4-section - 1 4. Canonical Ensembles and Molecular...

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1 4. Canonical Ensembles and Molecular Partition Functions A canonical ensemble is an ensemble consisting of a very large number N of “ equivalent” systems, where equivalent means that all have the same composition (i.e., the same numbers of particles of each type), the same volume V , and the same set of internal energy levels, and have the same β value! Note that as for microcanonical systems, specifying the composition (represented by the number of particles N ) and and the volume V eFectively uniquely speci±es the set of all possible energy states ε i of the particles comprising the system. 4.1 Systems of Distinct ( distinguishable )Part ic les Consider a system consisting of two independent particles, particle a whose distinct states labelled with the index i a have energies ε a ( i a ) particle b whose distinct states labelled with the index i b have energies ε a ( i b ) In general these indices i a and i b may represent a collection of quantum numbers associated with the various modes of motion of each particle, but initially we treat them collectively. ²or the overall system the total energy is clearly ²or the overall two-particle system, the partition function will be
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2 Stat Mech Summary . . . to date . . . All distinct microstates of a system occur with equal probability The (very) long-time average (or thermodynamic value) of a property of a system is equal to the average of that property over an ensemble of equivalent systems For large N ,l n ( N !) N ln( N ) N For a system composed of a very large number N of objects (atoms/molecules), the most probable distribution of population over the possible states of those objects is so overwhelmingly dominant that the probability of encountering any distribution deviating signi±cantly from the most probable one is negligible For a system composed of a very large number N of objects, at equilibrium, the the distribution of the population of those objects over their possible internal states (states i with energy ε i ) is governed by the “Maxwell-Boltzmann distribution law”: a i = N e βε i q ( N,V,β ) where q ( )= X all states i e i in which β is a non-negative constant. If W mp is the statistical weight (the number of distinct microstates) associated with a system in its most probable distribution, and a tiny amount of energy dE tot is input into that system, dW mp = βdE tot If two microcanonical systems in their most probably distribution, system A characterized by β A and system B characterized by β B , are brought into thermal contact (so that energy may flow, but particles may not), the fact that necessarily dW tot 0m e a n st h a t ( β A β B ) dE A 0 This leads us to associate the fundamental physical property “ β ”w i thth e inverse of the Kelvin temperature T (since experience tells up that heat spontaneously flows from hot to cold), and introducing the proportionality constant k B , we write β = 1 k B T
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3 For a system composed of N a distinguishable particles of type a , N b
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4-section - 1 4. Canonical Ensembles and Molecular...

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