D. Harvey - Modern Analytical Chemistry (794078), страница 22

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For example, a buret with scale divisions every 0.1 mL has an inherent indeterminate error of ±0.01 – 0.03 mL when estimating the volume to thehundredth of a milliliter (Figure 4.3). Background noise in an electrical meter (Figure 4.4) can be evaluated by recording the signal without analyte and observing thefluctuations in the signal over time.3031Figure 4.3Close-up of buret, showing difficulty inestimating volume. With scale divisions every0.1 mL it is difficult to read the actualvolume to better than ±0.01 – 0.03 mL.SignalEvaluating Indeterminate Error Although it is impossible to eliminate indeterminate error, its effect can be minimized if the sources and relative magnitudes of theindeterminate error are known.

Indeterminate errors may be estimated by an appropriate measure of spread. Typically, a standard deviation is used, although insome cases estimated values are used. The contribution from analytical instrumentsand equipment are easily measured or estimated. Indeterminate errors introduced by the analyst, such as inconsistencies in the treatment of individual samples,are more difficult to estimate.To evaluate the effect of indeterminate error onthe data in Table 4.1, ten replicate determinations ofthe mass of a single penny were made, with resultsshown in Table 4.7. The standard deviation for thedata in Table 4.1 is 0.051, and it is 0.0024 for thedata in Table 4.7. The significantly better precisionwhen determining the mass of a single penny suggests that the precision of this analysis is not limitedby the balance used to measure mass, but is due to asignificant variability in the masses of individualpennies.63TimeFigure 4.4Table 4.7Replicate Determinations of theMass of a Single United StatesPenny in CirculationReplicate NumberMass(g)123456789103.0253.0243.0283.0273.0283.0233.0223.0213.0263.024Background noise in a meter obtained bymeasuring signal over time in the absence ofanalyte.1400-CH04 9/8/99 3:54 PM Page 6464Modern Analytical Chemistry4B.3 Error and UncertaintyAnalytical chemists make a distinction between error and uncertainty.3 Error is thedifference between a single measurement or result and its true value.

In otherwords, error is a measure of bias. As discussed earlier, error can be divided into determinate and indeterminate sources. Although we can correct for determinateerror, the indeterminate portion of the error remains. Statistical significance testing,which is discussed later in this chapter, provides a way to determine whether a biasresulting from determinate error might be present.uncertaintyUncertainty expresses the range of possible values that a measurement or resultThe range of possible values for amight reasonably be expected to have.

Note that this definition of uncertainty is notmeasurement.the same as that for precision. The precision of an analysis, whether reported as arange or a standard deviation, is calculated from experimental data and provides anestimation of indeterminate error affecting measurements. Uncertainty accounts forall errors, both determinate and indeterminate, that might affect our result. Although we always try to correct determinate errors, the correction itself is subject torandom effects or indeterminate errors.To illustrate the difference between precision and uncertainty, consider the use of a class A 10-mL pipet for deTable 4.8 Experimentally Determinedlivering solutions. A pipet’s uncertainty is the range ofvolumes in which its true volume is expected to lie.

SupVolumes Delivered by a 10-mLpose you purchase a 10-mL class A pipet from a laboraClass A Pipettory supply company and use it without calibration. TheVolumeVolumepipet’s tolerance value of ±0.02 mL (see Table 4.2) repreDeliveredDeliveredsents your uncertainty since your best estimate of its volTrial(mL)Trial(mL)ume is 10.00 mL ±0.02 mL.

Precision is determined ex110.00269.983perimentally by using the pipet several times, measuring29.99379.991the volume of solution delivered each time. Table 4.839.98489.990shows results for ten such trials that have a mean of 9.99249.99699.988mL and a standard deviation of 0.006. This standard devi59.989109.999ation represents the precision with which we expect to beable to deliver a given solution using any class A 10-mLpipet. In this case the uncertainty in using a pipet is worsethan its precision.

Interestingly, the data in Table 4.8 allowus to calibrate this specific pipet’s delivery volume as 9.992 mL. If we use this volume as a better estimate of this pipet’s true volume, then the uncertainty is ±0.006.As expected, calibrating the pipet allows us to lower its uncertainty.errorA measure of bias in a result ormeasurement.4C Propagation of UncertaintySuppose that you need to add a reagent to a flask by several successive transfersusing a class A 10-mL pipet.

By calibrating the pipet (see Table 4.8), you know thatit delivers a volume of 9.992 mL with a standard deviation of 0.006 mL. Since thepipet is calibrated, we can use the standard deviation as a measure of uncertainty.This uncertainty tells us that when we use the pipet to repetitively deliver 10 mL ofsolution, the volumes actually delivered are randomly scattered around the mean of9.992 mL.If the uncertainty in using the pipet once is 9.992 ± 0.006 mL, what is the uncertainty when the pipet is used twice? As a first guess, we might simply add the uncertainties for each delivery; thus(9.992 mL + 9.992 mL) ± (0.006 mL + 0.006 mL) = 19.984 ± 0.012 mL1400-CH04 9/8/99 3:54 PM Page 65Chapter 4 Evaluating Analytical DataIt is easy to see that combining uncertainties in this way overestimates the total uncertainty.

Adding the uncertainty for the first delivery to that of the second deliveryassumes that both volumes are either greater than 9.992 mL or less than 9.992 mL.At the other extreme, we might assume that the two deliveries will always be on opposite sides of the pipet’s mean volume. In this case we subtract the uncertaintiesfor the two deliveries,(9.992 mL + 9.992 mL) ± (0.006 mL – 0.006 mL) = 19.984 ± 0.000 mLunderestimating the total uncertainty.So what is the total uncertainty when using this pipet to deliver two successivevolumes of solution? From the previous discussion we know that the total uncertainty is greater than ±0.000 mL and less than ±0.012 mL.

To estimate the cumulative effect of multiple uncertainties, we use a mathematical technique known as thepropagation of uncertainty. Our treatment of the propagation of uncertainty isbased on a few simple rules that we will not derive. A more thorough treatment canbe found elsewhere.44C.1 A Few SymbolsPropagation of uncertainty allows us to estimate the uncertainty in a calculated result from the uncertainties of the measurements used to calculate the result. In theequations presented in this section the result is represented by the symbol R and themeasurements by the symbols A, B, and C.

The corresponding uncertainties are sR,sA, sB, and sC. The uncertainties for A, B, and C can be reported in several ways, including calculated standard deviations or estimated ranges, as long as the same formis used for all measurements.4C.2 Uncertainty When Adding or SubtractingWhen measurements are added or subtracted, the absolute uncertainty in the resultis the square root of the sum of the squares of the absolute uncertainties for the individual measurements. Thus, for the equations R = A + B + C or R = A + B – C, orany other combination of adding and subtracting A, B, and C, the absolute uncertainty in R issR =s 2A + sB2 + sC24.6EXAMPLE 4.5The class A 10-mL pipet characterized in Table 4.8 is used to deliver twosuccessive volumes.

Calculate the absolute and relative uncertainties for thetotal delivered volume.SOLUTIONThe total delivered volume is obtained by adding the volumes of each delivery;thusVtot = 9.992 mL + 9.992 mL = 19.984 mLUsing the standard deviation as an estimate of uncertainty, the uncertainty inthe total delivered volume issR = (0.006)2 + (0.006)2 = 0.0085651400-CH04 9/8/99 3:54 PM Page 6666Modern Analytical ChemistryThus, we report the volume and its absolute uncertainty as 19.984 ± 0.008 mL.The relative uncertainty in the total delivered volume is0.0085× 100 = 0.043%19.9844C.3 Uncertainty When Multiplying or DividingWhen measurements are multiplied or divided, the relative uncertainty in the resultis the square root of the sum of the squares of the relative uncertainties for the individual measurements.

Thus, for the equations R = A × B × C or R = A × B/C, or anyother combination of multiplying and dividing A, B, and C, the relative uncertaintyin R issR=REXAMPLE22 sA  sB  sC   +  +  A BC24.74.6The quantity of charge, Q, in coulombs passing through an electrical circuit isQ=I×twhere I is the current in amperes and t is the time in seconds. When a currentof 0.15 ± 0.01 A passes through the circuit for 120 ± 1 s, the total charge isQ = (0.15 A) × (120 s) = 18 CCalculate the absolute and relative uncertainties for the total charge.SOLUTIONSince charge is the product of current and time, its relative uncertainty issR=R2 0.01  1  + 0.15  120 2= ±0.0672or ±6.7%.

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