Modern Analytical Cheymistry - Chapter 7

Chapter 7 Obtaining and Preparing Samples for Analysis When we first use an analytical method to solve a problem, it is not unusual to find that our results are of questionable accuracy or so imprecise as to be meaningless. Looking back we may find that nothing in the method seems amiss. In designing the method we considered sources of determinate and indeterminate error and took appropriate steps, such as including a reagent blank and calibrating our instruments, to minimize their effect. Why, then, might a carefully designed method give such poor results? One explanation is that we may not have accounted for errors associated with the sample. When we collect the wrong sample or lose analyte while preparing the sample for analysis, we introduce a determinate source of error. If we do not collect enough samples or collect samples of the wrong size, the precision of the analysis may suffer. In this chapter we consider how collecting samples and preparing them for analysis can affect the accuracy and precision of our results. 179 180 Modern Analytical Chemistry 7A The Importance of Sampling When a manufacturer produces a chemical they wish to list as ACS Reagent Grade, they must demonstrate that it conforms to specifications established by the Ameri-can Chemical Society (ACS). For example, ACS specifications for NaHCO3 require that the concentration of iron be less than or equal to 0.001% w/w. To verify that a production lot meets this standard, the manufacturer performs a quantitative analy-sis, reporting the result on the product’s label. Because it is impractical to analyze the entire production lot, its properties are estimated from a limited sampling. Sev-eral samples are collected and analyzed, and the resulting mean,X, and standard de-viation, s, are used to establish a confidence interval for the production lot’s true mean, m m = X ± ts 7.1 n where n is the number of samples, and t is a statistical factor whose value is deter-mined by the number of samples and the desired confidence level.* Selecting a sample introduces a source of determinate error that cannot be cor-rected during the analysis. If a sample does not accurately represent the population from which it is drawn, then an analysis that is otherwise carefully conducted will yield inaccurate results. Sampling errors are introduced whenever we extrapolate from a sample to its target population. To minimize sampling errors we must col-lect the right sample. Even when collecting the right sample, indeterminate or random errors in sam-pling may limit the usefulness of our results. Equation 7.1 shows that the width of a confidence interval is directly proportional to the standard deviation. 80 The overall standard deviation for an analysis, so, is determined by ran-70 dom errors affecting each step of the analysis. For convenience, we di-60 vide the analysis into two steps. Random errors introduced when collect-50 ing samples are characterized by a standard deviation for sampling, ss. 40 The standard deviation for the analytical method, sm, accounts for ran- 30 dom errors introduced when executing the method’s procedure. The re-20 lationship among so, ss, and sm is given by a propagation of random error 10 s2 = s2 +s2 7.2 0 0 0.5 1 1.5 sm/ss Figure 7.1 Percent of overall variance (so) due to the method as a function of the relative magnitudes of the standard deviation of the method and the standard deviation of sampling (sm/ss). The dotted lines show that the variance due to the method accounts for 10% of the overall variance when ss = 3 ´ sm. 2 Equation 7.2 shows that an analysis’ overall variance may be lim-ited by either the analytical method or sample collection. Unfortu- nately, analysts often attempt to minimize overall variance by im-proving only the method’s precision. This is futile, however, if the standard deviation for sampling is more than three times greater than that for the method.1 Figure 7.1 shows how the ratio sm/ss affects the percentage of overall variance attributable to the method. When the method’s standard deviation is one third of that for sampling, indeterminate method errors explain only 10% of the overall variance. Attempting to improve the analysis by decreasing sm provides only a nominal change in the overall variance. *Values for t can be found in Appendix 1B. Chapter 7 Obtaining and Preparing Samples for Analysis 181 EXAMPLE 7.1 A quantitative analysis for an analyte gives a mean concentration of 12.6 ppm. The standard deviation for the method is found to be 1.1 ppm, and that due to sampling is 2.1 ppm. (a) What is the overall variance for the analysis? (b) By how much does the overall variance change if sm is improved by 10% to 0.99 ppm? (c) By how much does the overall variance change if ss is improved by 10% to 1.9 ppm? SOLUTION (a) The overall variance is s2 = s2 + s2 = (1.1)2 + (2.1)2 = 1.21 + 4.41 = 5.62 » 5.6 (b) Improving the method’s standard deviation changes the overall variance to so = (0.99)2 + (2.1)2 = 0.98 + 4.41 = 5.39 » 5.4 Thus, a 10% improvement in the method’s standard deviation changes the overall variance by approximately 4%. (c) Changing the standard deviation for sampling so = (1.1)2 + (1.9)2 = 1.21 + 3.61 = 4.82 » 4.8 improves the overall variance by almost 15%. As expected, since ss is larger than sm, a more significant improvement in the overall variance is realized when we focus our attention on sampling problems. To determine which step has the greatest effect on the overall variance, both s2 and ss must be known. The analysis of replicate samples can be used to estimate the overall variance. The variance due to the method is determined by analyzing a stan-dard sample, for which we may assume a negligible sampling variance. The variance due to sampling is then determined by difference. EXAMPLE 7.2 The following data were collected as part of a study to determine the effect of sampling variance on the analysis of drug animal-feed formulations.2 % Drug (w/w) % Drug (w/w) 0.0114 0.0099 0.0105 0.0102 0.0106 0.0087 0.0100 0.0095 0.0098 0.0105 0.0095 0.0097 0.0105 0.0109 0.0107 0.0103 0.0103 0.0104 0.0101 0.0101 0.0103 The data on the left were obtained under conditions in which random errors in sampling and the analytical method contribute to the overall variance. The data on the right were obtained in circumstances in which the sampling variance is known to be insignificant. Determine the overall variance and the contributions from sampling and the analytical method. 182 Modern Analytical Chemistry SOLUTION The overall variance, s2, is determined using the data on the left and is equal to 4.71 ´ 10–7. The method’s contribution to the overall variance, s2, is determined using the data on the right and is equal to 7.00 ´ 10–8. The variance due to sampling, s2, is therefore s2 = s2 – s2 = 4.71 ´ 10–7 – 7.00 ´ 10–8 = 4.01 ´ 10–7 7B Designing A Sampling Plan sampling plan A plan that ensures that a representative sample is collected. A sampling plan must support the goals of an analysis. In characterization studies a sample’s purity is often the most important parameter. For example, a material sci-entist interested in the surface chemistry of a metal is more likely to select a freshly exposed surface, created by fracturing the sample under vacuum, than a surface that has been exposed to the atmosphere for an extended time. In a qualitative analysis the sample’s composition does not need to be identical to that of the substance being analyzed, provided that enough sample is taken to ensure that all components can be detected. In fact, when the goal of an analysis is to identify components present at trace levels, it may be desirable to discriminate against major components when sampling. In a quantitative analysis, however, the sample’s composition must accurately represent the target population. The focus of this section, therefore, is on designing a sampling plan for a quantitative analysis. Five questions should be considered when designing a sampling plan: 1. From where within the target population should samples be collected? 2. What type of samples should be collected? 3. What is the minimum amount of sample needed for each analysis? 4. How many samples should be analyzed? 5. How can the overall variance be minimized? Each of these questions is considered below in more detail. 7B.1 Where to Sample the Target Population Sampling errors occur when a sample’s composition is not identical to that of the population from which it is drawn. When the material being sampled is homoge-neous, individual samples can be taken without regard to possible sampling errors. Unfortunately, in most situations the target population is heterogeneous in either time or space. As a result of settling, for example, medications available as oral sus-pensions may have a higher concentration of their active ingredients at the bottom of the container. Before removing a dose (sample), the suspension is shaken to min-imize the effect of this spatial heterogeneity. Clinical samples, such as blood or urine, frequently show a temporal heterogeneity. A patient’s blood glucose level, for instance, will change in response to eating, medication, or exercise. Other systems show both spatial and temporal heterogeneities. The concentration of dissolved O2 in a lake shows a temporal heterogeneity due to the change in seasons, whereas point sources of pollution may produce a spatial heterogeneity. When the target population’s heterogeneity is of concern, samples must be ac-quired in a manner that ensures that determinate sampling errors are insignificant. If the target population can be thoroughly homogenized, then samples can be taken without introducing sampling errors. In most cases, however, homogenizing the Chapter 7 Obtaining and Preparing Samples for Analysis 183 target population is impracticable. Even more important, homogenization destroys information about the analyte’s spatial or temporal distribution within the target population. Random Sampling The ideal sampling plan provides an unbiased estimate of the target population’s properties. This requirement is satisfied if the sample is collected at random from the target population.3 Despite its apparent simplicity, a true ran-dom sample is difficult to obtain. Haphazard sampling, in which samples are col-lected without a sampling plan, is not random and may reflect an analyst’s uninten-tional biases. The best method for ensuring the collection of a random sample is to divide the target population into equal units, assign a unique number to each unit, and use a random number table (Appendix 1E) to select the units from which to sample. Example 7.3 shows how this is accomplished. random sample A sample collected at random from the target population. EXAMPLE 7.3 To analyze the properties of a 100 cm ´ 100 cm polymer sheet, ten 1 cm ´ 1 cm samples are to be selected at random and removed for analysis. Explain how a random number table can be used to ensure that samples are drawn at random. SOLUTION As shown in the following grid, we divide the polymer sheet into 10,000 1 cm ´ 1 cm squares, each of which can be identified by its row number and its column number. 0 1 2 98 99 0 1 2 98 99 For example, the highlighted square is in row 1 and column 2. To pick ten squares at random, we enter the random number table at an arbitrary point, and let that number represent the row for the first sample. We then move through the table in a predetermined fashion, selecting random numbers for the column of the first sample, the row of the second sample, and so on until all ten samples have been selected. Since our random number table (Appendix 1E) uses five-digit numbers we will use only the last two digits. Let’s begin with the fifth entry and use every other entry after that. The fifth entry is 65423 making the first row number 23. The next entry we use is 41812, giving the first column number as 12. Continuing in this manner, the ten samples are as follows: Sample Row Column 1 23 12 2 45 80 3 81 12 4 66 17 5 46 01 Sample Row Column 6 93 83 7 91 17 8 45 13 9 12 92 10 97 52 ... - tailieumienphi.vn