D. Harvey - Modern Analytical Chemistry (794078), страница 53
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A redox buffer contains an oxidizing agent and its conjugatereducing agent. Calculate the potential of a solutioncontaining 0.010 mol of Fe3+ and 0.015 mol of Fe2+. What isthe potential if sufficient oxidizing agent is added such that0.002 mol of Fe2+ is converted to Fe3+?6O SUGGESTED READINGSA lucid discussion of Berthollet’s discovery of the reversibility ofreactions is found inRoots-Bernstein, R.
S. Discovering. Harvard University Press:Cambridge, MA, 1989.The following texts and articles provide additional coverage ofequilibrium chemistry and the systematic approach to solvingequilibrium problems.Butler, J. N. Ionic Equilibria: A Mathematical Approach. AddisonWesley: Reading, MA, 1964.Butler, J. N.
Solubility and pH Calculations. Addison-Wesley:Reading, MA, 1973.Chaston, S. “Calculating Complex Equilibrium Concentrations bya Next Guess Factor Method,” J. Chem. Educ. 1993, 70,622–624.Fernando, Q.; Ryan, M. D. Calculations in Analytical Chemistry,Harcourt Brace Jovanovich: New York, 1982.Freiser, H.
Concepts and Calculations in Analytical Chemistry, CRCPress: Boca Raton, 1992.Freiser, H.; Fernando, Q. Ionic Equilibria in Analytical Chemistry,Wiley: New York, 1963.Gordus, A. A. “Chemical Equilibrium I. The ThermodynamicEquilibrium Concept,” J. Chem. Educ. 1991, 68, 138–140.Gordus, A. A. “Chemical Equilibrum II.
Deriving an ExactEquilibrium Equation,” J. Chem. Educ. 1991, 68, 215–217.Gordus, A. A. “Chemical Equilibrium III. A Few Math Tricks,”J. Chem. Educ. 1991, 68, 291–293.Gordus, A. A. “Chemical Equilibrium IV. Weak Acids and Bases,”J. Chem. Educ. 1991, 68, 397–399.Gordus, A. A.
“Chemical Equilibrium VI. Buffer Solutions,”J. Chem. Educ. 1991, 68, 656–658.Gordus, A. A. “Chemical Equilibrium VII. Precipitates,” J. Chem.Educ. 1991, 68, 927–930.Gordus, A. A. Schaum’s Outline of Analytical Chemistry. McGrawHill: New York, 1985.Olivieri, A. C. “Solution of Acid–Base Equilibria by SuccessiveApproximations,” J. Chem. Educ. 1990, 67, 229–231.Ramette, R. W.
Chemical Equilibrium and Analysis. AddisonWesley: Reading, MA, 1981.Thomson, B. M.; Kessick, M. A. “On the Preparation of BufferSolutions,” J. Chem. Educ. 1981, 58, 743–746.Weltin, E. “Are the Equilibrium Concentrations for a ChemicalReaction Always Uniquely Determined by the InitialConcentrations?” J. Chem. Educ. 1990, 67, 548.Weltin, E. “A Numerical Method to Calculate EquilibriumConcentrations for Single-Equation Systems,” J. Chem. Educ.1991, 68, 486–487.Weltin, E.
“Calculating Equilibrium Concentrations,” J. Chem.Educ. 1992, 69, 393–396.Weltin, E. “Calculating Equilibrium Concentrations for StepwiseBinding of Ligands and Polyprotic Acid-Base Systems,”J. Chem. Educ. 1993, 70, 568–571.Weltin, E. “Equilibrium Calculations are Easier Than YouThink— But You Do Have to Think!” J. Chem. Educ. 1993, 70,571–573.6P REFERENCES1. (a) Runo, J. R.; Peters, D. G.
J. Chem. Educ. 1993, 70, 708–713;(b) Vale, J.; Fernandez-Pereira, C.; Alcalde, M. J. Chem. Educ. 1993,70, 790–795.2. Van Slyke, D. D. J. Biol. Chem. 1922, 52, 525–570.3. (a) Bower, V. E.; Bates, R. G. J. Res. Natl. Bur. Stand. (U. S.) 1955, 55,197–200; (b) Bates, R. G. Ann. N. Y. Acad. Sci.
1961, 92, 341–356;(c) Bates, R. G. Determination of pH, 2nd ed. Wiley-Interscience: NewYork, 1973, p. 73.4. Lambert, W. J. J. Chem. Educ. 1990, 67, 150–153.5. Kolthoff, I. M.; Lingane, J. J. Phys. Chem. 1938, 42, 133–140.6. Davies, C. W. Ion Association. Butterworth: London, 1962.1400-CH07 9/8/99 4:02 PM Page 1797ChapterObtaining and PreparingSamples for AnalysisWhen we first use an analytical method to solve a problem, it isnot unusual to find that our results are of questionable accuracy or soimprecise as to be meaningless. Looking back we may find thatnothing in the method seems amiss.
In designing the method weconsidered sources of determinate and indeterminate error and tookappropriate steps, such as including a reagent blank and calibratingour instruments, to minimize their effect. Why, then, might a carefullydesigned method give such poor results? One explanation is that wemay not have accounted for errors associated with the sample. Whenwe collect the wrong sample or lose analyte while preparing the samplefor analysis, we introduce a determinate source of error. If we do notcollect enough samples or collect samples of the wrong size, theprecision of the analysis may suffer. In this chapter we consider howcollecting samples and preparing them for analysis can affect theaccuracy and precision of our results.1791400-CH07 9/8/99 4:02 PM Page 180180Modern Analytical Chemistry7A The Importance of SamplingPercentage of s 2o due to s 2mWhen a manufacturer produces a chemical they wish to list as ACS Reagent Grade,they must demonstrate that it conforms to specifications established by the American Chemical Society (ACS).
For example, ACS specifications for NaHCO3 requirethat the concentration of iron be less than or equal to 0.001% w/w. To verify that aproduction lot meets this standard, the manufacturer performs a quantitative analysis, reporting the result on the product’s label. Because it is impractical to analyzethe 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 deviation, s, are used to establish a confidence interval for the production lot’s truemean, µtsµ = X±7.1n20where n is the number of samples, and t is a statistical factor whose value is determined by the number of samples and the desired confidence level.*Selecting a sample introduces a source of determinate error that cannot be corrected during the analysis.
If a sample does not accurately represent the populationfrom which it is drawn, then an analysis that is otherwise carefully conducted willyield inaccurate results. Sampling errors are introduced whenever we extrapolatefrom a sample to its target population. To minimize sampling errors we must collect the right sample.Even when collecting the right sample, indeterminate or random errors in sampling may limit the usefulness of our results. Equation 7.1 shows that the width of aconfidence interval is directly proportional to the standard deviation.The overall standard deviation for an analysis, so, is determined by random errors affecting each step of the analysis.
For convenience, we divide the analysis into two steps. Random errors introduced when collecting samples are characterized by a standard deviation for sampling, ss.The standard deviation for the analytical method, sm, accounts for random errors introduced when executing the method’s procedure. The relationship among so, ss, and sm is given by a propagation of random error102s2o = sm+ s2s807060504030000.51sm /ss1.5Figure 7.1Percent of overall variance (s o2) due to themethod as a function of the relativemagnitudes of the standard deviation of themethod and the standard deviation ofsampling (sm/ss).
The dotted lines show thatthe variance due to the method accounts for10% of the overall variance whenss = 3 × sm.7.2Equation 7.2 shows that an analysis’ overall variance may be limited by either the analytical method or sample collection. Unfortunately, analysts often attempt to minimize overall variance by improving only the method’s precision.
This is futile, however, if the standarddeviation for sampling is more than three times greater than that for themethod.1 Figure 7.1 shows how the ratio sm/ss affects the percentage of overallvariance attributable to the method. When the method’s standard deviation isone third of that for sampling, indeterminate method errors explain only 10%of the overall variance. Attempting to improve the analysis by decreasing s mprovides only a nominal change in the overall variance.2*Values for t can be found in Appendix 1B.1400-CH07 9/8/99 4:03 PM Page 181Chapter 7 Obtaining and Preparing Samples for AnalysisEXAMPLE 7.1A 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 tosampling is 2.1 ppm.
(a) What is the overall variance for the analysis? (b) Byhow much does the overall variance change if sm is improved by 10% to 0.99ppm? (c) By how much does the overall variance change if ss is improved by10% to 1.9 ppm?SOLUTION(a) The overall variance is2 + s2 = (1.1)2 + (2.1)2 = 1.21 + 4.41 = 5.62 ≈ 5.6so2 = sms(b) Improving the method’s standard deviation changes the overall variance toso2 = (0.99)2 + (2.1)2 = 0.98 + 4.41 = 5.39 ≈ 5.4Thus, a 10% improvement in the method’s standard deviation changes theoverall variance by approximately 4%.(c) Changing the standard deviation for samplingso2 = (1.1)2 + (1.9)2 = 1.21 + 3.61 = 4.82 ≈ 4.8improves the overall variance by almost 15%.
As expected, since ss is largerthan sm, a more significant improvement in the overall variance is realizedwhen we focus our attention on sampling problems.2To determine which step has the greatest effect on the overall variance, both sm2and ss must be known. The analysis of replicate samples can be used to estimate theoverall variance. The variance due to the method is determined by analyzing a standard sample, for which we may assume a negligible sampling variance.
The variancedue to sampling is then determined by difference.EXAMPLE 7.2The following data were collected as part of a study to determine the effect ofsampling variance on the analysis of drug animal-feed formulations.2% Drug (w/w)0.01140.01020.01000.01050.00990.01060.00950.0095% Drug (w/w)0.01050.00870.00980.00970.01050.01030.01010.01090.01030.01010.01070.01040.0103The data on the left were obtained under conditions in which random errors insampling and the analytical method contribute to the overall variance. The dataon the right were obtained in circumstances in which the sampling variance isknown to be insignificant.
Determine the overall variance and thecontributions from sampling and the analytical method.1811400-CH07 9/8/99 4:03 PM Page 182182Modern Analytical ChemistrySOLUTIONThe overall variance, so2, is determined using the data on the left and is equal to4.71 × 10 –7 . The method’s contribution to the overall variance, s m2, isdetermined using the data on the right and is equal to 7.00 × 10–8.
The variancedue to sampling, s2s , is therefore2 = 4.71 × 10–7 – 7.00 × 10–8 = 4.01 × 10–7s2s = so2 – sm7B Designing A Sampling Plansampling planA plan that ensures that a representativesample is collected.A sampling plan must support the goals of an analysis. In characterization studies asample’s purity is often the most important parameter.
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