Indicator equation

Figure C2.2.11. (a) Splay, (b) twist and (c) bend defonnations in a nematic liquid crystal. The director is indicated by a dot, when nonnal to the page. The corresponding Frank elastic constants are indicated (equation(C2.2.9)).

The summations extend over sets of integers (ij,kj), (ij,k), (i,k), (0, respectively, which satisfy the indicated equations as well as the inequalities... 

Figure 3.16 Volcano plot for the hydrogen evolution reaction (HER) for various pure metals and metal overlayers. Values are calculated at 1 barof H2 (298K) and at a surface hydrogen coverage of either 0.25 or 0.33 ML. The two curved lines correspond to the model (3.24), (3.25) transfer coefficients (not included in the indicated equations) of 0.5 and 1.0, respectively, have also been added to the model predictions in the figure. The current values for specific metals are taken from experimental data on polycrystalline pure metals, single-crystal pure metals, and single-crystal Pd overlayers on various substrates. Adapted from [Greeley et al., 2006a] see this reference for more details.

It is shoiiin that X2(r l 2S indicating equation(20) is obviously more appropriate to the nonionic-ionic systems with and without added salt. If equation 13 is applied here, at 3D, 45 and 6D mNm l, X2o- values would be 0.47 0.44 and 0.43, respectively, which are quite different from X2(fValues, showing further the appropriateness of equation (207 (or equation (14)) for the system without added salt. 

Dialkyl methyl phosphonate derivatives 37a-c of mannopyranosides may be prepared from cyclic sulfate 36 by reaction with the appropriate lithiated methylphosphonate, prepared by titration of the corresponding methyl-phosphonate with butyllithium in the presence of 1,1-diphenylethylene as indicator (Equation 4) <2002TL4017>. 

The following results are found with the indicated equations ... 

Dialkylvinylaluminums transfer their vinyl substituent to stereodefined allylic acetates with Pd° catalysis with overall inversion of configuration. Metal attack of the organoaluminum is thereby indicated (equation 173).154... 

Comparing Equations 3.40, 3.43, and 3.44, it can be seen that equation coefLcients are not substantially changed when the number of compounds is reduced to half, indicating Equation 3.40 is robust. A prediction error of 0.272 in logarithmic scale for Equations 3.43 and 3.44 shows that, on the average, predicted values are accurate within a factor of 2. 

Fig. 1.16. Sequential evaluation of various electrochromatographic parameters from experimentally measured quantities in the two upper blocks by using the indicated equations of the present work.

Already at pH 10 and continuing down to pH 6, compounds 12 (X = H), 13 and 14 show a sharp decrease in k/s-1. Reaction is not buffer-catalyzed in this region, hence this is not due to rate-controlling protonation, by water, of a minority of enamine in equilibrium with its enammonium ion (equations 14 and 15). Nor can C-protonation by H30+ be rate-controlling since one would observe either an upward slope in the pH-rate profile (when [E] > [NH+]) or a plateau (when [NH+] > [E]). Although such characteristics in the rate profile were observed for compounds 1-3, they were not observed for the propiophenone enamines. Rather, a switch to rate-controlling nucleophilic attack on the iminium ion is indicated (equation 16) for which rate-law equation 22 is appropriate. 

Figure 1 Adsorption isotherms ofC in NaX (zeolite structure FAU), where n is thejamount absorbed and P is the pressure. Points indicate experimental data. Solid lines indicate Equation (1)...

This ellipse is also shown in Figure 9,10. In fact, 24 out of the 500 points, or 4.8% of the estimated parameters, lie outside this ellipse, which indicates Equation 9.11 is fairly accurate with this many random experimental trials. Note one often sees parameters reported with plus/minus limits. For this problem, one might report... 

Figure 4.2 Estimation of diffusion coefficients by autoradiography of tissue sections, (a) Digitized autoradiographic image of radiolabeled NGF released from a polymeric implant in a rat brain, reproduced from [18]. (b) The resulting concentration profile, obtained by scanning intensities from the autoradiographic image, and using calibrated standards to determine the local concentration the solid line indicates Equation 3-58.

The combination of the indicated equations allows to receive the following formula for calculation ... 

Thirdly, e decay rate change at Z) growth at D=. 5 can be attributed to qualitative changes of the dependence with certain reservations. This bending point is not accidental and can be characterized within the framework of fractal analysis [25], where intersections number of two arbitrary fractals with dimensions and can be estimated according to the Eq. (13) of Chapter 1. The indicated equation predicts the possibility of two fractals interpenetration (their transparency and opaqueness. Fractals penetrate freely one into another (are transparent), if decreases at growth, that is, if + /2. Assuming Df=Df and =3, let us obtain the transparence (leakiness) condition for macromolecular coils D <. 5. Thus, the repulsive interactions intensification of either macromolecule elements or macromolecule and solvent results in the end in fractals (macromolecular coils) transparence one for another. 

Let us note one more important aspect. The Eqs. (54) and (55) comparison allows to give one more definition of flexible- and rigid-chain polymers. The indicated equations suppose, that the greatest values of macromolecular coil fiactal dimension are equal to 2.0 (i.e., a coil in ideal 0-solvent [10]) for flexible-chain polymers and 1.667 (i.e., a coil in good solvent [10])—for rigid-chain ones. Hence, such polymers, chain rigidity of which (intramolecular interactions) allows to compensate it by... 

---