Extrema lines

There are certain rules, due to thermodynamic constrains, that these extrema (minima or maxima) lines must satisfy. For example, the and Cp lines must emerge from the LLCP. However, this is not necessarily the case for other extrema lines such as the p line (e.g., see Fig. 5a). In fact, thermodynamic arguments [45,46] indicate that the line must either merge with the p""" line or end at a spinodal line. In the first case, the line must merge with the p " line at a point in the P-T plane where the line has an extremum point see... 

There are also thermodynamic constrains that determine the points, in the P-T plane, at which the kj and p extrema lines may intersect one another. Specifically, the KT extrema line (i.e., the set of points belonging to either or Xyl "), may intersect the p extrema line only at the point where the slope of the p extrema line is infinite [47] (point A in Fig. 5a and b). Similar constrains between other thermodynamic extrema lines and the p extrema line can be obtained. In particular, it can be shown [43] that the line in the P-T plane defined by the condition dCp T, P)/dP)j = 0 must intercept the p extrema line at the point at which its slope is zero (point B in Fig. 5a and b). ... 

The presence of an LLCP and extrema lines in the supercritical region not only affects the thermodynamic properties of the liquid but also affects its dynamics. Recent computer simulations, based on atomistic [41], silica [39], and different water models [41], show that there is an intimate relationship between the C p line and the dynamic properties of the liquids, LDL and HDL. Specifically, it is found that in the more ordered liquid (i.e., the liquid with less entropy), the temperature dependence of the diffusion coefficient at constant pressure is given by D(T) exp(-EAT) (where Ep, is a constant), indicating that such a liquid is Arrhenius [51]. Instead, the less ordered liquid is found to be non-Arrhenius. Interestingly, in the supercritical region of the P-T plane, the dynamics of the fluid... 

Fig. 5. Pressure versus temperature phase diagram of the potential illustrated in Figure 1 for for rigid dumbbells with interatomic separation A/rr = 0.20 and A/rr = 0.50. The thin solid lines represent the boundaries between the fluid and the solid phases, the bold solid line, the TMD s lines, the dashed lines represent the D extrema lines and the dotted-dashed lines the...

An additional argument in favor of the presence of intercalation is the temperature dependence of the line width of the electron spin resonance signal. Although the starting metaanthracite shows a derivative extremum line width of 12.5 1.0 gauss (G) that is independent of temperature, the C8K product shows a line width that depends upon temperature ... 

For maximum ENDOR enhancement, the Zeeman modulation amplitude has to be about one half of the width of the EPR line which is saturated at an extremum of its first derivative. However, in an EPR spectrum with line widths of typically 1 mT this Zeeman modulation contributes 20 kHz to the width of a proton ENDOR line. It turns out that in many cases a remarkably better resolution of the spectra may be obtained with a single coding in which only the rf field is modulated. 

Figure 11. Top FCM and REM curves for Etrad2Mn2Cu3. The applied field is 20 Oe. The figure also shows the derivative dFCM/dT. Bottom plots of in-phase, /M, and out-of-phase, x m, against T for Etrad2Mn2Cu3. The vertical line corresponds to the extremum of dFCM/dT.

The state which corresponds to the minimum value of the detonation velocity possesses a number of remarkable properties in this state the extremum of the entropy is reached—a minimum on the Hugoniot adiabate and a maximum on the line joining the corresponding point with the initial one, pQ, v0, in the p, v plane. The detonation velocity in this state is equal to the sum of the velocity of the products and the velocity of sound in them. 

The wide line fluorine nuclear magnetic resonance of the intercalation compound "CifcBFV may be used not only to demonstrate the chemical identity of the inserted species but also to establish the translational freedom of this species. The chemical shift of the fluorine resonance is at (70+10) ppm vs. CF3COOH, consistent (11) with BF4 (71 ppm) but not with BF3 (54 ppm). (The neutral/ anion complex, B2 7> is also possible (12)). The derivative extremum llnewldth is narrow (0.02 mT = 800 Hz) at all temperatures between -168°C and 23°C. A simple calculation suggests that translation, and not rotation, is the cause of this narrow line. Assuming a first stage compound (as indicated by X-ray diffraction)... 

Whether a function is a maximum or a minimum at an extremum is, of course, determined by the sign of its second derivative or curvature at this point. The second derivative of a function/(x) at x (illustrated diagrammati-cally in Fig. 2.2) is the limiting difference between its two first derivatives or tangent lines which bracket that point... 

At the temperature corresponding to fig. 29.6, we have three phases a, e, j8 on a straight line. The system must therefore be in an indifferent state cf. fig. 29.1), and hence because of the Gibbs-Konovalow theorem the temperature of coexistence must pass through an extremum. If we suppose this to be a maximum, then the system will be represented by a diagram of the kind shown in fig. 29.7, which represents conditions of constant p, and in which temperature is plotted vertically. 

The calculated collision-induced absorption spectra associated with the 5D line are shown in Fig. 39.9(a) with the experimental one. The experimental peak in the blue side of this transition was attributed to the 6s2 5d2 transition of the collision complex. In the theoretical spectmm, R denotes, in the classical picture, the intemuclear distance of the collision complex at which the absorption occurs. The steep cusp exists in both experimental and theoretical spectra and is due to the extremum in the excitation energy... 

The solution to this three equations gives a set of x, y, that corresponds to an extremum. We obtain x —, y = —, e = Thus, we have obtained not only the position of the minimum x — y = —, but also the Lagrange multiplier e. The minimum value of E, which has been encountered when going along the straight line y = jx — is equal to... 

This method was widely used by Mangaray et al. The authors have presented [11] as the Gaussian function of (8j - 82). Therefore, dependence (1/ ()ln[Ti]j /= /(8,) can be expressed by a straight line intersecting the abscissa at a point for which 8 = 8j. For natural rubber and polyisobutylene, the paraffin solvents and ethers containing alkyl chains of a large molecular mass were studied. For polystyrene, aromatic hydrocarbons were used. For polyacrylates and polymethacrylates esters (acetates, propionates, butyrates) were used. The method was used for determination of 8j of many polymers. " In all cases, the authors observed extrema in the dependence of [tj] = f(8,), and the obtained values of 82 coincided well with the values determined by other methods. But for some polymers it was not possible to obtain extremum in dependence of [q] = f(8,). ... 

It is seen from Figure 3.97 that the maximum A (minimum B) of the critical line coincides with the maximum (minimum) of the upper (lower) cusp point line. Thi.s relation is valid generally. The extremum condition of the cusp point line requires dx, be equal to zero along the cusp line. Calculations involving Fxjiiations 14, 15, and 31 lead to an equation symmetrical with Equation 31... 

The dashed line in Fig. 7.62 shows at what point along each curve A, becomes zero. When X is zero, the extremum in the velocity profile occurs right at the screw surface. The dimensionless throughput in this case is ... 

This is the same value as the optimum dimensionless flow rate described by Eq. 7.258. It is interesting to note that this throughput is determined only by the power law index. The data above the dashed line have been determined with Eq. 7.265. In this case, Fr < s + 1 and no extremum occurs in the velocity profile. The data below the dashed line have been determined with Eq. 7.262. In this case, Fr > s + 1 and an extremum does occur in the velocity profile. 

The relation Eq. (42) between line tension and contact angle at given Xq is the condition for a local extremum of F. From Eqs. (41) and (42), one has for the value of the reduced free energy at this local extremum... 

An extremum of /c(x) in a fixed contour plane is calculable by setting its first derivative to zero. Thus, we arrive at third derivatives of the PES In the case of the simple test potential, the calculation leads to an implicitly given curvature line... 

Solving Equation (5.17) identifies the point that is both an extremum of / and also lies along the line x = y. This situation is different from that when x and y are independent. Then each of the partial derivatives equals zero (compare Equation (5.17) with Equation (5.12)). 

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