Property Products versus Averaged Properties

Althongh some prodncts of properties are related to physical qnantities (e.g., according to Conlomb theory, the prodnct of charges is related to the force between them), other prodncts of properties may not have a physical meaning. In this case, a variant of Eqnation 5.13 may be applied that nses the mean of properties  

FIGURE 5.3 RDF descriptor calculated with Cartesian distances and the partial atomic charge as dynamic property. The charge distribution affects the probability in both the positive and negative direction. The strong negative peaks correspond to atom pairs with charges 

The effects of exchanging the property product by a mean property are obvious A property product with particular small or large values in one of the properties will lead to a significant decrease or increase in peak intensities. This has an amplifying effect on those peaks that originate from atom pairs with strongly different properties — often pairs of heteroatoms with nonheteroatoms. In contrast to that, Equation 5.14 leads to an attenuative effect with those atom pairs becanse the properties are averaged. 

Pairs with Cl exhibit a similar shape in distribution with generally smaller prod-nets of properties than averages. Pairs with 08 lead to an inverse behavior of prod-nets and averages dne to the strong negative charge of the oxygen atoms. 

Conseqnently, the resulting RDF descriptors — here called the amplified and attennated RDF — exhibit quite different shapes. The attenuated RDF descriptors typically exhibit more, or significant, peaks than the amplified variant when dynamic properties are used. In particular, dynamic properties of carbon atoms are often small in comparison with one of the heteroatoms. Thus, peaks related to carbon atoms are also small, whereas the peaks related to the heteroatoms are intensified. 

---

A plot of In n versus L is a straight line whose intercept is In and whose slope is —l/Gt. (For plots on base-10 log paper, the appropriate slope correc tion must be made.) Thus, from a given product sample of known shiny density and retention time it is possible to obtain the nucleation rate and growth rate for the conditions tested if the sample satisfies the assumptions of the derivation and yields a straight hne. A number of derived relations which describe the nucleation rate, size distribution, and average properties are summarized in Table 18-5. 

The models presented in the previous section are evaluated by comparing their predictions to actual plant data and also to the predictions of correlations proposed by previous researchers. The models give predictions that are close to the actual data. The average relative percent error between the predicted values and the actual values ranges from 0 to 10% for the proposed models. Figure 2.5 represents cross plots of the predicted versus the actual product yields and properties. As can be seen, all data points lie close to the 45° line, indicating a good fit. 

---