o andu ₁ are the parameters. We are interested in the number of zeros as well as in the stability. We want to determine the stable region on the parameter plane, where the real parts of the roots off are negative. The D-curve (along which the discriminant off is zero) helps us. We applied the method to the cases of cubic and quartic equation, giving pictorial meaning to the root structure. In this respect, the R-curves and the I-curves (along which the sum or difference, respectively, of two zeros is constant) also have a significance. Using these concepts, we established a relation between the (n - 1)th Routh-Hurwitz condition and the Hopf bifurcation.