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). 相似文献
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 相似文献
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 = 1016.8 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 k11c, = 109.6 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. 相似文献
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 相似文献
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 kr(…) 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: 相似文献
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 ΔHf(trimethylene) ≤ 280 kJ/mol at 300°K. 相似文献
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)/kr(AA)kr(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. 相似文献
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. 相似文献
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. 相似文献
H2S increases the thermal isomerization of butene-2 cis (Bc) to butene-1 (B1) and butene-2 trans (Bt) 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 Bt formation results from the addition–elimination processes: . It is shown that the initiation step of pure Bc 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 k1', 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. 相似文献
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. 相似文献
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). 相似文献
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 k2c/k2a 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. 相似文献
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–CO2(g) including ZnO, ZnCO3, Zn5(OH)6(CO3)2, and Zn2+ is given. 相似文献
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 [1m] X [r-n] → [2s, 1t] with respect to the group structure with the Gel'fand chain of subgroups . 相似文献
Metastable N2(A3Σu+), 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(A3Σu+) υ′ = 1 as well. 相似文献
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 kexchange (CF3 + Sn(Me)4) ~ 107 ml m?1 s?1. 相似文献
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 Enm (N) at large N have the asymptotic behavior Nn–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 Enm (N) and E(Z, N) are also proposed. 相似文献
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 ? 103.36±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. 相似文献