P/<, values

We need to consider hypothetically what would happen if we investigated a treatment that had absolutely no effect. In particular, we calculate the risk that we might obtain a result as impressive as the one we actually observed. That likelihood is then quoted as the P value. 

A strict understanding of P needs to take account of two things  

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The existence of this situation (for nonporous solids) explains why the ratio test discussed above and exemplified by the data in Table XVII-3 works so well. Essentially, any isotherm fitting data in the multilayer region must contain a parameter that will be found to be proportional to surface area. In fact, this observation explains the success of Ae point B method (as in Fig. XVII-7) and other single-point methods, since for any P/P value in the characteristic isotherm region, the measured n is related to the surface area of the solid by a proportionality constant that is independent of the nature of the solid. 

The characteristic isotherm concept was elaborated by de Boer and coworkers [90]. By accepting a reference from a BET fit to a standard system and assuming a density for the adsorbed film, one may convert n/rim to film thickness t. The characteristic isotherm for a given adsorbate may then be plotted as t versus P/P. For any new system, one reads t from the standard r-curve and n from the new isotherm, for various P/P values. De Boer and co-work-ers t values are given in Table XVII-4. A plot of t versus n should be linear if the experimental isotherm has the same shape as the reference characteristic isotherm, and the slope gives E ... 

Use the data of Fig. XVII-1 to calculate qst for a range of p values and, in conjunction with Problem 11, plot st versus n/rim-... 

If //is 00 (very large) or T is zero, tire system is in the lowest possible and a non-degenerate energy state and U = -N xH. If eitiier // or (3 is zero, then U= 0, corresponding to an equal number of spins up and down. There is a synnnetry between the positive and negative values of Pp//, but negative p values do not correspond to thennodynamic equilibrium states. The heat capacity is... 

Assuming now that the power series expansion in F(p) can be terminated to keep /f(p) well behaved at large p values, it may be shown [95] that... 

The hydrophobic constant r is a measure of the contribution of a substituent X to the lipophilidty of compound R-X compared with R-H. The constant representing the solvent/solvent system, analogously to Hammett s p constant for the reaction type, was arbitrarily set to 1 for octanol/water and thus does not appear in Eq. (7). The lipophilidty constant ti allows the estimation of log P values for congeneric series of compounds with various substituents (see Eq. (8)). 

The obvious drawback of this method is that the parent solute, at least, has to be available or must be synthesized, and its log P value has to be determined experimentally. Nys and Rekker therefore developed a method known as the fragmental constant approach, which is based on the additivity of fragment contributions to the molecular lipophilidty [7] (see Eq. (9), where a( is the inddence of fragment i, fi the lipophilic fragment constant, Ci a corredion factor, and the frequency of Cm). 

In order to obtain the fragmental constants, sets of molecules for which experimental log P values are available are cut into the predefined fragments and the numerical contribution of each fragment is determined by multiple hnear regression analysis. 

These few examples are of course a small and arbitrarily chosen set of methods for the calculation of log P values. Nevertheless, it is hoped that they demonstrate some basic principles in the prediction of a physicochemical property. 

However, for screening large databases it is still necessary to use the easily calculated log P value. 

When, in a column headed M.p., a value is given in parenthesis, it indicates that the compound is liquid at room temperature and that the value given is consequently the boiling-point. Conversely in a column headed B.p., values given in parenthesis are those of the melting-point. A blank space indicates that the compound has not apparently been recorded. 

Having found the small- and large-p behaviors of S(p), we ean take S to have the following form to interpolate between large and small p-values ... 

The fact that good correlations are observed with d" rather than with a, is indicative of a strong infiuence of the substituent through a direct resonance interaction with a positive charge in the reacting system. The p-values are positive, which is expected for substituted dienophiles in a normal electron demand Diels-Alder reaction. Furthermore, the p-values do not exceed unity and are not significantly different from literature values reported for the uncatalysed reaction. It is tempting to... 

Table 2.4. Solvent effect on the Hammett p-values for the Diels-Alder reaction of2.4 with 2.5 catalysed by Cu(N03)2 at 25 - C.

Good to excellent Hammett plots were obtained using substituent constants (see Figure 2.6). Surprisingly, literature examples of good Hammett correlations of stability constants are rare The p-values are shown in Table 2.7. 

As anticipated, the complexation is characterised by negative p-values, indicating that the binding process is favoured by electron donating substituents. The order of the p-values for complexation of the different Lewis-acids again follows the Irving-Williams series. 

The effect of substituents on the rate of the reaction catalysed by different metal ions has also been studied Correlation with resulted in perfectly linear Hammett plots. Now the p-values for the four Lewis-acids are of comparable magnitude and do not follow the Irving-Williams order. Note tlrat the substituents have opposing effects on complexation, which is favoured by electron donating substituents, and reactivity, which is increased by electron withdrawirg substituents. The effect on the reactivity is clearly more pronounced than the effect on the complexation equilibrium. 

Table 2.7. Hammett p-values for complexation of 2.4a-e to different Lewis-adds and for rate constants (kcat) of the Diels-Alder reaction of 2.4a-e with 2.5 catalysed by different Lewis-acids in water at 2.00 M ionic strength at 25°C.

The susceptibility tensors give the correct relationship for the macroscopic material. For individual molecules, the polarizability a, hyperpolarizability P, and second hyperpolarizability y, can be defined they are also tensor quantities. The susceptibility tensors are weighted averages of the molecular values, where the weight accounts for molecular orientation. The obvious correspondence is correct, meaning that is a linear combination of a values, is a linear combination of P values, and so on. 

Streitwieser pointed out that the eorrelation whieh exists between relative rates of reaetion in deuterodeprotonation, nitration, and ehlorination, and equilibrium eonstants for protonation in hydrofluorie aeid amongst polynuelear hydroearbons (ef. 6.2.3) constitutes a relationship of the Hammett type. The standard reaetion is here the protonation equilibrium (for whieh p is unity by definition). For eon-venience he seleeted the i-position of naphthalene, rather than a position in benzene as the referenee position (for whieh o is zero by definition), and by this means was able to evaluate /) -values for the substitutions mentioned, and cr -values for positions in a number of hydroearbons. The p -values (for protonation equilibria, i for deuterodeprotonation, 0-47 for nitration, 0-26 and for ehlorination, 0-64) are taken to indieate how elosely the transition states of these reaetions resemble a cr-eomplex. 

The applicability of the two-parameter equation and the constants devised by Brown to electrophilic aromatic substitutions was tested by plotting values of the partial rate factors for a reaction against the appropriate substituent constants. It was maintained that such comparisons yielded satisfactory linear correlations for the results of many electrophilic substitutions, the slopes of the correlations giving the values of the reaction constants. If the existence of linear free energy relationships in electrophilic aromatic substitutions were not in dispute, the above procedure would suffice, and the precision of the correlation would measure the usefulness of the p+cr+ equation. However, a point at issue was whether the effect of a substituent could be represented by a constant, or whether its nature depended on the specific reaction. To investigate the effect of a particular substituent in different reactions, the values for the various reactions of the logarithms of the partial rate factors for the substituent were plotted against the p+ values of the reactions. This procedure should show more readily whether the effect of a substituent depends on the reaction, in which case deviations from a hnear relationship would occur. It was concluded that any variation in substituent effects was random, and not a function of electron demand by the electrophile. ... 

For solvolysis carried out at 25°C, the correlation gives the following values for p -6.15 for series 262 and -6.68 for series 263. The p values obtained for five other heterocycles and for benzene appear to be related... 

The relation between p values and Aq values is determined by the extent of substituent interaction with the charge developed in the transition state and thus is related to the pattern of charge distribution within the ring itself (386). 

Some Hammett values for reactions in thiazole and in nucleophile are reported in Table V-3. The observed p values for normal substitution processes (methoxy and thiophenoxysubstitution) are high and positive, indicating that the substituent plays an important role in modifying the stability of the intermediate anion. 

TABLE v-3. p VALUES FOR SOME SUBSTITUTION REACTIONS OF 2-HALOGENO-X-THIAZOLES WITH SUBSTirUTED NUCLEOPHILES... 

The Chauvenet tables relate the number of measurements to p values and assess the probability that the unusual value may be real or may be ignored. In the present example, the tables show that p = 1.73 for 6 measurements. Since the calculated value (1.80) is greater than 1.73, the result may be rejected. 

Does a P value of 1, 10, or 100 sec work best in Eq. (2.28) to describe the variation of 17 with 7 for this sample Note that no single-term version of the Eyring theory gives a totally acceptable fit. 

It should be observed that Eq. (3.102) may be viewed as a distribution function for relaxation times. In fact, if N,. is large enougli, integer increments in p may be approximated as continuous p values. This makes Tp continuous also. The significance of this is that Eq.(3.90) can be written as an integral in analogy with (3.62) if p is continuous ... 

Note that subtracting an amount log a from the coordinate values along the abscissa is equivalent to dividing each of the t s by the appropriate a-p value. This means that times are represented by the reduced variable t/a in which t is expressed as a multiple or fraction of a-p which is called the shift factor. The temperature at which the master curve is constructed is an arbitrary choice, although the glass transition temperature is widely used. When some value other than Tg is used as a reference temperature, we shall designate it by the symbol To. 

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