Laboratory Experiment pubs.acs.org/jchemeduc
Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
Exploring Exponential Decay Using Limited Resources Ed DePierro,† Fred Garafalo,*,† and Patrick Gordon‡,† †
MCPHS University, 179 Longwood Avenue, Boston, Massachusetts 02115, United States Cape Cod Community College (4 Cs), 2240 Iyannough Road (Route 132), West Barnstable, Massachusetts 02668, United States
‡
S Supporting Information *
ABSTRACT: Science students need exposure to activities that will help them to become familiar with phenomena exhibiting exponential decay. This paper describes an experiment that allows students to determine the rate of thermal energy loss by a hot object to its surroundings. It requires limited equipment, is safe, and gives reasonable results. Students record the decreasing temperature values displayed by a thermometer, which was initially heated by body contact. The procedure is easy to repeat, taking only 3 min. The students then convert temperature changes into thermal energy losses and plot graphs. Data analysis can be used to teach graphing and ratio interpretation. This experiment can be used by high school or first-year college students in situations where resources or space may be limited or where large student populations dictate a need for uncomplicated experimental procedures, particularly those that minimize the production of chemical waste.
KEYWORDS: First-Year Undergraduate/General, High School/Introductory Chemistry, Laboratory Instruction, Hands-On Learning/Manipulatives, Kinetics, Rate Law
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INTRODUCTION Phenomena as diverse as emptying a water tank, discharging a capacitor, cooling a hot object, radioactive decay, and some chemical processes all exhibit exponential decay behavior.1 The ubiquitous nature of this behavior suggests that science students should be given the opportunity to explore exponential decay and to learn what it entails. Several authors in this Journal have reported examples of activities in which students study the rate of cooling or of heating.2−5 For a number of years, first-year chemistry students at MCPHS University conducted an experiment that involved cooling warm water in a beaker by immersing it in an ice bath. However, as the number of these students grew over a 10-year period from under 200 to over 700, factors associated with the experiment became more problematic. The strain on the ice machine over as many as nine consecutive hours of use, the time involved in collecting sufficient data, and, at times, student inattention to details (such as the positioning of the thermometer or maintaining sufficient ice in the bath), led us to develop a less labor and equipment intensive experiment. This paper describes a quantitative cooling experiment that uses just a digital thermometer and a timing device. It builds on aspects of the qualitative experiments described in ref 4. The experiment is safe, gives reasonable results in terms of demonstrating exponential decay behavior, and can be performed using minimal equipment in a short period of time with no production of chemical waste. The experiment can be useful for high school or first-year college instruction when resources or time in the laboratory are limited or when large numbers of students dictate the need for relatively © XXXX American Chemical Society and Division of Chemical Education, Inc.
uncomplicated procedures. Data analysis can be used to teach graphing and ratio interpretation.
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THE EXPERIMENT The experiment has six parts (A−F), which are summarized here. The interested reader can access the complete experimental procedure, including an appendix on graphing, and notes on how the experimental conditions were optimized in the Supporting Information. Part A: Interpreting Graphs
Students are shown three graphs, in which temperature is plotted against clock reading for a hypothetical object that is cooling, and are asked to identify, with a partner, which one they think represents constant rate of cooling, increasing rate of cooling, and decreasing rate of cooling. They are asked to explain their reasoning and then to predict what they will see in the experiment they are about to perform. Part B: Experimental Procedure and Data Collection
Students work in pairs. A clean digital thermometer (Traceable, long stem) is suspended from a clamp attached to a stand. One partner clasps the metal thermometer tip between his/her hands until it registers a steady Celsius temperature in the mid 30s. (Rapidly, but carefully, rubbing the thermometer tip between one’s hands can speed up the process.) S/he then indicates this information to the second partner who records this temperature for clock reading zero. Partner two then Received: July 28, 2017 Revised: January 23, 2018
A
DOI: 10.1021/acs.jchemed.7b00571 J. Chem. Educ. XXXX, XXX, XXX−XXX
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announces “now” as s/he starts a timer, and partner one simultaneously releases his/her grip. At ten-second intervals, partner two announces “now,” and partner one then states the temperature, which partner two records on Data Sheet One (Part C). This process is repeated for 180 s.
Laboratory Experiment
HAZARDS This experiment avoids the risks associated with manipulating hot plates and hot liquids, but care must be taken when warming the thermometer, especially if the student rubs it between his or her hands, since the tip is pointed. It is also important to use a thermometer that is known to be clean, not contaminated by prior contact with potentially harmful substances.
Part C: Completing Data Sheet One
Students then complete Data Sheet One which has the following five column headings
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• t, (seconds)
DISCUSSION As mentioned earlier, this experimental approach simplified the procedure previously used by eliminating the need for hot water, ice, and various containers. It avoided concerns about students using sufficient insulation, positioning the thermometer to avoid contact with container walls, and leaving the apparatus completely undisturbed for relatively long periods of time. Concerns about unwanted cooling due to water evaporation and non-Newtonian effects below 10 °C were also avoided.3 With up to 700 first-year chemistry students currently in the class at MCPHS University in recent years, the transition to this procedure has improved the logistics of offering this type of activity for instructors and the laboratory manager, while not compromising the potential learning outcomes for students. Students plotted the two graphs by hand. Since data collection took relatively little time, the process of choosing appropriate scale units, sketching a curve (a piece of clear vacuum tubing works well here if a French curve is unavailable), picking data points, and calculating slopes could be practiced with the entire experiment still fitting into a 3 h period. This has become the second of two experiments that requires students to practice these skills in the first-semester laboratory at MCPHS University. At 4 Cs, in addition to performing the experiment, students (about 30) were given data sets from this type of experiment to practice their graphing skills in separate activities that formed the basis of a laboratory practical. Asking students to use graph papers with different grid patterns added to the challenge. The experiment was also conducted with a class of high school chemistry students, numbering about 40 at the John D. O’Bryant High School, Boston. One author (Gordon) has interacted with high school students numerous times as a volunteer in the ACS Science Coaches Program6 in an effort to kindle interest in science careers and to expose the students to learning activities they can expect to see in college. Treatment of the data via an Excel spreadsheet was also an option, but in all of these teaching environments, the authors chose to focus attention on graphing skills, since experience with earlier versions of the experiment had shown them that students were often unable to determine an appropriate scale unit for graphs. Graph creation provided students with opportunities to practice the important, and often underdeveloped critical thinking skill, ratio interpretation.7,8 Both high school and first-year college students, when confronted with a ratio such as 40 J/200 spaces on the graph paper often exhibited difficulty interpreting it to mean 0.2 J could be represented by one space on the graph paper. Also, many students did not recognize that the point where the abscissa and ordinate meet was not required to have the value 0,0. This led to underutilized space on the graph paper, with data points being compressed into just a small portion of the available area. An activity that guided students through the process of determining an appropriate scale unit for a graph is provided in
• T, (°C) • ΔT = T − TRT • TE = mCΔT, (joules) • ln TE ΔT is the difference between T at a given clock reading, t, and the room temperature, TRT, which was determined prior to data collection. TE is the excess thermal energy in the warm object, over and above the thermal energy it possesses at room temperature. In this simple experiment, the warm object is the outer stainless steel casing of the thermometer itself. Students use the specific heat capacity (C = 0.449 J/°C g) and density (d = 7.87 g/cm3) of iron, the major component of different stainless steels, along with a crude estimate of one cubic centimeter as the volume of material actually heated in the casing, to produce a value of 3.53 J/°C for mC in column four, which is multiplied by each ΔT to give the mCΔT values. ln TE is the natural logarithm of TE. The calculation involving volume uses 1.00 cm3 in order to justify the number of significant figures in the quantity, mC. Part D: Plotting Two Graphs
Each student in the pair prepares one of two graphs, either Graph 1: TE vs t, or Graph 2: ln TE vs t. Part E: Interpreting Graph 1
Using Graph 1, students are asked to fill in spaces on a second data sheet indicating the clock readings and excess thermal energy values corresponding to TE0 (at t = 0), TE0/2, TE0/4, and TE0/8. They are then asked several questions about the rate at which thermal energy is lost during the experiment and which graph back in Part A is consistent with their experimental results. Still using Graph 1, they repeat the process, but this time recording clock readings and excess thermal energy values for TE0, TE0(0.8), TE0(0.8)2, and TE0(0.8)3. The terms, halftime time and 0.8-lifetime are introduced. They are then asked to read a few paragraphs that reinforce prior classroom discussions of the equations TE = TE0 e−kt, ln (TE/TE0) = −kt, and ln TE = ln TE0 − kt (with its connection to the equation of a straight line, y = mx + b). This includes the effect of k on the shape of the mathematical curve and physical factors that can affect the magnitude of k. Based on the approximately constant experimental half-life times, students are asked how the half-life times can be constant while the rate of thermal energy loss decreases. Part F: Interpreting Graph 2
Students determine the slope of Graph 2 and the value of the rate constant, k, which is (−1)(slope). They compare their results with those obtained by other groups and are asked to consider what might lead to different experimenters obtaining different values. B
DOI: 10.1021/acs.jchemed.7b00571 J. Chem. Educ. XXXX, XXX, XXX−XXX
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generated in this way using data obtained by students or by instructors agreed with those generated by Excel. These optimization experiments led to the procedure currently used, which is described above in Part B. Under these conditions, for a given run, percent relative standard deviation (% rsd) for two and often three consecutive half-lives of 5% or less were common. Half-life times were about 46 s, with k values of 0.015(±0.001) s−1.9 Interested readers can find optimization efforts described in greater detail in a document in the Supporting Information, titled, Instructor Notes on Optimizing the Experiment. Although graphs plotted by hand (student and instructor) were not as accurate as those generated by Excel, they still provided data from which useful conclusions could be drawn. With help from scaffolding provided in Data Sheet Two in the experiment, most students could see that the rate of thermal energy loss decreased with time, but it was harder to generate nearly constant consecutive half-life times from Graph 1. This is one reason why readings and discussions in both the laboratory and classroom were added to show students the ideal behavior implied by the equation TE = TE0 e−kt. However, ln TE vs t plots in Graph 2 were usually linear over three half-lives, consistent with drawing the conclusion that the process obeys first-order decay behavior. This is a good example of the benefit that can often be obtained by turning a nonlinear plot into a linear one. For many years we have relied on an action research model, in which feedback obtained by observing students and interacting with them in the laboratory, classroom, and help sessions has guided improvement of classroom presentations and laboratory activities.8,10−12 The evolution undergone by this experiment over several years is a case in point. Although estimation of the excess thermal energy in the stainless steel shell of the thermometer is crude at best,16 we found that it was easier for students to think about an extensive property, energy, leaving the system as time passes (TE = TE0 e−kt) rather than thinking about how the difference in temperature between the room and the warm system decreases as time passes (ΔT = ΔT0 e−kt). The responses of some students to the graph interpretation activity in Part A of the experiment indicated that they were still struggling with simple ratio interpretation. In addition, logarithms have always been a challenge to many students,11 so it was deemed best to avoid confronting them with expressions like ln (ΔT/ΔT0), as was done in earlier versions of the experiment. The logarithm of a ratio of two quantities that are each in turn the difference of two quantities is not easily interpreted by most students. The authors found that when high school and first-year college students were mired in complicated calculations, this often led to meaningless symbol manipulation and a failure to gain a qualitative picture of what the calculations represent in terms of the behavior of the system under study. The equation describing exponential rate of warming of a cool object to a given temperature, (T − Tinitial) = ΔTmaximum[1− e−kt]), is even more challenging in terms of the mathematical manipulations that students must perform and how they should be interpreted. In the case of warming, for time intervals of a given duration, the temperature increase repeatedly decreases by a corresponding fixed fraction. This is why the authors chose not to develop a quantitative experiment based on the qualitative heating experiment described in ref 4. The curriculum has evolved such that classroom and laboratory activities prior to this experiment now lay the
an appendix after the experiment in the Supporting Information. Finally, care must be exercised when applying statistics in Excel. (For example, an R2 [correlation coefficient] of 0.9 can be obtained for a straight line driven through data exhibiting exponential curvature.5) Over the past several years, the authors have experimented with optimizing the temperature range and mode of heating used in this experiment. Exponential decay data (student and instructor generated) were graphed by the authors using an Excel program, but in order to mimic more closely the procedure followed by the students, the half-life times were estimated by eye from these graphs, not by computer calculation, and so were the % relative standard deviation values for consecutive half-life times. See Figures 1 and 2 for
Figure 1. Thermal energy dissipated by a warmed thermometer vs clock reading.
Figure 2. Ln of thermal energy dissipated by a warmed thermometer vs clock reading.
sample graphs plotted in Excel. Data collected by students produced graphs that were comparable to those produced using instructor-collected data. Half-life times and % rsd values C
DOI: 10.1021/acs.jchemed.7b00571 J. Chem. Educ. XXXX, XXX, XXX−XXX
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Laboratory Experiment
SUMMARY To summarize, this experiment provides an opportunity for students to learn about exponential decay using a simple procedure that can be employed when time and resources are limited. The short time needed to conduct it allows more time to engage students in interpreting data and symbolic representations. The fact that it is easily repeatable also makes it possible to serve as a way to introduce exploration of experimental error and reproducibility if so desired. An instructor can choose the extent to which s/he wishes to engage students in the mathematical descriptions employing exponents and logarithms that are described in the experiment. Students can also be asked to compare manual calculations with data generated by Excel. When chemical reagents or laboratory facilities are limited, this activity can substitute for a chemical kinetics experiment that demonstrates first-order behavior.
foundation for it. Four classes use experimental observations to introduce students to the concept of energy, its transformation from one type to another, the effects of mass and specific heat capacity on the extent of temperature change of various substances, and temperature and energy changes associated with objects coming to thermal equilibrium. Experimental evidence is given to support the idea that energy transfer between a hot and cold object is due to molecular motion transfer from the hot object to the cold object. In this way, the idea of energy transfer is made more concrete. (The more abstract concept of radiant energy transfer is covered later in the year.) These ideas are reinforced in a laboratory experiment that deals with the effect of mass on the extent of temperature change, and the determination of a specific heat capacity. Students are asked guiding questions in the classroom and laboratory, such as, “What besides temperature change is needed in order to determine an energy change associated with a sample of matter?” “Which requires more energy, increasing the temperature of an object 1 °C or 2 °C?” “Why?” These activities set the stage for the next step, exploring rate of energy loss. For the past two years, students have been asked to fill in Data Sheet Two in Part E, using data from Graph 1, before answering questions about the rate at which thermal energy was lost. (See the experimental procedure in the Supporting Information.) The authors found that this format, in which students had to put specific thermal energy and clock reading values into the table, helped to focus them on the manipulations (taking differences and making ratios) needed to answer the questions in Part E about what was happening to the rate of energy loss, and the relationship between this and the temperature difference between system and surroundings. This contrasts with just asking these questions without this additional scaffolding, as was done in prior years. Even though students were introduced to data sets and graphs and asked questions about rate of temperature decrease and rate of energy loss in two classes prior to doing this laboratory experiment, some still had trouble distinguishing temperature (and energy) decrease from rate of temperature (and energy) decrease until they did Part E of the experiment. Discussion of related homework problems and test questions in help sessions has indicated to the authors that the symbolic equations expressing the relationship between quantities in an exponential decay process are a perennial challenge for firstyear students.11 Algebraic manipulation of the symbols in quantitative problems is an issue for many (even when logarithms and exponents are not involved) and expressing their qualitative meaning in words challenges even the best symbol manipulators. This is another reason why additional qualitative descriptions were added to Parts E and F as well as to the reading materials and homework problems used for relevant classroom discussion. The relatively short time required by students to gather data for this experiment gave them yet another opportunity to grapple with these challenges. The authors have learned the importance of cycling back to a challenging concept to give students repeated exposure to it. To this end, the general ideas associated with exponential decay are also revisited in the second semester, when the students study first-order chemical reaction kinetics in the classroom and laboratory.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.7b00571. Instructor notes on optimizing the experiment (PDF, DOCX) Procedure for the experiment: Determining the Rate of Cooling (including an appendix about graphing) (PDF, DOC) Comments for instructors (PDF, DOC)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Fred Garafalo: 0000-0001-6192-4213 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Dr. Zelias Dias Kibaja and the students of the John D. O’Bryant High School, Boston. REFERENCES
(1) Shive, J. N.; Weber, R. L. Similarities in Physics; Wiley: New York, 1982; pp 25−35. (2) Birk, J. P. Coffee Cup Kinetics. J. Chem. Educ. 1976, 53 (3), 195− 196. (3) Casadonte, D. J., Jr. Kinetics in Thermodynamics Clothing: Fun with Cooling Curves. J. Chem. Educ. 1995, 72 (4), 346−350. (4) Bartholow, M. Cool! Rates of Heating and Cooling. J. Chem. Educ. 2007, 84 (3), 448A−448B. (5) Bartholow, M. A Class Inquiry into Newton’s Cooling Curve. J. Chem. Educ. 2007, 84 (10), 1684−1685. (6) Ainsworth, S. J. Science Coaches. Chem. Eng. News 2011, 89 (37), 51−53. (7) Arons, A. B. A Guide to Introductory Physics Teaching, 1st ed.; Wiley: New York, 1990; pp 3 − 8. (8) Kennedy-Justice, M.; Pai, S.; Torres, C.; Toomey, R.; DePierro, E.; Garafalo, F.; Cohen, J. Encouraging Meaningful Quantitative Problem Solving. J. Chem. Educ. 2000, 77 (9), 1166−1173. (9) Unlike a first-order chemical process with a well-defined k at a given temperature, the rate constant for these thermal energy loss experiments is not truly unique. The optimization experiments indicate that half-life times and k values can vary with the experimental D
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conditions. It is as if thermal energy were a reactant, and just as a reverse reaction can complicate the kinetics once the concentration of a product gets large enough, it seems that the buildup of dissipated thermal energy in the vicinity of the thermometer can complicate the situation in these experiments. (See Notes on Optimization for a more detailed discussion.) The average k value and standard deviation reported above merely indicate that within our optimized conditions, a percent relative standard deviation for k of about 7% can be expected. (10) Garafalo, F.; LoPresti, V. Evolution of an Integrated College Freshman Curriculum: Using Educational Research Findings as a Guide. J. Chem. Educ. 1993, 70 (5), 352−359. (11) DePierro, E.; Garafalo, F.; Toomey, R. Helping Students to Make Sense of Logarithms and Logarithmic Relationships. J. Chem. Educ. 2008, 85 (9), 1226−1228. (12) In the MCPHS University first-year chemistry curriculum, discussion of experimental observation precedes the introduction of concepts and underlying atomic/molecular explanations. Content is linked to the development of formal reasoning skills, such as reasoning using ratios. This curriculum has its roots in the seminal ideas espoused by early adapters of constructivist learning theory.13 The approach has some similarities to POGIL,14 but our action research has always involved working with large, diverse populations of at least 150 students, and this has grown during the last ten years to as many as 700 students. While active learning in the classroom continues to be helpful in identifying where students struggle, clarifying appropriate learning outcomes, and translating these into activities to help students develop the necessary skills, interactions with students in the laboratory and help sessions have now proven more useful in driving curricular change. However, with up to 20 different adjunct instructors teaching laboratory each semester, weekly meetings, including extensive minutes, and a detailed packet including grading rubrics and suggestions for Socratic approaches to guide students, are provided in an effort to ensure consistent instruction across the numerous laboratory sections. An increase in the range of freshman student abilities over the past five years has added to the challenge. Although no longer integrated with biology,10 the curriculum maintains a focus on key concepts like energy and equilibrium, which span and unify the natural sciences. To this end, our approach has similarities to the curriculum developed by Cooper.15 (13) Arons, A. B., Conceptual Difficulties in Science. In Undergraduate Education in Chemistry and Physics: Proceedings of the Chicago Conferences on Liberal Education, No.1; Rice, M. R., Ed; University of Chicago, Chicago, IL, 1986; pp 23−32. (14) POGIL (Process Oriented Guided Inquiry) Home Page. http:// pogil.org/ (accessed January 2018). (15) Cooper, M.; Klymkowsky, M. Chemistry, Life, the Universe, and Everything: A New Approach to General Chemistry, and a Model for Curriculum Reform. J. Chem. Educ. 2013, 90 (9), 1116−1122. (16) As mentioned earlier, 1.00 cm3 is used in the calculation involving volume in order to justify three significant figures in the quantity, mC. ΔT values contain either three or two digits. A typical ΔT range is 14.0 to 1.1 °C. The grid on a piece of graph paper allows students to record two digits with certainty and to estimate a third digit. With appropriate scale units, they are estimating where to place seconds on the abscissa for both graphs, where to place tenths of joules for TE on the ordinate for Graph 1 (a typical range is 49.7 to 3.9 J), and where to place hundredths (no unit) on the ordinate for ln TE in Graph 2 (a typical range is 3.91 to 1.36). Students are asked to carry more significant figures than is justified when calculating mCΔT values when temperature changes are small and then to truncate appropriately the value of the rate constant, k. Reporting only two significant digits for k (typically in the range of 0.014−0.016 s−1) is most easily justified based on many ΔT values containing digits only in the ones and tenths places. The subtleties of significant figures involving logarithms are mentioned,11 but no effort is made to burden students with rigorously determining the significant figures that should be reported for a k value they determine from a slope of the line in Graph 2, since this would distract them from the main purpose of the experiment. An uncertainty propagation calculation for the slope of the
logarithm plot, using the two terminal points of the estimated best straight line, supports reporting two significant digits (assuming uncertainties of ± 1 s and ± 0.1 °C).
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DOI: 10.1021/acs.jchemed.7b00571 J. Chem. Educ. XXXX, XXX, XXX−XXX
