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Unformatted text preview: © M. S. Shell 2009 1/15 last modified 10/13/2009 Legendre transforms and other potentials ChE210A Until now, we have discussed the entropy and energy versions of the fundamental equation, gG¡, ¢, £¤ and ¡Gg, ¢, £¤ . Notice that the independent variables for these functions are ¡, ¢, £ and g, ¢, £ , respectively. These are the natural variables that are constant in isolated systems. In laboratory conditions, however, things are rarely isolated, and it is often difficult to control precisely the energy, entropy, and volume of a system when outside forces are acting on it. Instead, it is easier to control so-called field parameters like temperature and pressure. Here, we will discuss the proper procedure for switching the independent variables of the fundamental equation for other thermodynamic quantities. In doing so, we will consider systems that are not isolated, but rather held at constant temperature, pressure, and/or chemical potential through coupling to various kinds of baths. We will find that, in such sys- tems, entropy maximization requires us to consider the entropy of both the system and its surroundings. Moreover, new thermodynamic quantities will naturally emerge in this analysis: additional so-called thermodynamic potentials. To achieve a system that is maintained at a constant temperature or pressure, one couples the system to a bath, at which point it is no longer isolated. As we have discussed before, a bath is essentially a large reservoir of energy, volume, or particles that can exchange with the system of interest. The bath is so large that exchanging amounts of volume/energy/particles with the system that change the system state are in fact so small for the bath that it is essentially always at the same state. Some examples: system BATH energy exchange (constant temperature) volume exchange (constant pressure) particle exchange (constant chemical potential) system BATH system BATH heat conducting wall piston membrane © M. S. Shell 2009 2/15 last modified 10/13/2009 Recall our earlier result that systems that can exchange g / G / ¡ reach ther- mal/mechanical/chemical equilibrium by finding the same ¢ / £ / ¤ . Here, the main difference with these previous results is that the bath is so much larger than the system that it stays at the same ¢ / £ / ¤ , and hence it sets the final equilibrium values of these parameters. We consider the bath to be ideal in that it is infinitely large. Constant temperature coupling to an energy bath Let us consider the specific case in which a system comes to thermal equilibrium with an ideal energy bath, i.e., when the system is held at constant temperature. In reaching equilibrium, the entropy of the world is maximized subject to energy conservation: ¥ world ¦ ¥ § ¥ bath ¨maximum at equilibrium© g world ¦ g § g bath ¨constant due to enerªy conservation© Here, the unsubscripted variables are those pertaining to the system. Since the bath is so large relative to the energy transfer between it and the system, we can write the integrated form of...
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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.
- Fall '08