In the Classroom
An Experimental Approach to Teaching and Learning Elementary Statistical Mechanics Frank B. Ellis* Department of Chemistry and Environmental Science, New Jersey Institute of Technology, Newark, NJ 07102; *
[email protected] David C. Ellis Westfield, NJ 07090
This paper tells how to use an easily constructed apparatus to give demonstrations that teach chemical principles using elements of statistical mechanics to students in high school and first-year university chemistry courses. Demonstrations are presented for exothermic and endothermic reactions, the dynamic nature of equilibrium, and Le Châtelier’s principle. These demonstrations show that temperature, energy, and entropy all play a role in determining the final equilibrium. The apparatus consists of an open vibrating box, typically divided into two parts, where excited beads bounce within each part and between the two parts (see Figure 1). Because its front wall is transparent, students can observe the motion of the energized beads. The parts have an adjustable relative height, or potential energy, and an adjustable area, or entropy. A barrier of adjustable height between the two parts is analogous to the activation energy. The average energy of the bouncing beads may be varied and is analogous to the average molecular kinetic energy. This setup permits a large variety of demonstrations where the group properties of the beads are investigated in light of their individual properties. In concept, this apparatus is similar to a device for teaching about entropy and a more recently reported equilibrium machine (1–2). In contrast to those devices, its results are more quantitative, and it allows a larger variety of demonstrations. This apparatus operates akin to the operation of the device presented by Plumb, where a vibrating base transmits energy to hard spheres (3).
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The statistical mechanics of beads is the same as the statistical mechanics of atoms or molecules to the extent that both beads and molecules have a Maxwell–Boltzmann energy distribution. In this case, the beads may be used to represent atoms or molecules and give insights into their properties. For example, in the simplest case the apparatus may be operated without any platform, starting with the beads grouped together and covering a small fraction of the total base area. When the apparatus is turned on the beads start to bounce in parabolic paths and—like a gas—spread out to fill all the space available to them. The bouncing beads are seen to have a distribution of energies indicated by the different heights to which they bounce as the vertical component of the kinetic energy is changed to potential energy, and by the different widths of their parabolic path, which is a product of the horizontal component of the velocity and the time in flight. This distribution is analogous to the thermal distribution of energies in an atomic system at some fixed temperature. By extension, the bead distribution of energies can be considered to define the “bead temperature”. Equipment and Its Operation The heart of the apparatus consists of a vibrating box that is generally divided into part A and B of different area and elevation (see Figure 1). Platform (a1) is removable and may
B Figure 1. Drawing of the apparatus during operation showing an equilibrium distribution of beads. The open box part is approximately 23 × 33 × 21 cm. In this setup, the difference in height between the two parts, Δh, is 5 cm and the adjustable barrier height, b, is 10 mm. The small circles represent glass beads. Note that more energetic beads are required to jump from part B to A than from part A to B.
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Background for Simulations
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Details: (a1) a removable platform; (a2) binder clip and vertical post to anchor platform; (c) adjustable spring length; (d) screw and nut(s) to unbalance fender washer; (e) electric drill, and (f) setup to measure drill speed, including light chopper, LEDs, and frequency counter.
Journal of Chemical Education • Vol. 85 No. 1 January 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom 0.25
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Figure 2. Plot of the bead ratio (number of beads in the upper level vs number of beads in the lower level at equilibrium) as a function of the speed of the spinning unbalanced shaft. The ratio increases as average bead energy increases because more beads have sufficient energy to jump to the upper level. In this setup, the areas of the two levels are approximately equal, the upper level is 5.0 cm higher, the barrier height is 4 mm, and the unbalancing mass is 8.79 g.
Figure 3. Schematic view of an apparatus for easy bead removal. (A) PVC male adaptor with one end a ¾-in. pipe thread and the other end a 1-in. socket; (B) A piece of vacuum cleaner bag material covering the socket end, held in place with masking tape (as are subsequent pieces); (C) Eight 1½-in. finishing nails; (D) Piece of tape holding strip (E) of vacuum cleaner bag material in place.
be replaced with other platforms of different height and area. Similarly, there is a removable barrier separating part A from part B that may be exchanged with barriers of different height. When the apparatus is turned on, the box is vibrated by a rotating washer unbalanced by an off-centered screw and one or more nuts (Figure 1d), driven by an electric drill (Figure 1e) connected to a speed controller (not shown). Sufficiently strong box vibrations cause the beads to bounce. Energy from the rotating unbalanced washer is most efficiently transferred to the box and from the box to the beads at the resonance frequency of the apparatus (see Figure 2). When the drill speed is far from the resonance frequency the beads barely bounce—less than a mm. Near the resonance frequency, as indirectly shown in Figure 2, energetic bouncing is possible and the most energetic bouncing occurs at the resonance frequency. Thus, the average bead energy may be controlled by the closeness of the driving frequency to the resonance frequency. In practice the driving frequency is kept less than or equal to the resonance frequency in order that the beads immediately settle when the controller is turned off. The resonance frequency is determined by the total vibrating mass (mainly the mass of the box and the platform) and the springs’ stiffness and length (see the online supplement). The barrier height for beads in part A to transfer to part B is b; the barrier height for beads in part B to transfer to part A is Δh + b. Beads whose vertical energy component exceeds mgb, where m is the mass of one bead and g is the acceleration due to gravity, may jump from part A to B. Those whose vertical component of energy exceeds mg(Δh + b) may jump from part B to A. The setup for measuring drill speed (Figure 1f ) is needed only when quantitative comparison between different demonstrations at the same average bead energy is desired. For the qualitative demonstrations discussed in this article, it is un-
necessary because the average bead energy may be adequately approximated by reproducing the envelope of typical maximum height to which the beads bounce. Glass seed beads or e-beads are used in these demonstrations. Seed beads are approximately spherical glass beads about 1 mm in diameter and with a small hole through the middle. The e-beads are similar to seed beads, but are about 3.5 mm in diameter. The beads should occupy less than 2% of the base area or they start to damp each other. The most energetic beads are the most sensitive to damping. For a uniform bead distribution over the total area, about 1200 seed beads or 100 e-beads can be used with little damping. The larger beads are easier to observe, but the smaller beads are preferred when obtaining statistical data. The beads may be removed by hand or scooped with a piece of paper. However, they are more easily removed using a filtering device attached to a vacuum (see Figure 3). The threaded end inserts into the nozzle of a vacuum cleaner and is held in place by the suction of the vacuum. Exothermic Reactions Using the setup shown in Figure 1, an exothermic reaction may be illustrated by starting with about five hundred seed beads in an elevated part A. (The large number of beads is not counted, but is approximated using a balance. The average mass per bead is ~13 mg. So, 500 beads have a mass of ~6.50 g.) The difference in elevation represents the difference in chemical potential energy between the reactants and products. The reactants and products are represented by the beads in part A and part B, respectively. The barrier height is equivalent to the activation energy of the reaction and reducing its height demonstrates the effect of catalysis.
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 1 January 2008 • Journal of Chemical Education
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In the Classroom
The drill speed controller is turned on to produce a low drill speed that is increased until the beads are bouncing to a height of ~1–3 mm maximum (drill speed is below the resonance frequency). It is observed that no “reaction” takes place because no beads have sufficient energy to jump the 5-mm barrier. Next, the drill speed is increased until some of the beads have sufficient energy to jump the barrier, but insufficient energy to return. In this case, the beads slowly, completely transfer to the lower level. The demonstration is repeated for somewhat more energetic beads. It is observed that the beads’ irreversible transfer to the lower part occurs at a faster rate. Reactions are often considered to take place because of two different driving forces. These demonstrations show that the first arises from the drive, urge, or tendency of a system to seek a state of minimum energy.
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Figure 4. Apparatus set up to be analogous to an endothermic reaction. The elevated part is 2.4 cm higher and has an area 50× greater than the area of the lower part. Only 20 e-beads are used. On the left, average bead energy is low; on the right it is high.
Equilibrium The demonstration is repeated using a drill speed closer to the resonance frequency such that the most energetic beads can jump the barrier from either part. The beads rapidly transfer to the lower part, although not all end up in the lower part because some beads jump from the lower to the upper part. At equilibrium the rate of crossing between the two parts is equal—the larger number of beads in the lower part times the small fraction with sufficient energy to jump the larger barrier, Δh + b, equals the smaller number of beads in the upper part times the larger fraction of beads with sufficient energy to jump the smaller barrier, b. This demonstration also shows the dynamic nature of equilibrium. For this average bead energy, one can show that the resulting equilibrium number of beads in each part is independent of the initial bead location; that is, they may all start in the lower part or upper part or some in each part. In each case, one observes that the beads move irreversibly towards the same equilibrium bead populations. Average Bead Energy The last demonstration is continued by varying the average bead energy. It is observed that the greater the average bead energy, the greater the fraction of beads in the upper part at equilibrium. Conversely, the lower the average bead energy, the greater the fraction of beads in the lower part at equilibrium. Thus, for an exothermic reaction, a low temperature favors a high yield. The demonstration makes it easy to see why this occurs. Specifically, many beads in the upper level have sufficient energy to jump the small barrier, b, and hence there is only a small increase in this rate as the average bead energy increases. However, as the average bead energy increases, a much greater fraction of beads in the lower part have sufficient energy to jump the large barrier, Δh + b. Thus, this rate dramatically increases as the average bead energy increases. Combining the two effects implies that the equilibrium shifts to a greater fraction of beads in the upper level as the average bead energy increases. Endothermic Reactions The setup shown in Figure 1 is also used to demonstrate endothermic reactions, but the beads start in the lower part.
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No beads transfer to the higher part until some of beads have sufficient energy to jump the high barrier, Δh + b. High average bead energies are needed for even a modest equilibrium number of beads in the upper part. The maximum yield for this setup is 50% and is approached as the average bead energy becomes large compared to mgΔh. An endothermic reaction capable of a large product yield is shown in Figure 4. The reactants and products are represented by the beads in part A and an elevated part B, respectively. Moving from part A to part B entails a large increase in area. As before, the average bead energy is increased or decreased by adjusting the drill speed toward or away from the resonance frequency. In Figure 4a, all the beads are in the lower part because the bead energies are less than mgΔh. As the average bead energy is increased some beads have sufficient energy to jump to the upper level and there is a small equilibrium number of beads at this level. As the average bead energy continues to be increased, the equilibrium population of beads in the upper level increases. In Figure 4b, the average bead energy is large enough that most of the time all the beads are in the upper part. These demonstrations for an endothermic reaction show that as the average bead energy increases, the equilibrium yield of product increases. In other words, the energy of the beads associated with the bead temperature makes an endothermic reaction possible. A higher bead temperature drives the reaction because the large area of part B becomes available to more of the beads—those whose energy is greater than mg(Δh + b). For these beads, the fraction in part B equals the area of part B divided by the total area. Thus, the large relative area of part B is also necessary for an endothermic reaction with a large yield. For real molecular systems, the area represents the numbers of ways a system can be found in a given state. Energy These demonstrations show that the equilibrium number of beads in the upper part decreases as the average difference in bead energy (ΔE) between parts A and B increases. The total energy of a bead is its bounce energy plus its potential energy because of the height of the platform or base of the box. As the beads have the same average bounce energy in each part,
Journal of Chemical Education • Vol. 85 No. 1 January 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom positional entropy
small area
large area thermal entropy
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Figure 5. Diagrams illustrating the two types of entropy. In each case, the state of greater disorder or entropy is shown on the right.
ΔE is the potential energy difference, that is, mgΔh where m is the mass of one bead, g is the acceleration due gravity, and Δh is the difference in height. The effect of ΔE on the equilibrium
number of beads in each part is demonstrated by using setups similar to the one shown in Figure 1. In this case, each demonstration uses a platform of the same area, but different height. In addition, the same average bead energy is used for each of the different platform elevations. Platform heights from 0.3 cm (the width of the board for the top of the platform) to 9.3 cm have been used (with greater heights too many beads start bouncing out of the apparatus). In one set of demonstrations where the areas of each part were equal and using about 560 seed beads, the number of beads in the elevated part was about 280, 160, 72, and 13 for elevations of 0 (no platform—just a barrier), 1.3, 2.9, and 5.0 cm, respectively. Thus, it is observed that as ΔE increases a rapid fall-off in the equilibrium number of beads in the upper part occurs. These results demonstrate that the more exothermic the reaction, the greater the product yield at equilibrium. Entropy Zumdahl shows that entropy has two components—position and energy (4). McQuarrie notes that entropy is a measure of the disorder of a substance and that the disorder is of two types—positional and thermal (5). It is sufficient to note that the larger the base area of part A or B, the larger the positional entropy for that state. (The actual positional entropy is proportional to the logarithm of the area—in analogy to the positional entropy of an ideal gas.) For example, beads restricted to move above a small base area must be in this very limited space, so there is little positional disorder or entropy. On the other hand, beads free to move above a large base area can be found anywhere in the space above this large area. Thus, this state has a large positional disorder or entropy (see Figure 5). The vertical space occupied by the beads is a measure of their thermal disorder. When the beads are gently bouncing (low temperature analogy) they are all confined to a space near the bottom of the box, that is, they have a small vertical displacement. When the beads are vigorously bouncing (high temperature analogy) they are much less confined, that is, they
may be anywhere within a large vertical displacement (see Figure 5). Thus, a higher temperature is seen to represent more disorder or greater entropy than a lower temperature. The effect of the area (positional entropy) on equilibrium can be shown by using platforms of the same height, but different area. Upper area/lower area ratios from 1/3–50/1 have been used. These demonstrations are done with a fixed average bead energy that is sufficient for some of the beads to jump to the upper part. One set of demonstrations with one part raised 2.9 cm and upper area/lower area ratios of 1/3, 7.2/1, and 50/1 yielded an average fraction of beads in the elevated part of 0.041, 0.28, and 0.76, respectively. (The number of seed beads used was about 800, 550, and 20, respectively. The 20-bead case was repeated 10 times and averaged.) It is observed that as the area of the upper part increases, the equilibrium fraction of beads in this part increases. Repeating these demonstration for lower and higher fixed average bead energies shows that both a large average bead energy and entropy (area) are important for a large equilibrium fraction of beads in the upper part. In particular, it is observed that a large entropy alone is not sufficient to have beads in the upper part at equilibrium. These demonstrations support the hypothesis that the second driving force in reactions is the product of the temperature and the change in entropy. It is the tendency of a system to seek a state of maximum entropy made possible by the thermal energy of the system. Le Châtelier’s Principle Le Châtelier’s principle states that if a stress is placed on a system at equilibrium (or moving towards equilibrium) the system adjusts to reduce the effect of the stress and to establish a new equilibrium. Le Châtelier’s principle may be demonstrated using the setup shown in Figure 1. About 400 seed beads are placed in the apparatus and the drill speed is adjusted such that around 10–40 beads are in the upper part at equilibrium. The system is perturbed by the addition of about 200 beads to the upper part. Thus, a large stress of excess beads is introduced to this part. It is observed that the stress of excess beads in the upper part is reduced by a net flow of beads to the lower part. Watching the process reveals why this takes place. Namely, before the equilibrium was disturbed, the rate of bead crossing between the two parts was equal. After the addition of beads to the upper part, the rate of crossing from this part increases and the system moves to establish a new equilibrium with fewer beads in the upper part. Alternatively, Le Châtelier’s principle may be demonstrated using the above setup with about 600 beads and having an equilibrium distribution of around 20–40 beads in the upper part. Beads from the upper part are repeatedly removed with a ladle and added to a transparent container. Even after it is clear that more beads have been removed than were originally present in the upper part, the number of beads in the upper part has only slightly decreased. In other words, there is a net transfer of beads from the lower part to the upper part to remove the stress of too few beads in the upper part. With regards to Le Châtelier’s principle and increasing the temperature, the traditional explanation is that the stress is excess energy, some of which the system moves to absorb by establishing a new equilibrium with more beads in the upper
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 1 January 2008 • Journal of Chemical Education
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In the Classroom
part. However, for many students the above explanation is hard to comprehend and is advantageously supplemented with a temperature demonstration. For example, still using the same setup and starting with about 600 beads in the lower part, the average bead energy is varied. It is observed that the greater the average bead energy, the greater the number of beads in the upper part because more beads have sufficient energy to jump to this part. Conclusion For many students, being introduced to new ideas through experimental data is more understandable than being introduced to them through abstract reasoning. This is not to say abstract reasoning is less important, rather that experimental data often provide an easier starting point. With this concept in mind, a two-state apparatus capable of a wide variety of demonstrations was presented. For classrooms that do not have this apparatus, gedanken (thought) experiments based on the apparatus may be effectively presented without its use for these three reasons:
• The process is simple—it involves only a two-state system with a lower part and an upper part
• The operation and behavior of the apparatus is easy for students to visualize
• A description of the results makes sense to students
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Acknowledgments The authors would like to express their appreciation to Joseph Bozzelli for comments and encouragement; to Grant Ellis for extensive editing; and to Randy Shirts for comments and suggestions regarding an early version of the manuscript. Literature Cited Plumb, R. C. J. Chem. Educ. 1964, 41, 254–256. Sawyer, D. J.; Martens, T. E. J. Chem. Educ. 1992, 69, 551–553. Plumb, R. C. J. Chem. Educ. 1966, 43, 648–651. Zumdahl, S. S. Chemical Principles, 5th ed.; Houghton Mifflin Co.: Boston, 2005; pp 413–416. 5. McQuarrie, D. A.; Rock, P. A. General Chemistry, 3rd ed.; W. H. Freeman and Co.: New York, 1987; pp 779–780.
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Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2008/Jan/abs78.html Abstract and keywords Full text (PDF) Links to cited JCE articles Supplement Equipment construction
Experimental development of the Boltzmann distribution law
Time-dependent demonstrations
Journal of Chemical Education • Vol. 85 No. 1 January 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
