Hammerstein integral equivalent of Riccati's equation

A transformation exists which allows the general Riccati equation to be written in a simpler form: The transformed equation has the equivalent nonlinear Hammerstein integral equation if the kernel N(r, r′) satisfies three conditions: and and A solution of the nonlinear integral equation is devised by repeatedly integrating the Hammerstein equation. During this procedure the kernel generates an equation that contains only coefficients of β(r)⁰ and β(r)¹. As a result, after truncating at the end of the nth cycle, it is a simple matter to write down a Padé-type approximation: all coefficients in this approximation are capable of being evaluated in terms of simple algebraic formulations of P(r), R(r), and integrals over P(r). The zeroes of the denominator of the Padé-type approximation define the points where singularities occur in β(r).     

Thermal stability of alcohols

3,3-Dimethylbutanol-2 (3,3-DMB-ol-2) and 2,3-dimethylbutanol-2 (2,3-DMB-ol-2) have been decomposed in comparative-rate single-pulse shock-tube experiments. The mechanisms of the decompositions are The rate expressions are They lead to D(iC₃H₇? H) – D((CH₃)₂(OH) C? H) = 8.3 kJ and D(C₂H₅? H) – D(CH₃(OH) CH? H) = 24.2 kJ. These data, in conjunction with reasonable assumptions, give and The rate expressions for the decomposition of 2,3-DMB-1 and 3,3-DMB-1 are and      

The kinetics of thermal dimerization of hexatriene

The thermal isomerization of cis-hexatriene (cHT) to cyclohexadiene (CHD) and the dimerization of CHD and trans-hexatriene (tHT) in the liquid phase in the temperature range 380 K-473 K are reported. The rate coefficients are: for the cHT to CHD isomerization for tHT dimerizationlog and for CHD dimerization; endo form exo form © 1993 John Wiley & Sons, Inc.     

The self-inhibited pyrolysis of isobutane

The pyrolysis of isobutane was investigated in the ranges of 770° to 855°K and 20 to 150 Torr at up to 4% decomposition. The reaction is homogeneous and strongly self-inhibited. A simple Rice-Herzfeld chain terminated by the recombination of methyl radicals is proposed for the initial, uninhibited reaction. Self-inhibition is due to abstraction of hydrogen atoms from product isobutene giving resonance-stabilized 2-methylallyl radicals which participate in termination reactions. The reaction chains are shown to be long. It is suggested that a previously published rate constant for the initiation reaction (1) is incorrect and the value k₁ = 10^16.8 exp (?81700 cal mol^?1/RT)s^?1 is recommended. The values of the rate constants for the reactions (4i) (4t) (8) are estimated to be and From a recalculation of previously published data on the pyrolysis of isobutane at lower temperatures and higher pressures, the value k_11c, = 10^9.6 cm³ mol^?1 s^?1 is obtained for the rate constant of recombination of t-butyl. A calculation which is independent of any assumed rate constants or thermochemistry shows that the predominant chain termination reaction is the recombination of two methyl radicals in the conditions of the present work and the recombination of two t-butyl radicals in those of our previous study at lower temperatures and higher pressures.     

Pyrolysis of 2,4-dimethylhexene-l and the stability of isobutenyl radicals

2,4-Dimethylhexene-l has been decomposed in single-pulse shock tube experiments. Rate expressions for the initial reactions are and sec^?1 at 1.5–5 atm and 1050°K. This leads to ΔH^°_f300 (CH₂ = C(CH₃)CH₂) = 124 kJ/mol, or an allylic resonance energy of 50 kJ/mol. Rate expressions for the decomposition of the appropriate olefins which yield isobutenyl radicals and methyl, ethyl, isopropyl, n-propyl, t-butyl, and t-amyl radicals, respectively, are presented. The rate expression for the decomposition of isobutenyl radical is (at the beginning of the fall-off region). For the combination of isobutenyl and methyl radicals, the rate constant at 1020°K is Combination of this number and the calculated rate expression for 2-methylbutene-1 decomposition gives S. (1100) = 470 J/mol °K. This yields It is demonstrated that an upper limit for the rate of hydrogen abstraction by isobutenyl from toluene is      

Thermal stability of primary amines

Tertiary-amyl amine has been decomposed in single-pulse shock-tube experiments. Rate expressions for several of the important primary steps are This leads to D(CH₃? H) – D(NH₂? H) = ?10.5 kJ and D[(CH₃)₃C? H] – D[(CH₃)₂NH₂C? H] = + 6 kJ. The present and earlier comparative rate single-pulse shock-tube data when combined with high-pressure hydrazine decomposition results-(after correcting for fall off effects through RRKM calculations) gives where k_r(…) is the recombination rate involving the appropriate radicals. This suggests that in this context amino radical behavior is analogous to that of alkyl radicals. If this agreement is exact, then Rate expressions for the primary step in the decomposition of a variety of primary amines have been computed. In the case of benzyl amine where data exist the agreement is satisfactory. The following differences in bond energies have been estimated:      

Thermal decomposition of cyclopentane and related compounds

Cyclopentane has been decomposed in comparative-rate single-pulse shock-tube experiments. The pyrolytic mechanism involves isomerization to 1-pentene and also a minor pathway leading to cyclopropane and ethylene. This is followed by the decomposition of 1-pentene and cyclopropane. The rate expressions over the temperature range of 1000°–1200° K are Details of the cyclopentane decomposition processes are considered, and it appears that if the trimethylene radical is an intermediate, then ΔH_f(trimethylene) ≤ 280 kJ/mol at 300°K.     

Thermal decomposition of 3,4-dimethylpentene-1, 2,3,3-trimethylpentane, 3,3-dimethylpentane,and isobutylbenzene in a single pulse shock tube

Several hydrocarbons have been pyrolyzed in a single pulse shock tube. Rate parameters for the main bond breaking step have been found to be In combination with similar studies carried out earlier and through application of the well-established experimental rule (k(AB)/k_r(AA)k_r(BB))^1/2 ～ 2 where A and B are radicals and the rate constants are for the combination of these radicals, rate parameters for the thermal decomposition of all the hydrocarbons formed from any pair of the following radicals: methyl, ethyl, isopropyl, t-butyl, t-amyl, allyl, methylallyl, and benzyl have been calculated. The available calculated and experimental values of the decomposition rate constants are in excellent agreement. It appears that, with the possible exception of reactions involving the ejection of methyl radicals, the frequency factors per bond are nearly constant, depending only upon the type of carbon–carbon bond that is being broken. These values are all lower than those expected from the radical recombination rates. Heats of formation of ethyl, t-amyl, benzyl, methylallyl, n-propyl, s-butyl, isobutyl, neopentyl, and 3-pentyl radicals have been derived. Rate parameters for the decomposition of some simple ketones and ethers have also been estimated.     

Kinetics,mechanism, and endo selectivity of Diels–Alder reactions of alkylmonosubstituted ethenes with cyclohexa-1,3-diene in the gas phase

The reactions where Y = CH₃ (M), C₂H₅ (E), i? C₃H₇ (I), and t? C₄H₉ (T) have been studied between 488 and 606 K. The pressures of CHD ranged from 16 to 124 torr and those of YE from 57 to 625 torr. These reactions are homogeneous and first order with respect to each reagent. The rate constants (in L/mol·s) are given by The Arrhenius parameters are used as a test for a biradical mechanism and to discuss the endo selectivity of the reactions.     

Thermal stability of intermediate sized acetylenic compounds and the heats of formation of propargyl radicals

4-Methylhexyne-1, 5-methylhexyne-1, hexyne-1, and 6-methylheptyne-2 have been decomposed in comparative-rate single-pulse shock-tube experiments. Rate expressions for the initial decomposition reactions at 1100°K and from 2 to 6 atm pressure are In combination with previous results, rate expressions for propargyl C? C bond cleavage are related to that for the alkanes by the expression These results yield a propargyl resonance energy of D(nC₃H₇-H) – D(C₃H₃-H) = 36 ± 2 kJ, in excellent agreement with a previous shock-tube study. They also lead to D(CH₃C≡CCH₂-H) – D(C₃H₃-H) = 0.6 ± 3 kJ, D(sC₄H₉-H) – D(iC₃H₇-H) = 0 ± 3 kJ, D(iC₄H₉-H) – D(nC₃H₇-H) = 2 ± 3 kJ, and D(nC₃H₇-H) – D(iC₃H₇-H) = 13.9 ± 3 kJ (all values are for 300°K). The systematics of the molecular decomposition process are explored.     

H2S-promoted thermal isomerization of butene-2 CIS to butene-1 or butene-2 trans around 500°C

H₂S increases the thermal isomerization of butene-2 cis (B_c) to butene-1 (B₁) and butene-2 trans (B_t) around 500°C. This effect is interpreted on the basis of a free radical mechanism in which buten-2-yl and thiyl free radicals are the main chain carriers. B₁ formation is essentially explainedby the metathetical steps: whereas the free radical part of B_t formation results from the addition–elimination processes: . It is shown that the initiation step of pure B_c thermal reaction is essentially unimolecular: and that a new initiation step occurs in the presence of H₂S: . The rate constant ratio has been evaluated: and the best values of k₁ and k_1', consistent with this work and with thermochemical data, are . From thermochemical data of the literature and an “intrinsic value” of E_?3 ? 2 kcal/mol given by Benson, further values of rate constants may be proposed: is shown to be E₄ ? 3.5 ± 2 kcal/mol, of the same order as the activation energy of the corresponding metathetical step.     

The rate constant for t-C4H9 → H + i-C4H8 and the thermodynamic parameters of t-C4H9

The rate of the reverse reaction of the system has been measured in the range of 584–604 K from a study of the azomethane sensitized pyrolysis of isobutane. Assuming the published value for the rate constant of recombination of t-butyl we obtain Combination with our published data for k₁ permits the evaluation We have modified a previously published structural model of t-butyl by the inclusion of a barrier to free rotation of the methyl groups in order to calculate values of the entropy and enthalpy of t-butyl as a function of temperature. Using standard data for H and for i-C₄H₈ we obtain We have obtained other, independent values of this quantity by a reworking of published data using our new calculations of the entropy and enthalpy of t-butyl. There is substantial agreement between the different values with one exception, namely, that derived from published data on the equilibrium which is significantly lower than the other values. We conclude that the value obtained from the present work and a reworking of published data which involves the use of experimental data on t-butyl recombination is incompatible with the result based on iodination data.     

Unexpectedly rapid vapor-phase reaction between bromine and methyl iodide

The overall reaction (1) occurs readily in the gas phase, even at room temperature in the dark. The reaction is much faster than the corresponding process and does not involve the normal bromination mechanism for gas phase reactions. Reaction (1) is probably heterogeneous although other mechanisms cannot be excluded. The overall reactions (1) (2) proceed, for all practical purposes, completely to the right-hand side in the vapor phase. The expected mechanism is (3) (4) (5) (6) (7) where reaction (3) is initiated thermally or photochemically. Reaction (4) is of interest because little kinetic data are available on reactions involving abstraction of halogen by halogen and also because an accurate determination of the activation energy E₄ would prmit us to calculate an acccurate value of the bond dissociation energy D(CH₃? I).     

The kinetics of free radical chain reactions in solutions of trichlorofluorethylene in cyclohexane

The kinetics of the gamma-radiation-induced free radical chain reaction in solutions of C₂Cl₃F in cyclohexane (RH) was investigated over a temperature range of 87.5–200°C. The following rate constants and rate constant ratios were determined for the reactions: In competitive experiments in ternary solutions of C₂Cl₄ and C₂Cl₃F in cyclohexane the rate constant ratio k_2c/k_2a was determined By comparing with previous data for the addition of cyclohexyl radicals to other chloroethylenes it is shown that in certain cases the trends in activation energies for cyclohexyl radical addition can be correlated with the C? Cl bond dissociation energies in the adduct radicals.     

Zur Thermodynamik der Metallcarbonate 3. Mitteilung [1]. Löslichkeitskonstanten und Freie Bildungsenthalpien von ZnCO3 und Zn5(OH)6(CO3)2 bei 25°

The solubilities of ZnCO₃ and Zn₅(OH)₆(CO₃)₂ have been investigated at 25°C in solutions of the constant ionic strength 0,2 M consisting primarily of sodium perchlorate. From experimental data the following values for equilibrium constants and GIBBS free energies of formation are deduced: A predominance area diagram for the ternary system Zn²⁺–H₂O–CO_2(g) including ZnO, ZnCO₃, Zn₅(OH)₆(CO₃₎₂, and Zn²⁺ is given.     

On the irreducible tensor operators and the unitarily invariant decomposition of hermitian operators

In this paper, the unitarily invariant decomposition of Hermitian operators is performed by means of irreducible tensor operators to give the explicit expression of the coupling coefficients for [1^m] X [^r-n] → [2^s, 1^t] with respect to the group structure with the Gel'fand chain of subgroups .     

Quenching of N2(A3Σu+) by O2, O,N, and H

Metastable N₂(A³Σ_u⁺), υ = 0, υ = 1, molecules are produced by a pulsed Tesla-type discharge of a dilute N₂/Ar gas mixture. Rate coefficients for quenching these metastable levels by O₂, O, N, and H were obtained by time-resolved emission measurements of the (0, 6) and (1, 5) Vegard–Kaplan bands. In units of cm³/mole · sec at 300°K and with an experimental uncertainty of ±20%, these rate coefficients for N₂(A³Σu⁺) are Within the limits of error these coefficients apply to quenching N₂(A³Σ_u⁺) υ′ = 1 as well.     

Reaction of CF3 radicals with group IV tetramethyls

Arrhenius parameters have been measured for the abstraction of hydrogen from the C Si, Ge, and Sn tetramethyls: The rate constants correlate with the proton chemical shift, which is related to a polar effect. In all cases except carbon, a hot-molecule β-fluorine rearrangement-elimination reaction occurs following radical combination: We suggest the occurrence of a radical exchange reaction for the Si, Sn, and Ge systems, with k_exchange (CF₃ + Sn(Me)₄) ～ 10⁷ ml m^?1 s^?1.     

Relativistic theory of binding energies of heavy positive ions

The self-consistent relativistic Thomas–Fermi theory of heavy positive ions with N electrons and nuclear charge Ze is shown to lead to a chemical potential μ which has the scaling property with ? = α²Z², α being the fine structure constant. Combining this with the Layzer–Bahcall expansion for the total energy E(Z, N), namely, it is proved that the coefficients E_nm (N) at large N have the asymptotic behavior N^n–2m/3#1/3. The corresponding result for the scaling of the relativistic Thomas–Fermi energy is Scaling properties of the higher order terms in E_nm (N) and E(Z, N) are also proposed.     

Shock tube study of cyanogen oxidation kinetics

Mixtures of cyanogen and nitrous oxide diluted in argon were shock-heated to measure the rate constants of A broad-band mercury lamp was used to measure CN in absorption at 388 nm [B²Σ⁺(v = 0) ← X²Σ⁺(v = 0)], and the spectral coincidence of a CO infrared absorption line [v(2 ← 1), J(37 ← 38)] with a CO laser line [v(6 → 5), J(15 → 16)] was exploited to monitor CO in absorption. The CO measurement established that reaction (3) produces CO in excited vibrational states. A computer fit of the experiments near 2000 K led to An additional measurement of NO via infrared absorption led to an estimate of the ratio k₅/k₆: with k₅/k₆ ? 10^3.36±0.27 at 2150 K. Mixtures of cyanogen and oxygen diluted in argon were shock heated to measure the rate constant of and the ratio k₅/k₆ by monitoring CN in absorption. We found near 2400 K: and The combined measurements of k₅/k₆ lead to k₅/k₆ ? 10^?3.07 exp(+31,800/T) (±60%) for 2150 ≤ T ≤ 2400 K.