Absolute experimental error

However, it is not simply because a great number of measurements are made or that each individual value is close to the mean that the mean x is close to the exact value x0. Systematic errors can occur. A statistical approach to the problem consists of considering each measurement x, as the real value x0 plus an absolute experimental error e,. The error in the /th measurement is thus expressed as ... 

Each measurement x must be considered as the sum of the true value Xq and an absolute experimental error value s. The absolute error e, of measurement i is thus expressed as ... 

Values in parantheses represent the absolute experimental errors... 

Osmotic pressure experiments provide absolute values for Neither a model nor independent calibration is required to use this method. Experimental errors can arise, of course, and we note particularly the effect of impurities. Polymers which dissociate into ions can also be confusing. We shall return to this topic in Sec. 8.13 for now we assume that the polymers under consideration are nonelectrolytes. 

The values of the apparent rate constants kj for each temperature and the activation enthalpies calculated using the Eyring equation (ref. 21) are summarized in Table 10. However, these values of activation enthalpies are only approximative ones because of the applied simplification and the great range of experimental errors. Activation entropies were not calculated in the lack of absolute rate constants. Presuming the likely first order with respect to 3-bromoflavanones, as well, approximative activation entropies would be between -24 and -30 e.u. for la -> Ih reaction, between -40 and - 45 e.u. for the Ih la reaction and between -33 and -38 e.u. for the elimination step. These activation parameters are in accordance with the mechanisms proposed above. 

In many analyses, fhe compound(s) of inferesf are found as par of a complex mixfure and fhe role of fhe chromatographic technique is to provide separation of fhe components of that mixture to allow their identification or quantitative determination. From a qualitative perspective, the main limitation of chromatography in isolation is its inability to provide an unequivocal identification of the components of a mixture even if they can be completely separated from each other. Identification is based on the comparison of the retention characteristics, simplistically the retention time, of an unknown with those of reference materials determined under identical experimental conditions. There are, however, so many compounds in existence that even if the retention characteristics of an unknown and a reference material are, within the limits of experimental error, identical, the analyst cannot say with absolute certainty that the two compounds are the same. Despite a range of chromatographic conditions being available to the analyst, it is not always possible to effect complete separation of all of the components of a mixture and this may prevent the precise and accurate quantitative determination of the analyte(s) of interest. 

As an example of the use of MIXCO.TRIAD, an analysis of comonomer triad distribution of several ethylene-propylene copolymer samples will be delineated. The theoretical triad Intensities corresponding to the 2-state B/B and 3-state B/B/B mixture models are given In Table VI. Abls, et al (19) had earlier published the HMR triad data on ethylene-propylene samples made through continuous polymerization with heterogeneous titanium catalysts. The data can be readily fitted to the two-state B/B model. The results for samples 2 and 5 are shown In Table VII. The mean deviation (R) between the observed and the calculated Intensities Is less than 1% absolute, and certainly less than the experimental error In the HMR Intensity determination. 

The above explanation of autoacceleration phenomena is supported by the manifold increase in the initial polymerization rate for methyl methacrylate which may be brought about by the addition of poly-(methyl methacrylate) or other polymers to the monomer.It finds further support in the suppression, or virtual elimination, of autoacceleration which has been observed when the molecular weight of the polymer is reduced by incorporating a chain transfer agent (see Sec. 2f), such as butyl mercaptan, with the monomer.Not only are the much shorter radical chains intrinsically more mobile, but the lower molecular weight of the polymer formed results in a viscosity at a given conversion which is lower by as much as several orders of magnitude. Both factors facilitate diffusion of the active centers and, hence, tend to eliminate the autoacceleration. Final and conclusive proof of the correctness of this explanation comes from measurements of the absolute values of individual rate constants (see p. 160), which show that the termination constant does indeed decrease a hundredfold or more in the autoacceleration phase of the polymerization, whereas kp remains constant within experimental error. 

Experimentally, Ao j/2 is generally considered equal to the midpeak potential, and is then directly deduced from the voltammograms. This does not generate experimental errors, since an ion of known standard transfer potential (for instance, tetramethyl ammonium, TMA+) must be used as an internal reference to transpose the experimental potential scale (noted E) to the absolute Galvani potential scale, so that is obtained by ... 

The challenge is then to achieve the same degree of accuracy in the derived values of the experimental electron density. Recent studies have shown that in some cases this is indeed within the reach of the present-day modelling techniques [3-5]. When the major sources of experimental error have been corrected for the typical root mean square electron density residual can reach values as low as 0.05 e A-3, with maxima below 0.20eA-3 in absolute value. The observed residuals are usually due to the... 

All the algebraic and geometric methods for optimization presented so far work when either there is no experimental error or it is smaller than the usual absolute differences obtained when the objective functions for two neighboring points are subtracted. When this is not the case, the direct search and gradient methods can cause one to go in circles, and the geometric method may cause the region containing the maximum to be eliminated from further consideration. 

Since the laws of symmetry require that all properties of enantiomers (except their interactions with other chiral systems) be exactly the same, these studies have profited by the application of an absolute test for the presence of impurities, a perennial problem in monolayer research. In every case, all measurements were repeated with both enantiomers. Unless the results agreed within experimental error, the compounds were purified repeatedly until they did agree. 

Table 27 (Y = F, Br, H, NH2, Cl, CH3CH3, OH, CH3, SiH3 and C(0)CH3) gives an average (absolute) MP2 error of 2.1 kcalmol-1 and an average MP4 error of 3.6 kcalmol-1. In this comparison, where there was a choice of experimental BDE, the one closest to the calculated values was chosen for the averages. This selectivity does not affect the relative results between MP2 and MP4, and the outcome here, like for the CH3—Y comparison above, is that the MP2 energy is generally closer to experiment.

The various sources of experimental error were also considered. From Eq. (6), we should expect an accuracy of approximately 2% for AD/D since the errors in pH, a and H3Cit cancel in considering the 0/ values for two elements in the same aqueous phase. The relative error in 0/ can be estimated to be 5 % from Eq. (6). It appears of course that the absolute error in 0/exceeds this value. 

After analysis nodes 42(25), 50(18), 53(9), 98(56), 125(40), 189(26) are kept (the mean absolute phase errors are in parentheses). For reference the correct map using experimental phases is shown in Figure 6, and Figure 7 shows the best centroid map (for node 53). For reference the correct map using experimental phases is shown in Figure 6. The map correlation coefficient is 0.94. 

Cd, K and Zn are not precisely determined. Previously reported (13) results for Identical split samples Indicates that most of this experimental error was due to analytical Imprecision rather than collection and handling. Many of the samples were near the detection limit for the five trace metals (As, Cd, Cu, Pb, Zn), To determine the effect of these measurement errors the PCA was repeated with uncertainty scaled data. (The data standard deviation used In autoscaling was replaced with the measurement absolute error.)... 

The two intermediate curves each represent a single dilatometer, each prepared according to Procedure I and irradiated at three dose rates. The slopes of these lines are 0.60 and 0.71, and the absolute values of the rates of polymerization are markedly higher than for the previous samples. Within experimental error, the slopes of these two lines are essentially the same and are less than the value given for die previously quoted, lower rates of polymerization. 

The primary act in a photochemical reaction is absorption of a quantum of radiation by the photoactive molecule. In a quantitative study, therefore, a radiation source of known intensity and frequency a suitable cel for the photolyte and an appropriate detector of light intensity are absolutely necessary for the determination of rates of reaction. To avoid experimental error due to geometry of the reaction cell, the best arrangement is to have a plane parallel beam of monochromatic radiation, incident upon a flat cuvette with proper stirring arrangement, as given in Figure 1.2. 

Most researchers have found pseudo-first-order behavior for the various steps, and so it is possible to match theoretical curves with data to obtain the best rate constant values. Unfortunately, in most instances, too few data points were obtained to generate a unique theoretical fit. It is absolutely imperative that data be obtained for at least four conversion levels that are well spaced in the conversion matrix and extend to over 95% conversion. The partially hydrogenated dibenzothiophene intermediates are most often never detected as their desulfurization rates are extremely high (fcD, and kn2). The cyclohexylbenzenes and bicyclohexyls can arise from two different routes, and the concentrations of their precursors (biphenyl and cyclohexyl-biphenyl, respectively) pass through maximum values that can easily be calculated from the relative values of the formation and conversion rate constants. However, unique values for these relative rates can only be predicted if data are available well prior to and well beyond the times of maximum concentrations for these intermediates, because minor experimental errors can confuse curve-fitting optimization. 

The point of real significance is that an absolute value of k can be calculated which is of the right order of magnitude. The theory allows a calculation of the velocity constant from an experimentally determined value of E. E is subject to certain experimental errors which, from the nature of the relation between E and k, appear as proportional errors not in k itself but in the logarithm of k. The accuracy, therefore, with which the logarithm of k can be calculated is a just test of the theory the accuracy with which k itself can be calculated an excessively severe test, having regard to the inevitable errors involved in the determination of E. 

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