In the Laboratory
A Student Experiment in Non-Isothermal Chemical Kinetics
W
Steven C. Hodgson, Lawrence N. Ngeh, John D. Orbell, and Stephen W. Bigger* School of Life Sciences and Technology, Victoria University of Technology, St. Albans Campus, P.O. Box 14428, MCMC, Melbourne, 8001, Australia
The topic of chemical kinetics has been explored extensively in the educational literature and techniques such as electrolysis (1), gas chromatography (2), high-performance liquid chromatography (2), nuclear magnetic resonance spectroscopy (2, 3), infrared spectroscopy (4), and UV–visible spectroscopy (5) have been used to monitor reaction rates under isothermal conditions. However, on considering the wide range of literature available in the area, it is clear that the traditional approach to teaching chemical kinetics to undergraduate students rarely involves the discussion of non-isothermal systems. This is despite the fact, for example, that many industrially important chemical processes are not performed isothermally (6 ) and that common thermoanalytical techniques such as thermogravimetry often utilize non-isothermal conditions (7). The case of non-isothermal kinetics was originally treated by van’t Hoff (8), and various developments and applications have since been reported in the scientific literature (7, 9–11). A detailed theoretical treatment of non-isothermal kinetics has also appeared in the educational literature (12, 13) along with a small number of non-isothermal kinetic experiments (14–19). These comprise student experiments in which (i) the activation energies for the decomposition of various cyclic alkenes are determined using a temperature-programmed flow reactor (14), (ii) the activation energy for the luminescence reaction occurring in a “lightstick” is determined from light intensity measurements made under a temperature gradient (15), and (iii) the activation energies for the decomposition of CaC2O4 and NaHCO3 are determined using differential thermal analysis (16 ). The application of non-isothermal kinetics for predicting the shelf-life of foods, drugs, and simple chemical solutions has also been discussed (17 ). Furthermore, Birk (18) and Cassadonte (19) have considered the cooling characteristics of water as an example of physical processes that can be analyzed kinetically. In this experiment, designed for the advanced physical chemistry laboratory, the student learns how to obtain from a single, non-isothermal kinetic run the activation energy of the reaction between bromine and formic acid. The experimental conditions are chosen so that the reaction follows pseudo-first-order kinetics with respect to bromine. A method for processing the non-isothermal absorbance–time data obtained in the laboratory is described. The activation energy obtained from the non-isothermal method is compared with that obtained from isothermal experiments and the timesaving advantage of the non-isothermal method over the isothermal approach is identified. *Corresponding author. W Supplementary materials for this article are available in JCE Online at http://jchemed.chem.wisc.edu/Journal/issues/1998/ Sep/abs1150.html.
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Theory
The Reaction The oxidation of formic acid by bromine is represented by eq 1. HCOOH(aq) + Br2(aq) → CO2(g) + 2H+(aq) + 2Br᎑(aq) (1) The reaction is believed to proceed by the following mechanism (20, 21): H+(aq) + HCOO᎑(aq)
HCOOH(aq)
(2)
k0
HCOO᎑(aq) + Br2(aq) → CO2(g) + H+(aq) + 2Br᎑(aq) (3) where k0 is the second-order rate constant for reaction 3. The following rate law can be readily derived from the proposed mechanism: ᎑d[Br2]/dt = k0Ka[γ /(γ +γ ᎑)][HCOOH][Br2]/[H+]
(4)
where Ka is the acidity constant for the dissociation of formic acid (reaction 2) and γ , γ+, and γ᎑ are the activity coefficients of HCOOH, H+, and HCOO᎑, respectively. The kinetics of the reaction can be followed spectrophotometrically at 400 nm by monitoring the decrease in the concentration of Br2 as a function of time (20). Since the rate of the reaction is dependent on the hydrogen ion activity (21) as indicated in eq 4, it can be controlled by the addition of HClO4 to the reaction mixture. In highly acidic media, the reaction is considered to be an overall second-order process whose temperature-dependent rate law can be written simply as follows: ᎑dc/dt = k(T )bc
(5)
where for convenience the concentrations of HCOOH and Br2 at time t are henceforth represented by the symbols b and c, respectively, and k(T ) is a second-order rate constant given by eq 6: k(T ) = k 0 Ka[ γ /( γ + γ ᎑)]/[H +]
(6)
If b0 and c0 are the initial concentrations of b and c, respectively, and experimental conditions are chosen such that b0 >> c0, then the concentration of HCOOH remains effectively constant and is approximately equal to b0 throughout the reaction. Under these conditions the rate law is further simplified to ᎑dc/dt = k1c (7) where k1 is a pseudo-first-order rate constant and k1 = b0k(T ).
Newton’s Law of Cooling The rate at which a body cools is proportional to the difference between its temperature and the temperature of the surroundings (19): ᎑dT/dt = α (T – T1)
Journal of Chemical Education • Vol. 75 No. 9 September 1998 • JChemEd.chem.wisc.edu
(8)
In the Laboratory
where T is the temperature of the body at time t, T1 is the temperature of its surroundings, and α is a constant. The integration of eq 8 between the limits of T0 and T, which correspond to the temperature of the body at times t = 0 and t = t, respectively, gives T (t) = T1 + (T0 – T1)exp(᎑ αt )
(9)
It is found that eq 9 adequately describes the cooling characteristics of a cuvette containing a mixture of HCOOH and Br2 where the reagents were heated before being mixed and placed in a spectrophotometer.
Determining the Activation Energy The variation of the rate constant with temperature is given by the Arrhenius equation: k(T ) = Aexp(᎑Ea/RT )
(10)
where A is a constant, E a is the activation energy and R is the ideal gas constant. For non-isothermal conditions eqs 7 and 10 can be combined and rearranged to give ln[᎑(dc/dt)/c] = ln(b0 A) – (E a/R) × [1/T (t)]
(11)
where T (t) is given by eq 9. Thus, a plot of ln[᎑(dc/dt)/c] versus 1/T (t) is a straight line with a slope of ᎑E a/R. The abscissa values for such a plot can be calculated from eq 9 if the cooling curve parameters T0, T1, and α have been determined previously. The ordinate values can be calculated by systematically selecting sets of three consecutive absorbance values for which each absorbance is separated in time by a constant sampling interval, δt (see Fig. 1). In this experiment, a sufficiently small value of δt is chosen in order to validate the approximation made in the calculation of ordinate values. A suitable value of δt is 6.00 s, a time interval that will also allow absorbance data to be recorded manually if an automated spectrophotometer is unavailable. The gradient (dA/dt)i of the non-isothermal absorbance–time curve at the point (ti , Ai) is approximately equal to the gradient of the straight line drawn between the points (ti-1, Ai-1) and (ti+1, Ai+1). This approximation becomes increasingly better in the limit as δt → 0. Thus (dA/dt)i ≈ (Ai+1 – Ai-1)/2δt
(12)
Since c ∝ A, then (dc/dt)i /ci = (dA/dt)i /A i ≈ (A i+1 – A i-1)/(2Ai δ t) (13) The calculation is repeated in an iterative process in which the index i is varied in the range 2 ≤ i ≤ n – 1 where, n is the number of absorbance values recorded. This process will enable n – 2 values of the ordinate ln[(dc/dt)/c] to be calculated.1 The Experiment CAUTION: The reagents used for this experiment should be handled with care.
Materials and Reaction Mixture Aqueous solutions of Br2 (2.00 × 10᎑2 M), HCOOH (2.00 × 10᎑1 M), and HClO4 (4.00 × 10 ᎑1 M) are made up using analytical grade reagents.2 Into a 50-mL volumetric flask is placed 10.0 mL each of the HCOOH and HClO4 solutions. A 10.0-mL aliquot of the Br2 solution is placed in a 10-mL volumetric flask and both flasks, together with a 1.00-cm-path-length cuvette, are
Figure 1. Schematic diagram showing how consecutive nonisothermal absorbance–time data are mathematically manipulated to produce the ordinate values in eq 11.
equilibrated in a water bath set to 55.0 °C. To commence a run, the Br2 solution is quickly added to the contents of the 50mL volumetric flask with thorough mixing. A timer is activated at the start of the mixing process. A sample of the reaction mixture is quickly transferred to the preheated cuvette, which is then placed in the spectrophotometer, the elapsed time since mixing is noted, and absorbance–time readings are begun immediately and taken at regular intervals (say, 6.00 s). The entire mixing procedure must be performed in the minimum amount of time to enable sufficient absorbance data to be collected early in the reaction. Many modern spectrophotometers such as the Varian Cary-1 instrument used in our laboratory enable absorbance readings to be recorded at regular intervals and stored directly on computer disk. An automated system such as this is ideal for performing the experiment because it renders negligible the error associated with the time measurement. Values of the time must, of course, be corrected for the small amount of time required for the mixing procedure (typically, 25.0 ± 3.0 s).
Cooling Curve Characteristics The temperature of a cuvette containing a freshly prepared reaction mixture is recorded as a function of time after the cuvette is transferred to the spectrophotometer.3 A thermocouple, connected to a chart recorder via a reference junction, may be used to record the cooling curve. The values of T0, T1, and α (see eq 9) are determined from the analysis of a series of replicate cooling curves obtained in the laboratory session. Average values of these parameters are calculated and are assumed to describe the cooling characteristics of the reaction mixture whose absorbance–time profile is recorded during a kinetic run.4 An accurate characterization of the cooling curve is essential in order to obtain a reliable value of the activation energy from this experiment. A typical fit of experimental data to Newton’s law of cooling is shown in Figure 2. This plot shows that Newtonian behavior persists for as long as ca. 500 s, which is much greater than the time required to collect sufficient data during a kinetic run (typically 300 s). From the analyses of 10 replicate experiments average values of the following parameters were obtained: T0 = 52.9 ± 3.6 °C, T1 = 19.0 ± 0.1 °C, and α = (᎑1.08 ± 0.06) × 10᎑2 s ᎑1. The fact that the upper limit of the range of T0 is greater than the set temperature of the water bath reflects the magnitude of the experimental error associated with the 10 replicate determinations.
JChemEd.chem.wisc.edu • Vol. 75 No. 9 September 1998 • Journal of Chemical Education
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In the Laboratory
Non-Isothermal Kinetics Figure 3 shows plots of the absorbance (measured at 400 nm) versus time for the reaction between Br2 and HCOOH where curve 䊊 a is the theoretically calculated isothermal profile for 20 °C and curve 䊊 b is the experimentally obtained non-isothermal profile. Such a plot can be used to demonstrate to students the striking differences between the isothermal and non-isothermal decay curves. Figure 4 is an analysis of the non-isothermal profile shown as curve 䊊 b in Figure 3 and is typical of the result that is obtained using the method. From the analyses of 4 replicate experiments, an average value of the activation energy was calculated to be 58 ± 7 kJ mol᎑1, which compares favorably with the literature value of 60 ± 8 kJ mol᎑1(22). The error in the non-isothermal result mostly reflects the sensitivity of the method to the cooling curve parameter T0 and is most likely a manifestation of the assumption that the cooling rate during each separate run is defined by the average cooling curve parameters.
Isothermal Kinetics A series of isothermal kinetic experiments was conducted in our laboratory on the Br2/HCOOH system and the observed pseudo-first-order rate constants in the temperature range 25– 40 °C are given in Table 1. The activation energy obtained from an Arrhenius plot of these data (see Fig. 5) is 69 ± 7 kJ mol᎑1, which is consistent with both the non-isothermal and literature (22) values given above.
The time-saving advantage of the non-isothermal technique in experimentally obtaining an activation energy can be demonstrated to students by means of a simple instructive exercise. First, students are asked to consider a set of rate constants obtained in a series of separate isothermal experiments, such as those given in Table 1. They are then asked to derive an expression for ttot, the total experimental run time required to obtain all these kinetic data assuming that each isothermal rate constant was obtained by monitoring a single experiment to the stage where the Br2 concentration decreased to r % of the original concentration. The value of ttot is calculated from eq 14:
t tot = ᎑ln r /100
n
Σ i =1
1/k i
(14)
where ki is the ith pseudo-first-order rate constant in the set of n rate constants. Choosing r = 2% and using the data given in Table 1, a value of ttot = 5.8 × 103 s is obtained. This can be compared with a run time of ca. 300 s required for a typical non-isothermal experiment, having first established the appropri-
Figure 2. Typical fit of cooling curve data to Newton’s law.
Figure 4. Non-isothermal kinetic analysis of the absorbance–time data shown in curve (b) of Fig. 3.
Figure 3. Absorbance versus time plot for Br2/HCOOH system under (a) isothermal and (b) non-isothermal conditions.
Figure 5. Arrhenius plot of isothermal data given in Table 1 for the Br2/HCOOH system.
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Journal of Chemical Education • Vol. 75 No. 9 September 1998 • JChemEd.chem.wisc.edu
Research: Science & Education
ate cooling curve characteristics, which takes approximately 500 s. Conclusions This experiment demonstrates to students the principles and advantages of the technique of non-isothermal kinetic analysis. The experiment can also be used to (i) exemplify a situation in which a given quantity is measured using more than one experimental technique and (ii) emphasize that the confirmation of a result obtained using different experimental methods strengthens one’s confidence in the result. Acknowledgment SWB and JDO are grateful to the ARC Small Grants Scheme for providing funding to investigate the kinetics of non-isothermal systems. Notes 1. A listing of a simple BASIC computer program that calculates from the experimental data the abscissa and ordinate values required for this plot is available in JCE Online at http://jchemed.chem.wisc.edu/ Journal/issues/1998/Sep/abs1150.html. The program reads an input file of absorbance values recorded at equal time intervals during a non-isothermal kinetic experiment. Values of the cooling curve parameters are set in the program along with the time interval between absorbance readings and the time delay associated with mixing and transferring the reaction mixture to the spectrophotometer. The program writes a series of records to an output file; each record comprises the calculated abscissa and ordinate values delimited by a “tab” character. The output file can be read into an auxiliary graphics-display utility such as Excel or CricketGraph, to obtain a plot of the processed data. 2. As the volume of reaction mixture required for spectrophotometric analysis is small, it is conceivable that this experiment may be reproduced successfully on the microscale. The experiment can be successfully performed by making up stock solutions of 0.233 M HCOOH and 0.467 M HClO4. These solutions can be conveniently prepared from 90% (w/w) HCOOH and 69% (w/w) HClO4 , which are available commercially.
3. In all experiments, the temperature of the “sample block” in the spectrophotometer (Varian, Cary-1) was maintained at 19.0 °C. 4. Absorbance readings are not taken during characterization of the cooling curve, although a more sophisticated experimental setup could be designed to enable the simultaneous recording of absorbance and temperature data.
Literature Cited 1. Osborne, P. M.; Frere, B. Educ. Chem. 1986, 23, 12. 2. Peterson, T. H.; Bryan, J. H.; Keevil, T. A. J. Chem. Educ. 1993, 70, A97. 3. Potts, R. A.; Schaller, R. A. J. Chem. Educ. 1993, 70, 421. 4. Dawber, J. G.; Crane, M. M. J. Chem. Educ. 1967, 44, 151. 5. Hurst, M. O.; Hill, J. W. J. Chem. Educ. 1993, 70, 429. 6. Koch, E. Non-Isothermal Reaction Analysis; Academic: New York, 1977. 7. Flynn, J. H. In Aspects of Degradation and Stabilisation of Polymers; Jellinek, H. H. G., Ed.; Elsevier: New York, 1978; Chapter 12, pp 573–603. 8. van’t Hoff, J. Studien zur chemischen Dynamik (from Vant’ Hoff, J. Etudes de Dynamique Chimique, 1984); revised by Cohen, E., with a Foreword by J. van’t Hoff; W. Engelmann: Leipzig, 1896; F. Muller: Amsterdam, 1896. 9. Flynn, J. H. In Thermal Analysis; Schwenker, R. K.; Garn, P. D., Eds.; Academic: New York, 1969; pp 1111–1123. 10. Ozawa, T. In Thermal Analyses—Comparative Studies on Materials; Kambe, H.; Gom, P. D., Eds.; Wiley: New York, 1974; pp 155–167. 11. Dollimore, D. Anal. Chem. 1996, 68, 65; 1994, 66, 19; 1992, 62, 45; 1990, 64, 147. 12. Brown, M. E.; Phillpotts, C. A. R. J. Chem. Educ. 1978, 55, 556. 13. Salvador, F.; Gonzalez, J. L.; Tel, L. M. J. Chem. Educ. 1984, 61, 921. 14. Chan, K. C.; Tee, R. S. J. Chem. Educ. 1984, 61, 547. 15. Bindel, T. H. J. Chem. Educ. 1996, 73, 356. 16. Wendlandt, W. W. J. Chem. Educ. 1961, 38, 571. 17. Labuza, T. P. J. Chem. Educ. 1984, 61, 348. 18. Birk, J. P. J. Chem. Educ. 1976, 53, 195. 19. Cassadonte, D. J., Jr. J. Chem. Educ. 1995, 72, 346. 20. Brusa, M. A.; Colussi, A. J. Int. J. Chem. Kinet. 1980, 12, 1013. 21. Cox, B. G.; McTigue, P. T. J. Chem. Soc. 1964, 3893. 22. Smith, R. H. Aust. J. Chem. 1972, 25, 2503.
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