BoltzmannRelationStatModelsLect13F2011

BoltzmannRelationStatModelsLect13F2011 - Purdue University...

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Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 501: Statistical Thermodynamics Lecture 13: The Boltzmann Relation, Introduction to Statistical Models Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana [email protected] , 765-494-5623 (Phone) September 28, 2011

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Patterned Border Template 2 Purdue University School of Mechanical Engineering Lecture Topics The Boltzmann relation: link between the particle properties and macroscopic properties. Entropy and the particle assembly. The role of the most probable distribution. Formulation of microstates and macrostates for Fermi-Dirac statistics. Formulation of microstates and macrostates for Bose- Einstein statistics. Formulation of microstates and macrostates for Maxwell- Boltzmann statistics. The Boltzmann or dilute limit.
Patterned Border Template 3 Purdue University School of Mechanical Engineering The Boltzmann Relation Connection between assembly thermodynamic properties and the properties of the constituent particles is given by the Boltzmann relation. • Molecular disorder for an assembly is defined as the total number of microscopic states that it may assume consistent with the constraints of total energy and particle number. • Molecular Disorder = Number of Microstates = W tot Connection between molecular disorder and entropy. • Assembly typically contains on the order of 10 23 molecules . • Molecules continually interact via collisions. These particles may be bosons (integral spin) or fermions (half-integral spin). No two fermions can occupy the same quantum state in the assembly. • Statistical mechanics provides the means of averaging, over time, these constantly changing molecular properties to determine thermodynamic properties.

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Patterned Border Template 4 Purdue University School of Mechanical Engineering The Boltzmann Relation Connection between assembly thermodynamic properties and the properties of the constituent particles is given by the Boltzmann relation. • Molecular disorder for an assembly is defined as the total number of microscopic states that it may assume consistent with the constraints of total energy and particle number. • Molecular Disorder = Number of Microstates = W tot Connection between molecular disorder and entropy. • Consider a box with a partition . Initially, volume A is empty and volume B is filled with an ideal gas mixture: A B
Patterned Border Template 5 Purdue University School of Mechanical Engineering The Boltzmann Relation Initially the gas in chamber B has a certain total energy E, which is unaffected by the removal of the partition. Prior to the removal of the partition, the translational energy levels are given by After the partition is removed, the total energy and the number of particles is the same, but the energy level spacing decreases because of the volume increase, trans   1   2   3 h 2 8 m tot B 2/3 ( n 1 2 n 2 2 n 3 2 )

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BoltzmannRelationStatModelsLect13F2011 - Purdue University...

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