Uncertainty in Measurement and Significant Figures

Both uncertainty in measurement and significant figures are illustrated in Figure 3.7, which has two captions. The first caption focuses on uncertainty, and the second shows how to apply significant figures to the same measurements. Study these captions carefully now, before proceeding to the next paragraph. 

I Apply rules for significant figures to express uncertainty in measured and calculated values. 

There are a few basic numerical and experimental tools with which you must be familiar. Fundamental measurements in analytical chemistry, such as mass and volume, use base SI units, such as the kilogram (kg) and the liter (L). Other units, such as power, are defined in terms of these base units. When reporting measurements, we must be careful to include only those digits that are significant and to maintain the uncertainty implied by these significant figures when transforming measurements into results. 

Measurements of Length, Volume, and Mass Uncertainty in Measurement Significant Figures Problem Solving and Dimensional Analysis Temperature Conversions ... 

It is clear that no significant comparison between theory and experiment can be made unless i is known. That values of i determined from gas-phase measurements are relevant to the properties of dense fluid mixtures is demonstrated in Figure 5 shown plotted are the values of i determined from virial data on mixtures vs. -values that give the best fit to the properties of liquid mixtures of the same substances. The gas-phase values are probably no more reliable than 0.01 to 0.02. In fitting data on liquid mixtures, variations in of 0.002 may be significant, but the -values derived depend on the theory chosen to represent the data. As a result, the uncertainties in liquid and gaseous values of i are probably comparable. 

Suppose that you measure the density of a mineral by finding its mass (4.635 0.002 g) and its volume (1.13 0.05 mL). Density is mass per unit volume 4.635 g/1.13 mL = 4.101 8 g/mL. The uncertainties in mass and volume are 0.002 g and 0.05 mL, but what is the uncertainty in the computed density And how many significant figures should be used for the density This chapter answers these questions and introduces spreadsheets—a powerful tool that will be invaluable to you in and out of this course. 

To illustrate the usefulness of the Information Index in determining the best time interval, let us consider the grid point (l, 0.20, 35). From Figure 12.7 we deduce that the best time interval is 25 to 75 h. In Table 12.4 the standard deviation of each parameter is shown for 7 different time intervals. From cases 1 to 4 it is seen that that measurements taken before 25 h do not contribute significantly in the reduction of the uncertainty in the parameter estimates. From case 4 to 7 it is seen that it is preferable to obtain data points within [25, 75] rather than after the steady state has been reached and the Information Indices have leveled off. Measurements taken after 75 h provide information only about the steady state behavior of the system. 

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