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7 - Legendre transforms and other potentials

Published online by Cambridge University Press:  05 April 2015

M. Scott Shell
Affiliation:
University of California, Santa Barbara
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Summary

New thermodynamic potentials from baths

We began our discussion of equilibrium by considering how the entropy emerges in isolated systems, finding that it is maximized under constant E, V, N conditions. In practical settings, however, systems are not often isolated and it is difficult to control their energy and volume in the presence of outside forces. Instead, it is easier to control so-called field parameters like temperature and pressure.

In this chapter, 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 non-isolated systems that are held at constant temperature, pressure, and/or chemical potential through coupling to various kinds of baths. In such cases, we find that 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 conditions of constant temperature, pressure, or chemical potential, one couples a system to a bath. As discussed before, a bath is a large reservoir that can exchange energy, volume, or particles with the system of interest. While exchanging volume/energy/particles alters the system state, such changes are so minuscule for the bath that it is essentially always at the same equilibrium condition.

Type
Chapter
Information
Thermodynamics and Statistical Mechanics
An Integrated Approach
, pp. 123 - 148
Publisher: Cambridge University Press
Print publication year: 2015

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References

Callen, H., Thermodynamics and an Introduction to Thermostatistics, 3rd edn. New York: Wiley (1985).Google Scholar
Denbigh, K., The Principles of Chemical Equilibrium, 4th edn. New York: Cambridge University Press (1981).CrossRefGoogle Scholar
Tester, J. W. and Modell, M., Thermodynamics and Its Applications, 3rd edn. Upper Saddle River, NJ: Prentice Hall (1997).Google Scholar

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