Triangular scheme

Figure 7-6 depicts the relative concentrations in two experiments, one starting with pure trans isomer, the other cis. In the first of these [m] actually rises higher than [trans] before both fall nearly to zero. Microscopic reversibility comes into play because the rate constants for this triangular scheme are related by... 

FIGURE 4.2 Stahl triangular scheme to determine proper chromatographic conditions. (Adapted from Hahn-Deinstrop, E., J. Planar Chromatogr., 5, 57-61, 1992. With permission.)... 

Figure 14.1 Triangular scheme of composition isobutane/oxygen/inert, showing the flammability area for mixtures at room temperature, and the feed composition claimed by several industrial companies.

The gas phase isomerisation of substituted halobenzenes occurs readily on zeolites. Similarities appear with the same reaction catalyzed by AlClj in the homogeneous phase, in particular the applicability of the Hammett equation. However, several differences stand in the mechanism and the reaction scheme. Part of the transformation of halobenzenes occurs by means of a radical dechlorination/chlorination mechanism. For the same reasons the formal reaction follows a triangular scheme which differs from the consecutive scheme occuring in the liquid phase. Moreover, selectivity can be strongly affected by the restriction to diffusion in the porous volume of the solid. 

If sufficient knowledge is available concerning the previous history and chemical nature of the sample, then it is possible to choose the type of stationary and mobile phase according to the triangular scheme [5,40] (Fig. 55). Silica gel and aluminimn... 

The interrelations among cyclohexane, cyclohexene, and benzene in dehydrogenation can be shown by the following triangular scheme ... 

The evolution of concentration with time clearly indicates whether a particular component is an end product or an intermediate product that undergoes further reaction even as it is formed. The concentration-time (t or 7) profiles for the series, series-parallel, Denbigh, and triangular schemes mentioned above are sketched in Figure 5.4. An important feature of an intermediate product is that it has a maximum at a specific time If that compound happens to be the desired product, it is best to operate the reactor at / ax- Relationships for [Rjmax terms of the kinetic parameters of the reactions are important in maximizing production. They can be derived from the solutions given in Table 5.4, and are summarized in Table 5.5 for a few selected schemes in which R is produced as an intermediate. 

Figure 5.4 Concentration-time profiles of the various components in series, series-parallel, Denbigh, and triangular schemes.

Increasing composition of A Increasing composition of B Figure 7.3. Fractional recrystallization of a solution triangular scheme)... 

If the starting material contained a unit quantity of component A, and each crystallization step resulted in the deposition of a proportion P of this component, the proportions of A which would appear at any given point in the triangular scheme (see Figure 7.4) would be given by a term in the binomial expansion... 

A modification of the triangular scheme is shown in Figure 7.5. In this case further quantities of the feedstock AB are added to the system by dissolving it in successive mother liquors on the right-hand side of the diagram. This scheme is particularly useful if component A has a high temperature coefficient of solubility. 

For the three sets of 3N coordinates, x X and a Q l), to be equivalent, the transformations between them must be bijective, i.e. non-singular. In other words, the Jacobians of the applications in the triangular scheme below, must be non-zero, except, possibly, in sets of zero measure. As far as the various applications are concerned, those that are actually not obtainable straightforwardly are indicated by question marks in the following scheme ... 

Meffroy-Biget (65) and Meyerhoff (66) proposed a triangular scheme which is a combination of precipitation and extraction. In the first step the polymer is separated into two fractions. The next step comprises further separation of either fraction into two portions. Before continuing in this way one may, or may not, combine those two portions out of the four which are closest together in average molecular weight. The two possible triangular fractionation schemes are illustrated in Fig. 26. 

To the best of our knowledge there is a paucity of data in the literature on gas-liquid (without boiling) heat transfer in micro-channels. Hetsroni et al. (2003a) studied this matter in a test section, that contains 21 parallel triangular micro-channels of d/h = 130 pm. A scheme of the test section used in that work is shown in Fig. 5.13 (see Sect. 5.4). 

Fig. 6.1 The fundamental structural unit found in the Chevrel phases (cluster MoeXg full circles Mo atoms) displayed in three ways to emphasize different views of the connectivity. In (a) an octahedron of molybdenums (Mo-Mo = 2.7 A) is encased in a cube of chalcogens (Mo-S 2.45 or Mo-Se 2.6 A). Scheme (b) exhibits the same cluster as consisting of an octahedron with its triangular faces capped by chalcogenides. In (c), the cluster has been reoriented so that a threefold axis is vertical. (Reproduced from [10])...

Mossbauer spectra are usually recorded in transmission geometry, whereby the sample, representing the absorber, contains the stable Mossbauer isotope, i.e., it is not radioactive. A scheme of a typical spectrometer setup is depicted in Fig. 3.1. The radioactive Mossbauer source is attached to the electro-mechanical velocity transducer, or Mossbauer drive, which is moved in a controlled manner for the modulation of the emitted y-radiation by the Doppler effect. The Mossbauer drive is powered by the electronic drive control unit according to a reference voltage (Fr), provided by the digital function generator. Most Mossbauer spectrometers are operated in constant-acceleration mode, in which the drive velocity is linearly swept up and down, either in a saw-tooth or in a triangular mode. In either case. 

Dy3+-Dy3+ ferromagnetic coupling has also been observed in a double chain based on the triangular motif (topologic mode (B) in Scheme 4.1) [74]. Slow relaxation is observed (A = 44.2K, r0 = 2.4 X 10-8 s) but its ID origin is not demonstrated, and the onset of 3D ordering is visible at low temperature. 

Figure 3.20. A lateral view of different stacking sequences of triangular nets. They correspond to some typical close-packed structures. The first layer sequence shown corresponds to a superimposition according to the scheme ABABAB... (equivalent to BCBCBC... or CACACA... descriptions) characteristic of the hexagonal close-packed, Mg-type, structure. With reference to the usual description of its unit cell, the full stacking symbol indicating the element, the relative position of the superimposed layers and their distance is Mg Mg. The other sequences correspond to the schemes ABC.ABC. (Cu, cubic), ABAC.ABAC. (La, hexagonal), ACACBCBAB. (Sm, hexagonal). For Cu the constant ch of the (equivalent, non-conventional) hexagonal cell is shown which may be obtained by a convenient re-description of the standard cubic cell (see 3.6.1.3). ch = cV 3, body diagonal of the cubic cell.

Here, L is a lower triangular matrix (not to be confused with L, the Cholesky factor of the matrix of nonlinear parameters A ), and D is a diagonal matrix. The scheme of the solution of the generalized symmetric eigenvalue problem above has proven to be very efficient and accurate in numerous calculations. But the main advantage of this scheme is revealed when one has to routinely solve the secular equation with only one row and one column of matrices H and S changed. In this case, the update of factorization (117) requires only oc arithmetic operations while, in general, the solution from scratch needs oc operations. 

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