Conditionally stable case equations

Allylic alcohols are reduced with lithium or sodium in ammonia, or low molecular weight amines either with or without alcohols. The thermodynamically more stable product is often formed, leading to rearrangement in some cases (equation 71). Methyl and cyclic ethers are similarly reduced (equations 72 and 73), as are allylic acetates, halides and epoxides (equation 74 and 75). 7.i08 Benzylic and allylic sulfides and sulfones are readily reduced to hydrocarbons using lithium or sodium in alcoholic solvents or in amines. " Allylic sulfones are reduced in a similar manner (Scheme 11)," either with or without migration of the double bond, depending on the reaction conditions used. 

The emulsions, even in the presence of hexadecane and polymer, are not stable per se if they are stable, this is a fortunate case where the conditions according to equations (8.16) and (8.17) are fulfilled. 

Normally when a small change is made in the condition of a reactor, only a comparatively small change in the response occurs. Such a system is uniquely stable. In some cases, a small positive perturbation can result in an abrupt change to one steady state, and a small negative perturbation to a different steady condition. Such multiplicities occur most commonly in variable temperature CSTRs. Also, there are cases where a process occurring in a porous catalyst may have more than one effectiveness at the same Thiele number and thermal balance. Some isothermal systems likewise can have multiplicities, for instance, CSTRs with rate equations that have a maximum, as in Example (d) following. 

In some cases there also occur semistable limit cycles (in this discussion the single term cycle is used wherever it is unambiguous or if no confusion is to be feared) characterized by stability on one side and instability on the other side. Figure 6-5(a), (b), and (c) illustrate these definitions. Physically, only stable cycles are of interest the unstable cycles play the role of separating the zones of attraction of stable cycles in the case when there are several cycles. It is seen from this definition that, instead of an infinity of closed trajectories, we have now only one such trajectory determined by the differential equation itself and the initial conditions do not play any part. In fact, the term initial conditions means just one point (x0,y0) of the phase plane as a spiral trajectory O passes through that point and ultimately winds itself onto the cycle 0, it is clear that the initial conditions have nothing to do with this ultimate closed trajectory C—the stable [Pg.329]

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

Comparison of this equation with (79) gives >c = 1/h" > yl /4, that is, the Du Fort-Frankel scheme is stable for any r and h. This is a way of producing an analog of this scheme for the case when L is any elliptic operator with x still subject to condition (80). 

This condition can easily be verified in the case of discrete schemes for equations of mathematical physics. It allows one to extract from the primary family of schemes the set of stable schemes, within which one should look for schemes with a prescribed accuracy, volume of computations, and other desirable properties and parameters. 

C-NMR spectroscopic studies on a-substituted tris(ethynyl)methyl cations 49 prepared from alcohols 50 (equation 18) provided evidence for the participation of resonance structures with allenyl cationic character38. The parent tris(ethenyl)methyl cation (49, R = H) cannot be generated under stable carbocation conditions (SbFs/FSOsH) presumably due to the highly reactive unsubstituted termini of the three ethynyl groups and the resulting low kinetic stability. The chemical shift data (Table 1) give evidence that in all cases Ca and CY are deshielded more than Cg (relative to their precursor alcohols). 

Solving the forward problem of the isotopic and chemical evolution of n reservoir exchanging a radioactive and its daughter isotope requires the solution of 3n— 1 differential equations (the minus one stems from the closure condition). The parameters are n (n — 1) independent flux factors k for the stable isotope N and n (n — 1) independent M/N fractionation factors D. In addition, the n values of R y the n values of Rh and the n—1 allotments x of the stable isotope among the reservoirs must be assumed at some time, preferably at the beginning of the evolution (e.g., 4.5 Ga ago), or in the modern times, in which case integration is carried out backwards in time. 

In case of an inhomogenous periodic field such as the above quadrupole field, there is a small average force which is always in the direction of the lower field. The electric field is zero along the dotted lines in Fig. 4.31, i.e., along the asymptotes in case of the hyperbolic electrodes. It is therefore possible that an ion may traverse the quadrupole without hitting the rods, provided its motion around the z-axis is stable with limited amplitudes in the xy-plane. Such conditions can be derived from the theory of the Mathieu equations, as this type of differential equations is called. Writing Eq. 4.24 dimensionless yields... 

In the case of silyl-substituted arylacetylenes, the regioselectivity was reversed and a stable a-silylalkenylzinc reagent was formed preferentially, the latter being configurationally stable under the reaction conditions (equation 31). 

A potential way to avoid the formation of undesired side products, like in 7.2., is the use of such boron compounds that have only one transferable group. In most cases boronic acids are the compounds of choice, as they are easy to prepare, insensitive to moisture and air, and usually form crystalline solids. In certain cases, however the transmetalation of the heteroaryl group might be hindered by the formation of stable hydrogen bonded complexes. In such cases the use of a boronate ester, such as in equation 7.4., provides better yields. For example pyridine-2-boronic acid dimethylester coupled readily with a bromoquinoline derivative under conditions similar to 7.3. (potassium hydroxide was used as base and tetrabutylammonium bromide as phase transfer catalyst).6... 

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