Condition equation

The point at which electron transfer takes place clearly corresponds to the condition equating equations (A2.4.132) for the states and/we find that... 

Equation (A3.3.73) is referred to as the Gibbs-Thomson boundary condition, equation (A3.3.74) detemiines p on the interfaces in temis of the curvature, and between the interfaces p satisfies Laplace s equation, equation (A3.3.71). Now, since ] = -Vp, an mterface moves due to the imbalance between the current flowing into and out of it. The interface velocity is therefore given by... 

Lety( l) = 0 by convention, then the Gibbs-Thomson boundary condition, equation (A3.3.73). becomes at r... 

To simplify the algebra, the error in the a variable will be considered negligible relative to the error in the y variable, although this is not a necessar y condition. Equation (3-54) describes a plane passing through the origin. Let us restrict it to positive values of Xi, a 2, and y. 

This integral is a gamma function and is readily solved using a table of integrals. In writing the last result uIq has been replaced by rQ the mean-square coil dimensions under 0 conditions. Equation (9.49) involves rjj" not r so we note that rQ = nlQ and replace n by I i - j I, the number of units separating units i and j, to obtain... 

For cakes of high permeabihty, the capillary drain height may be an insignificant fraction of cake thickness, and film drainage becomes the controlling factor in a centrifugal field (7). Under unsteady-state conditions, equation 18 represents the drainable Liquid left in the cake as a function of the centrifugal filtration parameters ... 

Aminoisoxazole has been synthesized in- high yield by the hydrolysis and ring closure reaction of cyclohexylideneaminoacrylonitrile under acidic conditions (equation 45) (69GEP1814116). 

That is. Equation (5.29) satisfies the differential equation. Equation (5.27), and the boundary conditions. Equation (5.28), so is the exact solution if... 

The solution to the governing differential equation, Equation (5.32), is not as simple as for specially orthotropic laminated plates because of the presence of D. g and D2g. The Fourier expansion of the deflection w. Equation (5.29), is an example of separation of variables. However, because of the terms involving D.,g and D2g, the expansion does not satisfy the governing differential equation because the variables are not separable. Moreover, the deflection expansion also does not satisfy the boundary conditions. Equation (5.33), again because of the terms involving D. g and D2g. 

Acyl fluorides [256] and formaldehyde [257] add to fluorinated olefins, such as difluoroethene, under Lewis or Brijinsted acid conditions (equation 54). 

On the contrary, tnfluoromethyl groups severely impede Sj l or reactions when located a to the reaction site. However, aminolysis of l-phenyl-2,2,2,-tri-fluoroethyl tosylate [Sd] and tnfluoroethyl chloride [ST] and its corresponding nonaflate [SS] can be performed under severe conditions (equation 76). 

Tetrabutylammonium iodide in trifluoroacetic anhydride is an effective reducing reagent [dS] This system can be used for direct reduction of arenesulfonic acids to the corresponding thiols or disulfides m moderate yields under mild conditions (equation 18) Alkanesulfonic acids are reduced by this system to disulfides with 30-57% yields [dfi]... 

It is an easy exercise to show that if Pn satisfies the Kolmogorov consistency conditions (equations 5.68) for all blocks Bj of size j < N, then T[N- N+LPN) satisfies the Kolmogorov consistency conditions for blocks Bj of size j < N + 1. Given a block probability function P, therefore, we can generate a set of block probability functions Pj for arbitrary j > N hy successive applications of the operator TTN-tN+i, this set is called the Bayesian extension of Pn-... 

Probabilities for blocks Bj with length j < N may be calculated by appealing to the Kolmogorov consistency conditions (equation 5.68). We now examine some low order LST approximations in more detail. 

A basis set of probabilities, B = p(i),P(2), >P(s) is selected for parameterizing arbitrary iV-block probabilities. It is a simple exercise to show that, because of the constraints imposed by the the Kolmogorov consistency conditions (equation 5.68, s -= 2 basis elements are necessary. 

The essence of the LST for one-dimensional lattices resides in the fact that an operator TtN->N+i could be constructed (equation 5.71), mapping iV-block probability functions to [N -f l)-block probabilities in a manner which satisfies the Kolmogorov consistency conditions (equation 5.68). A sequence of repeated applications of this operator allows us to define a set of Bayesian extended probability functions Pm, M > N, and thus a shift-invariant measure on the set of all one-dimensional configurations, F. Unfortunately, a simple generalization of this procedure to lattices with more than one dimension, does not, in general, produce a set of consistent block probability functions. Extensions must instead be made by using some other, approximate, method. We briefly sketch a heuristic outline of one approach below (details are worked out in [guto87b]). 

With this additional condition, equations 8.139 yield... 

We first observe that the compressibility condition (equation 9.4) remains unchanged ... 

For a chemical cell operated under standard state conditions, equation (9.91) becomes... 

In contrast to thiirane oxides, the electrophilic oxidation of thiirene oxides to thiirene dioxides is feasible, probably because both the starting material and the end product can survive the reaction conditions (equation 21). 

When 3-thiolene dioxide is treated with hydrogen peroxide, the corresponding epoxide is obtained313. The 3,4-trans-diols can be obtained by hydrolysis under acidic conditions (equation 120). 

One of the most important reactions of sulfones in general is the R amberg Backlund rearrangement, which involves the conversion of a-halosulfones to olefins with accompanying loss of hydrogen halide and sulfur dioxide under basic conditions (equation 48)151. 

Lucchi and coworkers68 have found a novel addition reaction of enolizable ketones to 1, l-bis(phenylsulfonyl)ethylene (102) under neutral conditions (equations 82 and 83). 

Unsymmetrical thiosulphinates and thiosulphonates are both oxidized by potassium superoxide in pyridine in the presence of 18-crown-6 ether to produce sulphinic and sulphonic acids and a disulphide, under mild conditions (equation 84)200,201. Sulphinic and sulphonic acids were produced from both the R and R substituents whilst the disulphide was derived only from the sulphenyl side of the reactant. Thus, the reaction mixture contained five products, making the reaction not synthetically useful. Pyrolysis of thiosulphinates also produces mixtures of products, one being the thiosulphonate again this is not a synthetically useful reaction202. 

For an irreversible process it may not be possible to express the relation between pressure and volume as a continuous mathematical function though, by choosing a suitable value for the constant k, an equation of the form Pv = constant may be used over a limited range of conditions. Equation 2.73 may then be used for the evaluation of / 2 v dP. It may be noted that, for an irreversible process, k will have different values for compression and expansion under otherwise similar conditions. Thus, for the irreversible adiabatic compression of a gas, k will be greater than y, and for the corresponding expansion k will be less than y. This means that more energy has to be put into an irreversible compression than will be received back when the gas expands to its original condition. 

For a given upstream pressure P] the maximum flow rate occurs when U2 = s/kPiih, the velocity of transmission of a pressure wave under these conditions (equation 4.38). 

The intramolecular Diels-Alder reaction of 78 was investigated during the synthesis of isoquinoline alkaloids [65ij. No reaction occurred when solid-phase conditions were used (Florosil in DCM and CaCli) or when a variety of Lewis acids were employed (SnCU, BF3, RAICI2, Ti(z — Pr)4-TiCl4). A 56 % yield of 79 was obtained by carrying out the cycloaddition in toluene in a sealed tube at 200 °C. jS-CD catalysis in water under milder conditions (Equation 4.11) improved the conversion to 84 %. 

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