In the Classroom edited by
JCE DigiDemos: Tested Demonstrations
Ed Vitz Kutztown University Kutztown, PA 19530
Demonstrating Energy Migration in Coupled Oscillators A Central Concept in the Theory of Unimolecular Reactions submitted by:
Ronald E. Marcotte Department of Chemistry, Texas A&M University-Kingsville, Kingsville, TX 78363;
[email protected] checked by:
James A. Zimmerman Department of Chemistry, Southwest Missouri State University, Springfield, MO 65804-0089
In teaching undergraduate physical chemistry, the theory of first-order gas-phase reactions is typically explained in the context of the Lindemann–Hinshelwood mechanism (1): A + A A*
k1 k−1 k2
A + A*
for the energy to migrate and accumulate in the critical coordinate(s) for reaction. The simple demonstration described below utilizes the well-known action of coupled oscillators (2) to physically animate this intramolecular energy migration process making it less abstract, easier to explore, and much more exciting. Apparatus
product
This article describes a classroom demonstration to enhance the discussion. Briefly, in this mechanism a reservoir of energized molecules, A*, is built up through a series of fortuitous collisions with an overall rate constant k1. Most of these molecules are lost in the reverse process by deactivating collisions with an overall rate constant k᎑1. A small portion of the energized molecules may survive long enough to form product in the rate-determining, unimolecular step with a rate constant k2. This step is rate determining because the energy required for reaction is dispersed throughout the molecule in the various vibrational modes and time is required
The apparatus, shown in Figure 1, consists of a heavy ring stand supporting two lead weights on strings with a third string for the variable coupling. The apparatus is capable of a great range of behaviors resulting from small configuration changes. The following configuration has been found satisfactory: masses 1 and 2 are 30 grams each and are suspended from 50-cm strings. The location of attachment points “a, a” and “b, b” and the length of coupling string (c) all have a great influence on the time constant for energy transfer. Coupling string is 12-cm long and attached 15 cm below the support rod. The distance between the attachment points “a, a” is normally 15 cm, but can be varied up to 25 cm or more to demonstrate the stronger coupling case and the corresponding increase in the energy transfer rate. Moving from 15 to 20 cm will approximately double the rate of energy transfer. Discussion
Figure 1. Pendulum motion as nearly enough energy for reaction has migrated from mass 2 into mass 1.
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The demonstration can be related to the theories of unimolecular kinetics by staging a simulated reaction that occurs when one oscillator reaches the critical displacement. A magic marker is placed about 12 cm in front of mass 1. Mass 1 is held momentarily while mass 2 is pulled straight back 15 cm and released. The class seems to enjoy the drama as the energy slowly begins to flow from mass 2 into mass 1, which increases in energy (amplitude) until it finally reaches the critical displacement, knocking the marker over. Secondary reactions may also be possible with the remaining energy. This can be demonstrated after the primary reaction by quickly placing a second magic marker in front of one of the pendulums while it is in its low-energy state after reaction. It may still be able to gain enough energy to tip over the second marker. The near-resonant energy transfer demonstrated above is typical of reactions involving the migration of energy in the relatively low-frequency skeletal vibrations. Other bonds
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In the Classroom
with high-vibrational frequencies may not contribute so readily to the energy migration owing to the difficulties of nonresonant energy transfer. Resonant-energy transfer can easily be shown to be much more effective than nonresonant energy transfer by simply tying a knot in one string below point “b”, thereby changing the frequency of one pendulum. Now they are nonresonant and only a fraction of one pendulum’s energy will be able to migrate to the other, and this may not be enough for the simulated reaction to occur. The demonstration would also be relevant for an advanced class on chemical kinetics since this type of energy migration is an important part of more advanced theories of unimolecular reaction rates such as the Rice–Ramsperger–Kassel– Marcus (3) theory.
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Acknowledgment The author wishes to thank the Robert A. Welch Foundation for its generous and continued support for our Chemistry Department. Literature Cited 1. Atkins, P.; de Paula, J. Physical Chemistry, 7th ed.; W. H. Freeman and Company: New York, 2002. 2. Physics Demonstration Experiments; Meiners, H. F., Ed.; Ronald Press Co.: New York, 1970. 3. Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and Recombination Reactions; Blackwell Publications: London, 1990.
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