The_canonical_partition_function - last modified 11/4/2009...

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Unformatted text preview: last modified 11/4/2009 M. S. Shell 2008 1/15 The canonical partition function ChE210A A review of statistical mechanical concepts thus far Earlier in the course, we discussed the microscopic basis for the entropy and Boltzmanns equation, g G ln . Some of these statistical mechnical concepts that we encountered in those discussions are listed below: A microstate is just one configuration of the system. In a classical system of spherically symmetric particles, one microstate is characterized by a list of the 3 positions and 3 momenta , for a total of 6 pieces of information. For any microstate or , we can calculate the total, potential, and kinetic energies. In a classical system, this is because a microstate specifies the posi- tions and momenta of all the atoms. The potential energy function depends on the positions and the kinetic energy function depends on the momenta . In an isolated system at equilibrium, the system visits each microstate consistent with the total energy with equal probability. That is, the system spends an equal amount of time in each of the ,, microstates. This is the rule of equal a priori probabilities . If two systems can share energy, volume, and/or particles, the number of micro- states for the combined system (which is isolated) is given by a simple multiplici- ty for choosing different microstates in each: , , G , , , , , , In this equation, we are summing over all possible values of , , and in the general case that the systems can exchange with fixed total , , and . Generally speaking, the sum for , , is extremely sharply peaked around some values , , and/or . That means, for these values, the num- ber of microstates is extremely large and the system spends almost all of its time in these states. Therefore, from a macroscopic perspective, we almost always see single, precise values of , , and because the fluctuations away from these dominant values are small and rare. last modified 11/4/2009 M. S. Shell 2008 2/15 The observed values g G , G , and G are given by the maximization conditions: ln G g G ln g ln G G ln ln G G ln These conditions ensure equality of temperatures, pressures, and chemical po- tentials between the two systems. In subsequent lectures, we considered the properties of the entropy at a macroscopic level....
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This note was uploaded on 12/29/2011 for the course CHE 210a taught by Professor Staff during the Fall '08 term at UCSB.

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The_canonical_partition_function - last modified 11/4/2009...

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