In the Classroom edited by
JCE DigiDemos: Tested Demonstrations
Ed Vitz Kutztown University Kutztown, PA 19530
Promoting Graphical Thinking: Using Temperature and a Graphing Calculator To Teach Kinetics Concepts submitted by:
José E. Cortés-Figueroa* Organometallic Chemistry Research Laboratory, Department of Chemistry, University of Puerto Rico, Mayagüez, PR 00681-9019; *
[email protected] Deborah A. Moore-Russo† Department of Mathematics, University of Puerto Rico, Mayagüez, PR 00681-9018
checked by:
Mark Case Emmaus High School, 110 Delaware Ave., Catasauqua, PA 18032
There are many classroom activities that link temperature with mathematics and science. Precollege and college students easily grasp the qualitative idea of temperature (1), even though it is a difficult concept to define formally (2). The use of the Calculator-Based Laboratory System (CBL) with a temperature sensor and graphing calculator promotes integration of graphical thinking with chemical and physical concepts in the classroom and has been in use for several years (3, 4). The use of this technology permits student-centered instruction and exploration where students construct mathematical models to explain observations and trends. Through inexpensive and safe classroom experiments and demonstrations students can obtain data from, or even share data with, other students or their professors. A recent article reported the use of a graphing calculator to estimate the infinity reading (the angle of optical rotation of polarized light, α∞) of an acid-catalyzed sucrose inversion using the Kezdy– Swinbourne method (5). The actual acid-catalyzed sucrose inversion experiment must be done in a laboratory setting as opposed to a classroom setting because of the use of a polarimeter and of safety considerations. Recent articles have addressed the use of time-lag methods to analyze kinetics data to estimate infinite time readings (6, 7). Many articles have appeared in this Journal discussing first-order kinetics using actual chemical reactions appropriate for the laboratory (8–15), evaluation of data for and simulation of first-order reactions (16, 17), and nonlinear curve-fitting of kinetics data using various types of software (18–21). In this article we offer a safe, in-class demonstration using the cooling (or heating) of a piece of metal (the metal bead of the CBL’s temperature sensor) to teach the concept of a first-order chemical reaction. This demonstration permits students to construct a first-order chemical reaction model, including determination of the parameters (e.g., rate constant value and infinity reading) for the model.
† Current address: Graduate School of Education, State University of New York at Buffalo, Buffalo, NY 14260.
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The mathematical function that best describes the spontaneous cooling of a metal is given by Tt − T∞ = exp ( −
kc t ) T0 − T∞
(1)
where Tt is the temperature at time t, T0 is the initial temperature (t = 0), T∞ is the temperature at time infinity (or temperature when the metal’s temperature asymptotically approaches the environment’s temperature), kc is a constant (positive and negative for heating and cooling processes, respectively). Equation 1 is analogous to the equation that describes a first-order chemical reaction when a physical property (Pt ) is used to monitor the reaction’s progress (22). Pt must be proportional to the concentration of the reacting species: Pt can be the absorbance at a fixed wavelength, the area of a NMR signal, the angle of optical rotation, and so forth. It can be shown for any chemical reaction,
A
k
B
which proceeds to completion where all the reactants are converted to products that (22)
[ A ]t [ A ]0
=
Pt − P∞ = exp ( −kr t ) P0 − P∞
(2)
where [A]t is the concentration of species A at time t, [A]0 is the initial concentration of species A, Pt is the value of the physical property at time t, P0 is the initial value of the physical property, and P∞ is the value of the physical property at time infinity or after 10 half-lives, kr is the reaction rate constant. Since eqs 1 and 2 have the same mathematical form, the behavior of a cooling metal (or heating of a metal) can be used as a model to present and explain the behavior of a first-order reaction and the concept of P∞. Since the estimation of the P∞ value was addressed in a previous article in this Journal (5), this article will focus on the use of a cooling metal as a demonstration to explain the behavior of a firstorder chemical reaction.
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In the Classroom
Figure 2. Plot of temperature versus time for the cooling metal bead. Abscissa is time in seconds ranging from 0 to 600 in increments of 50; ordinate (in ⬚C) ranges from 0 to 100 in increments of 10.
Figure 1. Photograph of a calculator mounted on to the CBL with a temperature probe connected.
Table 1. Temperature Values versus Time for the Cooling Metal Bead Time /s
Te mpe rat ure /°C
Time /s
Te mpe rat ure /°C
0
76.222
300
25.318
20
68.591
320
24.581
40
60.407
340
23.929
60
53.875
360
23.452
80
48.588
380
22.977
100
44.378
400
22.614
120
40.718
420
22.341
140
37.610
440
21.976
160
35.171
460
21.691
180
32.976
480
21.500
200
31.279
500
21.310
220
29.595
520
21.119
240
28.465
540
20.930
260
27.163
560
20.837
280
26.047
580
20.744
∞
20.581a
a T e mpe rat ure e s t imat e d us ing t he Ke zdy –Sw inbourne me t hod (25, 26).
The Activity The actual activity demonstrates the spontaneous cooling of an object (Newton’s law of cooling) and is described elsewhere (23). The equipment for this demonstration is shown in Figure 1. The CBL with its sensors and graphing calculator are user-friendly and explicit directions on use of this technology are readily available in the literature (24). The temperature of the metal as a function of time for this particular demonstration is presented in Table 1 and the graph of temperature versus time is given in Figure 2. Once the plot in Figure 2 is constructed (the students can construct 70
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Figure 3. Plot of temperature versus time showing the goodness of fit of an exponential regression equation (r = .9271). Abscissa is time in seconds ranging from 0 to 600 in increments of 50; ordinate (in ⬚C) ranges from 0 to 100 in increments of 10. A
B
Figure 4. (A) Plot of (T − T∞) versus time. (B) Plot of (T − T∞) versus time showing goodness of fit of an exponential regression equation (r = .9926). Abscissa is time in seconds ranging from 0 to 600 in increments of 50; ordinate (in ⬚C) ranges from 0 to 100 in increments of 10.
Figure 5. Plot of ln(T − T∞) versus time. Abscissa ranges from 0 to 600 in increments of 50; ordinate (in ⬚C) ranges from ᎑1 to 5 in increments of 1.
their own graph since the tabular data can be passed from the instructor’s calculator to the student’s calculator) the instructor should ask the students to describe the graph. This part of the demonstration should spark a vivid discussion about the mathematical function that best models the relation between temperature and time. For example, questions such as how to distinguish a decreasing exponential function from a transformed reciprocal relationship (y = a兾x + b, where a, b are constants) arise during the demonstration. The instructor should let the student use the regression capabilities of the graphing calculator to explore different mathematical models (as time allows). For example, if one tries an expo-
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In the Classroom
nential fit of the data, Figure 3 will result, obviously not a very good fit. Next, ask the students to plot (T − T∞) versus time and to try an exponential fit; the plots shown in Figure 4A and 4B will result respectively. By plotting ln (T − T∞) versus time (Figure 5) the students should be able to determine kc by calculating the slope of the linear regression equation (4). This activity can be used in a graduate-level chemical kinetics course to demonstrate the Kezdy–Swinbourne method to estimate T∞ (analogous to P∞) (25, 26). Acknowledgments This teaching idea was born during the undergraduate research group meetings of the project Synthesis, Reactivity, Structure, and Electrochemistry of Fullerene Transition Metal Complexes, co-funded by the National Science Foundation (grant CHE-0102167) and the Donors of The Petroleum Research Fund, administered by the American Chemical Society, (ACS–PRF # 36623-B3). The financial support by these agencies is gratefully acknowledged. Literature Cited 1. Moore, D. A. Teach. Child. Math. 1999, 5, 538–543. 2. Jaisen, P. G.; Oberem, G. E. J. Chem. Educ. 2002, 79, 889– 894. 3. Sales, C. L.; Ragan, N. M.; Murphy, M. K. J. Chem. Educ. 2001, 78, 694–696. 4. Cortés-Figueroa, J. E.; Moore, D. A. J. Chem. Educ. 1999, 76, 635–638. 5. Cortés-Figueroa, J. E.; Moore, D. A. J. Chem. Educ. 2002, 79, 1462–1464. 6. McNaught, I. J. Chem. Educ. 1999, 76, 1457.
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7. Hemalatha, M. R. K.; NoorBatcha, I. J. Chem. Educ. 1997, 74, 972. 8. Roser, C. E.; McCluskey, C. L. J. Chem. Educ. 1999, 76, 1514. 9. Arce, J.; Betancourt, R.; Rivera, Y.; Pijem, J. J. Chem. Educ. 1998, 75, 1142. 10. Casanova, J., Jr.; Weaver, E. R. J. Chem. Educ. 1965, 42, 137. 11. Sicilio, F.; Peterson, M. D. J. Chem. Educ. 1963, 40, 269. 12. Brice, L. K. J. Chem. Educ. 1962, 39, 634. 13. Baginski, E.; Zak, B. J. Chem. Educ. 1962, 39, 635. 14. Smith, S. G.; Stevens, I. D. R. J. Chem. Educ. 1961, 38, 574. 15. Kahn, M. J. Chem. Educ. 1957, 34, 148. 16. Muranaka, K. J. Chem. Educ. 2002, 79, 135. 17. Lo, G. V. J. Chem. Educ. 2000, 77, 532. 18. Denton, P. J. Chem. Educ. 2000, 77, 1524. 19. Henderson, G. J. Chem. Educ. 1999, 76, 868. 20. Vitz, E. J. Chem. Educ. 1998, 75, 1661. 21. Zielinski, T. J.; Allendoerfer, R. D. J. Chem. Educ. 1997, 74, 1001. 22. Espenson, J. H. Chemical Kinetics and Reaction Mechanisms, 2nd ed.; McGraw-Hill: New York, 1995; p 22. 23. (a) Chris, B.; Bower, B.; Antinone, L.; Brueningsen-Kerner, E. Real-World Math with the CBL System: Activities for the TI83 and TI-83 Plus; Texas Instruments Incorporated, Dallas, TX 2000; p 61. (b) Nichols, S. D. CBL Explorations in Calculus for the TI-82; Meridian Creative Group, a Division of Larson Texts, Inc.: Erie, PA, 1995; pp 31–36. 24. A good description of chemistry experiments using Vernier probes with CBL technology and a graphing calculator is found in: Holmquist, D. D.; Randall, J.; Voltz, D. L. Chemistry with CBL; Vernier Software: Portland, OR, 1995. 25. Kezdy, F. J.; Kaj, J.; Bruylants, A. Bull. Soc. Chem. Belges. 1958, 67, 687. 26. Swinbourne, E. S. J. Chem. Soc. 1960, 2371.
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