Chemistry Everyday for Everyone edited by
Resources for Student Assessment
John Alexander University of Cincinnati Cincinnati, OH 45221
The Arrhenius Law and Storage of Food in a Freezer Ilya A. Leenson* Department of Chemistry, Moscow State University, 119899 Moscow, Russia
The Arrhenius equation is usually applied to express the dependence of reaction rate on temperature. Laidler gave a brief historical account of the development of this equation, beginning with the pioneer work of Wilhelmy, who in 1850 was the first to propose an equation relating the reaction rate constant to temperature (1). Pacey showed that the main focus of the Arrhenius law, the activation energy (E a), is not a single quantity but a composite quantity even for a single reaction. The reaction chosen for detailed study was D + H2 → DH + H and the author compared different interpretations of Ea for this simple reaction (2). Later Logan discussed ways in which the status of the Arrhenius equation has changed since it was first proposed. He pointed out that in many instances “the Arrhenius activation energy” is a quantity with no (not even approximate) physical significance (3). Olah discussed the relationship between Arrhenius’ threshold energy and the phenomenologically defined “effective activation” energy which “lacks a clear thermodynamic interpretation” (4). Yet not only chemists but many scientists from other disciplines do apply a simple Arrhenius law to their results and thus obtain the so-called “effective” (or “apparent”) value of Ea. Then, basing their discussion on its value, they can speculate about a mechanism for the process involved or simply extrapolate the data to new conditions. In this connection it is necessary to mention some “unconventional applications of the Arrhenius law” (5). Laidler demonstrated how it could be applied to such phenomena as chirping of tree crickets (E a = 51 kJ/mol), creeping of ants (108 kJ/mol below 16 °C and 51 kJ/mol above that temperature), flashing of fireflies (51 kJ/mol), rate of terrapin heartbeat (77 kJ/mol from 18 to 34 °C and much higher below 18 °C), human alpha brain-wave rhythm (about 30 kJ/mol), and even human rates of counting and forgetting (100 kJ/mol for both). J. A. Campbell, the author of a remarkable popular book on chemical kinetics (6 ), presented other examples in his Eco-chem column in this Journal (7). The heartbeat rate of the water flea Daphnia increases with the temperature (Ea = 50 kJ/mol). A rapid change in the rate of bacterial hydrolysis of fish muscle as T varies near 0 °C yields Ea = 1420 kJ/mol! But this value is very uncertain because of a very narrow temperature range (3.3 °C). Perhaps even more important is the fact that the temperature range spans the water freezing point; so the rate of a bacterial process can change dramatically. This article presents another unconventional application of Arrhenius’ equation. A modern-day student might appreciate the Arrhenius law better if it were applied to an amusing process rather than to some dull chemical reaction. The *Email:
[email protected].
504
description of the process will be followed by some questions for students and acceptable answers and solutions. Italian Pizza and Its Storage Time Once I purchased “Baguette breads with herb sauce, mushrooms in cream sauce, cheese, mushrooms and herbs; deep-frozen”, imported to Russia from Italy. On the back cover of the box there was nutrition and storage information. The text is given six times—in German, English, French, Spanish, Dutch, and Danish. Below is the inscription (the Dutch and Danish versions are omitted and the Russian is substituted for the English so that the students could have an opportunity to practice in foreign languages). Problems Derived from the Inscription Being a kineticist I immediately noticed the incompatibility of storage times at the different temperatures. A simple test revealed that my observation was true. It might be interesting to present the students with the following problems.
Aufbewarung zu Hause: Im Kühlschrank (0 °C) *Fach oder Eiswürfelfach (¯ 6 °C) **Fach (¯12 °C) ***Fach oder Tiefkühltruhe (¯18 °C)
1 Tag 1 Woche 2 Wochen 9 Monate
(0 °C) (¯ 6 °C) (¯ 12 °C) (¯ 18 °C) Conservation: au réfrigérateur (0 °C) *compartiment à glace (¯ 6 °C) **congélateur (¯12 °C) ***congélateur (¯18 °C)
1 jour 1 semaine 2 semaines 9 mois
Conservación: frigorífico compartimento (0 °C) *congelador (¯ 6 °C) **congelador (¯12 °C) ***congelador (¯18 °C)
1 día 1 semana 2 semanas 9 meses
Journal of Chemical Education • Vol. 76 No. 4 April 1999 • JChemEd.chem.wisc.edu
Chemistry Everyday for Everyone
Maybe they will find similar information on other kinds of frozen food and check its correctness. PROBLEM 1. Read the storage information, find a supposed mistake and correct it (in English). PROBLEM 2. Calculate the temperature for 4 months storage of this product. HINT: you may assume that the spoiling of this product obeys ordinary kinetic laws, including the Arrhenius equation.
Preliminary Comments and Discussion of the Problems The Arrhenius equation works quite well with many chemical reactions both simple and complex. Since foods are very complex systems it is not always possible to isolate a clear-cut chemical reaction (or several reactions) which lead to the observed changes in quality and to have simple mathematical models that can describe the reaction rate properly. General modes of food deterioration include microbial decay, enzymatic and nonenzymatic chemical change, lipid oxidation, vitamin loss, and others (8). Among these the growth of foodborne pathogens and spoilage microorganisms are of primary importance in the loss of food quality. The value of Ea is very specific for each system and may vary from 40 kJ/mol for simple hydrolysis reactions to 600 kJ/mol for enzyme and microbial thermal destruction. Of course, the best method to obtain a shelf life for a given kind of food would be to do studies at the desired temperatures, but this is usually not possible because it would be both costly and time-consuming (8). Besides, there are some limitations of the Arrhenius model. For example, during freezing, reactants are concentrated in the unfrozen water; the major effect would be obtained just below the thaw point for frozen foods (9, 10). Nevertheless, the experimental data do often obey the simple kinetic law whereby the shelf life of a given kind of food at different temperatures varies according to the Arrhenius equation (in ref 8 one can find a valuable review of general approaches to the kinetics of food deterioration). Indeed, one can reasonably assume that the food should be edible up to the end of the storage period and therefore could not be spoiled as much as 5% (or probably even 0.1%). Thus, we deal with the very beginning of the “kinetic curve” of product quality. The kinetic curve is always straight at its onset what-
ever its kinetic order (first, second, fractional, etc.). This means that the “spoiling rate” v is constant at a given temperature T and is inversely proportional to the storage time t. So we must attempt to apply the Arrhenius equation t = 1/v = constant × e E/RT to the experimental data. If this equation is applicable we would obtain a straight line in Arrhenius coordinates (ln t, 1/T ) and its slope would give an apparent activation energy E a for the process (whatever its mechanism might be). Given this straight line, we could make some predictions about the shelf life of a product, these predictions being most reliable within the experimental temperature range whereas only cautious extrapolation could be made beyond this range. Acceptable Solution of the Problems Let’s assume that “experimental data” presented on the cover of pizza follow the Arrhenius equation. The chart below presents these data for a subsequent treatment. t/days
1
7
14
≈275
ln t
0
1.95
2.64
5.62
T/°C
0
{6
{12
{18
T/K
27 3
267
261
25 5
1000T {1/K {1
3.66
3.75
3.83
3.92
The second and last lines yield a plot (Fig. 1) from which it is obvious that three points at 0, { 6, and {18 °C fit perfectly in a straight line (this is rather surprising, considering that the data were rounded off to the nearest integer), while the fourth (for {12 °C) lies significantly below the line. According to the Arrhenius plot this temperature should correspond to ln t = 3.7 (thick dotted line), whence t = e 3.7 = 40.4 days (about 6 weeks or 1.5 months). So we can assume a mistake in the information on the package. Now, the answer to the second question is very simple. The 4 months (122 days) storage time corresponds to a value ln t = 4.80, whence 1000/K = 3.88 (thin dotted line on the plot) and T = 258 K = {15 °C. An apparent activation energy from the plot obtained is equal to 180 kJ/mol. From this figure it is difficult to draw conclusions about a specific mechanism (or mechanisms) of pizza deterioration on storage. According to Labuza’s review (8) the most probable mechanism is enzymatic and microbial degradation. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9.
Figure 1. Arrhenius plot of the storage time data provided on the frozen pizza package.
10.
Laidler, K. J. J. Chem. Educ. 1984, 61, 494–498. Pacey, P. D. J. Chem. Educ. 1981, 58, 612–614. Logan, S. R. J. Chem. Educ. 1982, 59, 279–281. Olah, K. Acta Chim. Hung. 1990, 127, 135–147. Laidler, K. J. J. Chem. Educ. 1972, 49, 343–344. Campbell, J. A. Why Do Chemical Reactions Occur? PrenticeHall: Englewood Cliffs, NJ, 1965. Campbell, J. A. J. Chem. Educ. 1974, 51, 119; 1975, 52, 390. Labuza, T. P. J. Chem. Educ. 1984, 61, 348–358. Douzou, P. Cryobiochemistry. An Introduction; Academic: New York, 1977. Franks, F. In Water. A Comprehensive Treatise; Franks, F., Ed.; Plenum: New York–London, 1982; Vol. 7, Chapter 3.
JChemEd.chem.wisc.edu • Vol. 76 No. 4 April 1999 • Journal of Chemical Education
505
