In the Laboratory
New Highlights on Analyzing First-Order Kinetic Data of the Peroxodisulfate–Iodide System at Different Temperatures
W
J. Yperman* Laboratory of Applied Chemistry, CMK, Hasselt University, Agoralaan – Building D, B-3590 Diepenbeek, Belgium; *
[email protected] W. J. Guedens Department SBG, Hasselt University, Agoralaan – Building D, B-3590 Diepenbeek, Belgium
Recently three articles (1–3) have been published examining the analysis problems of pseudo first-order kinetics and its use for undergraduates. The work by Hemalatha and NoorBatcha (1) compares three methods at one temperature using the peroxodisulfate–iodide system to determine the rate constant, k, and the absorption value at infinite time, A∞. The so-called “conventional method” (method I) requires measurements at time of completion if linear regression is used (for nonlinear regression this is not needed) and of course measurements at various time intervals. The two other methods are the so-called “time-lag methods” of Guggenheim (4) (method II) and a similar method proposed by a number of authors (5–7) (method III). The latter methods used grouped kinetic data, separated by a fixed time interval and no measurement at time of completion is required. This last requirement enables the lab to be completed within the lab period. In the present experiment we expand on the work by Hemalatha and NoorBatcha to include different temperatures. Students combine their results to determine the activation energy of the reaction using the Arrhenius equation. While conducting the experiment, the theoretical concepts of the different methods can be explained. Simultaneously, students can follow the evolution of the absorbance value as a function of time on their PC-screen (8). This experiment fits within the first-year general chemistry curriculum when the principles of chemical kinetics (rates and mechanisms of chemical reactions) and an introduction to spectrophotometrical methods (Beer’s law) are discussed. Experimental Procedure In the present study, a thermostat and a single-beam UV– vis spectrophotometer (Ultraspec 2000) are used. The closed plastic cuvette, with beam path of 1 cm, is also thermostated. The experiments are performed at six different temperatures: 20.00, 21.50, 23.00, 24.50, 26.00, and 27.50 ⬚C. Each experiment consists of at least two independent measurements so the students can asses the error propagation (9–11). Absorption values at infinite time (A∞) can be calculated for methods II (4) and III (5–7). The mean of these absorption values is used in method I (conventional method) to calculate the rate constant (required in the case of linear regression).
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Hazards K2S2O8(s) is harmful by inhalation and an eye irritant. Inhalation of KI dust may irritate respiratory tract. KI can act as a skin or an eye irritant and may cause sensitization or allergic reaction in people susceptible to KI. Solutions for the students are prepared by the technical staff starting from K2S2O8(s), KI(s), and KCl(s). In this way the solutions of K2S2O8, KI, and KCl and the formed quantities of I2 and K2SO4 are so diluted that they pose no immediate hazard. Theoretical Background and Data Analysis For method I, ln(A∞ − At ) versus t reveals a straight line. The value of A∞ used is the mean of the A∞ values obtained from methods II and III. The rate constant can be calculated from the slope and a new value of A∞ can be recalculated from the intercept. In method II, ln(At+τ − At ) versus t results in a straight line and the rate constant k is extracted from the slope. At time t = 0, and because of known t and k value, A∞ can be calculated by taking the intercept. In method III for At+τ versus At, a straight line is obtained. From the slope, the rate constant k can be calculated because t is known. From the intercept A∞ can be calculated, since t and k are known. A detailed workup of the kinetic equations is given in the Supplemental Material.W Results At a rather high temperature, that is, 30 ⬚C, the experimental results were subject to uncontrollable fluctuations and differences in rate constant were found if all data points were not used. This means that the result depends upon the number of data points used, which is unacceptable. In addition, it was observed that the 30 ⬚C data at initial time period were inaccurate and gave different results if they were used, even by using the time-lag methods II and III. This can also be seen with the data of Hemalatha and NoorBatcha (1). Another possible error can arise from the relative high partial pressure of the I2 formed. Thus experiments were performed at temperatures below 30 ⬚C.
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Journal of Chemical Education
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In the Laboratory
1.2 1.0
At
0.8 0.6 0.4 0.2 0.0 0
500
1000
1500
2000
2500
3000
3500
4000
Time / s
Figure 1. Experimental data of the absorbance (At ) versus time (t ) at 20 ⬚C.
The absorbance as a function of time is given for the 20 ⬚C experiment in Figure 1. The results for method II and method III are summarized in Table I and the 20 ⬚C data are plotted in Figures 2 and 3. Both methods give good linear fits and similar k values (error in the last digit is given between parentheses). Also, the corresponding A∞ values are calculated. One should expect, based on mathematical concepts, that both methods II and III should give exactly the same results for k and for A∞ for the same data points. However, this is sometimes not the case when comparing results obtained by a number of students during the past few years. At the moment, there is no valid explanation. For this reason we believe that the calculated errors are too optimistic.
Furthermore, the program (8) is constructed in such a way that the students are urged to interpret how k and A∞ can be calculated. Our experience with students tells us that this is not straightforward. Students have problems interpreting slope and intercept in a correct way and thus relating the unknowns in the equations with these values. Using the results for methods II and III (Table 1), for each temperature the mean A∞ can be calculated (its value is nevertheless equipment dependent) and used to make a linear fit according to method I. This is the only way to use method I, otherwise the completion of the reaction, which takes several hours, is needed. This is not ideal because the lab experiment is limited in time and didactically irrelevant if it takes too long. Heating up the reaction mixture for a further 20 min at 60 ⬚C, as suggested by Hemalatha and NoorBatcha (1) is not practical, results in extra errors, and the A∞ obtained differs significantly from the calculated value. Furthermore, it was found that the rate constant is sensitive towards the value of A∞. Additionally, the uncertainty of the measured value is high because (i) of a larger consumption of iodide and thus no longer constant [I−] concentration, (ii) possible side or subsequent reactions, (iii) the measured signal of the (didactic) equipment is not stable anymore (3), so another strategy was searched. The results for method I are given in Table 2 and the results for the 20 ⬚C data are plotted in Figure 4. A correlation coefficient for each temperature of almost 1 is found. The calculated k values and new calculated A∞ values correspond well with the value of method II and III. If students do not perform the experiment in a correct way, differences
-0.3 1.2 -0.4
y = 0.7191 + 0.6219x R = 1.00
At +τ
ln(At +τ − At )
1.1 -0.5
-0.6
1.0
0.9 -0.7
y = −0.3343 − 2.637 × 10ⴚ4 x R = −1.00
0.8
-0.8 0.7 0
200
400
600
800
1000 1200 1400 1600 1800 2000
0.0
0.1
Time / s
0.2
0.3
0.4
0.5
0.6
0.7
Figure 2. Linear fit of experimental data at 20 ⬚C according to method II.
Figure 3. Linear fit of experimental data at 20 ⬚C according to method III.
Table 1. Calculated Values at Different Temperatures for the Rate Constant and Absorbance Method II
T/°C
a
642
Method III
Mean
k/(10᎑4 s᎑1)
A∞
k/(10᎑4 s᎑1)
A∞
k/(10᎑4 s᎑1)
A∞
a
2.637(1)
1.8942(6)
2.6384(9)
1.9021(5)
2.6377(7)
1.8982(9)
21.50
2.936(3)
1.887(2)
2.937(3)
1.895(1)
2.937(2)
1.891(2)
23.00
3.244(5)
1.896(2)
3.245(5)
1.901(2)
3.245(4)
1.899(3)
24.50
3.602(4)
1.884(2)
3.599(3)
1.888(1)
3.601(3)
1.886(2)
26.00
4.033(9)
1.871(3)
4.035(9)
1.870(3)
4.034(6)
1.871(4)
27.50
4.405(7)
1.887(2)
4.400(6)
1.888(6)
4.403(5)
1.888(6)
20.00
0.8
At
Temperatures values are accurate to ± 0.01 °C.
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Vol. 83 No. 4 April 2006
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In the Laboratory Table 2. Calculated Values for the Rate Constant and Absorbance Method I ᎑4
᎑1
k/(10 s )
A∞a
20.00b
2.6472(2)
1.8901(1)
21.50
2.9461(4)
1.8836(2)
23.00
3.2515(7)
1.8926(4)
24.50
3.6061(6)
1.8837(2)
26.00
4.029(1)
1.8703(6)
27.50
4.403(1)
1.8864(4)
a
Calculated using the mean of A∞ of methods II and III. values are accurate to ± 0.01 °C.
ln(1.898 − A t )
T/°C
0.6
0.4
0.2
0.0
-0.2
ⴚ4
y = −0.6366 − 2.6472 × 10 R = −1.00
x
-0.4
b
Temperatures
0
500
1000
1500
2000
2500
3000
3500
4000
Time / s
-7.7 -7.8 -7.9 -8.0 -8.1
y = 12.4 − 6055.6x R = −0.9997
-8.2 -8.3
Conclusion
3.32
It is demonstrated that kinetic experiments for general chemistry undergraduates can be performed at different temperatures. Even if temperature differences are small, reliable values for the rate constants can be determined. These constants result in a linear Arrhenius plot. Furthermore, it can be concluded that a one hour experiment (or even less e.g., 40 minutes) is long enough to produce these reliable rate constants independent of the proposed method. In addition, absorption values at completion time can be calculated using method II and III. It is also demonstrated that during this undergraduate chemistry lab not only the rate constant can be determined at different temperatures for a pseudo-firstorder kinetic experiment but also that the activation energy of this reaction can be determined. Finally, it is found that method I is sensitive to the value of absorption at completion time. Acknowledgments The authors wish to thank H. Breemans, K. Van Vinckenroye, and J. Kaelen for the technical assistance in the development of this lab experiment. W
Figure 4. Linear fit of experimental data at 20 ⬚C using the mean value of A∞ according to methods II and III.
ln(k)
can be found for k and A∞ for method I as compared to methods II and III. At the moment, this is also not clear to us. The calculation done for method I is not independent as it is for methods II and III. The Arrhenius plot, using the mean k values from methods II and III at different temperatures is shown in Figure 5. A good correlation was found. An activation energy, Ea, of 50.3 kJ兾mol was calculated for this pseudo-first order reaction. It is our experience that the students are successful in completing the experimental problems of this lab exercise: no practical problems and more correct results. They understand the theoretical background and do not lose themselves in complicated, hectic, and error-producing activities related to an alternative classical lab procedure with a lot of pipetting, diluting, and titrating.
Supplemental Material
Notes for the instructor, including the details of the kinetic expressions, are available in this issue of JCE Online. www.JCE.DivCHED.org
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3.34
3.36
1 T
3.38
ⴚ3
(10
3.40
3.42
ⴚ1
K
)
Figure 5. Arrhenius plot.
Literature Cited 1. Hemalatha, M. R. K.; NoorBatcha, I. J. Chem. Educ. 1997, 74, 972–974. 2. McNaught, I. J. J. Chem. Educ. 1999, 76, 1457. 3. Urbansky, E. J. J. Chem. Educ. 2001, 78, 921–922. 4. Guggenheim, E. A. Philos. Mag. 1926, 2, 538–543. 5. Kezdy, F. J.; Kaz, J.; Bruylants, A. Bull. Soc. Chim. Belges 1958, 67, 687–706. 6. Mangelsdorf, P. G. J. Appl. Phys. 1959, 30, 442–443. 7. Swinbourne, E. S. J. Chem. Soc. 1960, 2371–2372. 8. LabVIEW User Manual, January 1998 Edition, National Instruments Corporation: Austin, Texas, 1998. 9. Guedens, W. J.; Yperman, J.; Mullens, J.; Van Poucke L. C. and Pauwels E. J. J. Chem. Educ. 1993, 70, 776–778. 10. Guedens, W. J.; Yperman, J.; Mullens, J.; Van Poucke L. C. and Pauwels E. J. J. Chem. Educ. 1993, 70, 838–840. 11. Guedens, W. J. Ph.D. Thesis, Limburgs Universitair Centrum, Diepenbeek, Belgium, 1999.
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