The_canonical_partition_function

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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 Boltzmann’s equation, g G ¡ ¢ lnΩ . 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 ª«¤ § ¬ 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 Ω«®,¯,£¬ 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: Ω«® ° ,¯ ° ,£ ° ¬ G ± Ω ² «® ² ,¯ ² ,£ ² ¬Ω ³ «® ° ´ ® ² , ¯ ° ´ ¯ ² ,£ ° ´ £ ² ¬ µ ¶ ,· ¶ ,§ ¶ In this equation, we are summing over all possible values of ® ² ,¯ ² , and £ ² in the general case that the systems can exchange with fixed total ® ° ,¯ ° , and £ ° . Generally speaking, the sum for Ω«® ° ,¯ ° ,£ ° ¬ is extremely sharply peaked around some values ® ² ¸ ,¯ ² ¸ , and/or £ ² ¸ . 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 ® ² ,¯ ² , and £ ² 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: ¤ lnΩ G ¤g G ¥ ¤ lnΩ ¦ ¤g ¦ ¤ lnΩ G ¤¢ G ¥ ¤ lnΩ ¦ ¤¢ ¦ ¤ lnΩ G ¤£ G ¥ ¤ lnΩ ¦ ¤£ ¦ 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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