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Old Rule of Thumb and the Arrhenius Equation Ilya A. Leenson Department of Chemistry, Moscow State University, Moscow 119899, Russia;
[email protected] In many textbooks one can find an empirical rule for how to evaluate quickly an increase in the rate of a chemical reaction when temperature changes. For example: “According to an old rule of thumb, the rate of a reaction approximately doubles with each 10 °C temperature rise” (1). Different authors sometimes add extra information to this rule. Thus, J. B. Russel (1), among others, pointed out that “unfortunately, the rule is so approximate as to be of limited value”. It has been suggested that a reaction must proceed in a homogeneous system (2, 3) or in solution (4, 5), or “at ordinary temperature” (5), “at room temperature” (6 ). Even mention of a reaction rate could be found: “This rule… applies only to reactions that last longer than a second or two” (2). Other authors specify the temperature coefficient itself. “The specific rate is usually increased by a factor of about two or three for every 10 °C rise of temperature” (3). “In reality, the factor is usually in the range from 1.5 to 4” (7 ). “As a rough working rule a 5 per cent to 10 per cent increase in rate per degree centigrade rise in temperature may be assumed” (6 ). This gives a factor from 1.0510 = 1.63 to 1.1010 = 2.6 for a 10° increase in temperature. On the other hand, it is now widely accepted that over a limited range of temperature the majority of the reactions studied can be adequately described with the Arrhenius equation rate = A exp({E /RT ) where the pre-exponential factor A and activation energy E are assumed to be constant. (It should be noted that in special circumstances, experimental data, especially those obtained with improved precision and over a wide range of temperature, may not fit into the Arrhenius equation [8, 9]; moreover, there are many reactions with negative temperature coefficients [10]). Assuming the validity of this equation we can see that the increase in a rate depends on both activation energy and absolute temperature. Thus, for the gas-phase reaction of hydrogen atoms with ethane H + C2H6 → H2 + C2H5 E = 40.6 kJ/mol in the temperature range 300–1100 K (11). Consequently, a “ten-degree” increase in a rate would be 1.69 in the range 300–310 K and only 1.04 in the range 1090– 1100 K, so at high temperatures the rate of this reaction is practically independent of temperature. For the reaction
A question arises about the origins of this rule and the meaning of all above-cited assumptions that concern conditions of its application. The answer is simple. The rule was put forward more than a century ago, when experimental techniques allowed study only of those reactions that proceed with rates convenient for measurement—that is, neither too fast nor too slow. In other words, a reaction should last rather longer than an hour or two (but not “longer than a second or two” as some authors [2] proposed). For example, a first-order reaction that proceeds almost to completion (say more than 99%) in two hours has a half-life of about 18 min. It is very convenient for a kinetic study with the conventional techniques. Only with these reactions is the empirical rule adequate. Many highly sophisticated modern experimental methods make it possible to measure rates of both extremely fast and very slow reactions, and the above-cited empirical rule is here of no importance. It would be quite valuable pedagogically to outline on a graph the area (i.e., the specific relationship between Arrhenius’ activation energy E and an empirical temperature coefficient f ), where the empirical rule is applicable. Let f be 2 to 3 (the most frequent cited values) for a 10° temperature increase. Thus, according to Arrhenius’ law we have f = rate(T + 10)/rate(T ) = exp[10E/RT (T + 10)] = 2 or 3 (for the same reaction pre-exponential values A should be identical for a narrow temperature interval and cancel)1. The latter equation could be rewritten as E = R ln f. T(T + 10)/10 = 0.83 ln f.T (T + 10) (Activation energy E is not any function of temperature; this dependence serves exclusively for the convenience of plotting the graph.) The last equation is the equation of a parabola with only positive values having physical meaning. The area on the graph is restricted with two branches of the parabola: with f = 2 we have E = 0.58T (T + 10) and with f = 3 we obtain E = 0.91T (T + 10). Those values of E and T which ensure the correctness of the empirical rule are located only between these branches. Below the lower branch f < 2 and above the upper branch f > 3.
H + C2H4 → C2H5 E = 3.4 kJ/mol at 463 K (12), and the increase in rate would be only 1.18 in the range from 453 to 463 K. Alternatively, for the reaction C2H6 + C2H4 → 2C2H5 E = 251 kJ/mol near 450 K (11). Here, when temperature rises from 440 to 450 K the rate increase would be 4.6. These simple calculations appear to be in contradiction with the empirical rule of the thumb.
Note 1. It was kindly pointed out by one of the reviewers that the same equation and reasoning can be found in Levine, I. N. Physical Chemistry, 3rd ed.; McGraw-Hill: New York, 1988; pp 539–540.
JChemEd.chem.wisc.edu • Vol. 76 No. 10 October 1999 • Journal of Chemical Education
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Literature Cited 1. Russel, J. B. General Chemistry, 2nd ed.; McGraw-Hill: New York, 1992; p 499. 2. Holtzclaw, H. F.; Robinson, W. R.; Odom, J. P. General Chemistry, 9th ed.; Heath: Lexington, MA, 1991; p 419. 3. Glasstone, S. The Elements of Physical Chemistry; Van Nostrand: New York, 1946; p 606. 4. Laidler, K. J. Chemical Kinetics; McGraw-Hill: New York, 1950; p 57. 5. The Encyclopedia of Chemistry, 3rd ed.; Hampel, C. A.; Hawley, G. G., Eds.; Van Nostrand: New York, 1973; p 952.
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6. Latham, J. L. Elementary Reaction Kinetics; Butterworths: London, 1962; pp 22, 31. 7. Atkins, P. W.; Beran, J. A. General Chemistry; 2nd ed.; Scientific American Books: New York, 1990; p 463. 8. Logan, S. R. J. Chem. Educ. 1982, 59, 279–281. 9. Laidler, K. J. J. Chem. Educ. 1984, 61, 494–498. 10. Leenson, I. A.; Sergeev, G. B. Uspekhi Khimii 1984, 53, 721–752; Russ. Chem. Rev. (Engl. Transl.) 1984, 53, 417. 11. Benson, S.W. Thermochemical Kinetics; Wiley: New York, 1968; Chapter III. 12. Kerr, J. A.; Parsonage, M. J. Evaluated Kinetic Data on Gas Phase Addition Reactions; Butterworths: Birmingham, 1972; p 19.
Journal of Chemical Education • Vol. 76 No. 10 October 1999 • JChemEd.chem.wisc.edu
