Discrepancy correlations

Combination of Eq. 7 or Eq. 8 with the Young-Dupre equation, Eq. 3, suggests that the mechanical work of separation (and perhaps also the mechanical adhesive interface strength) should be proportional to (I -fcos6l) in any series of tests where other factors are kept constant, and in which the contact angle is finite. This has indeed often been found to be the case, as documented in an extensive review by Mittal [31], from which a few results are shown in Fig. 5. Other important studies have also shown a direct relationship between practical and thermodynamic adhesion, but a discussion of these will be deferred until later. It would appear that a useful criterion for maximizing practical adhesion would be the maximization of the thermodynamic work of adhesion, but this turns out to be a serious over-simplification. There are numerous instances in which practical adhesion is found not to correlate with the work of adhesion at ail, and sometimes to correlate inversely with it. There are various explanations for such discrepancies, as discussed below. 

The MP2 treatment recovers the majority of the correlation effect, and the CCSD(T) results with the cc-pVQZ basis sets are in good agreement with the experimental values. The remaining discrepancies of 9cm , 13cm and lOcm are mainly due to basis set inadequacies, as indicated by the MP2/cc-pV5Z results. The MP2 values are in respectable agreement with the experimental harmonic frequencies, but of course still overestimate the experimental fundamental ones by the anharmonicity. For this reason, calculated MP2 harmonic frequencies are often scaled by 0.97 for comparison with experimental results. ... 

The HF level as usual overestimates the polarity, in this case leading to an incorrect direction of the dipole moment. The MP perturbation series oscillates, and it is clear that the MP4 result is far from converged. The CCSD(T) method apparently recovers the most important part of the electron correlation, as compared to the full CCSDT result. However, even with the aug-cc-pV5Z basis sets, there is still a discrepancy of 0.01 D relative to the experimental value. 

Few other reactions of series of substituted pyridines have been investigated extensively. Dondoni, Modena, and Todesco have measured the rate of N-oxidation of a limited series of pyridines and found a good correlation with normal u-values with a p-value of — 2.23. The A-alkylation of pyridines with alkyl iodides in nitrobenzene has been studied by Brown and Cahn and by Clarke and Rothwell. Unfortunately, the only data available are for the parent compound and for alkyl derivatives, and, since the a-values for the various alkyl groups in a given position are substantially constant, this leaves a correlation of only three independent points. However, the rates of A-alkylation of the j8- and y-alkyl derivatives are so nearly equal that it appears as if no correlation existed. Clarke and Rothwell have also studied the alkylation with allyl bromide in nitromethane at various temperatures, and in this case a more extensive series is available. The authors state that no overall Hammett correlation is obtained however, the j8-substituted derivatives fall on one straight line and the y-derivatives on another one with a different slope. The data are shown in Fig. 2. The line for the j8-compounds, p = — 2.53 0.31, r = 0.95, is seen not to be very good the line for the y-derivatives, p = — 1.42 0.06, r = 0.99, is much more satisfactory. It does not seem likely that the discrepancy is due to the intervention of resonance effects, since in this case one would expect the correlation for the y-derivatives to be poorer than that for the j8-analogs. More extensive studies with a wider variety of substituents would seem very desirable. 

Table 16-111 collects some data. The existence of a correlation between the ionization energy difference, Ei(X) — Ei( Y), and the bond energy discrepancy, AHxr — i(AHx, + is obvious. 

It has been suggested that the discrepancies between the value of k ikK observed and that predicted on the basis of simple statistics may reflect the greater sensitivity of combination to steric factors. Beckhaus and Rtichardt164 reported a correlation between log(A )/ ,<.,) (after statistical correction) and Taft steric parameters for a scries of alkyl radicals. 

Table 3 shows that the activation enthalpies determined by various authors can be very different. These differences cannot be correlated to discrepancies in reaction orders since, even when these are the same, activation energies can vary. Since the theoretical difference between activation enthalpy and activation energy is low (2RT = 3kJ mol"1) with regard to the differences found in experimental determinations, the values discussed below are either enthalpies or energies of activation (For more detailed information see Table 3). 

The connection that has been shown in Section VIII to exist between burn-out in a rod bundle and in an annulus leads to the question of whether or not a link may also exist between, for example, a round tube and an annulus. Now, a round tube has its cross section defined uniquely by one dimension—its diameter therefore if a link exists between a round tube and an annulus section, it must be by way of some suitably defined equivalent diameter. Two possibilities that immediately appear are the hydraulic diameter, dw = d0 — dt, and the heated equivalent diameter, dh = (da2 — rf,2)/ however, there are other possible definitions. To resolve the issue, Barnett (B4) devised a simple test, which is illustrated by Figs. 38 and 39. These show a plot of reliable burn-out data for annulus test sections using water at 1000 psia. Superimposed are the corresponding burn-out lines for round tubes of different diameters based on the correlation given in Section VIII. It is clearly evident that the hydraulic and the heated equivalent diameters are unsuitable, as the discrepancies are far larger than can be explained by any inaccuracies in the data or in the correlation used. 

A summary of a number of correlations proposed for volumetric mass-transfer coefficients and specific interfacial area is presented in Table II, which includes data additional to those of Westerterp et al. (W4). It is apparent that disagreement exists as to the numerical values for the exponents. This is due, in part, to the lack of geometric similarity in the equipment used. In addition, variation in operating factors such as the purity of the system (surfactants), kind of chemical system, temperature, etc., also contribute to the discrepancies. To summarize Table II ... 

Since the rate was independent of acidity even over the range where H0 and pH differ, and the concentration of free amine is inversely proportional to the acidity function it follows that the rate of substitution is proportional to h0. If the substitution rate was proportional to [H30+] then a decrease in rate by a factor of 17 should be observed on changing [H+] from 0.05 to 6.0. This was not observed and the discrepancy is not a salt effect since chloride ion had no effect. Thus the rate of proton transfer from the medium depends on the acidity function, yet the mechanism of the reaction (confirmed by the isotope effect studies) is A-SE2, so that again correlation of rate with acidity function is not a satisfactory criterion of the A-l mechanism. 

The main problem in Eas0 vs. correlations is that the two experimental quantities are as a rule measured in different laboratories with different techniques. In view of the sensitivity of both parameters to the surface state of the metal, their uncertainties can in principle result of the same order of magnitude as AX between two metals. On the other hand, it is rare that the same laboratory is equipped for measuring both single-crystal face is not followed by a check of its perfection by means of appropriate spectroscopic techniques. In these cases we actually have nominal single-crystal faces. This is probably the reason for the observation of some discrepancies between differently prepared samples with the same nominal surface structure. Fortunately, there have been a few cases in which both Ea=0 and 0 have been measured in the same laboratory these will be examined later. Such measurements have enabled the resolution of controversies that have long persisted because of the basic criticism of Eazm0 vs. 0 plots. 

Basically, there may be three reasons for the inconsistency between the theoretical and experimental friction factors (1) discrepancy between the actual conditions of a given experiment and the assumptions used in deriving the theoretical value, (2) error in measurements, and (3) effects due to decreasing the characteristic scale of the problem, which leads to changing correlation between the mass and surface forces (Ho and Tai 1998). 

Hartree-Fock (HF), molecular orbital theory satisfies most of the criteria, but qualitative failures and quantitative discrepancies with experiment often render it useless. Methods that systematically account for electron correlation, employed in pursuit of more accurate predictions, often lack a consistent, interpretive apparatus. Among these methods, electron propagator theory [1] is distinguished by its retention of many conceptual advantages that facilitate interpretation of molecular structure and spectra [2, 3, 4, 5, 6, 7, 8, 9]. 

This leaves us with a computed resuit in iess than satisfactory agreement with the experimental value of about 170 kcal/mol(57). The neglect of electron correlation and the limited basis set used are the most important sources of the discrepancy. In a previous study on monolayer graphite(56), basis set effects were found to lead to a significant underestimation of the cohesive energy. 

Harmonic IR spectra of C3H2 calculated at the RHF/6-311++G(d,p), MP2/6-31 1++G(d,p) and MP4/6-31 1++G(d,p) levels are reported in Table 3. The results are nicely converging as electronic correlation is progressively included in the wave function. Excellent agreement between theory and experiment is thus obtained at the MP4 level, which allows for a correct treatment of simultaneous correlation effects in coupled vibrations. The only discrepancies which could show up, would proceed from anharmonicity, as illustrated by the CH stretching vibrations which are found shifted to higher frequencies than anticipated. 

However, as already noted, the barite content in Kuroko ore inversely correlates to the quartz content and the occurrences of barite and quartz in the submarine hydrothermal ore deposits are different. The discrepancy between the results of thermochemical equilibrium calculations based on the mixing model and the mode of occurrences of barite and quartz in the submarine hydrothermal ore deposits clearly indicate that barite and quartz precipitated from supersaturated solutions under non-equilibrium conditions. Thus, it is considered that the flow rate and precipitation kinetics affect the precipitations of barite and quartz. 

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