Statistical equations

Some equations frequently used in evaluation of experimental data are given in this chapter. Their theoretical background and mathematical derivations can be found in cited literature. 

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For the second term, we use the same trick Leave n on the right-hand side of the vertical and use the Bayesian statistics equation to invert C and a ... 

Table 4.2 Comparison of the ideal gas entropy in J K mol 1 at a pressure of 101.32 kPa as calculated from the Third Law and from the statistical equations...

Limiting expressions of r for large numbers of coordinates in WEG, FAN, and PAR were calculated from Equation 3.B4 and expressions described in Appendix 3B for a family of (3i and p2 values and were tested by simulation. Specifically, 500 (x, y) random correlated coordinates were distributed over WEG, FAN, and PAR for specific Pi and p2> and rwas calculated from the usual statistics equation (Eq. 3.B 1 in Appendix 3B). The simulation was repeated 1000 times, and the average r was compared to the limiting expressions. 

The gel point is usually determined experimentally as that point in the reaction at which the reacting mixture loses fluidity as indicated by the failure of bubbles to rise in it. Experimental observations of the gel point in a number of systems have confirmed the general utility of the Carothers and statistical approaches. Thus in the reactions of glycerol (a triol) with equivalent amounts of several diacids, the gel point was observed at an extent of reaction of 0.765 [Kienle and Petke, 1940, 1941], The predicted values of pc, are 0.709 and 0.833 from Eqs. 148 (statistical) and 2-139 (Carothers), respectively. Flory [1941] studied several systems composed of diethylene glycol (/ = 2), 1,2,3-propanetricarboxylic acid (/ = 3), and either succinic or adipic acid (/ = 2) with both stoichiometric and nonstoichiometric amounts of hydroxyl and carboxyl groups. Some of the experimentally observed pc values are shown in Table 2-9 along with the corresponding theoretical values calculated by both the Carothers and statistical equations. 

Finally, we note that all the statistical equations of this chapter could have been borrowed directly from the kinetic theory of gases by simply changing the variables. We illustrate this now by going in the opposite direction. For example, if we replace the quantity 3/n 2 by m/ kBT and replace L by v in Equation (69), we obtain the Boltzmann distribution of molecular velocities in three dimensions. If we make the same substitutions in Equation (73), we obtain an important result from kinetic molecular theory ... 

Segregated or corpuscular models regard biomass as a population of individual cells. Consequently, the corresponding mathematical model is based on statistical equations. Such models are valuable for describing the variations in a given populations such as the age distribution amongst the cells. This approach is also useful for describing stochastic events, in which case probability and statistics are applied. 

Experimental data obtained by any of the assay methods must be evaluated by someone. Judgements are commonly based on the experience of the analyst and accumulated laboratory data or published results. The evaluations range from comparison with a simultaneous standard to highly sophisticated statistical equations requiring many calculations. Evaluation of juice content should be considered as an estimate in the context of placing the sample somewhere in the natural population distribution, and the probability of that estimate should be reported. Unfortunately, many literature reports fail to mention or minimize the uncertainty of the estimate. In samples where the presence of foreign substances is proven, one can state with absolute certainty that the juice has been adulterated. 

Several laboratories have measured solubilities and/or partition coefficients of solutes in higher molecular weight media, and their data provides a test of this approach to estimating polymer solubilities. Flynn and Yalkowsky (17,18) studied the transport and solubility properties of a series of p-aminobenzoate esters, p-HjNCjH COOR, R = methyl to hexyl, in poly(dimethylsiloxane) fluid (PDMS). We find that their values of log P(PDMS) correlate well with reported (33) values of log P values of the same series of solutes in oleyl alcohol/water, as illustrated by the plot in Figure 8 and the correlation statistics (Equation 17). 

Mean centering changes the number of degrees of freedom in a principal component model from k to k + 1. This affects the number of degrees of freedom used in some statistical equations that are described later. 

As argued for the PRESS statistic (Equation 6.21) in PCA, this RMSECV, statistic is also not suitable for use with contaminated data sets because it includes the prediction error of the outliers. A robust RMSECV (R-RMSECV) measure can be constructed in analogy with the robust PRESS value [61], Roughly said, for each PCR model under investigation (k = 1,. .., km3X), the regression outliers are marked and then removed from the sum in Equation 6.29 or Equation 6.30. By doing this, the RMSECV, statistic is based on the same set of observations for each k. The optimal number of components is then taken as the value kopl for which RMSECV, is minimal or sufficiently small. 

Both the classical and statistical equations [Eqs. (5.22) and (5.23)] yield absolute values of entropy. Equation (5.23) is known as the Boltzmann equation and, with Eq. (5.20) and quantum statistics, has been used for calculation of entropies in the ideal-gas state for many chemical species. Good agreement between these calculations and those based on calorimetric data provides some of the most impressive evidence for the validity of statistical mechanics and quantum theory. In some instances results based on Eq. (5.23) are considered more reliable because of uncertainties in heat-capacity data or about the crystallinity of the substance near absolute zero. Absolute entropies provide much of the data base for calculation of the equilibrium conversions of chemical reactions, as discussed in Chap. 15. 

The methods of conformational statistics, discussed so far, had as starting point the real polymer chain. The aim was to relate the dimensions of the coiled polymer molecule statistically to the mutual displaceability of the chain atoms. Nearly exact relationships are obtained for a large number of freely jointed or freely rotating elements. Under conditions of restricted movability, however, the statistical equations can generally not be solved and empirical factors like s, a and a are introduced. 

Liao (2000) derived a test statistic for single dispersion effects in 2" k designs. He applied the generalized likelihood ratio test for a normal model to the residuals after fitting a location model, which results in Bartlett s (1937) classical test for comparing variances in one-way layouts. The test is then applied, in turn, to compare the variances at the two levels of each of the k experimental factors. We caution that the test statistic (equation (3) in Liao) is written incorrectly. 

A. Favre, Statistical Equations of Turbulent Gases, in Problems of Hydrodynamics and Continuum Mechanics, M. A. Lavrent ev, ed.. Society for Industrial and Applied Mathematics, Philadelphia, 1969, 231-266. 

What we have available regarding statistical equations of state (appendix 2) is of limited vedue. These equations contain at most two parameters, one accounting for the molecular cross-section (=a ) and the other for lateral Interaction. Such... 

In the absence of any applied force, a molecule is nevertheless moving because of themial agitation. Yet owing to frequent collisions with the surrounding molecules, free motions occur only over atomic distances. The overall molecular movement then resembles a random succession of linear segments of approximately identical length , and duration t. Over a time period t = mr, the average square displacement A"" is determined via statistical equations to be A = [107]. Let us now suppose that a force is... 

Multiple linear regression analysis of Equation (1) can also be used and for this kxy is determined with as many different combinations of and self-interaction coefficients (p or p ) can be measured by fitting the Bronsted or Hammett data to a binomial expression (logA = a + bx + cx ) by regular statistical software packages or by a program based on the statistical equation in Appendix 1 (Section A 1.1.4.4). 

Use a least squares program such as that derived from statistical equations in Appendix 1, Section A 1.1.4.3) to fit the data to Equations (30) and (31) and deduce peq. 

If we also average over all speeds of electrons and neglect some of the finer points of statistics, Equation 13 becomes... 

The Ci term of the Mooney-Rivlin equation is often identified with the shear modulus of the statistical equation we see that both Cx s depend on Vr in the same manner [compare equation (6-92) with (6-87) or (6-93) with (6-88)]. 

Step 5 Compute the test statistic (Equation 8.5). First, we must find s, where... 

The statistical equation underestimates p whereas Equation 2.24 overestimates the experimental value. The Carothers equation leads to a high p beeause molecules larger than the observed exist in the mixture, and these undergo gelation before the predicted value is attained. This difficulty is overcome in the statistieal treatment. 

Thus, according to (31) with K < 1 and 8 upheld of 8t, an upheld shift in the proton spectrum of the -CH2D would be expected. The 5-3.47 shift observed for [Fe2(/Lt-CH2D)(/it-CO)(CO)2(/Lt-dppm)(/Lt-C5H5)]PF6 represents an upheld shift of 0.57 ppm. In the deuterium nmr spectrum, the resonance is observed at 5 -1.88, a downheld shift of 1.02 ppm showing approximately twice the effect required by the statistics (Equation (32) with K substituted for K). Equations (31) and (32) neglect an intrinsic... 

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