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THE VALUE OF ERROR PROBLEMS IN THE TEACHING OF QUANTITATIVE ANALYSIS C. C. MELOCHE University of Michigan, Ann Arbor, Michigan
TEACHERS of a science such as chemistry or physics have long maintained that the solution of numerical problems which illustrate scientific principles has an important function. Since much time is required to obtain first hand experience of an extensive character in the lahoratory any method which rapidly and effectively transmits the experience of other qualified workers in the field is to be commended. One of the most useful methods available in the study of quantitative analysis is the solution and discussion of problems illustrating the effect of errors and interfering substances on methods of analysis. Such problems properly constructed very effectively focus attention on the need for care in many important details of procedure and on the serious results of neglecting such precautions. The student who overlooks details in the laboratory is in this manner brought to realize the results of his carelessness and the need for better technique. Furthermore, he becomes better able to decide for himself just what the cause of his trouble has been. The type of problem mentioned requires the student to apply a knowledge of elementary inorganic chemistry and certain physicochemical principles with much more discernment than is ordinarily needed in stoichiometric problems. Thus the student who makes a careful study of the errors and limitations of analytical methods is in a much better position to judge critically as to the accuracy that may be expected of a given laboratory procedure. On account of the difficulties encountered from the student point of view more numerical problems along this line should be included and more should be explained carefully. With a background of this sort laboratory work by the student acquires in general much greater reliability and significance. The types and examples mentioned below illustrate the scope and varied nature of this method of instruction. Some of the more important considerations upon which the construction of useful error problems in acidimetry and alkalimetry may be based are the following: (1) the unohserved presence of a definite concentration of carbonic acid when phenolphthalein is used as indicator in the volumetric determidation of a specific moderately weak or strong acid with standard fixed alkali; (2) the mistaken use of some indicator such as methyl red which changes color a t a definite pH less than seven for the determination of the end point in the titration of a moderately weak acid of known ionization constant; (3) the fallacious assumption that the equivalence point in the titration of a moderately
weak monobasic acid of known ionization constant with standard strong alkali is reached a t a pH of exactly seven; (4) the mistaken use of phenolphthalein or some other indicator that changes color a t a definite pH greater than seven for the determination of the end point in the titration of a typical salt of a very weak monobasic acid of known ionization constant with standard strong acid; (5) the wrong assumption that the equivalence point in the titration of a salt of a very weak acid of known ionization constant with standard strong acid is reached at a pH of exactly seven; (6) the unohserved presence of a d e h i t e percentage of foreign acid, acid salt, or salt of a very weak hase when the determination of total acidity is taken as the accurate measure of the percentage of a given single acid; (7) failure to take account of the increase in the ionproduct constant of water when a moderately weak acid or a moderately weak base is titrated a t or near the boiling point of water. Likewise the corresponding series of problems based upon the presence of carbonate in the standard alkali, mistaken assumptions in the selection of an indicator in the titration of a moderately weak base or a salt of a very weak base, and the unobserved presence of foreign hase, basic salt, or salt of a very weak acid when a single base is being determined may all be constmcted. In volumetric precipitation reactions the percentage error due to the unobsemed presence of a definite weight or percentage of another substance simultaneously precipitated or titrated is readily calculated. .Another type of error problem in volumetric precipitation reactions is based upon the fact that due to the solubility of the precipitate and the nature of the indicator reaction the actual end point does not exactly coincide with the equivalence point as required by the solubility product princip1e.l In volumetric reactions that depend upon complex formation an inherent error results when the complex formed is relatively unstable, i. e., the instability constant is too large. For this reason the actual end point does not coincide with the true equivalence point. The amount of this discrepancy may usually be calculated if the sensitivity of the indicator reaction-that is, the amount or concentra'SMITI~,T. B., ''Analytioill Processes," 2nd ed., Edward .4rnold and Co., London, 1940, pp. 153-8. "Volumetric AnslyKOLTEOFF,I. M., AND V. A. STENGER, sis," 2nd ed. Interscience Publishers, Inc., New York, 1942, Vol. I, pp. 154-8. SMITH, T. B., ibid., pp. 246-52.
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tion of standard reagent required to affect the indicator visually-the concentration of the active indicator agent, and the necessary instability constants or instability constant and solubility product are all known. This is when the reaction is assumed to be stoichiometric and not when the errors have been compensated. A still different problem results when a standard solution of a complexing agent reacts with a definite amount of accessory constituent as well as with the constituent intended. In certain cases the use of an excessive amount of indicator leads to a large blank, the amount of which may be calculated if the amount of indicator is known. The calculation of the inherent error in volumetric oxidation and reduction reactions is based upon the calculation of the concentration of the residual unreacted constituents a t equilibrium when equivalent amounts of the reacting substances have been mixed and allowed to react. This is as a rule easily accomplished by first calculating the equilibrium constant (K) of the volumetric reaction from the normal electrode potentials of the partial reactions involved using
purities. He will then obsenre that the presence of a relatively small amount of impurity which has a very small equivalent may lead to serious error, but that the presence of a considerably larger amount of titratable impurity with a ver.y large equivalent may have a nearly negligible effect on the final results. A problem may sometimes be constructed on the assumption that a definite part of the volumetric oxidizing agent or reducing agent is reduced or oxidized to a state different from the one intended. The calculation of the titration error as defined by Kolthoff and Stengerg includes allowance for the sensitivity of the indicator and therefore brings the calculated error into closer agreement with the actual experimental error. If it is thought that such a calculation is too complicated for the beginner, it is usually possible to have the student calculate the residual concentration of the titrated substance a t equivalence regardless of the particular indicator employed. The term "inherent error" has been used by the writer in referring to the results of such calculations. Problems illustrating other errors in volumetric analysis may be n based upon change in temperature of the standard sothe relation, log K = -- (Em - EoU)at 25%. lution, errors in calibration, and the inconsistent use of 0.0~~~.591 Knowing from mass-law considerations the ratio of the various units of volume. The gross error that results concentrations (or activities) of oxidant and reductant in all classes of volumetric reactions when the student in terms of K and knowing the sum of the concentra- uses the wrong normality, the wrong equivalent, the tions (or activities) of oxidant and reductant it is easily wrong titer, or the wrong logarithm may also be calpossible to calculate the residual concentration (or culated. In gravimetric analysis the error due to solubility of activity) of either oxidant or r e d u ~ t a n t . ~By , ~ this means it is possible to decide whether the reaction the precipitate may as a rule be easily calculated on the goes so nearly to completion at equilibrium that it may basis of the solubility product principle but the error be considered highly satisfactory, whether the reaction due to contamination or decomposition of the ignited falls short of completion only a small amount that may precipitate may be difficult for the amateur to evaluate. be safely compensated on the empirical basis, or Gravimetric error problems may in large part be classiwhether the reaction falls so far short of completion fied according to the cause of error as follows: (1) that compensation is difficult or impossible. No infor- some error in the assigned value of one or more of the mation in regard to the speed of the reaction is obtained. weights exists or the true sum of the weights has been The calculation of inherent error in this manner is of a misread. (2) The precipitate weighed or separated is fundamental nature. The concept of formal oxidation- soluble to an appreciable extent. (3) The precipitate reduction potentials6 may lead to even more practical weighed is contaminated with a relatively inert nonresults when such calculations are undertaken. With volatile impurity of a foreign nature. (4) The precipielementary students it is usually preferred to work in tate is in part volatile as such when heated for unduly terms of molar concentrations rather than activities' prolonged periods or a t temperatures above those customarily prescribed. (5) The precipitate is in part and limit the discussion to the simplest cases. In the titration of a reducing agent with a standard decomposed on ignition and one of the decomposition oxidizing agent it is easy to calculate the error caused products is volatile. (6) The precipitate weighed is by the unobserved presence of a definite amount of contaminated with a foreign anion coprecipitated or some other reducing agent. Likewise in the titration post precipitated by the same cation as that contained of an oxidizing agent with a standard reducing agent. in the precipitate. (7) The precipitate weighed is If the student has properly learned the calculation of contaminated by occlusion or adsorption of a foreign equivalents8 he will seldom be mistaken in the calcu- anion. (8) The precipitate weighed is contaminated lation of the error in volumetric methods due to im- with a foreign cation coprecipitated or post precipitated by thesame anion that i s contitined in the KoLTH I, M,, AND V, A. STENGEE, ibid., pp. 73-86, 6 M~LLER E., , "Die Elektrometri~oheMassana~yse,77 6th ed. cipitate. (9) The precipitate weighed is contaminated Theodore Steinkopff, Dresden and Leipzig, 1942, pp. 24-34. by occlusion or adsorption of a foreign cation. (10) SMITH,G. F.,Trans. Illinois State Acad. Sci., 36, 132 (1943). The deposit obtained in electrolytic analysis may con' WILLARD,H. H., AND N. H. FURMAN,"Elementary Quantitative Analysis," D. V m Nostrand Company, Ino., New York, 3rd tab all or a part of some other metal or metals present, especially if the constant current method has been used. ed., 1940, pp. 215-8. MELOCEE, C. C., J. CAEM.EDUC., 24, 475 (1947).
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