In the Laboratory
An Undergraduate Physical Chemistry Experiment on the Analysis of First-Order Kinetic Data M. R. K. Hemalatha and I. NoorBatcha* Department of Chemistry, The American College, Madurai 625002, Tamilnadu, India
The analysis of first-order kinetic data to estimate the kinetic parameters is a common exercise in undergraduate physical chemistry laboratory (1). Several methods are available to estimate the rate constant from first-order kinetic data using linear regression techniques. Conventional methods (method I) require measurements at initial time (t = 0) and completion times (t = ∞) in addition to the measurements at various time intervals. However, these measurements at t = 0 and t = ∞ are either difficult to make or accompanied by large uncertainties. The Guggenheim method (2) (method II) or a similar method proposed by Kezdy et al. (3), Mangelsdorf (4), and Swinborne (5) (method III) does not require measurements at t = 0 and t = ∞. The latter methods are known as time-lag methods because they involve grouping kinetic data separated by a fixed time interval, τ. Even though the time-lag methods are well known in the literature (1–7), very little effort has been made to introduce first-order kinetic experiments utilizing these methods to the undergraduate students. As a result, students get the impression that initial and final measurements of a kinetic run are absolutely necessary and the approximations resulting from such measurements are unavoidable. Schwarz and Gelb (6, 7) have made use of a simulated data set to evaluate alternative approaches to the analysis of first-order kinetic data. In the present work we describe how time-lag methods can be easily implemented in an undergraduate physical chemistry laboratory by following the kinetics of a simple chemical reaction. By using this approach the students can be made to realize the extent of error caused by “infinite time” measurements and how this can be circumvented. We make use of the reaction
HITACHI 200-10 double beam UV-vis spectrophotometer is employed. The kinetics is monitored by following the variation of absorbance at 350 nm, which is the wavelength of maximum absorbance of I3{ ions. All reagent solutions are prepared in double distilled water. The following solutions are prepared using standard procedures: [K2 S2 O8 ] = 0.001 M; [KI] = 0.1 M; [KCl] = 0.2 M. The pseudo-first-order condition is maintained by having the concentration of KI at least 20 times greater than that of K 2S2 O8 . Ionic strength is maintained at 0.1 M using KCl solution. The absorbances are recorded every 60 s for 40 min. The infinite time reading is taken after warming the reaction mixture for a further 20 min at 60 °C and then cooling it to room temperature. In the present work, the kinetics is studied by following the concentration of the product (I 3{ ) of the reaction. However, the methods discussed here can also be applied to studies in which the pseudo-first-order kinetics is followed by measuring the concentration of the limiting reagent (S2O 82{ in the present case). Data Analysis In terms of the absorbance of I3{, the integrated first order rate expression is (A∞ – At ) = (A∞ – A0) e{kt
In method I, a linear plot of ln (A∞ – At) versus t is made and the pseudo-first-order rate constant is extracted from the slope. At time equal to t + τ, (A∞ – At + τ ) = (A∞ – A0) e{k(t + τ)
Experimental Procedure The kinetics of the persulphate–iodide reaction can be studied by measuring the absorption of I 3{ formed during the course of this reaction (9). In the present work, a *Corresponding author. Present address: Department of Chemistry, Universiti Malaya, 59100 Kuala Lumpur, Malaysia.
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(2)
Combining these two equations we get
S 2O 82{ + 2I{ → 2 SO42{ + I2 to compare different approaches to evaluating pseudo-firstorder rate constants. The kinetic behavior of this reaction is well known (8) and the reaction is typically carried out in undergraduate physical chemistry laboratory to investigate the effect of ionic strength. The kinetics of this reaction is studied by standard methods. The pseudo-first-order rate constant, k, is calculated using conventional and time-lag methods. It is shown that the three methods can give results of similar accuracy. However, time-lag methods are very useful when it is difficult or impossible to obtain initial or infinite time readings. Time-lag methods can also be used to predict the infinite time readings.
(1)
ln (At + τ – At ) = ln {(A∞ – A0) (1 – e{kτ)} – kt
(3)
This equation forms the basis of method II. A plot of ln (A t +τ – At ) vs. t should give a straight line with the slope equal to {k. A comparison of eqs 1 and 2 leads to At + τ = At e{kτ +A∞ (1 – e{kτ)
(4)
In method III, a plot of At + τ vs. At is made and the rate constant is extracted from the slope. It should be noted that in methods II and III, knowledge of initial and final time readings is not required; and in method III it is the timedependent property that is plotted in both axes, rather than time itself. Method III is also easier to use, as it requires fewer arithmetic operations. To implement methods II and III the absorbances recorded in a single kinetic run are divided into two groups: Ati , …, Atn , and Ati + τ , …, Atn +τ . The choice of τ is crucial. If τ = T1/2, where T1/2 = 1/2 T and T is the entire time period over which the kinetic data are recorded, then all the data are used only once. If τ < T1/2, then some data are used more than once. If τ > T1/2, then the analysis will not use all the
Journal of Chemical Education • Vol. 74 No. 8 August 1997
In the Laboratory data: data in the range of (T – τ) to τ will be ignored. Hence a choice of τ ≠ T1/2 results in unequal emphasis being placed on different data points.
Table 1. Typical Data for Persulphate–Iodide Reactiona Time (s)
At
At + τ
60
0.070
0.695
120
0.100
0.715
180
0.140
0.735
240
0.180
0.760
Results and Discussion
A typical data set for the persulphate–iodide reaction 300 0.220 0.775 is given in Table 1. Typical plots for all three methods 360 0.260 0.795 are shown in Figure 1. The k 420 0.295 0.815 values are calculated by the 480 0.330 0.830 linear least squares method 540 0.365 0.840 and the reported errors correspond to the 95% confidence 600 0.400 0.860 interval. The values of k cal660 0.430 0.875 culated using different τ val720 0.460 0.890 ues are given in Table 2. There are significant differ780 0.490 0.900 ences in the k values ob840 0.520 0.910 tained using the three meth900 0.550 0.925 ods. The k values obtained using method II decrease 960 0.575 0.935 slowly, whereas the k values 1020 0.605 0.945 calculated using method III 1080 0.625 0.955 increase slowly with increas1140 0.650 0.965 ing τ values. This trend is due to the different emphasis 1200 0.670 0.975 placed on the data points, as a [K 2S 2O8 ] = 7.67 × 10 {5 M; described above, when differ[KI] = 0.075 M; temperature = ent τ values are used. If τ = 30 °C; ionic strength = 0.1 M; τ = 1200 s. T1/2, then all data points are given equal emphasis and the results from methods II and III agree with each other. Hence in the absence of any other guidelines, the use of τ = T1/2 seems to be the better choice. The k values calculated using methods II and III are less than the k value obtained (k = [7.51 ± 0.05] × 10 {4 s{1) using method I. This can be attributed to the use of experimentally obtained A∞ in the conventional method. The A∞ value can be estimated using method II as A∞ = {exp (intercept) / (1 – exp ({kτ)} + A0
(5)
and using method III as A∞ = intercept / (1 – slope)
(6)
The A∞ values thus calculated are given in Table 2. They are different from the experimentally obtained value of 1.17. If the A∞ value estimated as above is used in method I instead of the experimental value, then the k values calculated from all three methods become equal. The reported Table 2. k and A ∞ Calculated Using Methods II and IIIa, b τ (s)
Method II
k
(10{4 s{1)
Method III
A∞
k
(10{4 s{1)
A∞
300
6.76 ± 0.22
1.21 ± 0.04
6.36 ± 0.16
1.25 ± 0.04
600
6.66 ± 0.16
1.24 ± 0.04
6.40 ± 0.14
1.25 ± 0.04
1200
6.52 ± 0.12
1.24 ± 0.08
6.52 ± 0.14
1.24 ± 0.04
1600
6.45 ± 0.14
1.24 ± 0.07
6.83 ± 0.15
1.12 ± 0.06
method I, k = (7.51 ± 0.05) × 10 {4s{1, and the experimentally obtained A ∞ = 1.17. The time period T = 2400 s. b[K S O ] = 7.67 × 10{5 M; [KI] = 0.075 M; temperature = 30 °C; 2 2 8 ionic strength = 0.1 M. aUsing
Figure 1. Typical plots of the data for the persulphate–iodide reaction, using the three methods. [K 2S2 O8] = 7.67 × 10{5 M; [KI] = 0.075 M; temperature = 30 °C; ionic strength = 0.1 M.
Vol. 74 No. 8 August 1997 • Journal of Chemical Education
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In the Laboratory errors in the calculated values of k are smaller for method I than for the other two methods. However, in the calculation of k using method I the measured A∞ values are assumed to be known accurately. The experimental determination of A∞ is often unreliable for some reactions owing to several factors related to the nature of the reactions and the instrumental procedures involved. The measurement of impossible or inconvenient A∞ can be completely avoided using time lag methods, and these methods are also useful for the predicting infinite time readings. In summary, all three methods can provide results of similar accuracy, if accurate initial and infinite time readings are available. The present work clearly brings out the approximation involved in assuming a measurement made at finite time as an “infinite time reading”. Time lag meth-
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ods are preferable because they do not require initial and infinite time readings. Between the two time lag methods investigated here, method III is easier to implement. Literature Cited 1. Schoemaker, D. P.; Garland, C. W.; Steinfeld, J. I. Experiments in Physical Chemistry, 3rd ed.; McGraw Hill: New York, 1974. 2. Guggenheim, E. A. Philos. Mag. 1926, 2, 538. 3. Kezdy, F. J.; Kaz, J; Bruylants, A. Bull. Soc. Chim. Belges 1958, 67, 687. 4. Mangelsdorf, P. G. J. Appl. Phys. 1959, 30, 442. 5. Swinborne, E. S. J. Chem. Soc. 1960, 2331. 6. Schwartz, M. L.; Gelb, R. I. Anal. Chem. 1978, 50, 1592. 7. Schwartz, M. L. Anal. Chem. 1981, 53, 206. 8. King. C. V.; Jacobs, M. B. J. Am. Chem. Soc. 1931, 51, 1704. 9. Moya, M. L.; Izquierdo, C.; Casodo, J.; J. Phys. Chem. 1991, 95, 6001.
Journal of Chemical Education • Vol. 74 No. 8 August 1997
