Chain condition

Methylcyclopropane shows strikingly different reactivity toward chlorination and bromination under radical-chain conditions. With chlorine, cyclopropyl chloride (56%) is the major product, along with small amounts of 1,3-dichlorobutane (7%). Bromine gives a quantitative yield of 1,3-dibromobutane. Offer an explanation for the difference. 

The N,0- and N,S-heterocyclic fused ring products 47 were also synthesized under radical chain conditions (Reaction 53). Ketene acetals 46 readily underwent stereocontrolled aryl radical cyclizations on treatment with (TMSlsSiH under standard conditions to afford the central six-membered rings.The tertiary N,0- and N,S-radicals formed on aryl radical reaction at the ketene-N,X(X = O, S)-acetal double bond appear to have reasonable stability. The stereoselectivity in hydrogen abstractions by these intermediate radicals from (TMSlsSiH was investigated and found to provide higher selectivities than BusSnH. 

Although under canonical radical chain conditions these silanes are poor reducing agents, there are some interesting applications, in particular, using Ph2SiH2 [100]. For example, the reaction of a variety of substituted adamantanes... 

By far the most generally useful synthetic application of allyltributyltin is in the complementary set of transition metal- and radical-mediated substitution reactions. When the halide substrates are benzylic, allylic, aromatic or acyl, transition metal catalysis is usually the method of choice for allyl transfer from tin to carbon. When the halide (or halide equivalent) substrate is aliphatic or alicyclic, radical chain conditions are appropriate, as g-hydrogen elimination is generally not a problem in these cases. 

AS is the entropy difference between crystal and liquid and is related to the increase in disorder when a crystal melts, or with the probabilities of the chain conditions in both phases. This follows from ... 

The intermolecular radical addition of alkyl iodides to terminal alkynes has been used in cyclization cascades performed under radical chain conditions (Scheme 2.3). For example, Oshima and coworkers applied triethylborane as radical initiator to... 

In this sequence, NO3 can formally be regarded as a synthon for O atoms in solution. Because the released NOa is a comparatively stable radical that does not initiate a radical chain process under the applied reaction conditions, this sequence is termed a self-terminating oxidative radical cyclization —the cascade-initiating radical contains the unreactive leaving group that is released in the final step of the reaction. In contrast to reactions performed under classical radical chain conditions and that are nearly always associated with a loss in overall functionality of the molecule, in self-terminating radical cyclizations the net amount of functional groups is retained in the system. 

One of the earliest radical cyclization cascades initiated by addition of S-centered radicals to alkynes was reported in 1987 by Broka and Reichert (Scheme 2.25). Thiophenyl radicals, PhS, which were generated under radical chain conditions, undergo addition to the terminal end of the C = C triple bond in enyne 138. The resulting vinyl radical 141 can undergo cychzation in both 6-endo (preferred) and 5-exo fashion, and reduction of the radical intermediates 142 and 143 leads to the final observed products 139 and 140, respectively. 

Under long chain conditions and maintenance of the equilibrium, the rate of COCI2 formation is... 

In the references cited/ - it was fully recognized that products differ greatly between chain and non-chain conditions. The purpose of this analysis is to provide an overall viewpoint of this effect and convenient tables for ready location of chain and non-chain regions. 

The chain condition is the general description of the evolution of the probability density for an infinitely divisible stable process and the solution to which has the most general form of a Markov probability density. When the range of the variate is unbounded 0= (—00,00) and the process under consideration has translational invariance, so the probability density is independent of the origin of the coordinate system P(x, t xo, to) = P(x — xo, t — to), the chain condition becomes... 

The stationary chain condition (159) is more simply expressed in terms of characteristic functions, the Fourier transform of the probability density, as the product... 

The symmetric solution to the chain condition is obtained by setting the constant in the exponential to be independent of a, b(a) = b. The most general solution is obtained using... 

Definition 4. A topological space X is noetherian if its closed sets satisfy the descending chain condition (d.c.c.). It is equivalent to require that all open sets be quasi-compact (= having the Heine-Borel covering property, but not necessarily T2). 

Proposition 2. If X is a prevariety, then the closed sets of X satisfy the descending chain condition, i.e., X is a noetherian space. 

More generally, let R be any commutative ring with descending chain condition. Then R is the direct sum of its primary subrings ... 

Definition 1. A scheme X is noetherian if for all open sets U C X, the partially ordered set of closed subschemes of U satisfies the descending chain condition. 

The chlorination has been compared with bromination by BrCClj carried out under radical chain conditions. In this reaction, cyclohexane is about one-fifth as reactive as toluene, but in the chlorination by trichloromethanesulfonyl chloride, cyclohexane is about three time more reactive that toluene. Does this information permit a choice between the chain sequences you have written in part (a) ... 

Methylcyclopropane shows strikingly different reactivity toward chlorine and bromine under radical chain conditions in CHjClj solution. The main product with chlorine is chloromethylcyclopropane (56%), along with smaller amounts of 1,3-dichlorobutane and l,3-dichloro-2-methylpropane. Bromine gives only... 

When the diamine used is aromatic, only low-molar-mass polymer is formed because of the lower-reaction rates. To produce longer chains, conditions must be readjusted so that both phases are polar and miscible, and vigorous high-speed stirring of the system is necessary. This is used in the reaction between isophthaloyl chloride and /n-phenylene diamine. These aromatic polyamides are particularly versatile materials and of considerable interest. 

Causality models based on event chains (or dominos or layers of Swiss cheese) are simple and therefore appealing. But they are too simple and do not include what is needed to understand why accidents occur and how to prevent them. Some important limitations include requiring direct causality relationships, subjectivity in selecting the events to include, subjectivity in identifying chaining conditions, and exclusion of systemic factors. 

Therefore, in an attempt to obtain simple analytic expressions for the distribution functions of stiff polymer chains, condition (5.2a) is relaxed. The relaxation of this condition is in the original spirit of the use of Wiener integrals. If this condition were imposed for flexible polymer chains, the Wiener measure would be 2[t s)] exp (—3L/2/) and would give equal weight (measure) to all continuous configurations of the polymer. Thus the use of (5.2a) would not yield the correct gaussian distribution for flexible chains. 

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