The statistical factor

Br0nsted s correlation was developed some years before Hammett s. Here we present a formulation that shows how both LFERs are related for general acid catalysts. For the sake of simplicity, we ignore the statistical factors of p and q. The Hammett equation applied to the values of bh is... 

In order to assess the analysis given in Figures 1 and 2 in terms of the components 1 and 2, the theoretically predicted abundances of the two components were calculated at 30 K, using the ZPVE values from Table 3. Including the statistical factors, the ratio obtained from the ZPVE s is 0.31 0.69, implying that substitution at the H2 position is slightly favoured compared to the purely statistical probability, as expected from the difference in bond strengths bewtween Cj—H and C2—H2. In a comparision with the... 

In the case of R = C6H5, the P/C phenyl shift dominates over the C/C shift to the extent of 2 1, the statistical factor already having been accounted for1U. 

Neglect of the statistical factor of2for L L is irrelevant to the present discussion. 

However, the behaviour near m = raB needs some other explanation. My proposal involves the specific solvation of the backside of the carbenium ion by the strong dipole of the solvent this displaces the monomer molecule which is located there in the absence of the solvent, so that the 7t-bond to the monomer at the front is weakened and the unimolecular isomerization-propagation becomes accelerated, despite the statistical factor which, alone, would produce a deceleration, as explained at the end of Section 3a. As the dilution proceeds from m = raB downwards, the polymerization goes through a dieidic phase, in... 

If the statistical factor is taken into account for each ion, then one obtains ... 

In chain reactions involving three termination steps (two uncrossed and one crossed) the quantity = 1c /(1c 1c )1/2 is frequently used to interrelate the cross-termination constant with the two uncrossed termination constants. For many different types of radical < is found to be about 1 (or alternatively, if the statistical factor of 2 favoring the crossed termination process is ignored in the definition of the rate constants, < 2). In the present reaction system —3-6, in agreement with the value obtained by Russell at 90°C. (26). The crossed termination constant itself is somewhat less than half the value found for kt. This seems reasonable since only one hydrogen atom will be available for transfer in the crossed termination, compared with the two that are available in the self-reaction of two tetralylperoxy radicals. In addition, steric hindrance to reaction should be greater for the crossed termination than for Reaction 8. The products are presumably cumyl alcohol, a-tetralone [3,4-dihydro-l(2H)naphthalenone], and oxygen (28). 

An analysis of the [Co-( )-pn3]3+ system may be carried out if the statistical term is considered solely an entropy effect and the conformational term an enthalpy contribution. Also since the four tris and four mixed species are not differentiated statistically, only the equilibrium constant k = tris/mixed is considered. For (4-)-pn/(—)-pn= 1 and assuming the ligands are distributed binomially around the metal ion, the statistical factor gives fc = 0.33 (J7/=0 assumed) which leads to TAS= -0.66 kcal/mole at 25°. 

By considering the symmetry of the normal modes of transition states Murrell and Laidler showed that problems encountered when calculating the statistical factors of transition states (which are needed to calculate the partition function in transition state theory) were associated with configurations of too high a symmetry to be transition states (61, 62). 

The introduction of the last term in Equation 27, the statistical factor for the number of ways of arranging i P molecules and (n — i)W molecules in the... 

For the n-butenes the activation energy profile shows no difference among the top of the barriers linking the three isomers (Table II). According to the model the theoretical selectivities determined only by the statistical factors should be k2i/kn ku/h2 kn/k2S = 1 3 3. The corresponding experimental selectivities were 1.2 2.9 2.4 and were temperature independent. The relative reactivities predicted by the model compared with those found experimentally are 1-butene cis-2-butene ran -2-butene = 1.0 (0.38 0.06) (0.12 0.03) vs. 1 0.37 0.18, respectively. 

Can we predict the pKa values in Table 7-1 With an appropriate dielectric constant chosen Eq. 2-8 can be applied. The difference between the two successive log Kf values reduced by 0.6 (the statistical factor) is a... 

In the MWC treatment, the assumption is made that the mixed dimer AB can be neglected entirely. However, a more general treatment requires that we consider all dimeric forms. The formation constants KAA, Km, and KAB are defined as follows13 113 (Eqs. 7-32 to 7-34 note the statistical factor of 2 relating KAB to the association constant Kt) ... 

In the shock tube investigation of 52, a series of experiments was performed using mixtures of THF and THF-da and also partially deuterated (3,3,4,4-D4) reactant, which enabled clear conclusions with regard to the cleavage of individual bonds. In this way it was shown that ethene is formed by elimination from the THF ring at the C-2-C-3 (C-4-C-5) (path B) and C-3-C-4 positions (path C) in a ratio of =2.2 1, which indicates that the rates of these paths are practically identical except for the statistical factor of 2. 

Figure 8.1 Shape of the statistical factor for 3 decay, which represents the expected shape of the electron momentum distribution before distortion by the Coulomb potential.

Exercise 4-15 Use the data given above for the percentages of the monochlorides formed in the vapor-phase chlorination of 2-methylbutane at 300° and take into account the statistical factors for the different numbers and kinds of hydrogens in answering the following ... 

Source Data from Spike and Parry, Thermodynamics of chelation I. The statistical factor in chelate ring formation, J. Amer. Chem. Soc. 75 2726, 1953. 

In connection with transition-state theory, one will also occasionally meet the concept of a statistical factor [13]. This factor is defined as the number of different activated complexes that can be formed if all identical atoms in the reactants are labeled. The statistical factor is used instead of the symmetry numbers that are associated with each rotational partition function (see Appendix A.l) and, properly applied, the... 

The specific rates of chloro-aliphatic derivatives range from 4 x 108 m-1 see-1 for compounds carrying a deactivated chlorine atom, e.g. chloro-ethanol, to the diffusion-controlled CF3C1, an activated chloroderivative (Anbar and Neta, 1967a). Polychloro-derivatives, e.g. CHC1S or CC14, react at diffusion-controlled rates (k = 3x 1010 m-1 sec-1) (Hart et al., 1964a). This is expected in view of the inductive effect of adjacent chlorine atoms in addition to the statistical factor in polychloro-derivatives. 

The di-f-butylhydroxylamine and di-/-butylhydroxylamine ethers probably result from DTBN scavenging of radicals produced by hydrogen atom abstraction from the olefins by excited 3-ethoxyisoindolenone (50). The observed destruction of DTBN as a function of olefin structure is consistent with this mechanism. Based upon allyl radical stability and the statistical factor, excited 3-ethoxyisoindolenone should abstract hydrogen atoms more rapidly from tetraroethylethylene than from ds-2-butene. 

Relative rates at position 5 of thiophene ring. The values are corrected for the statistical factor. b Reaction with Br2 in anhydrous acetic acid at 25°.85... 

The only available data concern the effects of alkyl groups on the trifluoroacetylation reaction (Table XXVI). A methyl group in position 2 increases (taking into account the statistical factor) the reactivity of the other a position by a factor of 23.8 (compared with factors of 380 and 1720 observed in thiophene and furan, respectively).143... 

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