In the Classroom
A Mechanical Apparatus for Hands-On Experience with the Morse Potential Michael A. Everest Department of Biology and Chemistry, George Fox University, Newberg, Oregon 97132
[email protected] Several phenomena in chemistry may be explained by invoking an interparticle (atomic or molecular) potential that is repulsive at short distances and attractive at long distances. The simplest example is a harmonic potential in which the restoring force is proportional to the displacement from the equilibrium separation. This potential, which treats chemical bonds as ideal Hooke's law springs, is sufficient to account for many features of molecules. However, a simple spring model does not account for all the features of chemical bonds. For example, unlike ideal springs, chemical bonds break when extended too far. That is, at large interatomic distances, the restoring force becomes gradually weaker, and eventually the potential does not increase at all as the atomic separation is increased. The bond has broken. One chemically important potential that exhibits bond breaking is the Morse potential (1). The Morse potential energy function, V(x), can be expressed as V ðxÞ ¼ De ½1 - e - aðx - x0 Þ 2
ð1Þ
in which De is the well depth, a is related to the width of the well, x is the distance between the nuclei of bonded atoms, and x0 is the equilibrium internuclear distance. The Morse potential is depicted graphically in Figure 1. The Morse potential is frequently used as a model for the interatomic potential in chemical bonds because the quantum energy levels calculated from this potential are a good approximation for the energy levels observed in molecular vibrational spectroscopy. The Morse potential is widely taught in chemistry classes and has been a frequent topic of contributions to this Journal (2-5). Because of the importance of this potential, and other potentials that exhibit bond breaking, most chemists have internalized several physical features of these potentials. Specifically, atoms involved in a stable chemical bond would be (if they could be grasped with the hands) very difficult to compress much shorter than the equilibrium bond distance, they would provide some resistance to extension beyond the equilibrium position, and the resistance to extension would become weaker and weaker as the bond was extended to very long lengths. Eventually, there would be no resistance to further extension, as the bond would be broken. Whereas the experienced chemist can perform this thought experiment simply by inspecting a graph such as Figure 1 (or perhaps even by considering eq 1 alone), the student has no direct experience of what a chemical bond would feel like if it could be pushed and pulled while grasped with the hands. Below we describe an apparatus that gives students that opportunity. Our Apparatus An apparatus designed to give students a hands-on experience of a model potential must provide forces both to the left and
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Figure 1. The Morse potential from eq 1. For simplicity, x0 is chosen to be zero.
to the right, depending on the position of an object that represents one of the atoms in the chemical bond (or, more precisely, whose position represents one particular interatomic distance). The required forces, F(x), are determined from the potential according to dV ¼ - 2aDe e - ax þ 2aDe e - 2ax ð2Þ F ðxÞ ¼ dx where V is from eq 1 in which x0 has been set to zero for simplicity. The force is graphically depicted in Figure 2. The negative term represents attractive forces (pulling toward shorter bond length;the left in our apparatus, consistent with Figures 1 and 2), and the second term represents repulsive forces (pushing to longer bond length or to the right). The apparatus only needs to provide the net force as a function of the position of an object. This net force could be attained in many different ways. For example, a constant force (e.g., from a suspended mass) or a Hooke's law force (from a spring) could be applied in one direction, and a more complicated force in the other direction could be designed to give the desired net force. We chose to break the force into two parts, one for each term on the right-hand side of eq 2. This choice permits being able to feel just the repulsive or attractive forces apart from the presence of the other. Two specifically designed pulleys can be used to build an apparatus that can generate an arbitrary force as a function of distance, so long as the force can be written as the sum of two terms, each of which is a monotonic function of distance. If the two terms have opposite signs (as is the case in eq 2), one pulley may be placed on the left and is designed to pull in that direction
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r 2010 American Chemical Society and Division of Chemical Education, Inc. pubs.acs.org/jchemeduc Vol. 87 No. 10 October 2010 10.1021/ed100389c Published on Web 08/30/2010
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In the Classroom
Figure 2. The force from eq 2. Notice that the force is positive (repulsive) at x < 0 and negative (attractive) at x > 0. The minimum of -Dea/2 occurs at x = (ln 2)/a.
(the “attractive” force), whereas another pulley is placed on the right and is designed to pull in that direction (the “repulsive” force). A schematic diagram of an apparatus we constructed to model the Morse potential is shown in Figure 3. Each side consists of a circular pulley rigidly attached to an irregular pulley. A weight is suspended by a string from each irregular pulley, and the circular pulleys are attached to each other by a third string. The set of pulleys on each side are attached to a rigid board with bearings that permit free rotation about the point coincident with the center of the circular pulley. A plastic ball (from a molecular model kit) is fixed to the string attaching the two circular pulleys. The pulleys are designed so the force required to hold the plastic ball stationary at a certain position is determined from the Morse potential. A second plastic ball with a hole through the axis is fixed to the mounting board with the string running through it. These two balls aid the student in visualizing a human-scale diatomic molecule. For our apparatus, the circular pulley on each side has a 5 cm radius. The irregular pulley on the attractive pulley (left side) has a radius r(θ) = 5.7e-0.588θ þ 2.5 and the irregular pulley on the repulsive side(right side) has a radius r(θ) = 8.1e-1.175θ þ 2.5, where the lengths are in centimeters and θ is in radians. The derivation of these equations is available in the supporting information. When 500 g masses are suspended from the irregular pulleys on each side, the result is a potential with a fwhm (full width at half minimum) of approximately 15 cm and a maximal force equivalent to suspending a 100 g mass in the gravitational field. A computer-aided design (CAD) model of pulleys described in the preceding paragraph was made, and pulleys were constructed of ABS plastic using a 3D printer (Dimension SST 1200), a machine capable of producing plastic parts directly from computer files. Our pulleys are approximately 1 cm thick. The irregular pulleys could also be made by making a polar graph of the equations and using these graphs as templates to cut pulleys out of wood or acrylic sheet plastic. These would need to be attached to circular pulleys constructed in a similar fashion. A narrow groove also must be made around the perimeter of each pulley to guide the string. Precision ball bearings (0.75 in. diameter, similar to those used in skateboards) purchased at the local hardware store were pressed into a hole at the center of each pulley, and the pulleys were mounted 1072
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Figure 3. A schematic diagram of an apparatus to model the Morse potential. The student grasps the solid circle (a plastic ball fixed to the string) and moves it toward and away from the split circle (a plastic ball through which the string freely passes). The pulley on the left provides the attractive part of the potential and the pulley on the right provides the repulsive part of the potential.
on an approximately 2 ft 3 ft piece of particle board with standard bolts, nuts, and washers. Heavy braided fishing line (other strong string would also work) was used to connect the two circular pulleys to each other and to suspend 500 g masses from the irregular portion of each pulley. Design details including the CAD files of our pulleys and step-by-step instructions in pulley construction are available as supporting information. The apparatus demonstrates all the qualitative features one would expect from an interatomic potential including the steep repulsive wall, a minimum at short displacements, a maximal force as the bond is being broken, and a leveling off of the potential at very large displacements. Two Learning Goals Introductory Chemistry The macroscopic model of a covalent bond that we have constructed can be used to achieve several learning goals at different levels of chemical education. Perhaps the most widely applicable learning goal that students may achieve from using this model is that it always takes energy to break a covalent bond, and there is always energy released when a covalent bond is formed. Students frequently have a false intuitive sense that energy is released when chemical bonds are broken. We suspect that this misunderstanding comes from the common observation that wood burning in a fire clearly gives off energy and seems to
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In the Classroom
involve a net breaking of chemical bonds. Of course, the strong CO2 and H2O bonds that are formed are not as immediately accessible to the senses. The misconception that breaking bonds releases energy is also propagated by misleading wording in textbooks discussing the hydrolysis of ATP that describe the process as releasing energy when a “high-energy bond” is broken. We believe that this common misconception can be readily corrected when students are given the opportunity to experience a physical model of a covalent bond. This learning goal is appropriate for high school chemistry students, as well as students in introductory and nonmajor college chemistry classes.
We had also hoped to demonstrate the relationship between the interatomic potential and some simple chemical reaction dynamics. For example, if dynamical phenomena could be performed with the model, we could demonstrate the impossibility of the formation of a diatomic molecule directly from atom combination in a biatomic collision (e.g., 2O f O2 cannot happen in the atmosphere but 2O þ M f O2 þ M does.) This demonstration was also difficult with the current apparatus because of the lateral swing of the masses.
Advanced Chemistry
Additional learning goals that may be appropriate for upperdivision chemistry students would be the ways in which this classical model does not accurately represent a covalent bond. Specifically, because it is purely classical, it cannot represent phenomena that are quantum mechanical in nature. For example, whereas it is possible to apply a static force and hold the classical model at a particular position, this is not possible for an atom in a molecule; the concept of force is not commonly used in quantum chemistry. Second, any discussion of trajectories of atoms in an anharmonic potential would be quasi-classical in nature, and further discussion would be needed to help students understand the applications and limitations of this approach.
More advanced students are in a position to learn more subtle features of interatomic potentials by using a physical model potential of a chemical bond. For example, appropriate learning goals for more advanced students include (i) the equilibrium bond length is the balance of attractive and repulsive effects, (ii) the potential becomes independent of bond length at large displacements, and even that (iii) the potential is steeper on the repulsive side than the attractive side. Although these do not have the same wide-ranging significance as the learning goal discussed above, they are important aspects of interatomic potentials that ought to be part of any chemist's intuition. If students are familiar with these features of chemical bonds, they are in a better position to understand the shapes of vibrational wave functions of an anharmonic oscillator and Franck-Condon effects in electronic spectroscopy. These learning goals are most appropriate for physical chemistry and other upper-division college chemistry classes.
Limitations
Acknowledgment We thank Steve Petzold and Phillip Christiansen for their assistance in the construction of the pulleys. This project was supported in part by George Fox University Grant GFU08G0006.
Improvements in the Apparatus
Literature Cited
We had hoped initially that the model could be used to demonstrate dynamical phenomena such as anharmonic oscillation. We intended to demonstrate this by starting the mobile atom at some displacement far from the equilibrium position and letting it go to move freely. This demonstration works for one or two cycles of oscillation. However, because the suspended masses are free to move left and right as well as up and down, the motion quickly becomes chaotic as the suspended masses swing laterally and the force they apply on the pulley deviates from what was assumed in the design of the pulleys. We hope to improve the model by replacing the free-hanging masses with constant-force springs attached to the board.
1. Morse, P. M. Phys. Rev. 1929, 34, 57–64. 2. Zuniga, J.; Bastida, A.; Requena, A. J. Chem. Educ. 2008, 85, 1675– 1679. 3. Zielinski, T. J. J. Chem. Educ. 1998, 75, 1191. 4. Pettitt, B. A. J. Chem. Educ. 1998, 75, 1170–1171. 5. Lessinger, L. J. Chem. Educ. 1994, 71, 388–391.
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Supporting Information Available Additional design details including the derivation of the equations for the shapes of the pulleys, the CAD files of our pulleys, and step-bystep instructions in pulley construction. This material is available via the Internet at http://pubs.acs.org.
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