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Boltzmann Factor e-E/kT
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There are two models to explore with these revision questions. ladder2.mdl and bltz2.mdl.
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The e-E/kT function
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Running bltz2.mdl model allows the isothermal atmosphere idea to be explored. The screen shows the function discussed in the A2 question on the isothermal atmosphere. The vertical scale is height in scaled units because k is set to 1 in the model for ease. The temperatures can be varied in the different cases but also in mid run using the slide controls. The horizontal scale shows the relative probability of a particle being at a certain height. The model re-runs when it gets to h=144. This allows you to play with T and g variables on a repeat run.
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The gravitational potential energy of a molecule can increase randomly by that molecule borrowing energy from the other particles around it and the heat bath at temperature T. A complex process is perfectly described through the action of the exponential Boltzmann factor. Of course the real atmosphere is not isothermal.
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1. Play with the T and g parameters and record the effects on the distributions. You can do this using the edit/copy window function in Modellus.
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| Try to explain mathematically why their shape changes as indicated in the animation window.
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2. Set T2 = 100 then adjust "g" to get the same "red" distribution as before T2 was adjusted. Explain why the value of g you chose works. Think again about the Boltzmann factor.
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Jumping around on an energy level ladder
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This is a screen from ladder2.mdl. The slide control allows adjustment of a temperature parameter. The energy level ladder and a population bar chart are also shown. The particle is in equilibrium with a heat bath at temperature T. In the spirit of the dice game from chapter 14 in Advancing Physics A2 the particle has to borrow energy from the heat bath to jump up the energy levels. This is increasingly difficult according to the Boltzmann factor rules. The model just uses a simple dice game to model the random jostling played out in nature between the green particle and its neighbours in the "heat bath".
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The scenario above is replayed at higher T parameter. The differences are clear.
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You can explore the scenarios included in the model. The E/kT parameter is varied in the simulation and the nature of the energy level/population function can be explored.
- Use the numerical data in the column to see if the population numbers are related exponentially. Use the constant ratio definition of exponential to help you do this. You can copy the table into Excel to help analysis.
- Explore the model and determine how E/kT controls the exponential factor between populations on the different energy level ladders at different temperatures. Try to relate your findings back to the e-E/kT factor.
- Look at the green particle shuffling at one temperature and get a feeling for the levels it is accessing. Then increase the temperature. Does its new behaviour make sense? When you change T you must allow some time for the models to settle down again.
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Download this Resource
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Package:
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boltzmann.exe (81Kb)
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Contents:
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bltz2.mdl, ladder2.mdl and boltzman.doc
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Instructions:
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Download the self extracting ZIP archive. Locate the file using Windows Explorer and double click. The self extractor will then start. The default installation path is C:\AP Revision. Having set the path press the Unzip button. After the files have been extracted change directory to C:\AP Revision. To start the models double click on bltz2.mdl and ladder2.mdl
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