General Statistical Correlation

Since the assumptions from which Eqs. (3.101) and (3.104) are derived cannot be more than rough approximations which may be expected to break down at high sorbate concentrations, an alternative approach based on Eq. (3.105) has been adopted for the correlation and analysis of equilibrium data for strongly adsorbed species such as the xylenes on X and Y zeolites. By integration of the Gibbs equation one may calculate the spreading pressure (or surface potential ) as a function of equilibrium vapor pressure or sorbate concentration, directly from an experimental isotherm [Eq. (3.50)]. It follows from Eqs. (3.90), (3.94), and (3.99) that 

TABLE 42. Correlation of Equilibrium Isotherms for Hydrocarbons on Zeolite NaX and NaY According to Eq. (3.105) 

If there are no significant nonideal interactions in a mixed adsorbed phase then a comparison of the equilibrium pressures for the two pure sorbates at the same temperature and spreading pressure (or the same value of 0 provides 

FIGURE 4.7. Experimental equilibrium data for hydrocarbons in NaX and NaY zeolite showing conformity of the isotherms to Eq, (4.6). 

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CONCEPT OF BASIC INITIATING PARAMETER. Since no direct theoretical method of predicting detonation incidence rate apparently exists, the only practical method of solution appears to be recourse to a statistical correlation of the controlled experimental data with frequency of occurrence data in the desired field application, using the best linking parameter that can be determined to establish the characteristic constants of an appropriate form of generalized empirical reaction rate equation. Past attempts to obtain a general statistical correlation have frequently been hampered by an erroneous assumption that the total impact energy constituted the sole independent experimental test parameter. The inadequacy of this as sumption is clearly shown in Fig, 2, which is based on test results reported by Lucas [5]. 

In a very broad overview of the structural categories one can state several statistical correlations with type of function. Hemes are almost always bound by helices, but never in parallel a//3 structures. Relatively complex enzymatic functions, especially those involving allosteric control, are occasionally antiparallel /3 but most often parallel a//3. Binding and receptor proteins are most often antiparallel /3, while the proteins that bind in those receptor sites (i.e., hormones, toxins, and enzyme inhibitors) are most apt to be small disulfide-rich structures. However, there are exceptions to all of the above generalizations (such as cytochrome cs as a nonhelical heme protein or citrate synthase as a helical enzyme), and when one focuses on the really significant level of detail within the active site then the correlation with overall tertiary structure disappears altogether. For almost all of the dozen identifiable groups of functionally similar proteins that are represented by at least two known protein structures, there are at least... 

The random walk model may be generalized by introducing a statistical correlation between two successive steps, in such a way that the probability a for a step in the same direction as the previous step differs from the probability ft for a step back ( random walk with persistence ). In this case... 

Rigorously, fully uncorrelated configurations (C(k) 0) require an infinite interval (k—>-00). In general, for an interval larger than 2r, the statistical correlation is less... 

Hyperlipidaemias are common 66% of the adult UK population have a plasma cholesterol concentration in excess of 5.2 mmol/1, the lowest concentration generally associated with cardiovascular risk (in fact, statistical correlation can be shown with cholesterol concentrations well below this value). 

The functional dependence expressed in Equations 4.29 and 4.31 governs the behavior of convective heat transfer. In some cases the functionality can be determined analytically, but in most cases it can be determined only as a statistical correlation of experimental data. Dimensionless groups are used to generalize empirical correlations for convective heat transfer. These groups can be determined from the parameters in Equations 4.29 and 4.31. They are the Reynolds, Nusselt, Grashof, and Prandtl numbers, respectively, defined as ... 

Fig. 3.2. Generalized field correlations of porosity, permeability and density with depth and distribution in the Oued el-Mya and Ghadames Basins (based on statistical processing of reservoir petrophysical data)...

Proportion of variance explained in general statistical programs provide information of the eigenvectors and, in some cases, the correlation between the original variables and the principal components. However, these correlations can be calculated from the eigenvectors in the following formula (Pla, 1986) ... 

The same is true of the classical Myers-Prausnitz theory with activity coefficients introduced in order to account for nonideality of the adsorbed phase and of the general statistical model [Eq. (4.17)] with the cross coefficients retained as parameters. Since the cross coefficients cannot, as yet, be predicted theoretically from the single-component isotherms, this reduces somewhat the predictive value of these models. However, it has been shown that, for the system N2-O2-CO-IOX, the vacancy solution theory with the cross coefficients evaluated from limited binary data provides a good prediction of the ternary equilibrium data. The same approach may be extended to multicomponent systems provided data for all constituent binaries are available. The vacancy solution theory thus provides a practically useful means of data correlation and makes possible the prediction of multicomponent equilibrium behavior from binary data. The potential for the application of classical solution theory or of the statistical models in a similar way has not yet been investigated to the same extent. 

In unpublished work the generalized statistical model [Eq. (4.17)] has been successfully applied to the correlation of liquid phase adsorption equilibrium data for Cg aromatics on faujasite zeolites. For these systems the saturation limit corresponds to approximately three molecules/cage, and at equilibrium with the liquid the adsorbent is essentially saturated so that each cage can be assumed to contain three sorbate molecules. This simplifies the model since only the terms corresponding to / + y = 3 in Eq. (4.17) need be retained, and the expression for the separation factor, assuming an ideal binary fluid phase, becomes... 

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