Approximate formulae for calculation of the integral òT0Tmexp(-E/RT)dT

Summary In this paper two approximate formulae have been developed for calculation of the integral ò^T₀T^mexp(-E/RT)dT by using integration-by-parts approaches. They are in the following forms: I(m,T) = (RT^m+2)/(E+(m+2)RT)exp(-E/RT) I(m,T) = (RT^m+2)/(E+(m+2)(0.00099441E+0.93695599RT)exp(-E/RT) The validity of the two formulae has been confirmed and their accuracies have been tested with data from numerical calculating. In contrast to existing other integral methods, both the present approaches are simply used, accurate, and can be used for arbitrary values of m.     

Algorism for the solution of the exponential integral in non-isothermal kinetics at linear heating

The solution of the exponential integral at linear heating for the general case that the activation energy linearly depends on temperature according toE(T)=E ₀+RBT is

Kinetics of the reactions of the hydroxyl radical with dimethyl ether and diethyl ether

Absolute rate coefficients for the reactions of the hydroxyl radical with dimethyl ether (k₁) and diethyl ether (k₂) were measured over the temperature range 295–442 K. The rate coefficient data, in the units cm³ molecule^?1 s^?1, were fitted to the Arrhenius equations k₁ (T) = (1.04 ± 0.10) × 10^?11 exp[?(739 ± 67 cal mol^?1)/RT] and k₂(T) = (9.13 ± 0.35) × 10^?12 exp[+(228 ± 27 kcal mol^?1)/RT], respectively, in which the stated error limits are 2σ values. Our results are compared with those of previous studies of hydrogen-atom abstraction from saturated hydrocarbons by OH. Correlations between measured reaction-rate coefficients and C? H bond-dissociation energies are discussed.     

Photon correlation spectroscopy of polystyrene as a function of temperature and pressure

Photon correlation spectroscopy is employed to study the slowly relaxing density and anisotropy fluctuations in bulk atactic polystyrene as a function of temperature from 100 to 160°C and pressure from 1 to 1330 bar. The light-scattering relaxation function is well described by the empirical function ?(t) = exp[?(t/τ)^β], where for polystyrene β = 0.34. The average relaxation time is determined at each temperature and pressure according to 〈τ〉 = (τ/β)Γ(1/β) where Γ(x) is the gamma function. The data can be described by the empirical relation 〈τ〉 = 〈τ〉₀ exp[(A + BP)/R(T ? T₀)] where R is the gas constant and T₀ is the ideal glass transition temperature. The empirical constant A/R is in good agreement with that determined from the viscosity or the dielectric relaxation data (1934 K). The empirical constant B can be interpreted as the activation volume for the fundamental unit involved in the relaxation and is found to be comparable to one styrene subunit (100 mL/mol). The quantity B appears to be a weak function of temperature. The use of pressure as a tool in the study of light scattering near the glass transition now has been established.     

The kinetics and mechanisms of gas phase elimination of primary,secondary, and tertiary 2‐hydroxyalkylbenzenes

2‐Phenylethanol, racemic 1‐phenyl‐2‐propanol, and 2‐methyl‐1‐phenyl‐2‐propanol have been pyrolyzed in a static system over the temperature range 449.3–490.6°C and pressure range 65–198 torr. The decomposition reactions of these alcohols in seasoned vessels are homogeneous, unimolecular, and follow a first‐order rate law. The Arrhenius equations for the overall decomposition and partial rates of products formation were found as follows: for 2‐phenylethanol, overall rate log k₁(s⁻¹)=12.43−228.1 kJ mol⁻¹ (2.303 RT)⁻¹, toluene formation log k₁(s⁻¹)=12.97−249.2 kJ mol⁻¹ (2.303 RT)⁻¹, styrene formation log k₁(s⁻¹)=12.40−229.2 kJ mol⁻¹(2.303 RT)⁻¹, ethylbenzene formation log k₁(s⁻¹)=12.96−253.2 kJ mol⁻¹(2.303 RT)⁻¹; for 1‐phenyl‐2‐propanol, overall rate log k₁(s⁻¹)=13.03−233.5 kJ mol⁻¹(2.303 RT)⁻¹, toluene formation log k₁(s⁻¹)=13.04−240.1 kJ mol⁻¹(2.303 RT)⁻¹, unsaturated hydrocarbons+indene formation log k₁(s⁻¹)=12.19−224.3 kJ mol⁻¹(2.303 RT)⁻¹; for 2‐methyl‐1‐phenyl‐2‐propanol, overall rate log k₁(s⁻¹)=12.68−222.1 kJ mol⁻¹(2.303 RT)⁻¹, toluene formation log k₁(s⁻¹)=12.65−222.9 kJ mol⁻¹(2.303 RT)⁻¹, phenylpropenes formation log k₁(s⁻¹)=12.27−226.2 kJ mol⁻¹(2.303 RT)⁻¹. The overall decomposition rates of the 2‐hydroxyalkylbenzenes show a small but significant increase from primary to tertiary alcohol reactant. Two competitive eliminations are shown by each of the substrates: the dehydration process tends to decrease in relative importance from the primary to the tertiary alcohol substrate, while toluene formation increases. © 1999 John Wiley & Sons, Inc. Int J Chem Kinet 31: 401–407, 1999     

Kinetic model of induced codeposition of Ni-Mo alloys

The kinetic model of induced codeposition of nickel-molybdenum alloys from ammoniun citrate solution was studied on rotating disk electrodes to predict the behavior of the electrode-position. The molybdate (MoO42-) could be firstly electro-chemically reduced to MoO2, and subsequently undergoes a chemical reduction with atomic hydrogen previously adsorbed on the inducing metal nickel to form molybdenum in alloys. The kinetic equations were derived, and the kinetic parameters were obtained from a comparison of experimental results and the kinetic equations. The electrochemical rate constants for discharge of nickel, molybdenum and water could been expressed as k1(E) = 1. 23 × 109 CNi exp( - 0.198FE/RT) mol/(dm2·s), k2(E) =3.28× 10-10 CMoexp( - 0. 208FE/ RT) mol/(dm2·s) and k3(E) = 1.27 × 10-6exp( - 0.062FE/ RT) mol/(dm2 · s), where CNi and CMo are the concentrations of the nickel ion and molybdate, respectively, and E is the applied potential vs. saturated calomel electrode (SCE). The codeposition p     

Metal Ions Drive Thermodynamics and Photochemistry of the Bis(styryl) Macrocyclic Tweezer

UV/Vis and NMR spectroscopy were used for the structural elucidation and thermodynamic and photochemical studies of the metal‐coordinated crown‐containing macrocyclic tweezer (E,E)‐ 1 . The bis(styryl) tweezer (E,E)‐ 1 formed two types of complexes with magnesium(II): a 1:1 intramolecular asymmetric sandwich complex [(E,E)‐ 1 ]?Mg²⁺ and a 1:2 complex [(E,E)‐ 1 ]?(Mg²⁺)₂. In the former case, there is direct cation intramolecular exchange (0.299 s^?1, ΔG^≠=69.4 kJ mol^?1) between two parts of the bis(styryl) tweezer (E,E)‐ 1 . Addition of barium(II) to the bis(styryl) tweezer (E,E)‐ 1 led to an intramolecular centrosymmetric sandwich 1:1 complex [(E,E)‐ 1 ]?Ba²⁺. Irradiation of [(E,E)‐ 1 ]?Ba²⁺ afforded reversible intramolecular [2π+2π] photocyclization with excellent stereoselectivity and quantitative yield. In contrast, irradiation of [(E,E)‐ 1 ]?(Mg²⁺)₂ resulted in reversible stepwise E,Z‐isomerization.     

Fourier-transform infrared studies of polypropylene during mechanical deformation

Quantitative Fourier-transform infrared (FTIR) measurements of frequency shifts Δν and absorbance profile asymmetry are reported for various polypropylene samples as a function of uniaxial stress σ. Generally, it was found that the frequency shift coefficient α_χ, defined by Δν = α_χσ, depended on stress rate \documentclass{article}\pagestyle{empty}\begin{document}$\dot \sigma$\end{document}, draw ratio, λ, molecular orientation f, tensile modulus E, and annealing conditions. With annealing, α_χ decreased with increasing shrinkage in the case of highly oriented isotactic PP. The α_χ values for the “helix bands” were less affected than those for the “liquid bands.” With increasing \documentclass{article}\pagestyle{empty}\begin{document}$\dot \sigma$\end{document}, generally α_χ increased to an apparent asymptotic limit. With increasing λ, f, or E, α_χ also increased from α_χ ? 0 for λ = 1 (spherulitic) to maximum values for highly oriented isotactic PP. The observed variations in α_χ can be interpreted in terms of the changes in the peak position and shape of the nonuniform molecular stress distribution. Analogous behavior with x-ray diffraction peaks obtained for polymers under stress is discussed.     

Ozone–olefin reactions in the gas phase 1. Rate constants and activation energies

Reactions of ozone with simple olefins have been studied between 6 and 800 mtorr total pressure in a 220-m³ reactor. Rate constants for the removal of ozone by an excess of olefin in the presence of 150 mtorr oxygen were determined over the temperature range 280 to 360° K by continuous optical absorption measurements at 2537 Å. The technique was tested by measuring the rate constants k₁ and k₂ of the reactions (1) NO + O₃ → NO₂ + O₂ and (2) NO₂ + O₃ rarr; NO₃ + O₂ which are known from the literature. The results for NO, NO₂, C₂H₄, C₃H₆, 2-butene (mixture of the isomers), 1,3→butadiene, isobutene, and 1,1 -difluoro-ethylene are 1.7 × 10^{?1 4} (290°K), 3.24 × 10^?17 (289°K), 1.2 × 10^{?1 4} exp (–4.95 ± 0.20/RT), 1.1 × 10^{?1 4} exp (–3.91 ± 0.20/RT), 0.94 × 10^{?1 4} exp ( –2.28 ± 0.15/RT), 5.45 ± 10^{?1 4} exp ( –5.33 ± 0.20/RT), 1.8 ×10^?17 (283°K), and 8 × 10^?20 cm³/molecule ·s(290°K). Productformation from the ozone–propylene reaction was studied by a mass spectrometric technique. The stoichiometry of the reaction is near unity in the presence of molecular oxygen.     

A Study of Magnetic Relaxation in Dysprosium(III) Single-Molecule Magnets

Although the development of single-molecule magnets (SMMs) is rapid, there are only two families of high energy barrier (U_eff) dysprosium(III) SMMs known so far: the cyclopentadienyl (Cp) family with a sandwich structure and the pentagonal-bipyramidal (PB) family with D_5h symmetry. These high-barrier SMMs, which usually possess U_eff>500 cm⁻¹ allow the separate study of the four magnetic relaxation paths, namely, direct, quantum tunnelling, Raman and Orbach processes, in detail. Whereas the first family is chemically more challenging to modify the Cp rings, it is shown herein that the latter family, with the common formulae [DyX¹X²(L_eq)₅]⁺, such as X¹/X²=⁻OCMe₃, ⁻OSiMe₃, ⁻OPh, Cl⁻ or Br⁻; L_eq=THF/pyridine/4-methylpyridine, can be readily fine-tuned with a range of axial and equatorial ligands by simple substitution reactions. This allows unambiguous confirmation that the U_eff mainly depends on the identity of X¹ and X², rather than on L_eq. More importantly, the fitted parameters are barrier dependent. If X¹ is an O donor and X² is a halide, 500<U_eff<600 cm⁻¹, log τ_0avg (s)=−10.66, log C_avg (s⁻¹ K⁻ⁿ)= −5.05, n_avg=4.1 and T_H=9 K (in which τ₀ is the pre-exponential factor for the Orbach relaxation process, C and n are parameters used to describe Raman relaxation, and T_H is the highest temperature at which magnetic hysteresis is observed). For cases in which both X¹ and X² are O donors, 900<U_eff<1300 cm⁻¹, log τ_0avg (s)=−11.63, log C_avg (s⁻¹ K⁻ⁿ)= −6.03, n_avg=4.1 and 18<T_H<25 K. Based on these results, it can be further concluded that U_eff not only has a linear correlation to the axial Dy−X bond lengths, but also to T_H for these PB SMMs. This represents the first systematic study of a family of lanthanide SMMs and derives the first magneto-structural correlation in Dy SMMs.     

Kinetic and thermochemical characteristics of the dissociation of Mo(CO)6 and W(CO)6

The thermal dissociation of gaseous Mo(CO)₆ and W(CO)₆ in an argon carrier gas, Mo(CO)₆ → Mo(CO)₅ + CO (1) and W(CO)₆ → W(CO)₅ + CO (2), is studied over temperature ranges of ∼585–685 K for (1) and ∼690−810 K for (2) at a total gas concentrations of 4 × 10⁻⁶ and 4 × 10⁻⁵ mol/cm³ by using the shock tube technique in conjunction with absorption spectrophotometry. The measured rate constants are extrapolated to the high-pressure limit by means of a newly developed procedure, with the resultant expressions for the indicated temperature ranges reading as k_d1,∞(T),[s⁻¹] = 10^{16.12 ± 0.68}exp[(−148.8 ± 8.1 kJ/mol)/RT] and k_d2,∞(T),[s⁻¹] = 10^{15.93 ± 0.63}exp[(−171.7 ± 8.9 kJ/mol)/RT]. Comparison of the high-pressure dissociation rate constants with the published data revealed a considerable discrepancy, a tentative explanation of which is given. Based on the obtained high-pressure dissociation rate constants and the available data on the high-pressure room-temperature rate constants for the reverse reaction of recombination, the first bond dissociation energies for these molecules are evaluated and compared with previous determinations, both theoretical and experimental. The enthalpies of formation of Mo(CO)₅ and W(CO)₅ are determined: Δ_fH°(Mo(CO)₅, g, 298.15 K) = −644.1 ± 5.6 kJ/mol and Δ_fH°(W(CO)₅, g, 298.15 K) = −581.9 ± 6.6 kJ/mol. Based on the enthalpies of formation of Mo(CO)₅, W(CO)₅, Mo(CO)₆, and W(CO)₆, and the published molecular parameters of these four species, their thermochemical functions are calculated and presented in the form of NASA seven-term polynomials.     

Excess molar volumes of (cyclohexanone+trichloromethane,or 1,2-dichloroethane,or trichloroethene,or 1,1,1-trichloroethane,or cyclohexane) at T=308.15 K,and of (cyclohexanone+dichloromethane) at T=303.15 K

Excess molar volumes V_m^E have been measured using a dilatometric technique for mixtures of cyclohexanone (C₆H₁₀O) with trichloromethane (CHCl₃), 1,2-dichloroethane (CH₂ClCH₂Cl), trichloroethene (CHClCCl₂), 1,1,1-trichloroethane (CCl₃CH₃), and cyclohexane (c-C₆H₁₂) at T=308.15 K, and for cyclohexanone+dichloromethane (CH₂Cl₂) at T=303.15 K. Throughout the entire range of the mole fraction χ of C₆H₁₀O, V_m^E has been found to be positive for χ C₆H₁₀O+(1−χ)c-C₆H₁₂, and negative for χ C₆H₁₀O+(1−χ)CH₂Cl₂, χ C₆H₁₀O+(1−χ)CHClCCl₂, χ C₆H₁₀O+(1−χ)CHCl₃, and χ C₆H₁₀O+(1−χ) CCl₃CH₃. For χ C₆H₁₀O+(1−χ)CH₂ClCH₂Cl, V_m^E has been found to be positive at lower values of χ and negative at high values of χ, with inversion of sign from positive to negative values of V_m^E for this system occurring at χ∼0.78. Values of V_m^E for the various systems have been fitted by the method of least squares with smoothing equation, and have been discussed from the viewpoint of the existence specific interactions between the components.     

Interface microstructure and diffusion kinetics in diffusion bonded Mg/Al joint

The interface microstructure, formation of diffusion bonded joint and regulation of atom diffusion were studied by means of scanning electron microscope (SEM), energy dispersion spectroscopy (EDS) and electron probe microanalyser (EPMA). The experimental results indicated that an obvious interfacial transition zone was formed between Mg and Al, and there are three intermetallic layers Mg₁₇Al₁₂, MgAl and Mg₂Al₃ in this zone. Diffusion activation energy of Mg and Al in the above layers was lower than that in the Mg and Al base metals. The thickness (x) of each layer can be expressed as x ² = 4.14exp(−28780/RT)(t−t ₀), x ² = 31.4exp(−25550/RT)(t−t ₀) and x ² = 0.6exp(−22600/RT)(t−t ₀) corresponding to Mg₁₇Al₁₂, MgAl and Mg₂Al₃ with heating temperature (T) and holding time (t).     

Kinetics of polymerization of styrene initiated by substituted benzoyl peroxides. IV. Decomposition induced by substituted diethylanilines

Equilibrium complex formation between symmetrically substituted benzoyl peroxides and ring-substituted N,N-diethylanilines has been quantitatively measured. From the results the rate constant for the unimolecular decomposition of the complexes, producing an initiating free radical, is found to be approximately 10¹² exp {(–25 X 10³)/RT}. The data obtained are consistent with a previously postulated mechanism for radical production in the presence of vinyl monomers.     

Kinetics of hydroxyl radical reactions with formaldehyde and 1,3,5-trioxane between 290 and 600 K

Absolute rate constants are measured for the reactions: OH + CH₂O, over the temperature range 296–576 K and for OH + 1,3,5-trioxane over the range 292–597 K. The technique employed is laser photolysis of H₂O₂ or HNO₃ to produce OH, and laser-induced fluorescence to directly monitor the relative OH concentration. The results fit the following Arrhenius equations: k (CH₂O) = (1.66 ± 0.20) × 10^?11 exp[?(170 ± 80)/RT] cm³ s^?1 and k(1,3,5-trioxane) = (1.36 ± 0.20) × 10^?11 exp[?(460 ± 100)/RT] cm³ s^?1. The transition-state theory is employed to model the OH + CH₂O reaction and extrapolate into the combustion regime. The calculated result covering 300 to 2500 K can be represented by the equation: k(CH₂O) = 1.2 × 10^?18 T^2.46 exp(970/RT) cm³ s^?1. An estimate of 91 ± 2 kcal/mol is obtained for the first C? H bond in 1,3,5-trioxane by using a correlation of C? H bond strength with measured activation energies.     

Pyrolysis of vinylacetylene between 300 and 450°C

Vinylacetylene was pyrolyzed at 300–450°C in a packed and an unpacked static reactor with a pinhole bleed to a quadrupole mass spectrometer. The reactant and C₈H₈ products were monitored continuously during a reaction by mass spectrometry. In some runs, the products were also analyzed by gas chromatography after the run. In these runs CH₄, C₂H₆, C₃H₆, and C₂H₄ were also detected. The reaction for vinylacetylene removal and C₈H₈ formation is homogeneous, second order in reactant, and independent of the presence of a large excess of N₂ or He. However, C₈H₈ formation is about half-suppressed by the addition of the free-radical scavengers NO or O₂. The rate coefficient for total vinylacetylene removal is 1.7 × 10⁶ exp(?79 ± 13 kJ/mol RT) L/mol · s. The major reaction for C₄H₄ removal is polymerization. In addition four C₈H₈ isomers, carbon, and small hydrocarbons are formed. The three major C₈H₈ isomers are styrene, cyclooctatetraene (COT), and 1,5? dihydropentalene (DHP). The C₈H₈ compounds are formed by both molecular and free-radical processes in a second-order process with an overall k ? 3 × 10⁸ exp(?122 kJ/mol RT) L/mol · s (average of packed and unpacked cell results). The molecular process occurs with an overall k = 8.5 × 10⁷ exp (?118 kJ/mol RT) L/mol · s. The COT, DHP, and an unidentified isomer (d), are formed exclusively in molecular processes with respective rate coefficients of 4.4 × 10⁴ exp(?77 kJ/mol RT), 1.7 × 10⁵ exp(?89 kJ/mol RT), and 3.1 × 10⁹ exp(? 148 kJ/mol RT) L/mol · s. The styrene is formed both by a direct free-radical process and by isomerization of COT.     

[(Ph)2(NMe2)C(OLi)⋅THF]2: Crystal Structure of the Tetrahedral Intermediate Formed in the Reaction of N,N-Dimethylbenzamide and Phenyllithium

Information on the reaction path for the 1,2-eliminiation of LiNMe₂ to form benzophenone is provided by the X-ray crystal structure analysis of the tetrahedral adduct [(Ph)₂(NMe₂)C(OLi)⋅THF]₂ (a portion of the structure is shown schematically), which is prepared from N,N-dimethylbenzamide and phenyllithium. A N1−Li1 interaction, which is not observed, would lead to loss of the anomeric effect (n_N→σ^*_C−O) as well as high conformational strain along the C1−N1 bond.     

Calculation of the Integral ∫T0T′mexp(–E/RT′)T′

A technique has been developed for the calculation of the ∫^T ₀ T′^mexp(–E/RT′)T′. The accuracy of the method is tested by comparing its predictions with numerical results and those of a method due to Quanyin and Su. This revised version was published online in July 2006 with corrections to the Cover Date.     

Water-absorption properties of an acetal copolymer

The kinetics of sorption from the liquid phase to equilibrium and desorption were studied over the temperature range 0–80°C. Equilibrium uptake was found to increase linearly with concentration in this range. Sorption-desorption kinetics showed the diffusion coefficients to decrease with increasing concentration, although the extent of this dependence did not appear in itself to be temperature-dependent. The apparent diffusion coefficient obeyed the law D = D₀ exp {? E/RT} over the temperature range studied, giving E = 9.9 kcal./mole and D₀ = 0.45 cm.² sec.^?1. These values are compared with corresponding values for other polymers.