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Orbital Shaped Standing Waves Using Chladni Plates doi.org/10.26434/chemrxiv.10255838.v1 Orbital Shaped Standing Waves Using Chladni Plates Eric Janusson, Johanne Penafiel, Andrew Macdonald, Shaun MacLean, Irina Paci, J Scott McIndoe Submitted date: 05/11/2019 • Posted date: 13/11/2019 Licence: CC BY-NC-ND 4.0 Citation information: Janusson, Eric; Penafiel, Johanne; Macdonald, Andrew; MacLean, Shaun; Paci, Irina; McIndoe, J Scott (2019): Orbital Shaped Standing Waves Using Chladni Plates. ChemRxiv. Preprint. https://doi.org/10.26434/chemrxiv.10255838.v1 Chemistry students are often introduced to the concept of atomic orbitals with a representation of a one-dimensional standing wave. The classic example is the harmonic frequencies which produce standing waves on a guitar string; a concept which is easily replicated in class with a length of rope. From here, students are typically exposed to a more realistic three-dimensional model, which can often be difficult to visualize. Extrapolation from a two-dimensional model, such as the vibrational modes of a drumhead, can be used to convey the standing wave concept to students more easily. We have opted to use Chladni plates which may be tuned to give a two-dimensional standing wave which serves as a cross-sectional representation of atomic orbitals. The demonstration, intended for first year chemistry students, facilitates the examination of nodal and anti-nodal regions of a Chladni figure which students can then connect to the concept of quantum mechanical parameters and their relationship to atomic orbital shape. File list (4) Chladni manuscript_20191030.docx (3.47 MiB) view on ChemRxiv download file SUPPORTING INFORMATION Materials and Setup Phot... (6.79 MiB) view on ChemRxiv download file MathSI.pdf (74.45 KiB) view on ChemRxiv download file Chladni plates 2 - Petite.mov (1.56 MiB) view on ChemRxiv download file 1 1 Orbital shaped standing waves using Chladni plates 2 3 Eric Janusson, Johanne Penafiel, Shaun MacLean, Andrew Macdonald, Irina Paci*, J. Scot 4 McIndoe* 5 6 Department of Chemistry, University of Victoria, P.O. Box 3065 Victoria, BC V8W3V6, Canada. 7 Fax: +1 (250) 721-7147; Tel: +1 (250) 721-7181; E-mail: [email protected], [email protected] 8 2 9 10 11 12 3 13 Abstract 14Chemistry students are often introduced to the concept of atomic orbitals with a representation 15of a one-dimensional standing wave. The classic example is the harmonic frequencies which 16produce standing waves on a guitar string; a concept which is easily replicated in class with a 17length of rope. From here, students are typically exposed to a more realistic three-dimensional 18model, which can often be difficult to visualize. Extrapolation from a two-dimensional model, 19such as the vibrational modes of a drumhead, can be used to convey the standing wave concept 20to students more easily. We have opted to use Chladni plates which may be tuned to give a two- 21dimensional standing wave which serves as a cross-sectional representation of atomic orbitals. 22The demonstration, intended for first year chemistry students, facilitates the examination of 23nodal and anti-nodal regions of a Chladni figure which students can then connect to the concept 24of quantum mechanical parameters and their relationship to atomic orbital shape. 25 26 Graphical Abstract 27 28 Keywords 4 29First-Year Undergraduate / General, Demonstrations, Hands-On Learning / Manipulatives, Atomic 30Properties / Structure. 31 32 33 34 35 36 5 37 Introduction 38 39 Understanding that an electron in an orbital can be thought of as a three-dimensional 40standing wave is a tough concept to wrap one’s head around. A one dimensional standing wave 41is easy to visualize and indeed demonstrate in a lecture environment - a length of rope held at 42each end by volunteers can be induced into a standing wave, which if energetic enough, will 43readily display nodes. 1,2 However, making the jump to a three-dimensional wave is much 44tougher, perhaps because our minds run out of dimensions to consider. Length, breadth, and 45depth are conceivable; but where does the amplitude go? The meanings of the different regions 46of the wave can be even more confusing given the multi-dimensionality of the problem. If we 47back off one dimension, and consider a two-dimensional standing wave, it becomes easier - we 48can use the third dimension to convey the amplitude, and the standing waves, if chosen 49carefully, can illustrate the cross-sectional shape of the orbital. A physical demonstration of wave 50behaviour in two dimensions, prior to three-dimensional illustration through computer-based 51models, allows for direct student involvement and easier conceptualization of a complex 52problem in first-year Chemistry. 53 The shape of orbitals has been taught to students using media as diverse as circular 54magnets, 3 Styrofoam models, 4,5 beakers of coloured water, 6 websites, 7 and balloons. 8 Various 55simple models have been used to introduce the concept of standing waves to students in the 56context of atomic (and/or molecular) orbitals. Instructors have tapped the rim of a coffee cup to 57relate nodes to the energies of aromatic molecular orbitals, 9 have used Kundt’s tubes (devices 58that set up acoustic standing waves in different gaseous media as a characterization tool), 10 and 6 59have reproduced Wiener’s historic experiment on a standing light wave 11 using a laser pointer. 12 60Computer-based orbital illustrations are an extremely valuable advance, allowing direct 61visualization of the quantum mechanical solutions for atomic orbitals. There exist excellent 62examples in the literature of two- and three-dimensional visualizations of atomic orbitals, 63making use of custom or proprietary software in order to produce plots of atomic and molecular 64orbitals and examine the orbital’s shape based on a given change in parameters. 13–20 65 Understanding and correctly interpreting these illustrations has become a fundamental 66learning goal in introductory quantum atomic structure. However, orbital shapes can be 67conceptually disconnected from the ideas of wave behavior in quantum chemistry, in the same 68way Styrofoam models are. Demonstrations of wave mechanics, preferably with direct student 69involvement, are an effective way to illustrate and promote the correct conceptualization of 70wavefunction shapes, their properties and the quantum mechanical wave equation. 21–23 An 71appealing and effective way to perform this illustration is by using the Chladni experiment 72discussed in the following pages. It should be noted here that the two-dimensional acoustic 73waves that produce Chladni paterns have been used previously to illustrate theoretically wave 74behaviour: The two-dimensional wave equation is discussed at some length in Donald 75McQuarrie’s standard textbook “Quantum Chemistry” 22 and pictures of a drum with low- 76frequency Chladni paterns are given on page 63. A series of analogies including Chladni 77paterns, jellyfish and viscous glop are used in a fascinating account of quantum chemical 78equivalencies by Dylan Jayatilaka from the University of Southern Australia. 24 The purpose of the 79current article is to illustrate how Chladni paterns can be used to physically, interactively 80demonstrate wave behaviour in the first-year Chemistry class. 7 81 A two-dimensional standing wave can be generated using common percussion 82instruments. When the appropriate harmonic frequency is applied to a drumhead the wave 83patern remains in one position as the amplitude of the wave fluctuates. Above the fundamental 84harmonic frequency (first harmonic), the drumhead becomes segmented by nodal lines in which 85there is no change in amplitude; conversely, in between these nodal lines are antinodes where 86the change in amplitude is at a maximum. 25,26 The standing wave formed on a drumhead is a 87useful analogy; however, these waves are not detectable to the unassisted naked eye, and we 88can more easily control and visualize two-dimensional standing waves using devices known as 89Chladni plates. Ernst Chladni (1756-1827) was a German physicist and musician most well-known 90for his study of vibrational modes produced on metal plates. In Chladni’s experiment, excitation 91was achieved by drawing the bow of a violin across the edge of the plate. Sprinkling sand across 92the surface of the plate while doing so caused the sand to setle in specific paterns shaped by 93the nodal regions of the vibrational mode, now referred to as Chladni figures. These paterns 94appear as the sand is tossed away from regions of the plate where a strong vibration is occurring 95(primarily the antinodal regions) and setles into regions where nodal lines are present (there is 96a particularly striking music video involving a variety of visualized standing waves by N. J. 97Stanford in which a Chladni plate plays a prominent role). 27 The modern method of generating 98Chladni figures is achieved with a frequency generator, a voice coil and metal plate. This in-class 99demonstration enables students to participate in active learning by producing a tangible 100representation of an atomic orbital, which they can control. The process of creating these 101beautiful cymatic figures is a unique way to intrigue and educate students. 102 8 103 Materials 104 The source of the excitation signal used was an Keysight (Agilent) 33522A function- 105generator. A custom power-amplifier, consisting of a Canakit (cat. # CK003) 10-Wat single- 106channel audio amplifier circuit, volume (amplitude) control and associated power-supply, was 107used to boost the signal to the levels required to drive the voice-coil that vibrates the plate. The 108voice-coil used was a PASCO SF-9324 mechanical wave driver. The function-generator was 109connected to the power-amplifier by a BNC cable. The power-amplifier positive and negative 110outputs were connected to the voice-coil by banana-connector cables. The plates used were 111round and square aluminium metal plates were used with measurements of 24 cm in diameter, 112and 24×24 cm, respectively. Each plate was 0.1 cm in thickness.
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