(91e) On the Thermodynamics of Systems Under the Influence of Gravity

Corti, D. S. - Presenter, Purdue University
Given Pablo Debenedetti’s fondness of, in addition to his nearly unparalleled understanding of, classical thermodynamics, it is therefore fitting to present in his honor on a topic in thermodynamics to which he himself has made important contributions. While there are many examples to choose from, I will limit my discussions here to two aspects of the thermodynamics of systems subject to a gravitational field.

First, I revisit Pablo’s earlier work on the thermodynamic stability of fluids under the influence of gravity (1988, J. Chem. Phys. 89, 6881). In that work, the standard (i.e., gravity-free) thermodynamic stability criteria were shown to be unaffected by the presence of gravity. Thus, the single-phase fluid is thermodynamically stable when exposed to a gravitational field only if the standard stability criteria are satisfied everywhere throughout the fluid. While the validity of this conclusion is not in question, I nonetheless present a new derivation of some of the results contained in Pablo’s earlier work. This new approach makes use of a novel method for obtaining the extrema of a multivariable function in which the initially independent variables are subject to a given set of constraints, and which also avoids the use of Lagrange multipliers. Furthermore, and because Lagrange multipliers are not required, explicit expressions for the second “constrained derivatives” are obtained. Hence, the nature of the constrained extrema (e.g., maxima or minima) can be readily determined. By extending this constrained analysis from the discrete variable case to the continuum variable limit, thereby yielding constrained functional derivatives, thermodynamic extrema principles (based on the minimization of various thermodynamic potentials subject to appropriate constraints) are then invoked to obtain the conditions of equilibrium and stability for fluids in the presence of a gravitational field.

Second, I reconsider a recent analysis of a solid sphere located within a gravitational field (De Palma and Sormani, 2015, Am. J. Phys. 83, 723). First, after initially resting on a platform, the temperature of the sphere is increased, which serves to raise its center of mass due to the resulting thermal expansion. Next, the top of the sphere is then attached to a taut string while (temporarily) removing the platform. Thus, a subsequent decrease in the temperature of the sphere serves to raise yet again its center of mass. Finally, work can be generated by letting the sphere fall back down to the platform (now returned to its initial location). Consequently, a cyclic process has been created that (apparently) violates the second law of thermodynamics. The claimed violation only appears, according to the authors’ analysis, when the internal energy is assumed to be independent of the gravitational field, as is normally done for such systems. No thermodynamic inconsistencies presumably arise if the internal energy is instead chosen to be a function of the strength of the gravitational field. Such a conclusion does not, however, immediately follow from this previous analysis, as certain work interactions during the expansion/contraction of the sphere were not rigorously determined. To show this more clearly, I consider in lieu of the sphere a similar cyclic process generated using an ideal gas in a container with a movable piston (and with a similar net upward movement of the center of mass of the gas). By accounting properly for the pressure variations that develop at equilibrium in a gravitational field, as well as comparing the process to the appropriate Carnot cycle, I show that this cyclic process does not in fact violate any laws of thermodynamics. Thus, the standard assumption of the independence of the internal energy from the gravitational field can still be safely invoked.