Laboratory Experiment pubs.acs.org/jchemeduc
Measuring the Gas Constant R: Propagation of Uncertainty and Statistics Robert J. Olsen*,† and Simeen Sattar‡ †
School of Natural Sciences and Mathematics, The Richard Stockton College of New Jersey, Galloway, New Jersey 08205-9441, United States ‡ Physics Program, Bard College, Annandale-on-Hudson, New York 12504-5000, United States S Supporting Information *
ABSTRACT: Determining the gas constant R by measuring the properties of hydrogen gas collected in a gas buret is well suited for comparing two approaches to uncertainty analysis using a single data set. The brevity of the experiment permits multiple determinations, allowing for statistical evaluation of the standard uncertainty uR within a laboratory period, while calculating R from the several measured quantities (pressure, volume, etc.) invokes a number of different methods of estimating the individual uncertainties involved in the calculation of the combined standard uncertainty uc(R). The two uncertainties, as obtained by several classes of upper-level college students, are typically within a factor of two or three. In the course of the data analysis, students interpret a q-q plot, identify outliers using Dixon’s test, and apply the F test to the ratio of the two variances. Thus, the experiment gives students experience with a variety of methods for evaluating data they have gathered themselves. The Supporting Information contains an Excel workbook containing student data and all the above-mentioned calculations. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Laboratory Instruction, Computer-Based Learning, Inquiry-Based/Discovery Learning, Hands-On Learning/Manipulatives, Gases, Student-Centered Learning
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quick to execute. A small class can produce a data set that is large enough for statistical analysis if each student performs two or three trials (each trial requires about 30 min). Depending on the length of a laboratory period and the extent to which the laboratory and classroom components of the course are integrated, students can analyze the data in the remainder of the laboratory period, in a subsequent classroom period, or as part of the process of writing the report. Through doing the data analysis, students learn how to propagate uncertainty, a technique whose importance is affirmed by its inclusion in all current physical chemistry laboratory textbooks and its regular appearance in this Journal.7−9 Of most significance, students produce their own experimental evidence in support of the validity of propagation of uncertainty. They extend their knowledge of statistics through use of Dixon’s test and the F test, both of which arise naturally from the data analysis and both of which are standard topics in physical chemistry laboratory textbooks. This experiment can be used to introduce students to a more sophisticated language of statistics and data analysis, with the instructor having considerable flexibility in determining the appropriate level of detail. Likewise, the instructor has considerable flexibility in choosing how the propagation of uncertainty is carried out: students may use analytical derivatives,2−4 numerical derivatives,7−9 or a Monte Carlo approach.9
ritical evaluation of experimental data and quantification of uncertainty are fundamental to the practice of science.1 Owing to their importance, these concepts and skills are developed from the start of the college chemistry curriculum in general chemistry and reach their maximal sophistication in analytical and physical chemistry. Of the two main approaches to estimating uncertainty in a derived result, statistical analysis and propagation of individual uncertainties, the first is given more attention in analytical chemistry, whereas the second falls to the physical chemistry laboratory. Physical chemistry laboratory textbooks discuss both approaches in considerable detail.2−4 Coming at the beginning of the course, these treatments can overwhelm students. When should each method be applied? What are their advantages? How do they compare? Starting from a common general chemistry experiment5,6 in which the gas constant R is determined, we have developed an experiment for the physical chemistry laboratory that allows students to focus on these two approaches to uncertainty analysis and answer these questions. Taken together, data collection and analysis requires between three and four hours.
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EXPERIMENTAL OVERVIEW The gas constant depends on several measured quantities whose uncertainties must be estimated. Propagating these uncertainties produces an estimate of the uncertainty in R. The laboratory component involves no new physical concepts or unfamiliar instrumentation, so it is conceptually simple and © XXXX American Chemical Society and Division of Chemical Education, Inc.
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dx.doi.org/10.1021/ed3005374 | J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
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Laboratory Experiment
EXPERIMENTAL PROCEDURE This experiment has been tested by five classes of physical chemistry students; the refined procedure described here has been tested by a group of upper-level college chemistry and physics majors. A small volume of hydrochloric acid solution is poured into a gas buret. The remainder of the buret is filled with distilled water. A strip of magnesium is suspended by a thread a few centimeters below the surface of the water. The buret is then inverted in a large beaker of water, bringing the denser hydrochloric acid solution to the top. The acid flows downward, reaches the magnesium, and reacts with it, generating hydrogen gas, which collects at the top of the buret. After the reaction is complete, the following data are recorded: t, the Celsius temperature of the air near the beaker; tsoln, the Celsius temperature of the stirred solution in the beaker; V, the volume of the gas; pdiff, the height difference between the levels of solution in the buret and beaker; and patm, the barometric pressure.
test, both used subsequently. A procedure for making the plot and a spreadsheet template are included with the Supporting Information. In the next class meeting, during which students have access to computers, potential outliers are first identified using the q-q plot. Of the numerous methods for rejecting outliers, Dixon’s test was selected because students might be familiar with it in its simplest form (the Q test) and because it is conceptually easy to understand and apply. The critically reviewed data set is posted and is the single data set from which all students work. Students assess their data in two ways. For the statistical analysis, they calculate the mean (R̅ ) and its standard uncertainty uR, which is Sm, the standard deviation of the mean. This result is reported in the form R̅ ± uR. Next, students calculate the combined standard uncertainty uc(R) for their own trials and report their individual results as R ± uc(R). The instructor posts the values of uc(R) for the entire data set so that students can check their calculations. Students determine whether the accepted value of R lies in these intervals; discrepancies between the observed values of R and its accepted value lead to consideration of sources of systematic error. The uncertainty analysis reveals that the uncertainties in the mass of Mg and volume of gas in the buret are the dominant contributions to uc(R). In the final stage of the analysis, students first compare uR and uc(R). The two measures of uncertainty are of the same order of magnitude (typically within a factor of 3). As a next step, students compare the relative uncertainties, uR/R̅ and uc(R)/R; these are typically less than 10−3, indicating that any difference in the uncertainties is of limited practical significance.11 For a quantitative comparison, an F test12 may be applied to the ratio uR2(R)/uc(R)2. The null hypothesis is uc(R)2 = uR2; it is sometimes accepted and sometimes rejected at the 95% confidence level, owing to the sensitivity of the ratio to small changes in the widths of the uncertainties of the measured quantities (particularly the mass of Mg and the volume of the gas in the buret). That uR2/uc(R)2 for the pooled data is nearly always greater than one not only draws attention to the spread in the individual R values, which is not unexpected from a group of students with varying skill levels, but also suggests that the estimates of the uncertainties in the individual measurements, specifically V, might be too small. In contrast with titrations, where the buret is read twice, in this experiment the buret is read only once. In the former case, where students locate the meniscus is unimportant, so long as they locate it consistently, because the volume dispensed is found by subtraction. In this experiment, the differing judgments of students do matter and lead to a wider variability than one would expect based on the graduations marked on the buret. The ability to carry out a “what-if” analysis with the goal of finding the range of uncertainties in V leading to acceptance of the null hypothesis is a virtue of using a spreadsheet. However, the results of such an analysis must be checked against reasonable limits derived from laboratory experience. The Supporting Information further describes the steps taken in the analysis and illustrates them with a spreadsheet containing one complete set of student data, calculation of R, the q-q plot, application of Dixon’s test, calculation of the standard and combined standard uncertainties, and application of the F test. The mean and standard uncertainty of the full data set (N = 28) are R̅ = (0.081859 ± 0.000090) L atm mol−1 K−1. A q-q plot shows generally good agreement between the
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HAZARDS A 6 M HCl solution is corrosive and can be dangerous to the eyes and skin; eye protection and gloves should be worn. Spills should be neutralized with sodium bicarbonate. The resulting sodium chloride and excess sodium bicarbonate can be disposed of in the sink. The solution that remains at the end of the experiment (
