Insertion probability

The insertion probability falls below 10 , well before the freezing transition at q 0.49. Similar estimates can be made for the Leimard-Jones fluid. The lower this factor becomes, the poorer the statistics, and the more... 

Figure B3.3.8. Insertion probability for hard spheres of various diameters (indieated on the right) in the hard sphere fluid, as a fiinetion of paeking fraetion p, predieted using sealed partiele theory. The dashed line is a guide to the lowest aeeeptable value for ehemieal potential estimation by the simple Widom method.

Another triek is applieable to, say, a two-eomponent mixture, in whieh one of the speeies. A, is smaller than the other, B. From figure B3.3.8 for hard spheres, we ean see that A need not be particularly small in order for the test partiele insertion probability to elimb to aeeeptable levels, even when insertion of B would almost always fail. In these eireumstanees, the ehemieal potential of A may be detemiined direetly, while that of B is evaluated indireetly, relative to that of A. The related semi-grand ensemble has been diseussed in some detail by Kofke and Glandt [110]. 

The observed trend - a decrease in the number of branches with increased steric bulk - is quite surprising. One could expect the opposite trend, since the steric effects increase the ratio between 1,2- and 2,1-insertions, and intuitively, this should lead to an increase in the number of branches (less 2,1-insertions - less removed branches). However, for the systems 3, 4, 5, and 7 the insertions at the secondary carbons happen with relatively large frequencies for the systems 3, 4, and 5 the probabilities of the insertion starting from the secondary carbon are c.a. 0.4-0.5, and for the system 7 - c.a. 0.25-0.33. Since every insertion into the secondary carbon by definition adds a branch, the global number of branches for the systems 3-5 and 7 is larger than for the more bulky catalyst 6, for which there are practically no insertions from the secondary carbons. An increase in the steric bulk leads to a decrease in the secondary-insertion probability, and... 

Reaction 7, which involves silica insertion, probably occurs at elevated temperatures since structure stabilization is observed mainly after high-temperature calcination. 

Ocular inserts probably represent one of the oldest ophthalmic formulation approaches. In 1948 the British Pharmacopoeia described an atropine-in-gelatin wafer and ever since then numerous systems have been developed applying various polymers and different release principals. However, the difficulty of insertion by the patient, foreign-body sensation, and inadvertent loss of inserts from the eye make these systems less popular, especially among the elderly. Furthermore, the high cost involved in manufacture prevented the insert market from taking off [197],... 

Oxidative addition of methyl iodide to the coordinalively unsaturated cobalt (I) species (1) gives the methyl complex (2) which undergoes CO insertion, probably via methyl migration. Elimination of iodine from the acetyl complex (3) and oxidative addition of hydrogen gives (5). Reductive elimination of the primary product acetaldehyde leads to the unsaturaied complex (6) which oxidatively adds iodine. The catalytic cycle is closed by the elimination of hydrogen iodide from (7), which is consumed by reaction with methanol to give methyl iodide. 

Consider a liquid solvent composed of one atomic type, e.g., liquid N2, and the probability density AnX pDi (A) of the distance A to the nearest atomic center of an arbitrarily chosen point. For the case of a spherical distinguished solute, use the notation that px (0) is the insertion probability for a solute... 

Figure 6.1 Mayer-Montroll expansion for the insertion probability p(0 X"). The notation here is fairly standard (see, for example, Hansen and McDonald, 1976 Andersen, 1977). The solid lines indicate factors of Mayer / functions introduced in Eq. (6.2) and are further discussed as Ursell functions beginning on p. 126. The inclusion-exclusion interpretation for hard-core cases is that the second term - assesses the m molecular volumes excluded to the...

Hexene addition also caused the appearance of significant amounts of internal unsaturation when the catalyst was Cr/AIPO4. Shown in Table 52, this result suggests that increased 2,1-insertion occurs, followed by elimination (Scheme 34C). Phillips catalysts are not known for chain walking, and therefore 2,1-insertion probably stops polymerization until the chain can be eliminated. This pathway may account for the fact that 1-hexene tends to inhibit the activity of Cr / AIPO4 but not that of Cr/silica. 

Although a model was proposed in which olefin insertion occurred to place the metal on the terminus of the alkene ( 1,2-addition ) [10, 26], based upon subsequent mechanistic and synthetic studies of the hydrosilylation reaction of styrenes (see below), this model would appear to be incorrect [27]. Thus an irreversible, stereochemically determinant 2,1-insertion probably initiates the reaction, with subsequent o-bond metathesis completing the process. Most remarkable is the fact that, if correct, this model demands that the olefin insertion takes place to orient the highly hindered metal center at a tertiary carbon center, and that apparently little, if any, P-hydride elimination occurs from the resultant organometallic. 

The ensemble average in the Widom formula, exp -p v g )), is sometimes loosely referred to as the insertion probability . It becomes very low for dense fluids. For example, for hard spheres, we can use the scaled-particle theory [104] or the Camahan-Starling equation of state [105] to estimate it (see figure B3.3.8. ... 

The polymerization reaction is a sequence of different events, such as monomer insertions, site isomerizations, and chain release reactions. The polymer chain can be seen as a permanent picture of the sequence of these events, and it is possible to use a statistical approach to study their distribution along the chain to increase our knowledge on polymerization mechanisms. As a consequence, a mathematical model of the polymerization can be built by assigning a probability at each event in our system. In the case of propene homopolymerization, this approach is (largely) used to study the mechanisms governing the stereoselectivity of the catalyst from the NMR spectrum of the polymer. In fact, the type and the relative amount of the stereosequences present in the chain are obtained from the methyl region of the spectrum and are usually determined at the pentad level (see section II.G). This distribution can be studied using insertion probabilities for propene enantiofaces, which depend on the type of stereocontrol mechanism active for the catalytic... 

The same is true for re-re and si-si options. The statistical modeling of experimental pentad distribution can be performed by considering a first-order Markovian model. The insertion probability for a si... 

These considerations, although not being a proof, make the cis-migration mechanism a very attractive model for olefin insertion, at least in octahedral complexes. Nevertheless, this conclusion should not be generalized without caution. In square-planar complexes RML3 (R = alkyl group or H), for instance, olefin insertion probably takes place at the site of the ligand R, in order to maintain the square-planar... 

The differential Eq. (14) relates the insertion probability with the pressure. When integrated for P, it gives a continuous analogue of Eq. (12) ( osmotic equation of... 

State ) that allows the calculation of PVT properties from a knowledge of the density-dependence of the insertion probability. When int rated for p , it provides an expression for the insertion probability if the equation of state is knowiL... 

Equation (12) is an exact expression for the pr ure in terms of the insertion probabilities. It is a form of the osmotic equation of state. The underlying physical picture is one of building up the system of N chains by adding chains one by one in the volume V. To make Eq. (12) practically useful, we need a way to calculate the insertion probabilities p (i,V,T). 

Etefining the repulsive insertion probability by exact analogy to Eq. (10),... 

The excluded volume interactions can be included via the correction factor or insertion probability for each bead, c [12] ... 

Nucleophilic attack at remote positions of coordinated ligands is observed when these are activated by strongly electron-withdrawing groups. For instance, PEts displaces a fluoride atom from complex 131 affording a phospho-niovinyl complex (Equation (66)). A related reaction is the hydrolysis of coordinated 2,4,6-trifluoropyrimidine, to afford a 4,6-difluoropyrimidin-2-one complex. An unusual reaction is the formal insertion (probably electrophilic attack) of heterocumulenes (GS2 or CO2) into a coordinate Ni-P bond of 70 (Equation (67)). 

Four principal theories describing the transformation of the isotropic fluid into a spontaneously organized nematic fluid, the I N transition, have been developed (1) Onsager s density expansion of the free energy of anisometric particles, (2) Rory s estimate of the insertion probability for a rod-like (multisite) solute into a lattice, (3) Maier and Saupe s construction of apotential of mean torque experienced by mesogens (or solutes) in a nematic environment, and (4) de Gennes transposition of Landau theory to the I N transition. We briefly examine each of these in reverse chronological order because each is relevant to more-recent theoretical descriptions of polymer mesophases. 

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