Parabolic

Equation V-64 is that of a parabola, and electrocapillary curves are indeed approximately parabolic in shape. Because E ax tmd 7 max very nearly the same for certain electrolytes, such as sodium sulfate and sodium carbonate, it is generally assumed that specific adsorption effects are absent, and Emax is taken as a constant (-0.480 V) characteristic of the mercury-water interface. For most other electrolytes there is a shift in the maximum voltage, and is then taken to be Emax 0.480. Some values for the quantities are given in Table V-5 [113]. Much information of this type is due to Gouy [125], although additional results are to be found in most of the other references cited in this section. 

The rate law may change with temperature. Thus for reaction VII-30 the rate was paralinear (i.e., linear after an initial curvature) below about 470°C and parabolic above this temperature [163], presumably because the CuS2 product was now adherent. Non-... 

In a study of tarnishing the parabolic law, Eq. VII-30, is obeyed, with kj = 0. The film thickness y, measured after a given constant elapsed time, is determined in a... 

If the small temis in p- and higher are ignored, equation (A2.5.4) is the Taw of the rectilinear diameter as evidenced by the straight line that extends to the critical point in figure A2.5.10 this prediction is in good qualitative agreement with most experiments. However, equation (A2.5.5). which predicts a parabolic shape for the top of the coexistence curve, is unsatisfactory as we shall see in subsequent sections. 

The leading tenn in equation (A2.5.17) is the same kind of parabolic coexistence curve found in section A2.5.3.1 from the van der Waals equation. The similarity between equation (A2.5,5t and equation (A2.5.17) should be obvious the fomi is the same even though the coefficients are different. 

As in tire one-fluid case, the experimental sums are in good agreement with the law of the rectilinear diameter, but the experimental differences fail to give a parabolic shape to tlie coexistence curve. 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

Non-parabolic barrier tops cause the prefactor to become temperahire dependent [48]. In the Smoluchowski... 

The solution flow is nomially maintained under laminar conditions and the velocity profile across the chaimel is therefore parabolic with a maximum velocity occurring at the chaimel centre. Thanks to the well defined hydrodynamic flow regime and to the accurately detemiinable dimensions of the cell, the system lends itself well to theoretical modelling. The convective-diffiision equation for mass transport within the rectangular duct may be described by... 

If a compact film growing at a parabolic rate breaks down in some way, which results in a non-protective oxide layer, then the rate of reaction dramatically increases to one which is linear. This combination of parabolic and linear oxidation can be tenned paralinear oxidation. If a non-protective, e.g. porous oxide, is fonned from the start of oxidation, then the rate of oxidation will again be linear, as rapid transport of oxygen tlirough the porous oxide layer to the metal surface occurs. Figure C2.8.7 shows the various growth laws. Parabolic behaviour is desirable whereas linear or breakaway oxidation is often catastrophic for high-temperature materials. 

Figure C2.8.7. Principal oxide growth rate laws for low- and high-temperature oxidation inverse logarithmic, linear, paralinear and parabolic.

The fonn of the classical (equation C3.2.11) or semiclassical (equation C3.2.11) rate equations are energy gap laws . That is, the equations reflect a free energy dependent rate. In contrast with many physical organic reactivity indices, these rates are predicted to increase as -AG grows, and then to drop when -AG exceeds a critical value. In the classical limit, log(/cg.j.) has a parabolic dependence on -AG. Wlren high-frequency chemical bond vibrations couple to the ET process, the dependence on -AG becomes asymmetrical, as mentioned above. 

By following Section II.B, we shall be more specific about what is meant by strong and weak interactions. It turns out that such a criterion can be assumed, based on whether two consecutive states do, or do not, form a conical intersection or a parabolical intersection (it is important to mention that only consecutive states can form these intersections). The two types of intersections are characterized by the fact that the nonadiabatic coupling terms, at the points of the intersection, become infinite (these points can be considered as the black holes in molecular systems and it is mainly through these black holes that electronic states interact with each other.). Based on what was said so far we suggest breaking up complete Hilbert space of size A into L sub-Hilbert spaces of varying sizes Np,P = 1,..., L where... 

All Np states belonging to the Pth sub-space interact strongly with each other in the sense that each pair of consecutive states have at least one point where they form a Landau-Zener type interaction. In other words, all j = I,... At/> — I form at least at one point in configuration space, a conical (parabolical) intersection. 

Before we continue and in order to avoid confusion, two matters have to be clarified (1) We distinguished between two types of Landau-Zener situations, which form (in two dimensions) the Jahn-Teller conical intersection and the Renner-Teller parabolical intersection. The main difference between the two is... 

In other words, the quantization that was encountered for the non-adiabatic coupling terms is associated with the quantization of the intensity of the magnetic field along the seam. Moreover, Eq. (154) reveals another feature, namely, that there are fields for which n is an odd integer, namely, conical intersections and there are fields for which is an even integer, namely, parabolical intersections. 

---