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04_562ln08

# 04_562ln08 - MIT OpenCourseWare http/ocw.mit.edu 5.62...

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MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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5.62 Lecture #4: Microcanonical Ensemble: Replace {P i } by . Q vs. . To this point, we have worked with the CANONICAL ENSEMBLE: P(E) = Ω (N,V,E)e E/kT Q(N,V,T) • probability of finding an assembly state with energy E in the ensemble • probability of finding the “gas” with energy E A physical picture that describes the canonical framework is Heat Bath Gas N is constant V is constant T is constant E fluctuates (HOW MUCH?) The energy of the gas fluctuates (with time or for different states within the ensemble). Extra energy is withdrawn from the heat bath or is deposited in the heat bath so that the temperature of the gas remains constant. A simpler ensemble that is also quite useful is the microcanonical ensemble The MICROCANONICAL ENSEMBLE is a collection of assemblies in states in which N, V, and E are fixed. Since all states of a microcanonical ensemble have same energy, E α = E β = E γ = … E, all assembly states are degenerate. (N,V,E) = degeneracy [e.g. particles in cube: (n x ,n y ,n z ) = (211), (121), (112)] = number of distinguishable assembly states with N, V, and E fixed = total number of assembly states in microcanonical ensemble.
5.62 Spring 2008 Lecture 4, Page 2 A physical picture of microcanonical framework is SYSTEM IS ISOLATED Gas • E is constant • T fluctuates Why have different ensembles? • Some physical situations more closely correspond to one ensemble or another (there are more than these two). • Some problems are easier to solve in the context of one ensemble or another • Results for macroscopic properties are independent of which type of ensemble is used. MICROCANONICAL ENSEMBLE: CALCULATION OF THERMODYNAMIC PROPERTIES (Macroscopic Observables from Microscopic Properties) As in Lecture #2, we want the set of assembly state probabilities which maximizes entropy subject to the normalization constraint. Ω

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04_562ln08 - MIT OpenCourseWare http/ocw.mit.edu 5.62...

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