In the Classroom Teaching with Technology
Using CBL Technology and a Graphing Calculator To Teach the Kinetics of Consecutive First-Order Reactions José E. Cortés-Figueroa* Organometallic Chemistry Resesarch Laboratory, Department of Chemistry, University of Puerto Rico, Mayagüez, PR 00681-9019; *
[email protected] Deborah A. Moore Department of Mathematics, University of Puerto Rico, Mayagüez, PR 00681-9018
Presenting the kinetics behavior of complex chemical reactions to undergraduates is a difficult task. Because of the mathematical complexity, students (and professors) often become so involved determining exact solutions that they miss the chemical concepts. One way to rectify this situation is to use technology to incorporate live examples of the kinetics of complex behaviors into lectures. In the experiments described below, Calculator-Based Laboratory technology (CBL) and a projectable TI-83 graphing calculator were used to demonstrate the kinetic behavior of a complex reaction. This type of activity addresses the frequent criticism that physical chemistry courses consist only of dull mathematical abstractions. By incorporating technology, students can be liberated from tedious manipulations and immersed in the physicochemical concepts. In this article we will explain how the CBL is used to review first-order reactions and how the graphing calculator is used to determine the rate constants for a series of consecutive first-order reactions. The kinetics system used for the experiment was a series of first-order reactions involving ligand substitution in organometallic complexes; the change of the absorbance intensity at 400 nm was recorded as a function of time (1). What makes this system challenging is that the behavior is biphasic (2). By biphasic we mean that the absorbance at a given time is affected by both consecutive reactions. Moreover, all the species in solution absorb at the same wavelength, and the rate constants governing the two consecutive reactions are of very similar magnitude. The experiment can be performed by the students in the laboratory or by the instructor as a prelaboratory demonstration. Description of CBL Technology The CBL system is a data collection instrument that feeds tabular and graphic information directly into either a graphing calculator or a computer. The CBL itself is a portable, hand-held device used to collect “real-world” data and a graphing calculator or a computer to analyze these data. With inexpensive sensors, this device can measure force, temperature, light intensity, pH, gas pressure, heart rate, magnetic field, etc. A good description of chemistry experiments using Vernier probes with CBL technology and the graphing calculator is found in the book by D. D. Holoquist, J. Randall, and D. L. Voltz, Chemistry with CBL™ (3). The CBL system can be used for classroom demonstrations with a projectable calculator or in the laboratory with cooperative groups with a regular graphing calculator. The cost of the CBL system (including the CBL, the sensors most frequently
used with chemistry experiments, and a graphing calculator) runs well under $1,000. Monitoring First-Order Reactions Using CBL Technology Since exact solutions of simple first-order reactions are relatively easy to obtain, a first-order reaction is demonstrated and the rate constant is determined during class. This allows students to review the behavior of a first-order reaction while becoming familiar with the CBL and the graphing calculator before they move to the more complex system. The pseudo-first-order reaction of the blue food colorant FD&C Blue #1, N-ethyl-N-[4-[[4-ethyl(3-sulfophenyl)methylamino]phenyl(2-sulfophenyl)methylene]-2,5cyclohexadien-1-ylidene]-3-sulfobenzenemethanaminium hydroxide inner salt (4), in an aqueous solution of sodium hypochlorite (e.g., Clorox) is an excellent experiment to demonstrate the mixing of reagents and the monitoring of the reaction’s progress using the CBL.1 This is a pseudo-firstorder reaction because the first-order reaction with respect to the colorant is carried out with a large excess of NaOCl, so that [NaOCl] >> [colorant] and the concentration of NaOCl remains virtually constant during the reaction. The reaction is monitored by observing the decrease of absorbance at 629 nm using the colorimeter and the CBL. CBL technology allows students to observe the decreasing exponential plot of absorbance as a function of time. The CBL and the graphing calculator work together so that the absorbance versus time values are simultaneously collected, displayed, and stored in tabular form in the memory of the calculator. Using the displayed graph and the trace feature of the calculator, it is easy to determine where the reaction becomes asymptotic. This information about absorbance at time infinity (A∞) (or absorbance after approximately 10 halflives) is used to determine the relative difference between the absorbance at a given time (A t ) and the final absorbance value (A∞). Since instrumental methods of analysis are used rather than direct analysis of concentration, the integrated rate law for first-order kinetics is (At – A∞) = (A0 – A∞)exp({kt). The derivation of this equation is nicely presented by J. H. Espenson (5). The calculator is used to create a table of ln(At – A∞), which is accompanied by a corresponding table of time elapsed. These data are used to construct a graph of ln(At – A∞) versus time, which is linear to at least two half-lives. Even with a limited statistical background, it is now quite simple to use the graphing calculator to determine the linear regression equation for ln(At – A∞). With a few calculations, it is easy
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In the Classroom Table1. Values of A for Dissociation of Piperidine (pip) from cis-(P(cy)3)(pip)W(CO)4 in Chlorobenzene
Figure 1. Plot of At at 400 nm vs time for two consecutive firstorder reactions. Abscissa is time in seconds ranging from 1 to 7200 in increments of 500; ordinate ranges from 1.1 to 1.7 in increments of 0.05.
to remind students that the negative of the slope of the regression line is the rate constant for the reaction. Consecutive First-Order Reactions Determinations of the rate constants that govern the steps of consecutive first-order reactions are well documented (eq 1) (2, 6 ). In these reactions, k1 and k2 govern steps 1 and 2, respectively. k1
k2
A → B → C
(1)
It is simple to determine the values of k1 and k2 if k1 is either much larger or much smaller than k2. If the progress of the reactions can be monitored by an observable change in a physical property of the species and each species in solution manifests that property without interference from the other species, then determination of k1 and k2 is again a simple task. However, if k1 and k 2 have close values and the physical property being used to follow the progress of the reaction is common to the species involved, then determining k1 and k2 becomes more difficult. For example, if the progress of the reaction in eq 1 is followed by observing the absorbance of the mixture where all species absorb at the same wavelengths with k1 ≈ k2, then the measured absorbance of the solution is equal to the sum of the absorbance of each individual species in solution (eq 2). At = A t,A + At,B + At,C + Asolv
(2)
In this equation A t,A, A t,B, At,C , and Asolv represent the values of the absorbances due to species A, B, C, and the solvent at time t.
Figure 2. A: Plot of ln (A∞ – A t) vs time for two consecutive first-order reactions. The last section of this plot is linear and is being ascribed to B → C. B: Plot of ln (A ∞ – At) vs time for two consecutive first-order reactions showing the goodness of fit of the regression equation. Abscissa is time in seconds ranging from 1 to 7200 in increments of 500; ordinate ranges from {1.8 to {0.8 in increments of 0.1.
636
Time/s
A
A
Time/s
A
78
1.400
237
1.294
1701
1.349
81
1.396
255
1.291
1851
1.359
87
1.391
300
1.275
1902
1.361
99
1.382
417
1.256
1992
1.370
108
1.371
550
1.259
2425
1.394
117
1.367
651
1.264
2625
1.407
129
1.353
750
1.270
2925
1.431
138
1.344
80 1
1.274
3625
1.486
147
1.339
999
1.290
4825
1.543
159
1.329
1101
1.301
5125
1.556
168
1.322
1299
1.314
5625
1.575
177
1.319
1401
1.323
6025
1.589
204
1.306
1500
1.332
7125
1.590
Time/s
In the experiment being described, the values of k1 and k2 are close, and the extinction coefficients have the following order of values: εC > εA > εB (1). So the absorbance at first decreases as the reaction progresses, reaches a minimum value, and then increases until it becomes asymptotic.2 This is shown in Figure 1. If the professor and students are using the same type of calculator, the professor can transfer the data to the students’ calculators for them to work with directly. The first section of this plot was ascribed to the reaction A → B and the second segment of the plot to the reaction B → C (1). It can be shown that (2) (A∞ – At) + β exp({k2t) = α exp({k 1t)
(3)
α = (εB – εA)k1 + (εA – εB)k2 [A] 0 /(k 2 – k 1)
(4)
β = (εC – εB)k1[A]0 /(k2 – k1)
(5)
Analysis of Consecutive First-Order Reactions Using CBL Technology Since both reactions in eq 1 are first order, both have exponential graphs of absorbance versus time. It is difficult to determine by simple inspection when one of the reactions ends and the other begins. In fact, during most of the experiment they overlap. To analyze each reaction, we first analyze the graph of At versus time, shown in Figure 1 (data from Table 1), where the first reaction is almost completed and the second reaction is predominantly occurring (2). This is decided by locating the last part of the graph where the curve seems to be following exponential growth. In this section the absorbance’s changes are due mainly to the second reaction B → C governed
A
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B
In the Classroom
by k2. This can be seen in Figures 2A and 2B. The linear segment of Figure 2A represents the increasing section of the data in Figure 1. Using both the graph and the table containing the values of At , the asymptote for the exponential function can be determined. This value is A∞, the value of the absorbance at time ∞. (Time ∞ is the time after at least 10 half-lives.) Next, the relative difference between absorbance and absorbance at time ∞ (A∞ – At) is calculated. After 2 half-lives the values of (A∞ – At) are extremely small, subject to large relative error, and should be eliminated from the newly created table. (A word of caution: a first-order reaction may appear to be biphasic and vice versa if the value of A∞ is incorrectly determined.) Next the logarithms of (A∞ – At ) are calculated and placed in another table. The ln(A∞ – At) is calculated instead of ln (At – A∞) because A∞ is greater than At at any time. These values are plotted against time using the graphing calculator. This plot has two sections. The linear section toward the end represents the part of the experiment where the first reaction is almost finished and it is mainly the second reaction B → C that is occurring (Fig. 2A). Using the statistical capabilities of the calculator, a linear regression equation for the last section of ln(A∞ – At) (the section dominated by the second reaction, where the contribution from the first reaction is negligible) can be determined. From eq 3, the linear regression equation is then ln(A ∞ – A t ) = ln({ β) – k 2t (see Fig. 2B.) The values of k2 and ln({β) can be determined from the slope and the intercept, respectively, of this plot. Now it is possible to account for the second reaction’s effect in the early part of the experiment where both reactions are occurring. To do this, generate (or extrapolate) Aext values from Aext = A∞ + β exp({k2t), treating the second reaction as if it were the only one occurring. Place these extrapolated values in a new table. A comparison of the plot of these Aext versus time values with the corresponding plot of the original data shows a good fit only toward the end of the plots (see Fig. 3). As expected, the fit is not good in the first part of the
Figure 4. A: Plot of calculated values of ln({[(A∞ – At) + b exp({ k2t )]) vs time for the first section of the biphasic plot. B: Plot of ln({[(A∞ – At ) + b exp({ k 2t )]) vs time for the first section of the biphasic plot showing the goodness of fit of the regression equation. This section is being ascribed to the reaction A → B. Abscissa is time in seconds ranging from 1 to 500 in increments of 50; ordinate ranges from {4 to {1.5 in increments of 0.5.
Figure 5. A: Plot of experimental values of At at 400 nm vs time for two consecutive firstorder reactions. B: Plot of At at 400 nm vs time for two consecutive first-order reactions showing the goodness of fit of eq 6. Abscissa is time in seconds ranging from 1 to 4000 in increments of 500; ordinate ranges from 1.1 to 1.7 in increments of 0.05.
Figure 3. Plots of At vs time for two consecutive first-order reactions including the equation that models the behavior of the second reaction. Abscissa is time in seconds ranging from 0 to 4000 (two half-lives) in increments of 500; ordinate ranges from 1.1 to 1.7 in increments of 0.05.
plot where the first reaction is also occurring. The time when the two plots overlap is the time when the contribution from the first reaction to the absorbance is negligible. To determine the value of k1, only data before the time when the two sections of the biphasic plot overlap should be considered. Now the quantity {[(A∞ – At) + β exp({k2t)] is calculated and placed in another table. According to eq 3 the plots of ln({[(A∞ – At) + β exp({k2t)]) versus time are expected to be linear. The linear plot of ln({[(A∞ – At) + β exp({k2t)]) versus time is constructed in the graphing calculator and shown in Figure 4A. The linear regression equation is then determined. It has slope equal to {k1 and intercept equal to ln α as shown in Figure 4B. Now that both rate constants k1 and k2 are known and α and β are also known, it is easy to obtain an equation for the biphasic process (eq 6). At = α exp({k1t) + β exp({k 2t) + A∞ (6) Using the graphing calculator, the plot of the original data for At versus time of and the plot of the modeling equation can be viewed together on the screen to visually determine how good the fit of the model is. The plot constructed with the original data is shown in Figure 5A and the plot from the modeling eq
A
B
A
B
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In the Laboratory
6 is given in Figure 5B. The experimental values of k1, k2, α , β, and R 2 are given in Table 2. Conclusion CBL technology and the graphing calculator offer the opportunity to introduce live examples in the classroom. The statistical capability of the graphing calculator permits analysis of complex kinetics behavior in classroom and prelaboratory sessions. The equipment is inexpensive and user friendly. Acknowledgments This work was financially supported by NASA-MASTAP through the grant Using CBL-Technology to Improve Science Education in Puerto Rico (NAGW - 4614). Valuable suggestions and comments from Betty Ramírez, Keith Wayland, and Nairmen Mina are gratefully acknowledged. Notes 1. This experiment was suggested by Josefina Arce from the University of Puerto Rico, Río Piedras. 2. All figures in this paper come directly from the screen of a TI-83. One of the benefits of using the TI-83, or many of the other graphing
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Table 2. Experimental Values of Parameters α β k2/s {1 R2
k1/s {1
0.00627
0.000295
0.3418
{ 0.4095
.999
calculators, is that it is possible to transfer tabular information rapidly using a cable that connects two calculators.
Literature Cited 1. Laboy, O.; Parés, E. I.; Cortés-Figueroa, J. E. J. Coord. Chem. 1995, 36, 273. 2. A good discussion of biphasic behavior is given in: Espenson, J. H. Chemical Kinetics and Reaction Mechanisms, 2nd ed.; McGrawHill, New York, 1995, pp 71–76. 3. A good description of chemistry experiments using Vernier probes with CBL technology and the graphing calculator is found in: Holoquist, D. D.; Randall, J.; Voltz, D. L. Chemistry with CBL™; Vernier Software: Portland, OR, 1995. 4. Thomasson, K.; Lofthus-Merchman, S.; Humbert, M.; Kulesvsky, N. J. Chem. Educ. 1998, 75, 231 5. Espenson, J. H. Op. cit., pp 22–28. 6. An article about software for modeling kinetic phenomena reviews the capabilities of several packages: Nash, J. C.; Quon, T. K. Am Stat. 1996, 50, 368.
Journal of Chemical Education • Vol. 76 No. 5 May 1999 • JChemEd.chem.wisc.edu
