One-step expansion

One-Step Expansion—No Work. If mass Ml is removed from the pan, the gas will expand, moving the piston to the right end of the cylinder. After expansion the gas occupies a volume V2 = 4Vj and pressure P2 = Pi/4. 

One-Step Expansion. Now consider an experiment with the gas initially at state 1 where the mass is replaced by a mass MJA. The gas will now expand against the pressure (Pextemai) ... 

For example, in the one-step expansion and compression the net work done is... 

The first step is to use tp) and tp) to create a fitted charge density that has the same value and slope as the true charge density on the hard core spheres. This is thus a continuous, differentiable fit. As input we need the value and slope of the charge density on the a spheres. This is directly obtainable form the one center expansion. 

A bold step by universities is needed if their chemical engineering departments are (1) to help the United States achieve the preeminent position of leadership in new technologies and (2) to keep the highly successful U.S. chemical processing industries at the forefront of world markets for established technologies. The universities should conduct a one-time expansion of their chemical engineering departments over the next 5 years, exercising a preference for new faculty capable of research at interdisciplinary frontiers. 

The first analytical estimation of the value of the Markstein number in the presence of realistic gas expansion was given by Clavin and Williams [8]. In the approximation of a one-step Arrhenius reaction with a high activation energy P, they found fhe following expression for Ma ... 

Create a potential for production expansion by being one step ahead of environmental regulations... 

This simple, one-step ring expansion is the only available method for the preparation of 2-hydroxy-3-alkylisocarbostyrils in good yield from the corresponding 2-alkyl-l-indanones. Table I lists five new hydroxyisocarbostyrils prepared in this manner. 

Kametani and co-workers used a previously demonstrated ring expansion method to construct the isopavine skeleton (144,145). The method was successfully applied to the synthesis of ( )-reframidine (27) (Scheme 23) (145). Treatment of the 3-aryl-3,4-dihydroisoquinolinium iodide 115 with diazomethane furnished aziridinium iodide 116. On standing in 6 N hydrochloric acid, crude 116 underwent a one-step ring expansion-ring closure to afford ( )-reframidine in 20% yield. The same product could be obtained via benzazepine 118 depending on the reaction conditions. It has been postulated that the aziridinium iodide 116 may have formed a transitory quinonoid intermediate 117 which is attacked... 

In the specific case of the MMCC(2,3)/PT approximation, we go one step further and, after rewriting each I l p), Eq. (89), in the form of the Cl expansion relative to the reference determinant ) used in the CCSD and EOMCCSD calculations whose results we want to improve. 

At the restoration stage, a one-center expansion in the spherical harmonics with numerical radial parts is most appropriate both for orbitals (spinors) and for the description of external interactions with respect to the core regions of a considered molecule. In the scope of the discussed two-step methods for the electronic structure calculation of a molecule, finite nucleus models and quantum electrodynamic terms including, in particular, two-electron Breit interaction may be taken into account without problems [67]. 

All molecular spinors 4>p can be restored as one-center expansions in the cores using the nonvariational one-center restoration (NOCR) scheme [26, 27, 90, 92, 93, 94, 19, 97, 98] that consists of the following steps ... 

Exercise. Write the Kramers-Moyal expansion for one-step processes using (5.2). Exercise. Construct the Fokker-Planck approximation for the M-equation (VI.9.12) and use it to find [Pg.209]

Corey states that their experience shows the above procedure for conversion of (26) - (29) is the method of choice for ring expansion of a,/ -unsaturated ketones such as (26) and that this procedure may be generally useful. However, it is recommended that the operationally simpler one-step Johnson homologation procedure9 be tried first to prepare cycloheptenones. 

The combination of a cluster expansion reaction with removal of a mononuclear fragment results in a net metal exchange reaction retaining the overall cluster nuclearity. This simple process inevitably has a very complex mechanism if performed as a one-step synthesis. Nevertheless such reactions have been found for trinuclear clusters and systematized for three different kinds of reagents (16,178). 

It is of importance to point out that if the right-hand side is truncated after two terms (diffusion approximation), the last relation leads to an expression similar to the familiar Fokker-Planck equation (4.116). The approximation of a master equation of a birth-death process by a diffusion equation can lead to false results. Van Kampen has critically examined the Kramers-Moyal expansion and proposed a procedure based on the concept of system size expansion.135 It can be stated that any diffusion equation can be approximated by a one-step process, but the converse is not true. 

Whatever the case, the formation of the cyclooctenone 320 from the cyclobutanone 317 constituted the first example of a one-step C4 — C8 ring expansion. In the same way, the spirocyclobutanone 266 63,73), was treated with vinylmagnesium bromide to generate a mixture of the diastereomeric cyclobutanols 322 and 323 in 78% yield (ratio 21/79). Each diastereomer, separated by HPLC, individually subjected to KH in THF at room temperature, rearranged cleanly to bicyclo[5.3. l]undec-l(l l)-en-4-one in 80% yield, Eq. (87)161>. 

Suggest a mechanism for this ring expansion in which fragmentation is one step. 

Styrenes and styrene oxides can be combined in a highly chemo- and regioselective fashion to yield 2,4-bis-aryl-substituted tetrahydrofurans using an iron catalyst <2005CC1996>. This tetrahydrofuran synthesis developed by Hilt et al. opens an unprecedented way for the one-step synthesis of racemic calyxolane A and calyxolane B with moderate diastereoselectivities. The iron-catalyzed ring-expansion reaction of epoxyalkenes was considerably... 

The formation of norcaradiene derivatives with naphthalene [reaction (22)] lends some support to this scheme. This mechanism resembles a bimolecular two-step process suggested for the reaction of chloromethyl-aluminum compounds with olefins (199-201). On the other hand, a bimolecular one-step methylene transfer mechanism is generally accepted for the formation of cyclopropane derivatives by the reaction of halo-methylzinc compounds with olefins. This difference between the mechanism proposed for the cyclopropane formation from olefin and that for the ring expansion of aromatic compound may be ascribable to the difference in the stability of intermediates the benzenium ion (XXII) may be more stable than an alkylcarbonium ion (369). 

---