Monotonic series

I, Hg, H, Pb, Bi, Po, and At. However, due to the automatism in some cases, the values of the fitted parameters do not follow the intuitive monotonous series along the rows of the Periodic Table. E.g. the parameters for phosphorus fall out from the monotonous sequence along the row. More recent analysis indicates that this happens due to the fact that experimental values for phosphorus compounds used in the parametrization turned out to be wrong by 140 kcal/mole. 

A major contribution to the rational organization of contact angle data was made by Zisman and co-workers. They observed that cos 6 (advancing angle) is usually a monotonic function of 7l for a homologous series of liquids. The proposed function was... 

Figure A2.5.6 shows a series of typical p, Fisothemis calculated using equation (A2.5.1). (The temperature, pressure and volume are in reduced units to be explained below.) At sufficiently high temperatures the pressure decreases monotonically with increasing volume, but below a critical temperature the isothemi shows a maximum and a minimum.

In addition to the dependence of the intennolecular potential energy surface on monomer vibrational level, the red-shifting of the monomer absorption as a fiinction of the number of rare gas atoms in the cluster has been studied. The band origin for the Vppp = 1 -t— 0 vibration in a series of clusters Ar -HF, with 0 < n < 5, was measured and compared to the HF vibrational frequency in an Ar matrix (n = oo). The monomer vibrational frequency Vp p red shifts monotonically, but highly nonlinearly, towards the matrix value as sequential Ar atoms are added. Indeed, roughly 50% of the shift is already accounted for by n = 3. 

FIGURE 3.2 Possible results of increasing the order of Moller-Plesset calculations. The circles show monotonic convergence. The squares show oscillating convergence. The triangles show a diverging series. 

Shindo studied the hydrogen-bonding ability of a fairly long series of substituted pyridine 1-oxides with methanol in chloroform solution and found that the OH frequency of the hydrogen-bonded OH group in methanol is well correlated with the <7-values. For four compounds, the intensity of the same band is also well correlated. In a similar study the OH frequencies of phenol vary monotonically with the CT-values, but not in a linear fashion. 

Another simple approach assumes temperature-dependent AH and AS and a nonlinear dependence of log k on T (123, 124, 130). When this dependence is assumed in a particular form, a linear relation between AH and AS can arise for a given temperature interval. This condition is met, for example, when ACp = aT" (124, 213). Further theoretical derivatives of general validity have also been attempted besides the early work (20, 29-32), particularly the treatment of Riietschi (96) in the framework of statistical mechanics and of Thorn (125) in thermodynamics are to be mentioned. All of the too general derivations in their utmost consequences predict isokinetic behavior for any reaction series, and this prediction is clearly at variance with the facts. Only Riietschi s theory makes allowance for nonisokinetic behavior (96), and Thorn first attempted to define the reaction series in terms of monotonicity of AS and AH (125, 209). It follows further from pure thermodynamics that a qualitative compensation effect (not exactly a linear dependence) is to be expected either for constant volume or for constant pressure parameters in all cases, when the free energy changes only slightly (214). The reaction series would thus be defined by small differences in reactivity. However, any more definite prediction, whether the isokinetic relationship will hold or not, seems not to be feasible at present. 

Polymerization temperature is one of the most important parameters in both scientific and technological senses. A series of polymerizations were carried out with cat(2,4-Me)/EAS system over a temperature range between 10 and 70 °C and the results are summarized in Table 1. The catalyst system showed only negligible activity at 10 °C and the highest activity at 50 °C. As the temperature increase, the molecular wd t decreases monotonously,... 

More specifically, the reduced variable Kj = kjA is defined on [0, ti]. Generally, the error in an FD approximation (or rather its dispersion relation, Eq. (36)) increases with Kj. The Taylor series approach outlined above, which leads to the standard Lagrangian FD approximations, is essentially perfect for very small Kj but quickly deviates from the correct, quadratic dependence, Eq. (37). The generic behavior is that the error increases monotonically with Kj. Instead of requiring that the fit be perfect in the limit of very small Kj, we require that the error be no greater than s from Kj = 0 up to some... 

In each series the oxidations show monotonic cathodic shifts, while the reductions show monotonic anodic shifts as the heteroatom increases in atomic number. Thus it is easier both to oxidize and reduce the heavier diheteroferrocenes. 

For a reaction whose rate-concentration curve rises monotonically (any nth-order reaction, n > 0) the reactors should be connected in series. They should be ordered so as to keep the concentration of reactant as high as possible if the rate-concentration curve is concave (n > 1), and as low as possible if the curve is convex n < 1). 

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