Statistical corrections

The value of k /k can be determined by measuring the ratio of the products 1-chlorobutane 2-chlorobutane during the course of the reaction. A statistical correction must be made to take account of the fact that the primary hydrogens outnumber the secondaiy ones by 3 2. This calculation provides the relative reactivity of chlorine atoms toward the primary and secondary hydrogens in butane ... 

When a Br nsted plot includes acids or bases with different numbers of acidic or basic sites, statistical corrections are sometimes applied in effect, the rate and equilibrium constants are corrected to a per functional group basis. If an acid has p equivalent dissociable protons and its conjugate base has q equivalent sites for proton addition, the statistically corrected forms of the Br insted relationships are... 

However, the kc values of monoimidazole complexes are larger than those of bisimidazole complexes although each of the former complexes have two hydroxyl groups so that statistical correction by factor 2 may be necessary for comparison. 

It has been suggested that the discrepancies between the value of k ikK observed and that predicted on the basis of simple statistics may reflect the greater sensitivity of combination to steric factors. Beckhaus and Rtichardt164 reported a correlation between log(A )/ ,<.,) (after statistical correction) and Taft steric parameters for a scries of alkyl radicals. 

However, it is not proper to apply the regression analysis in the coordinates AH versus AS or AS versus AG , nor to draw lines in these coordinates. The reasons are the same as in Sec. IV.B., and the problem can likewise be treated as a coordinate transformation. Let us denote rcH as the correlation coefficient in the original (statistically correct) coordinates AH versus AG , in which sq and sh are the standard deviations of the two variables from their averages. After transformation to the coordinates TAS versus AG or AH versus TAS , the new correlation coefficients ros and rsH. respectively, are given by the following equations. (The constant T is without effect on the correlation coefficient.)... 

The above expressions for the CO l (k ) and of are valid, if the statistically correct choice of the weighting matrix Q, (i=1,...,N) is used in the formulation of the problem. Namely, if the errors in the response variables (e, i=l,...,N) are normally distributed with zero mean and covariance matrix,... 

Given an EoS, the objective of the parameter estimation problem is to compute optimal values for the interaction parameter vector, k, in a statistically correct and computationally efficient manner. Those values are expected to enhance the correlational ability of the EoS without compromising its ability to predict the correct phase behavior. 

Most of the log Khb values were obtained from M. Berthelot. A statistical correction has been applied to obtain the values in parentheses. All of the data in Table 3.2 have been reported in reference 91. 

The statistical correction to log KHB is applied to those molecules having N indistinguishable atoms that can accept a hydrogen bond. Then KHB = "%Ni=l KHBi =, VKI B where Khb is the corrected value, and log KHB = log KHB - log N. 

Theoretically however, the statistically corrected values, K, are all the same, so that... 

We now have a sufficient number of conditions to determine the relationship between the measured equilibrium constants, K), K2,-, Km, and the statistically corrected constants, K), K2, ..., Km. The statistical corrections to be applied to the experimental stability constants are shown in Table 19.1 for the most common coordination numbers. 

Note that in the first case, the Cl ion is approaching an aqua complex that carries a + 3 overall charge. In the second case, the Cl is approaching as aquapentachloro complex that already carries a negative charge (—2), which is electrostatically unfavorable. Therefore, even after statistical correction of the stability constants is made, there is a great deal of difference in the likelihood that [M(H20)5C1]2+ and MCI 3 will form (the values of Kt and K6). 

These equilibria give directly only acidity differences between RH and R H and can vary with solvent and counterion. The corresponding — log f values have been converted to pK scales by choosing one compound as the standard and referring others to it. The standard chosen for tetrahydrofuran (THF) solutions is fluorene and it is assigned a pK of 22.9, its value in the DMSO scale (statistically corrected per hydrogen for fluorene the measured pK is 22.6)3,4. 

The product ratio a /b (statistically corrected for the number of competing H migrants), gives the relative migration rate of Ha vs. Hb, or k /kub- The rate constant for the migration of Ha corresponds to the intrinsic migratory aptitude of Ha (M[H]) multiplied by the bystander assistance factor for Y, B[Y], The carbon atom that bears Hb has no bystander substituent, so that km, is simply A/[H], We thus obtain Eq. 24. 

In the same vein is the observation that the lifetime of dipropylcarbene (59) in CH2C12 or cyclohexane is 0.3 ns,84 which, after statistical correction is 48 times less than the lifetime ( 21 ns) of Me2C in pentane.22 This reflects promotion by the propyl bystander groups of 59 of the 1,2-H shift to Z- and E-3-heptene.84 (Dipropylcarbene can be photolytically generated from either an oxadiazoline (diazoalkane)84 or diazirine85 precursor, but RIES lowers the efficiency of carbene production in either case.) Recently reported LFP lifetimes for Et2C and MeCEt in cyclohexane or benzene are 0.6-3 ns (cyclohexane) or 1-5 ns (benzene),14 in accord with the lifetimes of S822 and S9.84 The rate constants for carbene disappearance in cyclohexane ( 3 x 108 to 2 x 109 s 1) are presumably limited by 1,2-H shifts.14... 

The question of the choice of the appropriate symmetry number for con-formationally labile molecules is a subtle one and is at best an approximation. It is the opinion of this author that, if a = 18 for the n-alkanes is accepted as correct, then for the sake of consistency the a-values for all of the cycloalkanes have to be set at 2n, as shown by the following example. Consider the hypothetical reaction (36) for which a statistical correction is clearly... 

Avogadro s number and aRi is the symmetry number of the xr-meric cyclic oligomer, which represents a statistical correction for the number of equivalent bonds that can undergo the reverse ring-opening reaction. For Gaussian chains (57) takes the form (58) which is identical to (56), apart from the pres-... 

The Q test, suggested by Dean and Dixon (1951) is statistically correct and valid, and it may be applied easily as stated below ... 

Hydration and dehydration reactions have proved particularly convenient for studying catalysis by acids and bases over a wide range of structures and catalytic power. The results usually show a good correlation between acidic and basic catalytic constants and k ) and the of the catalyst, according to the usual Bronsted relations kjp = Ga qK /p) , ki, q = Gi, pjqKf, where p and q are statistical corrections. (These statistical corrections have not always been applied consistently, but the discrepancies thus introduced are not serious.) The information so far obtained is summarized in Table 3, and comments on the individual reactions follow. 

After allowing for a statistical correction for the number of protons that may be lost from the cation, the order of acid strengths (per NH-proton) of these cations remains unchanged. The initial trend of decreasing acidity with methyl substitution is reversed in the trimethylammonium ion, because with a decreasing number of protons in the cation, its hydration stabilization by hydrogen... 

Glaze, L.E, (2003, August). Bureau of Justice Statistics Bulletin Probation and Parole in the United States, 2002. (U.S, Department of Justice Office of Justice Programs. NCJ 201135.) http //www.ojp.usdoj.gov/bjs/pub/ pdf/ppus02.pdf h full text) In 2002,6.7 million people were on probation, in jail or prison, or on parole at yearend 2002 - 3.1% of all U.S. adult residents or 1 in every 32 adults. (Bureau of Justice Statistics, Corrections Statistics, Retrieved April 9, 2004 from http //www.ojp.usdoj.gov/bjs/ correct.htm)... 

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