Hypervirial theorem and matrix elements for the Coulomb potential

It is well known that hydrogen like atoms may be represented as Morse oscillators. Our aim in this article was to show how this result and the hypervirial theorem lead to a computational method with which the matrix elements for the Coulomb potential can be easily calculated. Also, a closed analytical expression for 〈n₂l₂|r^k|n₁l₁〉 is obtained. © 1995 John Wiley & Sons, Inc.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

STUDIES ON THE KINETICS OF PROPYLENE POLYMERIZATION WITH THE COMPLEXTYPE TiCl_3 CATALYST

The active center concentration C_p, the rate constant k_p, and the activation energy of chain propagation E_p in the polymerization of propylene with complex-type TiCl_3-(C_2H_5)_2AlCl catalyst system were studied. The Mn was corrected by (?) value determined by GPC. The values thus obtained for C_p, k_p, and E_p at 50℃were 3.01 mol/mol Ti, 6.27 1/mol·sec, and 5.10 Kcal/mol respectively.The kinetic parameters were compared with those obtained from conventional TiCl_3·AlCl_2 catalyst, showing that the higher activity of the complex-type catalyst over the conventional catalyst is not only due to the higher C_p of the former, but to a greater extent due to the increase of the k_p value.     

Molecular weight distributions and moments for condensation polymerizations characterized by two rate constants

Molecular weight distributions (MWD) and moments have been computed for condensation polymerizations of ARB-type monomers wherein the reactions involving the monomer are characterized by a rate constant k₁₁ and those involving other species, by k_p. For k₁₁/k_p < unity, the MWD curves are found to split into two, one for even n and one for odd values of n. Substantial amounts of unreacted monomer are found in the reaction mass for this case and so moments and the polydispersity index computed without the monomer are better representations of the distributions. For k₁₁/k_p > unit, conventional MWDs are observed with dispersions different from those representing polymerizations satisfying the equal reactivity hypothesis.     

Calculating the acidity constants of homologues and isomers of organic acids by means of recurrence relations

It is noted that the pK _a values of organic acids can be calculated using the unique recurrence relation pK _a(n + 1) = apK _a(n) + b from the pK _a values of other (usually the simplest and, consequently, better characterized) homologues of the same series. It is shown that this relation is valid within two taxonomic groups: insertion homologues of the ω-substituted acids X(CH₂) _n CO₂H (n ≥ 1) and isomers that differ in the position of substituents X in their alkyl fragments, k-X(C _n H_2n )CO₂H (n ≥ 1, 1 ≤ k ≤ n + 1). It is concluded that this algorithm is a consequence of the unique mathematical properties of recurrence relations.     

Free‐Radical Propagation Rate Coefficients via n‐Pulse Periodic Polymerization

Aspects of applying n‐pulse periodic initiation in pulsed laser polymerization/size‐exclusion chromatography (PLP/SEC) experiments are studied via simulation of molecular weight distributions (MWDs). In n‐pulse periodic PLP/SEC, sequences of n laser pulses at successive time intervals Δt₁ up to Δt_n are periodically applied. With the dark time intervals being suitably chosen, n‐modal MWDs with n well separated peaks occur. The n‐pulse periodic PLP/SEC method has the potential for providing accurate propagation rate coefficients, k_p. Among several measures for k_p, the differences in molecular weights at the MWD peak positions yield the best estimate of k_p under conditions of medium and high pulse laser‐induced free‐radical concentration. Deducing k_p from n dark time intervals (corresponding to n regions of free‐radical chain length) within one experiment at otherwise identical PLP/SEC conditions allows addressing in more detail a potential chain‐length dependence of k_p. Simulations are compared with experimental data for 2‐pulse periodic polymerization of methyl methacrylate.

Measured MWD (solid line) and associated first derivative curve (dotted line) for a 2‐pulse periodic bulk polymerization experiment of MMA at 20 °C.     

The production of organic nitrates from various anthropogenic volatile organic compounds

Production of organic nitrates from OH reaction with cyclohexane, cyclohexene, n‐butane, 1‐bromopropane, and p‐xylene in the presence of NO was studied. The total organic nitrate yields for cyclohexane and n‐butane were determined to be 17 ± 4 and 7 ± 2% respectively, which is in good agreement with previous determinations. Total yields for cyclohexene, 1‐bromopropane, and p‐xylene were 2.5 ± 0.5, 1.2 ± 0.3, and 3.2 ± 0.7 respectively. The yield for cyclohexene was five times smaller than that for cyclohexane. The 1‐bromopropane yield is three times smaller that that for n‐propane, but similar to that for propene, indicating that the effect of Br substitution in the reactant may be similar to that for OH substitution. The only nitrooxy product detected for p‐xylene was 4‐methylbenzylnitrate, which was formed following H abstraction from either methyl group. No organic nitrate was detected for peroxy radicals produced from OH addition to the ring, which accounts for 90% of the OH oxidation of p‐xylene. The calculated k_3b/k₃ value for p‐methyl benzyl peroxy radicals (0.32) was slightly smaller than for n‐octyl peroxy radicals (0.39). These data imply that substituent inductive effects impact the k_3b/k₃ ratios. We found no significant difference in the k_3b/k₃ ratios for primary vs. secondary peroxy radicals of the same carbon chain. © 2005 Wiley Periodicals, Inc. Int J Chem Kinet 37: 675–685, 2005     

Microwave spectrum,barrier to internal rotation and dipole moment in 5-methyl-pyrimidine

The microwave rotational spectrum of 5-methyl-pyrimidine has been investigated in the region from 8 to 27 GHz, the three types of lines to be expected for a molecule of this symmetry and with a very low sixfold barrier hindering internal rotation of the methyl top have been found: m = 0; |m| ≠ 0,3; |m| = 3n. From the m = 0 (a-type transitions) the rotational constants A′ (less methyl top) = 6108.41, B = 2642.198, C = 1844.196 MHz and the dipole moment μ = 2.881D have been determined. From the wide splitting of the lines |m| = 3, |k| = 1 the potential barrier has been derived as V₆ = 11.73 cal/mole.     

Modeling the Effects of Primary Radicals in Free‐Radical Polymerization

The effects of non‐ideal initiator decomposition, i.e., decomposition into two primary radicals of different reactivity toward the monomer, and of primary radical termination, on the kinetics of steady‐state free‐radical polymerization are considered. Analytical expressions for the exponent n in the power‐law dependence of polymerization rate on initiation rate are derived for these two situations. Theory predicts that n should be below the classical value of 1/2. In the case of non‐ideal initiator decomposition, n decreases with the size of the dimensionless parameter α ≡ (k_tz /k_dz) √r_ink_t, where k_tz is the termination rate coefficient for the reaction of a non‐propagating primary radical with a macroradical, k_dz is the first‐order decomposition rate coefficient of non‐propagating (passive) radicals, r_in is initiation rate, and k_t is the termination rate coefficient of two active radicals. In the case of primary radical termination, n decreases with the size of the dimensionless parameter β ≡ k_t,s r_in^1/2/k_p,s M r_t,l^1/2, where k_t,s is the termination rate coefficients for the reaction of a primary (“short”) radical with a macroradical, k_t,l is the termination rate coefficients of two large radicals, k_p,s is the propagation rate coefficient of primary radicals and M is monomer concentration. As k_t is deduced from coupled parameters such as k_t /k_p, the dependence of k_p on chain length is also briefly discussed. This dependence is particularly pronounced at small chain lengths. Moreover, effects of chain transfer to monomer on n are discussed.     

THE PHOTOEXCITED TRIPLET STATE OF TETRAPHENYL CHLORIN, MAGNESIUM TETRAPHENYL PORPHYRIN AND WHOLE CELLS OF CHLAMYDOMONAS REINHARDI. A LIGHT MODULATION-EPR STUDY

Abstract— The photoexcited triplet states of frozen solutions of tetraphenyl chlorin (TPC), magnesium tetraphenyl porphyrin (MgTPP) and whole cells of Chlamydomonas reinhardi have been studied by light modulation-EPR spectroscopy. The porphyrins were chosen to be studied as model compounds for chlorophyll molecules, From EPR spectra the zero field splitting parameters (ZFS) were calculated. For TPC, |D| = 0.0364 ± 0.0002 cm^-1, |E| = 0.0063 ± 0.0002 cm^-1. For MgTPP, |D| = 0.0310 ± 0.0002 cm^-1. For chloroplasts, |D| = 0.0280 ± 0.0004 cm^-1, |E| = 0.0032 ± 0.0004 cm^-1. In all compounds studied, except MgTPP, electron spin polarization (ESP) was observed. From the analysis of the kinetic curves at each canonical orientation we evaluated the spin lattice relaxation rate W, the depopulation rate constants k_p, and the ratio between the population rate constants, A_p, at zero magnetic field. For TPC in ethanol-toluene (5:1) k_x= (0.70 ± 0.10) × 10³ s^-1, k_y= (0.40 ± 0.07) × 10³ s^-1, k_x= (0.24 ± 0.05) × 10³ s^-1; A_x:A_y:A_z? 1.0:0.6:0.4; W= (2.60 ± 0.40) × 10³ s^-1. For MgTPP, only the total decay rate constant, k_T, was calculated: (1.5 ± 0.2) × 10 s^-1 in n-octane and (4.8 ± 0.8) × 10 s^-1 in ethanol. The results for TPC and MgTPP are compared to those reported previously for chlorophyll. It is concluded that the dynamics of the photoexcited triplet state in chlorophylls are mainly governed by the chlorin macrocycle. From the EPR spectrum and ZFS parameters of chloroplasts, we propose that both chlorophyll a and chlorophyll b are the main constituents of the EPR spectrum. From the analysis of the kinetic curves we obtain separately the kinetic parameters for chlorophylls a and b, k^a_x= (1.30 ± 0.20) × 10³ s^-1, k^a_y;= (0.85 ± 0.15) × 10³ s^-1k^a_x= (0.32 ± 0.05) × 10³ s^-1; A^a_x:A^a_y:A^a_z? 1.0:0.7:0.2; W^a= (1.20 ± 0.20) × 10³ s^-1; k^b_x= (0.56 ± 0.09) × 10³ s^-1, k^b_y= (0.30 ± 0.04) × 10³ s^-1, k^b_z= (0.06 ± 0.01) × 10³ s^-1; A^b_x:A^b_y:A^b_x? 1.0:0.6:0.1; W^b= (5.00 ± 0.80) × 10³ s^-1. These results are very close to those found separately for chlorophyll a and chlorophyll b oligomers in vitro.     

An Extended Method for the Practical Evaluation of the Standard Rate Constant from Cyclic Voltammetric Data

The Nicholson's treatment for the evaluation of the heterogeneous standard rate constant k⁰ of electron transfer from cyclic voltammetric (CV) data is combined with the Klingler and Kochi method suitable to totally irreversible systems in order to use very large ΔE_p x n values, where n is the number of electrons involved in a simple electron‐transfer process O+n e^??R. An empirical relationship between the dimensionless parameter Ψ=k⁰ [πDnvF/(RT)]^?1/2 and ΔE_p x n which practically replaces the Nicholson's working curve up to about 200 mV is reported.     

Living Carbocationic Polymerization. LVII. Kinetic Treatment of Living Carbocationic Polymerization Mediated by the Common Ion Effect

Abstract

The effect of anion concentration on the apparent rate constant of polymerization k^A _p of isobutylene (IB) induced by the 2-chloro-2,4,4-trimethylpentane (TMPCl)/TiCl₄ initiating system using the CH₂Cl₂/nC₆H₁₄ (60/40 v/v) solvent system at ?40 and ?80°C was studied by the use of nBu₄NCl. Computer simulation has shown that k^A _p decreases several orders of magnitude upon the addition of even a very small amount of common anion TiCl^?- ₅ to the charge. The rate of change is reduced in the concentration range of experimental interest. It was concluded that the decrease of k^A _p with increasing TiCl ^?- ₅ concentration is mainly due to the decreasing contribution of propagation by free ions. The contribution (%) of propagation by free ions to the apparent rate of propagation was calculated.     

Solvent effect on the radical polymerization of di-n-butyl itaconate

Solvent effect on the polymerization of di-n-butyl itaconate (DBI) with dimethyl azobisisobutyrate (MAIB) was investigated at 50 and 61°C. The solvents used were found to affect significantly the polymerization. The polymerization rate (R_p) and the molecular weight of the resulting polymer are lower in more polar solvents. The initiation rate (R_i) by MAIB, however, shows a trend of being rather higher in polar solvents. The stationary state concentration of propagating poly(DBI) radical was determined by ESR in seven solvents. The rate constants of propagation (k_p) and termination (k_t) were evaluated by using R_p, R_i, and the polymer radical concentration observed. The k_p value decreases fairly with increasing polarity of the solvent used, whereas k_t is not so influenced by the solvents. The solvent effect on k_p is explained in terms of a difference in the environment around the terminal radical center of the growing chain. Copolymerization of DBI with styrene (St) was also examined in three solvents with different physical properties. The poly(DBI) radical shows a lower reactivity toward St in a more polar solvent.     

Influence of pyridine oximate and quaternized pyridinium oximate ions on the hydrolysis of phosphate esters in cationic microemulsions

Kinetic studies have been performed to understand the hydrolytic potencies of oximate (2- and 4-pyridinealdoxime) and its functionalized oximate (4-(hydroxyiminomethyl)-1-alkylpyridinium bromide) ions (alkyl?=?C₁₀H₂₁ (4-C₁₀PyOx^-); alkyl?=?C₁₂H₂₅ (4-C₁₂PyOx^-)) in the cleavage of phosphate esters, diethyl p-nitrophenylphosphate (Paraoxon) and p-nitrophenyl diphenyl phosphate (PNPDPP) in a cationic (O/W) microemulsion system (ME) over a pH range 7.5 to 11.0 at 300?K. The k_obs values for the reaction of paraoxon with oximate and its functionalized oximate were determined in different microemulsion composition and the kinetic rate data shows that k_obs values increases with increasing water content. The specificity of different chain length of alcohols (n-butanol, n-pentanol, n-hexanol and n-octanol) was also investigated in hydrolytic reactions of paraoxon for different microemulsion composition.     

Augmentation of the Generalised n×n Eigenvalue Equation to a Generalised (n+1)×(n+1) Eigenvalue Equation

Generalised n×n eigenvalue equation B| _i = _i S ^b | _i (i=1,...,n) where B and S ^b are n×n Hermitian matrices while S ^b is in addition positive definite is considered. This equation is augmented to a generalised (n+1)(n+1) eigenvalue equation H| _k = _k S| _k (k=1,...,n+1) where Hermitian matrices H and S represent matrices B and S ^b , respectively, augmented by one additional row and one additional column. It is shown how the eigenvalues _k and the eigenvectors | _k of the augmented eigenvalue equation can be expressed in terms of the eigenvalues _i and the eigenvectors | _i of the original eigenvalue equation. Operation count to obtain by this method all augmented eigenvalues and eigenvectors is of the order O(n ²). Unless matrices involved are of some special kind such as sparse matrices or alike, this operation count is one order of magnitude smaller than operation count required by other presently known methods. In many practical cases operation count to obtain a single selected eigenvalue and/or eigenvector by this method is of the order O(n). In the case of the generalised eigenvalue equation, all other methods usually require again O(n ³) operations, even if only a single eigenvalue and/or eigenvector is required. Thus in many cases of interest operation count to obtain a selected eigenvalue and/or eigenvector by this method is two orders of magnitude smaller than operation count required by other methods.     

Comparison of rate coefficients for rydberg electron and free electron attachment

New experimental results on attachment reactions involving free electrons at sub-meV resolution allow for the first time a conclusive comparison of measured rate coefficients for Rydberg electron attachment with those calculated from the measured free electron cross sections on the basis of the quasi-free electron model for Rydberg electron collisions. Using classical velocity distributions for the highn Rydberg electrons and our measured free electron attachment cross sections, we calculate Rydberg electron attachment rate coefficientsk _n for the two cases SF₆ and HI for Rydberg binding energies |E _n | of 0.1–40 meV. We find a significant increase ink _n towards lower binding energies, especially for HI, which is due to the deviation of the free electron cross section from the limitings-wave behaviour ₀E ^–1/2. The increase at |E _n |2 meV is in qualitative agreement with our highn Rydberg data (n80) if -mixing due to residual electric fields is taken into account. For low , Rydberg rate coefficientsk _n(|E _n |) are significantly larger than free electron rate coefficientsk _e (E=|E _n |), while for circular orbits (=n–1) they agree. On average, attachment reactions of Rydberg electrons in low orbits proceed with an effective collision energy substantially smaller than the binding energy |E _n |.     

The Importance of Chain-Length Dependent Kinetics in Free-Radical Polymerization: A Preliminary Guide

The effect of chain-length dependent propagation at short chain lengths on the observed kinetics in low-conversion free-radical polymerization (frp) is investigated. It is shown that although the values of individual propagation rate coefficients quickly converge to the high chain length value (at chain lengths, i, of about 10), its effect on the average propagation rate coefficients, 〈k_p〉, in conventional frp may be noticeable in systems with an average degree of polymerization (DP_n) of up to 100. Furthermore it is shown that, unless the system is significantly retarded, the chain-length dependence of the average termination rate coefficient, 〈k_t〉, is not affected by the presence of chain-length dependent propagation and that there exists a simple (fairly general) scaling law between 〈k_t〉 and DP_n. This latter scaling law is a good reflection of the dependence of the termination rate coefficient between two i-meric radicals, k, on i. Although simple expressions seem to exist to describe the dependence of 〈k_p〉 on DP_n, the limited data available to date does not allow the generalization of these expressions.     

Relationship between kinetic chain length and radical polymerization rate

In the copolymerization of monomers M₁ and M₂ which form polymer radicals of chain length n of N¹_n with electron on a M₁ type and N²_n with one on a M₂ type, it is assumed that the specific rates of termination between N¹_n and N¹_n and N¹_s, N¹_n and N²_s, and N²_n and N²_s are k_α(ns)^?a, k_β(ns)^?a, and k_γ(ns)^?a, respectively, where k_α, k_β, and k_γ are the rate constants of reaction between segment radicals in the respective termination, and a is constant. The relation between kinetic chain length n? and polymerization rate R_p is derived as: 1/n? = 1/n?₀ + const. (R_p)^A(a), where n?₀ is the kinetic chain length of the polymer formed by transfer and A (a) is unity (predominance of transfer) and 1/(1–2a) (no transfer). In the copolymerization between methyl methacrylate (M₁) and styrene (M₂) at 60°C, when R_p → 0, k_r12/k₁₂ + k_r21/k₂₁ = 5.9× 10^?5 is obtained, where k_r12 and k_r21 are the rate constants of transfer of N¹ to M₂ and N² to M₁, and k₁₂ and k₂₁ are the rate constants of propagation of N¹ to M₂ and N² to M₁. In the absence of transfer, the a value is found to be 0.065 ± 0.008, from the relation between n? and R_p, regardless of the monomer composition. Such a value is also estimated by setting b = 0.72 in a = 0.153 (2b–1), where b is the constant in the Mark-Houwink equation. Further, the value of k_β is found to be 1.18 × 10⁹l./mole-sec, which is comparable with the diffusion-controlled rate of reaction between small molecules. The rate of reaction between segment radicals is fivefold larger than the polymer-polymer termination when transfer predominates.     

Hydrolyse acide du p-nitrophényl-diazométhane dans les mélanges H2O?D2O: analyse des effets isotopiques

The velocity of the hydrogen ion catalysed hydrolysis of p-nitrophenyl-diazo-methane (I) has been measured in H₂O? D₂O mixtures, giving an isotopic α_i = 0.49. The product isotope effect r = 5.1, determined from product analyses, combined with the (overall) solvent isotope effect k_H/k_D = 2.81, yields the primary kinetic isotope effect (k_H/k_D)^I = 3.8, and the secondary kinetic isotope effect (k_H/k_D)^II = 0.75. The CICH₂COOH-catalysed hydrolysis of I in H₂O? D₂O mixtures gave a straight-line plot of k_n/k_H versus the atomic fraction n of deuterium. With four carboxylic acids, as catalysts, values of about 4.3 for the kinetic (overall) isotope effects were observed.     

Relaxation Studies of Emulsion Polymerization of Styrene Initiated by Ultraviolet Light (using α-sodium anthraquinone sulfonate/triethyl amine as photosensitizer)

Abstract

This paper presents the results of relaxation studies of the seeded emulsion polymerization of styrene initiated by UV-light with a water-soluble photosensitizer. The relaxation kinetic runs were performed at 45, 50, 55, and 60°C using a small diameter rotary dilatometer with an inner magnetic agitator. The entry rate coefficient of thermally induced free radicals into the latex particles, p ₀; the exit rate coefficient of free radicals from latex particles, k; and the average number of free radicals per particle in the thermally induced background polymerization, n _ss(thermal) were determined. The propagating rate coefficient kp was determined with the data of monomer concentration in a particle, C _M, obtained from experiments with chemical initiation. The Arrhenius formula with p ₀, k, and k_p was also obtained.