Probability reaction

One may justify the differential equation (A3.4.371 and equation (A3.4.401 again by a probability argument. The number of reacting particles VAc oc dc is proportional to the frequency of encounters between two particles and to the time interval dt. Since not every encounter leads to reaction, an additional reaction probability has to be introduced. The frequency of encounters is obtained by the following simple argument. Assuming a statistical distribution of particles, the probability for a given particle to occupy a... 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

A3.11.5.3 EXACT QUANTUM EXPRESSIONS FOR THE CUMULATIVE REACTION PROBABILITY... 

Zhang D H and Light J C 1996 Cumulative reaction probability via transition state wave packets J. Chem. Phys. 104 6184-91... 

Figure B3.4.6. Reaction probabilities for the initial-state-selected process H2(v = 0,J = 0)+OH(v,y = 0) — H2O+H, for zero total angular momentum. Taken from [75] with pennission.

The methodology presented so far allows the calculations of state-to-state. S -matrix elements. However, often one is not interested in this high-level of detail but prefers instead to find more average infomiation, such as the initial-state selected reaction probability, i.e. the probability of rearrangement given an initial state Uq. In general, this probability is... 

N E) is called the cumulative reaction probability. It is directly related to the themial reaction rate k T) by... 

A major achievement [71, 82, 83, 84, 85, 86, 87 and 88] was the development of a simple quantum ( flux-flux ) expression for the cumulative reaction probability, N E), with the final result... 

The flux-flux expression and its extensions have been used to calculate reaction probabilities for several important reactions, including H2+02 H + H2O, by explicit calculation of the action of G in a grid representation with absorbmg potentials. The main power of the flux-flux fomuila over the long mn will be the natural way in which approximations and semi-classical expressions can be inserted into it to treat larger systems. 

Figure B3.4.10. Schematic figure of a ID double-well potential surface. The reaction probabilities exliibit peaks whenever the collision energy matches the energy of the resonances, which are here the quasi-bound states in the well (with their energy indicated). Note that the peaks become wider for the higher energy resonances—the high-energy resonance here is less bound and Teaks more toward the asymptote than do the low-energy ones.

Figure B3.4.11. (a) Reaction probability for a 4D study of the dissociation of incident U2 on CO. The probability exhibits sharp peaks whenever the energy matches that of a resonance wavefiinction. (b) Plot of...

Seideman T and Miller W H 1992 Quantum mechanical reaction probabilities via a discrete variable representation-absorbing boundary condition Green function J. Chem. Phys. 97 2499... 

Zhang D H and Zhang J Z H 1995 Quantum calculations of reaction probabilities for HO + CO and bound states of HOCO J. Chem. Phys. 103 6512... 

Chemical reactions can be studied at the single-molecule level by measuring the fluorescence lifetime of an excited state that can undergo reaction in competition with fluorescence. Reactions involving electron transfer (section C3.2) are among the most accessible via such teclmiques, and are particularly attractive candidates for study as a means of testing relationships between charge-transfer optical spectra and electron-transfer rates. If the physical parameters that detennine the reaction probability, such as overlap between the donor and acceptor orbitals. 

The Arbusov Reaction probably takes the following course. 

The mechanism of the reaction probably involves the following stages ... 

The mechanism of the reaction probably involves the production, by into -action of the aldehyde with hydroxide ions, of two reducing anions, the first (I) more easily than the second (II). Either of these anions may transfer a hydride ion to a carbonyl carbon atom in another aldehyde molecule ... 

The mecluLnism of this base-catalysed reaction probably involves the intermediate formation of an aldol ... 

The reaction probably proceeds as follows. Crotonaldehyde is first formed by oondensation of the depolymerised acetaldehyde in the presence of acid ... 

The mechanism of the reaction probably involves the production of bivalent carbon during the initial loss of nitrogen the group R shifte from an adjacent position to this carbon leading to the production of a keten the latter then reacts with the solvent to give an acid, an amide or an ester. 

The reaction probably proceeds by an initial cyclisation of the acylamlnoacetic acid, followed by a Perkin type of condensation of the aldehyde with the active methylene unit ... 

The mechanism of the Michael reaction probably follows the following course. Writing RCH =CHCOOC,Hs for CjHgOOCCH=CHCOOCjHs and B for OCjHj or (CjHsljNH for the sake of simplicity, we have ... 

The reaction probably proceeds through oxalacetic acid as an intermediate HOOCCH(OH)CH OH)COOH HOOCC(OH)=CHCOOH ... 

Despite this overwhelming body of evidence, two-step mechanisms have been suggested for the Diels-Alder reaction, probably inspired by special cases, where highly substituted dienes and/or dienophiles have been found to react through zwitterionic or biradicalintermediates (Scheme 1.2). 

Mechanistic investigations have focused on the two pericyclic reactions, probably as a consequence of the close mechanistic relation to the so successful aqueous Diels-Alder reaction. A kinetic inquest into the effect of water on several 1,3-dipolar cycloadditions has been performed by Steiner , van... 

Nate 2. The reaction probably proceeds via the complex of CuCN and KCN, affording a mixture of propargyl cyanide and cyanoallene. The acetylene isomerizes under the influence of the slightly alkaline KCN. If the latter is added at too fast a rate, resinification of the allene may occur. Care should therefore be taken that some CuCN is always present. 

Similarly, the A-mesitylbydrazone of acetophenone gives 2-phcnyl-4,5,7-trimethylindole as a minor product (10%)[5]. 2,4,6-Trialkylphenylhydrazones have also been observed to give 5,6,7-trialkylindoles as the result of a formal 3a 6 shift[6]. These reactions probably occur via a 1,5-shift followed by a 1,2-shift. 

Indoles can also be alkylated by lactones[l4]. Base-catalysed reactions have been reported for (3-propiolactone[15], y-butyrolactone[10] and 5-valerolac-tone[10]. These reactions probably reflect the thermodynamic instability of the N -acylindole intermediate which would be formed by attack at the carbonyl group relative to reclosure to the lactone. The reversibility of the JV-acylation would permit the thermodynamically favourable N-alkylation to occur. 

The oxidation of 3-substituted indole to oxindoles can also be done with a mixture of DMSO and cone, hydrochloric acid[6-9]. This reaction probably involves a mechanism similar to the halogenation with a protonated DMSO molecule serving as the electrophile[10]. 

The reaction of propiolic acid or its esters with 2-aminothiazole yields 7H-thiazolo[3.2o]pyTimidine 7-one (109) (Scheme 74) (273), The reaction probably proceeds by initial nucleophilic attack of 2-aminothiazole on the sp C followed by intramolecular addition of ring nitrogen to spC. 

The electrophilic character of the C-2 atom is more clearly evident in A-4-thiazoline-2-ones than in A-4-thiazoline-2-thiones. 3-Mcthyl-A-4-thiazoline-2-one is cleaved in alcaline medium to give methylaraine (36). This reaction probably starts with the nucleophilic attack of OH on C-2. 

The reaction probably must involve two steps, aldolization followed by crotonisation. and depends among other factors on the specific reactivity of the aldehyde and of the methyl, which is influenced both by the nature of the substituents on the ring and bv the nature of the anion (Scheme 35) (57. 58. 657)... 

The reaction probably proceeds through the acyclic intermediate (109), which can be isolated when the reaction is carried out in cold acetone and can be then cyclized by heating. 

Partial but not complete loss of optical activity m S l reactions probably results from the carbocation not being completely free when it is attacked by the nucleophile Ionization of the alkyl halide gives a carbocation-hahde ion pair as depicted m Figure 8 8 The halide ion shields one side of the carbocation and the nucleophile captures the carbocation faster from the opposite side More product of inverted configuration is formed than product of retained configuration In spite of the observation that the products of S l reactions are only partially racemic the fact that these reactions are not stereospecific is more consistent with a carbocation intermediate than a concerted bimolecular mechanism... 

Mechanistically these reactions probably proceed through the hydrate of the aldehyde and follow a course similar to that of alcohol oxidation... 

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