Experimental uncertainties

The largest errors in predicted compositions occur for the systems acetic acid-formic acid-water and acetone-acetonitrile-water where experimental uncertainties are significantly greater than those for other systems. 

If there is sufficient flexibility in the choice of model and if the number of parameters is large, it is possible to fit data to within the experimental uncertainties of the measurements. If such a fit is not obtained, there is either a shortcoming of the model, greater random measurement errors than expected, or some systematic error in the measurements. 

In many process-design calculations it is not necessary to fit the data to within the experimental uncertainty. Here, economics dictates that a minimum number of adjustable parameters be fitted to scarce data with the best accuracy possible. This compromise between "goodness of fit" and number of parameters requires some method of discriminating between models. One way is to compare the uncertainties in the calculated parameters. An alternative method consists of examination of the residuals for trends and excessive errors when plotted versus other system variables (Draper and Smith, 1966). A more useful quantity for comparison is obtained from the sum of the weighted squared residuals given by Equation (1). 

The third virial coefficient C(7) depends upon tliree-body interactions, both additive and non-additive. The relationship is well understood [106. 107. 111]. If the pair potential is known precisely, then C(7) ought to serve as a good probe of the non-additive, tliree-body interaction energy. The importance of the non-additive contribution has been confimied by C(7) measurements. Unfortunately, large experimental uncertainties in C (7) have precluded unequivocal tests of details of the non-additive, tliree-body interaction. 

The definition of initial conditions is generally limited in precision to within experimental uncertainties A and A p, more fiindamentally related by the Fleisenberg principle A q A= li/4ji. Therefore, we need to... 

In general, we know bond lengths to within an uncertainty of 0.00.5 A — 0.5 pm. Bond angles are reliably known only to one or twx) degrees, and there arc many instances of more serious angle enxirs. Tn addition to experimental uncertainties and inaccuracies due to the model (lack of coincidence between model and molecule), some models present special problems unique to their geometry. For example, some force fields calculate the ammonia molecule. Nlln to be planar when there is abundant ex p er i m en ta I evidence th at N H is a 11 i g o n a I pyramid. 

A number of factors limit the accuracy with which parameters for the design of commercial equipment can be determined. The parameters may depend on transport properties for heat and mass transfer that have been determined under nonreacting conditions. Inevitably, subtle differences exist between large and small scale. Experimental uncertainty is also a factor, so that under good conditions with modern equipment kinetic parameters can never be determined more precisely than 5 to 10 percent (Hofmann, in de Lasa, Chemical Reactor Design and Technology, Martinus Nijhoff, 1986, p. 72). 

The temperature dependence of A predicted by Eq. (5-11) makes a very weak contribution to the temperature dependence of the rate constant, which is dominated by the exponential term. It is, therefore, not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted dependence of A is observed experimentally. Uncertainties in estimates of A tend to be quite large because this parameter is, in effect, determined by a long extrapolation of the Arrhenius plot to 1/T = 0. 

The average error is the difference between the calculated and experimental AHf. In this comiection it should be noted that the average error in the experimental data for the hydrocarbons is 0.40 kcal/mol, i.e, MM2 essentially reproduces the experiments to within the experimental uncertainty. 

Examples of these are shown for the saturation data in Figure 4.2. At first glance, these transformations may seem like ideal methods to analyze saturation data. However, transformation of binding data is not generally recommended. This is because transformed plots can distort experimental uncertainty, produce compression of data,... 

It has been pointed out that analysis of terpolymerization data or copolymerization with chain transfer could, in principle, provide a test of the model. 5 However, to date experimental uncertainty has prevented this. 

AOS 2024 adsorption increases continuously on going from fresh water to 1% NaCl to 4% NaCl. Aqueous 1% NaCl was studied as an analog to aqueous 1% sodium sulfate. The last two entries of Table 19 show that increased AOS 2024 adsorption in the presence of sodium sulfate could mitigate any reduction in calcium ion-promoted surfactant precipitation. However, the larger than usual experimental uncertainty in the sodium sulfate results means that the... 

Fig. 1.25. Temperature-dependence of rotational relaxation cross-section from [81], For the lowest temperature point the experimental uncertainty is indicated, the latter being the biggest one over the whole set of measurements.

Effects of Temperature on Ionic Reactions in TD/D2 CH4/ CD4. Observation that the methanium ion proton (deuteron) transfer sequence fails to exhibit a temperature coefficient within experimental uncertainties leads unavoidably to the conclusion that none of the reactions from 1 to 12 requires thermal activation between —78° and 25°C. From Equations I, II, III, appropriate steady state assumptions, and representing both neutralization steps by kX2, we find that... 

HzPO . The values are 0.24, 0.21, 0.16 and 0.13, respectively. The values span a range of a factor of two which must be admitted to be a little larger than the experimental uncertainty and also easily within the differences among the anions in their probability of occupancy of the crucial outer sphere site adjacent to the leaving water molecule. All are nearly a factor of five below the water exchange rate. These results conform neatly to the predictions. 

Mills has concluded in his review article on molten slags that (1) most viscosity measurements were subject to experimental imcertainties of 25% (2) in some cases experimental uncertainties could be > 50% and (3) experimental uncertainties as low as 10% could be achieved by careful calibration of viscometers with high and low temperature reference materials. 

The only state whieh eould be seen in the 2000-6000 A window is the 1 B valenee state. The fact that this state was not seen in spite of its strong transition moment may well be due to the experimental uncertainty of 10% at the limit of the window. 

There are to be found lists of chemical substances in handbooks for each of which log P = f (T), and whose coefficients are to be inserted, are given. These lists are limited but nevertheless provide solutions for the most common chemical substances. When there are several experimental estimates of vapour pressures it is possible to estimate the importance of the experimental uncertainty from the standard deviation of the measurements. The relevance of the values can be verified from a series of different sources (to be rigorous, checking that it is a Gaussian sequence would be required). 

A brief study of the available data related to limits of inflammability in Part Two shows that these parameters are subject to high experimental uncertainty. For a large number of substances, the experimental values are widely dispersed. When they are submitted to quality estimation using statistical tools, in many cases they reveal that it is impossible to use them with confidence. The examples of difficulties raised by the statistical analysis of the LEL data can be multiplied. 

Figure 1.116. Lead isotopic variation in Japanese Neogene ores. The majority of data fall in a relatively narrow range which is no more than twice the experimental uncertainty indicated by the replicate analyses of NBS-SRM-981 standard (Sasaki et al., 1982).

Figure 1.177. Comparison between the stannite-sphalerite geothermometer after Nekrasov et al. (1979) and one after Nakamura and Shima (1982). Crossbars indicate experimental uncertainties (Shimizu and Shikazono, 1985).

Tab. 4.5 Kurfurst elemental homogeneity factors for selected RMs determined from experimental uncertainties with an INAA procedure using short-lived indicator nuclides. The sample masses ranged from 0.5 mg to 2.5 mg the number of determinations were 12 for each material...

---