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Other_ensembles - Other ensembles T he isothermal-isobaric...

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© M. S. Shell 2007 Other ensembles ChE210A The isothermal-isobaric ensemble In the last few lectures, we derived the microscopic probabilities for systems that are at con- stant temperature, i.e., coupled to an infinitely large energy bath. We will now find the same probabilities at both constant temperature and pressure, when the system is coupled to an infinitely large energy and volume bath. In doing so, we will also start to see some general features of partition functions and ensembles. Conceptually, the isothermal-isobaric derivation proceeds in a similar fashion to the canonical ensemble approach. We consider the system plus bath to constitute an isolated system, with constant total energy and volume. The probability of a microstate g in the system is given by the number of times a combined microstate gG appears, where G is any microstate of the bath. Since each combined microstate gets an equal amount of time according to the law of equal a priori probabilities, the probability that the system is in g is proportional to the number of bath microstates that are compatible with g ’s energy and volume: ± ² ³ Ω ´ µ¶ ´ ´ ¸ ³ Ω ´ µ¶ ¹ º ¶ ² ¹ º · ² ¸ ³ exp»¼ ´ µ¶ ¹ º ¶ ² ¹ º · ² ¸/½ ´ ¾ Note that here, a microstate g in the system is characterized by both a specific configuration (and hence energy) and a value of system volume. We now Taylor-expand the logarithm of the bath density of states in both ´ and · ´ and take the limit where the bath is infinitely large: ± ² ³ exp¿ ¼ ´ µ¶ ¹ ¹ ¸ ½ ´ º ² ½ ´ À¼ ´ µ¶ ¹ ¹ ¸ À¶ ´ · ² ½ ´ À¼ ´ µ¶ ¹ ¹ ¸ À· ´ Á terms of order  1/à ´ Ä We can absorb the first term in the exponential into the constant of proportionality, since it does not depend on the microstate g , but only on constant properties of the bath and total system. We can also simplify the bath entropy derivatives by recognizing that they relate to the bath temperature and pressure. Our final expression for the probability of a microstate in the system is: ± ² ³ exp¿º ² ½ ´ Å Æ· ² ½ ´ Å Ä ³ exp»ºÇ¶ ² º ÇÆ· ² ¾
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© M. S. Shell 2007 Relative to the canonical ensemble, we get an additional term in the Boltzmann factor that involving the pressure g (specified by the bath) and the volume associated with microstate G . We know that the probabilities of all of the microstates in the system must sum to one. This fact enables us to find a normalizing factor for ± ² : ± ² ³ exp´µ¶· ² µ ¶g¸ ² ¹ Δº»,g,¼½ Δº»,g, ¼½ ¾ ¿ ¿ exp´µ¶· À µ ¶g¸¹ all À at Á, Á Here, we have introduced the partition function Δ which serves as a normalizing factor for the microstate probabilities. Δ
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