Statistical Evaluation of Acid/Base Indicators A First Experiment for the Quantitative Analysis Laboratory David T. Harvey DePauw University, Greencastle. IN 46135 Manv current textbooks on quantitative analvtical chemistry begin with a discussion of experimental errors and the statistical treatment of data (1-3). Typically, students are introduced to the important differences between determinate and indeterminate errors, between accuracv and ~ r e c i sion, and are presented with variety of statis&al tools to aid in the interpretation of experimental data. Topics covered usually include the reporting of means and standard testing.(t-test and F-test), deviations, as well as hypothesis .. and outlier rejection. Although the statistics presented in the quantitative analysis course are relatively easy to calculate, their interpretation is often much more difficult. For example, the fact that precision and accuracy are independent measures of two different and unrelated sources of experimental error is often confused hv students. This orohlem is. uerhaus. exacerhated by the s u h l e relationshipbetween p;ecisiona"d arruracy which is inherent in hvuothesis testine. In usinn the t test, for instance, to compare an experimental meanto the true mean, the confidence level a t which errors in accuracy can be established is dependent on the precision. I t is not surprising, therefore, that problems are often encountered
a
Presented, in part, at the 10th Biennial Conference on Chemical Education held at Purdue University. July 31-August 4, 1988.
in which students confuse orecision and accuracv hv assuming that precise work mu;, of necessity, be accuiatk. Berinning the quantitative analysis lahorarory with an expe&entthat emphasizes the statistical analysis of data is one way to help students appreciate the difference between precision and-accuracy. Many of the laboratory manuals available as supplements to, or as part of, current quantitative analytical textbooks, begin with an experiment emphasizing the statistical analysis of data. Most of these experiments. however. involve little more than calculatine mean valuesand standard deviations for large data sets (y, 3, 4 ) . Several exoeriments oublished in this Journal 15-9) orovide a broader exposure to statistical methods of an~lysi$,'including the use of hypothesis testing; however, none of these experiments involves quantitative procedures common to the beginning analytical laboratory. In the experiment described here, students standardize a stock solution of HCI against a weak base using different acidbase indicators. ~ndividualdata and pooled data are then subject to an extensive statistical analysis, allowing comparisons between the indicators. Procedure The class was provided with 10 L of a stock solution of approximately 0.1 M HCI to he standardized against the primary standard Tris using one of five indicators assigned by the instructor. The indicators used were bromothymol blue (BB), methyl red (MR),
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hromocresol green (BG), methyl orange (MO)and erythrosine B [EB). . . An initial "roueh" titration was uerforrned to allow the students to ohserve the color crsnsition and to establish an spproximate endpoint volume. Sample sizes of Tria were then adjusted to keep the final volume of titrant between 3 6 4 0 mL. Students were then instructed to gather as much data as they felt was necessary, although a minimum of six titrations were required. After examining their data for outliers, using the Q-test at the 95% confidence level and in consultation with the instructor, each etudent reported his or her experimental results along with his or her mean value for the HCI concentration and his or her standard deviation. A table -~ summarizing t he class results was then di~tributedto the class along with a J P & ~ of questions involving the statistical analyair of the data. After turning in them reports, the class met to further discuss the statisticalanalysis of the titration data
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Results and Discussion
This lab has been used several times in the author's introductorv course on auantitative analvtical chemistrv. Although results vary from class to class;the data have &ways sewed as a useful wav t o introduce statistical calculations and statistical reanon&. A representative data set from the spring of 1988 will serve as an illustrative example of typical ~. ciass data. The statistical analysis begins with the calculation of a pooled data set, hy indicator, of mean HCI concentrations and precisions. Results are shown in Table 1. The pooled
1
0
I0
20
30
40
50
so
Volume of HCI (ml) Table 1. Pooled Data fromSprlng 1988
Indicatora
rP
S
Mean [HCI] (M)
%*
%#
BE
28 18 29 19 18
5 3 4 3 3
0.09585 0.08686 0.08641 0.08641 0.08166
0.00225 0.00098 0.00113 0.00065 0.00308
0.00109 0.00090 0.00109 0.00026 0.00217
MR BG MO EB
'Indicator abbreviationsas in text. b T ~ t B Inumber of measwements. iNumber of mx(snt3. dRepr~ducibilityof all analyses: degrees of heedom = n 1. pooled sfandam deviation for S sfuden~:degrees of freedom = n
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mean were calculated hv averaeine the trials - concentrations ~ ~ ~ ~ ~ of all students using the same indicator. Precisions were calculated two ways. The standard deviation of all measured concentrations using a single indicator (s.) is a measure of the ability of different students to titrate to acommon visual endpoint color. A traditional pwled standard deviation (s,) for each indicator, which is a measure of the ahilitv of students to titrate to a self-consistent visual endpoint color, also was calculated. Mean concentrations were compared, a t the 95% confidence level, using an appropriate t-test depending on a Ftest analysis of thes, precisions (10).The mean HCI concentrations determined using BB and EB were found t o be significantly different from the results of the other three indicators. No significant differences were found between the mean concentrations determined using MR, BG, and -
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The reproducibility of all analyses using a single indicator (s,) was compared using the F-test evaluated a t the 95% confidence level. Again, both BB and EB were found to give significantly different results from the other three indicators. No sienificant differences were found between MR, BG, and MO. Pooled standard deviations (s,) also were compared using the F-test. The precision of students using MO was found to he significantly better than precisions obtained by students using other indicators, while the precision of students using E B was found t o he significantly poorer. 330
Journal of Chemical Education
Thewetical titration curve calculated far the titration of 0.4000 g of Tris with 0.08650MHCI. Therangeof solution conditions tor Me wlor transition of each Indicator is also shown. The theoretical titration curve for this experiment is e d the titration curve are shown in the fimre. S u ~ e r i m ~ o s on the approximate rangeif sol&ion conditions over which the indicators are expected to undergo their color change ( I ) . The indicators chosen for this experiment fall into two categories, those that change color near the equivalence point of the titration curve (MR, BG, and MO), and those that change color a t either the beginning or the end of the equivalence point region (BB and EB). Using the titration curve, the students are able to explain the results of the statistical analvses described above. The fact that significant differences were not found hetween the mean HCI concentrations determined with MR, BG, and MO is not surprising since the volume error between these indicators should he small. In addition, the narrow range of volumes over which the color transition occurs for these indicators is consistent with the fact that significant differences were not found between their respective s, precisions. The significant differences between the mean HC1 concentrations determined using BB and EB relative to the other indicators is due to the large volume errors associated with their respective endpoints. The significantly poorer s, precisions for these indicators can he explained by the suhstantial range of volumes over which the color transition occurs, and differences in the sensitivity of individual students to the perception of color. standard deviations (s,), in conjunction with The the theoretical titration curve, provide a useful opportunity to emphasize the difference between accuracy and precision. It is clear to students in this experiment that accuracv is a function of choosing the correct indicator so that the eiperimentallv determined visual endpoint has a minimal volume error with respect to the true equivalence point. Individual experimental precisions, on the other hand, are a function of the ability of kach student to titrate reproducihly to a selfconsistent endpoint color and are independent of the appropriateness of the indicator for accurate work. In fact BB (an inappropriate indicator) and MR and BG (both appropriate
Table 2. Pooled Standard Devlallons for All ClaWS lndlcatore BE MR BG MO EB
i +
17
9
76 52 73 46 54
4 3 4 3 3
12 6 11 6 6
%r'
%xe
0.00276 0.00079 0.00086 0.00062 0.00246
Indicata abbreviation^ 8s In ted. a Total number of measure men to. rn~umberof classes. d N ~ m b e 01 r studan*. P O O Istandard ~ deviation f a C classes; degraea 01 heedom = n ' p r n l d standard deviation la Ssiudank; degrees of freedom = n
0.00110 0.00076 0.00080 0.00029 0.00197
-C -S
indicators) show no apparent significant differences in their s, precisions. Several other interesting observations can he made from this pooled data set. A comparison of s, and s, precisions for EB shows no apparent significant difference at the 95% confidence level. This suggests that not only is there a substantial variability between students over the selection of the proper endpoint color but also that the ability of individual students to reproducibly titrate to a self-consistent visuo i n~tcolor was poor. Students using this indicator al -- e - nd - ~~ ~ commented that they found the color change extremely difficult to monitor. This is, perhaps, a consequence of both a difficult color change (red to orange) and the broad range of volumes of titrant over which the color transition occurs. A comparison of the s, and s, precisions for BB shows a significant difference at the 95% confidence level. In this case it appears that, as with EB, the range of volumes over which the color transition occurs is large enough to cause anhat,antial variahilitv between the visual endpoint colors of ~different students. &like EB, however, the precision for RR sueeests that individual students had little difficulty -titrating to a self-consistent visual endpoint color. Perhaps the color transition for BB (blue to yellow) is more definitive. Finally, thes,precision for MO is significantly better than the s, precisions of the other indicators suggesting that students using this indicator had no difficulty titrating to a selfconsistent visual endpoint. Near the end of the class discussion, the students are instructed that caution should be used in making statistical interpretations from a limited amount of data. For example, the conclusions regarding the significantly poorer s, precision for EB or the significantly better s, precision for MO, may he due to a limited amount of data since, in both cases, only data from three students are included. It is quite conceivable that the experimental s, and sp precisions are not indicative of the "true" precisions for these indicators. The data from several classes, however, can be pooled together into a single data set. This is shown in Table 2 where, again, precisions are reported in two ways. The ~~
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pooled standard deviation of s, precisions (s,) is a measure of the ability of students in different classes, using the same indicator, to titrate to a common visual endpoint color. The pooled standard deviation of all students using a common indicator (s,,) is, as in Tahle 1, a measure of the ability of students to titrate to a self-consistent endpoint color. Although the molarity of HC1 does vary from class to class (relative standard deviation of 5% based on BG) it is anticipated that this will have little effect on the s, and s, preci;ions for different classes. Uncertainties related to the volume of titrant (volume readings and endpoint determination) are expected to dominate individual student precisions in this experiment. Maintaining an approximately constant enuivale~ice volume for all~ titrations, - > - ~ -~ mint ~ ~ ~ ~ reeardless of indicator or HCI molarity, ensures that reportea standard deviations ~rimarilvwill reflect differencesin the consistency of the visual endpoint determination. The results shown in Tahle 2 are consistent with the principle interpretations of the spring 1988 data. A comp&ison of s,, precisions, for example, shows that students using BB and EB have a significantly greater difficulty titrating to a common visual endpoint color. A comparison of s, precisions shows that students using EB have a significantly greater difficulty titratiug to a self-consistent visual endpoint, while students using MO have a significantly easier time titrating to a self-consistent visual endpoint. ~~~
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Conclusion The experiment described in this paper has served as an excellent introduction to the use of statistical methods in analytical chemistry. Students have the opportunity to use the Q-test for rejecting outliers, the t-test for comparing mean values. and the F-test for comparing precisions. Students also have an opportunity t o c a l ~ u l a t ~ ~ o odata l e d nets. The class discussion of the data and its interpretation in light of the theoretical titration curve helps the students gain a better perspective on the statistical interpretation of data. Acknowledgment The author expresses his gratitude to the 47 students who have collected the data and participated in its interpretation. Literature Cued 1. Harris, D. C. Quanfifofiv~Chemical Anoly~is,2nd ed.: W. H. Freeman: New York,
1987. 2. Skoug, D. A ; West, D. M.; Holler, F. J. Fundomenlola of Anolytirol Chemistry. 5th ed.; Saundem: New York, 1988. 3. Day, R. A..Jr.: Undemmd,A. L.Quantitative Analysis, 5th ed.: Prentiee-Hall: New Jersey, 1986. 4. Kennedy, J. H . Anolyl;eol Ch~mislry:Ploetica; Hareourt Brace Jovanovich: San
7. P h k , k. A.J Chem.Educ. 1585.62.536. 8. O'Reilly. J. E. J. Chem. Educ. 1986,63,694-896. 9. Gordus,A.A. J. Ch~m.Educ.1987,64,376377. lo. Miller, J. C.; Miller. J. N.Stotisrics for Anolyriroi Chemistry; Horwwd: Chicheater, England. 19%
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