Percent correlation

For reasons discussed below, data points for the 4-SCH3 and 3-SCH3 derivatives were omitted from the fits of Equations 10-12. Hence n, the number of data points included in the fits, equals 13 instead of 15. Examination of Equations 10-12 indicates, however, that even though log (I/LD90) exhibits a fairly strong dependence on <7, the data are not particularly well correlated by the Hansch equation. Addition of r2 and tt terms to Equation 10 raises the percent correlation (r2) from 61 to 67%, but F is decreased and the standard error SE increased in the process. Hence, the correlation is not statistically improved. 

In Equations 4 and 5, r is the multiple correlation coefficient, r2 is the percent correlation, SE is the standard error of the equation (i.e the error in the calculated error squares removed by regression to the mean sum of squares of the error residuals not removed by regression. The F-values were routinely used in statistical tests to determine the goodness of fit of the above and following equations. The numbers in parentheses beneath the fit parameters in each equation denote the standard error in the respective pa-... 

Referring to Equation 4b, it is evident that there is a statistically significant parabolic relationship between apara and x for the 4-substituted TFMS derivatives. Both x2 and x terms are necessary in the optimum correlation equation, as evidenced by the fact that the percent correlation (r2) jumps from 2% in Equation 4a to 90% in Equation 4b upon stepwise addition of the linear term in x- No simple linear correlation between apara and x was found. Fits of the form o-para = f°r ex ... 

Substrates and reduction products were identified on a Hewlett Packard HP6890 gas chromatograph equipped with an FID detector and employing a 30m Restek RTX-200 eapillary column (trifluoropropylmethyl polysiloxane pacing). Yields of the resulting alcohols from these reductions were based on area percents correlated to known quantities of high purity standards (Aldrich). 

Fig. 1 Absolute deviation of ground-state energy of the variational 2-RDM method with DQG and DQGT constraints and the percent correlation defined as (Srdm - Em>)/ Efa - Ebf) shown...

In the temperature retraction test (ASTM D 1329), the test specimen is stretched to a specified elongation and then frozen at -70°C. Then it is gradually warmed until it begins to retract toward its original unstretched dimensions. The temperature at which the specimen retracts by 10 percent correlates with the brittleness temperature. 

The most widely computed quantity by DMC approaches is the total energy. Table 2 lists these energies and percent correlation energy recovered by the DMC method for first-row atoms, diatomic hydrides, and homonuclear diatomic molecules. Comparison of these results with values from ab initio expansion approaches discussed in this volume and elsewhere provide a clear indication of the high accuracy of the DMC method. As discussed elsewhere in this volume, the... 

At pressures above the highest real data point, the extrapolated data were generated by the correlation of Lyckman et al. (1965), modified slightly to eliminate any discontinuity between the real and generated data. This modification is small, only a few percent, well within the uncertainties of the Lyckman method. The Lyckman correlation was always used within its recommended limits of validity--that is, at reduced temperatures no greater than 1.5 to 2.0. 

Sulfur is the heteroatom most frequently found in crude oils (see Table 1.5). Sulfur concentrations can range from 0.1 to more than 8 weight percent moreover, this content is correlated with the gravity of the crude oil and, therefore, its quality (light or heavy). 

We consider first some experimental observations. In general, the initial heats of adsorption on metals tend to follow a common pattern, similar for such common adsorbates as hydrogen, nitrogen, ammonia, carbon monoxide, and ethylene. The usual order of decreasing Q values is Ta > W > Cr > Fe > Ni > Rh > Cu > Au a traditional illustration may be found in Refs. 81, 84, and 165. It appears, first, that transition metals are the most active ones in chemisorption and, second, that the activity correlates with the percent of d character in the metallic bond. What appears to be involved is the ability of a metal to use d orbitals in forming an adsorption bond. An old but still illustrative example is shown in Fig. XVIII-17, for the case of ethylene hydrogenation. 

Fig. XVIII-17. Correlation of catalytic activity toward ethylene dehydrogenation and percent d character of the metallic bond in the metal catalyst. (From Ref. 166.)...

Density, mechanical, and thermal properties are significantly affected by the degree of crystallinity. These properties can be used to experimentally estimate the percent crystallinity, although no measure is completely adequate (48). The crystalline density of PET can be calculated theoretically from the crystalline stmcture to be 1.455 g/cm. The density of amorphous PET is estimated to be 1.33 g/cm as determined experimentally using rapidly quenched polymer. Assuming the fiber is composed of only perfect crystals or amorphous material, the percent crystallinity can be estimated and correlated to other properties. 

In 1963 a classification of coals by rank (differing from the ECE scheme) was pubUshed by the International Committee for Coal Petrology (Table 2) (9). This includes a classification of brown coal that correlates a number of important properties including the percent reflectance of vitrinite in the coal. This is a simpler version of that used in German practice, which further subdivides soft brown coals into foHaceous and earthy. Most brown coals belong to the latter group. 

Fig. 1. Correlation of Dk with percent water (log-linear plot). 1 barrer = 10 cm O2 (at STP)-cm/(cm -s-mm Hg).

About 50% of copper in food is absorbed, usually under equitibrium conditions, and stored in the tiver and muscles. Excretion is mainly via the bile, and only a few percent of the absorbed amount is found in urine. The excretion of copper from the human body is influenced by molybdenum. A low molybdenum concentration in the diet causes a low excretion of copper, and a high intake results in a considerable increase in copper excretion (68). This copper—molybdenum relationship appears to correlate with copper deficiency symptoms in cattle. It has been suggested that, at the pH of the intestine, copper and molybdate ions react to form biologically unavailable copper molybdate (69). 

For prediction of vapor density of pm e hydi ocaihon and nonpolar gases, tbe corresponding states method of Pitzer et al. is tbe most accurate method, witb errors of less than 1 percent except in tbe critical region where errors of up to 30 percent can occur. Tbe method correlates tbe compressibibty factor by Eq. (2-75), after which tbe density can be calculated by Eq. (2-75) ... 

The activity coefficient (y) based corrector is calculated using any applicable activity correlating equation such as the van Laar (slightly polar) or Wilson (more polar) equations. The average absolute error is 20 percent. 

Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but 0 g and were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good an average absolute error of 6.4 percent, considering the complexity of some of... 

Lee-Thodos presented a generahzed treatment of self-diffusivity for gases (and liquids). These correlations have been tested for more than 500 data points each. The average deviation of the first is 0.51 percent, and that of the second is 17.2 percent. 8 = PyVr, s/cm and where G = (X - X)/(X - 1), X = p,/T h and X = p /T evaluated at the solid melting point. 

Lee and Thodos expanded their earlier treatment of self-diffusivity to cover 58 substances and 975 data points, with an average absolute deviation of 5.26 percent. Their correlation is too involved to repeat here, but those interested should refer to the original paper. 

Sun-Chen examined tracer diffusion data of aromatic solutes in alcohols up to the supercritical range and found their data correlated with average deviations of 5 percent and a maximum deviation of 17 percent for their rather hmited set of data. 

Catchpole-Kinp examined binaiy diffusion data of near-critical fluids in the reduced density range of 1 to 2.5 and found that their data correlated with average deviations of 10 percent and a maximum deviation of 60 percent. They observed two classes of behavior. For the first, no correction fac tor was required R = 1). That class was comprised of alcohols as solvents with aromatic or ahphatic solutes, or carbon dioxide as a solvent with ahphatics except ketones as solutes, or... 

Many more correlations are available for diffusion coefficients in the liquid phase than for the gas phase. Most, however, are restiicied to binary diffusion at infinite dilution D°s of lo self-diffusivity D -. This reflects the much greater complexity of liquids on a molecular level. For example, gas-phase diffusion exhibits neghgible composition effects and deviations from thermodynamic ideahty. Conversely, liquid-phase diffusion almost always involves volumetiic and thermodynamic effects due to composition variations. For concentrations greater than a few mole percent of A and B, corrections are needed to obtain the true diffusivity. Furthermore, there are many conditions that do not fit any of the correlations presented here. Thus, careful consideration is needed to produce a reasonable estimate. Again, if diffusivity data are available at the conditions of interest, then they are strongly preferred over the predictions of any correlations. 

Wilke-Chang This correlation for D°b is one of the most widely used, and it is an empirical modification of the Stokes-Einstein equation. It is not very accurate, however, for water as the solute. Otherwise, it apphes to diffusion of very dilute A in B. The average absolute error for 251 different systems is about 10 percent. ( )b is an association factor of solvent B that accounts for hydrogen bonding. 

Siddiqi-Lucas In an impressive empirical study, these authors examined 1275 organic liquid mixtures. Their equation yielded an average absolute deviation of 13.1 percent, which was less than that for the Wilke-Chang equation (17.8 percent). Note that this correlation does not encompass aqueous solutions those were examined and a separate correlation was proposed, which is discussed later. 

Chen-Chen Their correlation was based on diffusion measurements of 50 combinations of conditions with 3 to 4 replicates each and exhibited an average error of 6 percent. In this correlation, = Vg/[0.9724 (V, /g -I- 0.04765)] and = the liquid molar volume at the... 

Hayduk-Laudie They presented a simple correlation for the infinite dilution diffusion coefficients of nonelectrolytes in water. It has about the same accuracy as the Wilke-Chang equation (about 5.9 percent). There is no explicit temperature dependence, but the 1.14 exponent on I compensates for the absence of T in the numerator. That exponent was misprinted (as 1.4) in the original article and has been reproduced elsewhere erroneously. 

Siddiqi-Lucas These authors examined 658 aqueous liqiiid mixtures in an empirical study. They found an average absolute deviation of 19.7 percent. In contrast, the Wilke-Chang equation gave 35.0 percent and the Hayduk-Laudie correlation gave 30.4 percent. 

Dilute Binary Hydrocarbon Mixtures Hayduk-Minhas presented an accurate correlation for normal paraffin mixtures that was developed from 58 data points consisting or solutes from C5 to C32 and solvents from C5 to Cig. The average error was 3.4 percent for the 58 mixtures. 

Riazi-Whitson They presented a generahzed correlation in terms of viscosity and molar density that was apphcable to both gases and liqmds. The average absolute deviation for gases was only about 8 percent, while for liquids it was 15 percent. Their expression relies on the Chapman-Enskog correlation [Eq. (5-194)] for the low-pressure diffusivity and the Stiel-Thodos correlation for low-pressure viscosity ... 

Vigne.s empirically correlated mixture diffusivity data for 12 binary mixtures. Later Ertl et al. evaluated 122 binary systems, which showed an average absolute deviation of only 7 percent. None of the latter systems, however, was veiy nonideal. 

Siddiqi-Lucas suggested that component volume fractions might be used to correlate the effects of concentration dependence. They found an average absolute deviation of 4.5 percent for nonpolar-nonpolar mixtures, 16.5 percent for polar-nonpolar mixtures, and 10.8 percent for polar-polar mixtures. 

Lockhart and Martinelh (ibid.) correlated pressure drop data from pipes 25 mm (1 in) in diameter or less within about 50 percent. In general, the predictions are high for stratified, wavy, ana slug flows and low for annular flow The correlation can be applied to pipe diameters up to about 0.1 m (4 in) with about the same accuracy. 

Some judgment is required in the use of these correlations because of construction features of the condenser. The tubes must be supported by baffles, usually with maximum cut (45 percent of the shell diameter) and maximum spacing to minimize pressure drop. The flow of the condensate is interruptea by the baffles, which may draw off or redistribute the liqmd and which will also cause some splashing of free-falling drops onto the tubes. 

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