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Why the Arrhenius Equation Is Always in the “Exponentially Increasing” Region in Chemical Kinetic Studies Harvey F. Carroll Department of Physical Sciences, Kingsborough Community College, The City University of New York, 2001 Oriental Boulevard, Brooklyn, NY 11235
In most general chemistry texts, the rate constant is said or implied to be an exponentially increasing function of temperature (1–3). The Arrhenius equation (4, 5) k = Ae᎑Ea/RT where k is the rate constant, A is the pre-exponential or frequency factor, Ea is the activation energy, R = 8.315 J mol᎑1 K᎑1 is the gas constant, and T is the thermodynamic temperature, is then stated as mathematically showing this effect. If you’ve ever tried to sketch the properties of the Arrhenius equation for your students, you probably used the simple exponential equation that is the model for it: y = e᎑1/x
ing function. But, as shown in Figure 1, this function asymptotically approaches one as x approaches infinity and does not look at all like an exponentially increasing function for the numerical values illustrated. However, when the equation is plotted with x going from 0 to 1, the curve does seem to have an “exponentially increasing”1 region and an inflection point, as is shown in Figure 2. Looking at smaller values of x, the curve clearly seems to be “exponential”, as shown in Figure 3. The first derivative, which has a maximum at the inflection point, is y′ = x᎑2e᎑1/x
You may have assumed that this is an exponentially increas-
Plotting y′ vs x (Fig. 4) shows that the derivative clearly has a maximum around 0.5, indicating an inflection point. The
Figure 1. A plot of y = e ᎑1/x for large values of x . The curve is asymptotic to y = 1.
Figure 2. A plot of y = e ᎑1/x for values of x between 0 and 1. Notice the appearance of an inflection point around x = 0.5.
Figure 3. A plot of y = e ᎑1/x for values of x between 0 and 0.3. This portion of the curve clearly appears to be “exponentially increasing”.
Figure 4. A plot of the first derivative of y = e ᎑1/x . Notice that there is a maximum at around x = 0.5.
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Journal of Chemical Education • Vol. 75 No. 9 September 1998 • JChemEd.chem.wisc.edu
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Figure 5. A plot of the second derivative of y = e ᎑1/x . The curve crosses the x-axis at 0.5, which is the x value of the inflection point.
Figure 6. The temperature value of the inflection point as a function of activation energy. It is highly unlikely that chemical kinetics experiments could ever be done in a regime where the Arrhenius equation is not “exponentially increasing”.
second derivative will give the exact value of the inflection point.
an activation energy of 60 kJ/mol (14 kcal/mol) would need a temperature of around 3600 K. It is clear that chemical kinetic studies are done at a low enough temperature in relation to the activation energy so that the Arrhenius equation is always in the “exponentially-increasing” region.
y′′ = x ᎑4e᎑1/x(1 – 2x) Setting 1 – 2x = 0 and solving for x gives x = 0.5, which is the x value of the inflection point (Fig. 5). Now to relate this to the Arrhenius equation and the world of chemical kinetics. For simplicity, let E a /R = a. The second derivative will give the exact temperature at the inflection point. k′′ = aT ᎑4 e᎑a/T(a – 2T) The inflection point is again determined by setting a – 2T = 0, giving T = a/2 = E a /2R This function is plotted in Figure 6. The plot can be interpreted to show that, at any activation energy, an extremely high temperature is needed to reach the regime where the Arrhenius equation is not “exponential”. For a reaction with an activation energy of 200 kJ/mol (48 kcal/mol) a temperature of around 12,000 K would be needed. A reaction with
Note 1. I use quotation marks around the term “exponentially increasing” when discussing the properties of the function y = e᎑1/x because it doesn’t increase in a true exponential fashion. Only a function of the form y = e x has a true exponentially increasing character.
Literature Cited 1. Zumdahl, S. S. Chemistry, 4th ed.; Houghton Mifflin: Boston, 1997; p 578. 2. Chang, R. Chemistry, 6th ed.; McGraw-Hill: New York, 1997; p 528. 3. Brown, T. L.; Lemay, H. E., Jr.; Bursten, B. E. Chemistry The Central Science, 7th ed. Prentice Hall: Upper Saddle River, NJ, 1997; p 508 4. Laidler, K. J. J. Chem. Educ. 1984, 61, 494. 5. Logan, S. R. J. Chem. Educ. 1982, 59, 279.
JChemEd.chem.wisc.edu • Vol. 75 No. 9 September 1998 • Journal of Chemical Education
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