Abstract
The Carnot cycle and the attendant notions of reversibility and entropy are examined. It is shown how the modern view of these concepts still corresponds to the ideas Clausius laid down in the nineteenth century. As such, they reflect the outmoded idea, current at the time, that heat is motion. It is shown how this view of heat led Clausius to develop the entropy of a body based on the work that could be performed in a reversible process rather than the work that is actually performed in an irreversible process. In consequence, Clausius built into entropy a conflict with energy conservation, which is concerned with actual changes in energy. In this paper, reversibility and irreversibility are investigated by means of a macroscopic formulation of internal mechanisms of damping based on rate equations for the distribution of energy within a gas. It is shown that work processes involving a step change in external pressure, however small, are intrinsically irreversible. However, under idealised conditions of zero damping the gas inside a piston expands and traces out a trajectory through the space of equilibrium states. Therefore, the entropy change due to heat flow from the reservoir matches the entropy change of the equilibrium states. This trajectory can be traced out in reverse as the piston reverses direction, but if the external conditions are adjusted appropriately, the gas can be made to trace out a Carnot cycle in P-V space. The cycle is dynamic as opposed to quasi-static as the piston has kinetic energy equal in difference to the work performed internally and externally.
Highlights
The Carnot cycle is the foundation upon which modern thermodynamics has been built, but we owe our modern understanding not to Carnot but to Clausius
As Planck made clear in 1926 [2], even the commonly quoted form of the second law is due to Clausius: there is a property of thermodynamic systems called entropy that is either constant or increases in all changes of the system
Attention has been drawn to the idea that an increase in entropy means that some property with the units of energy, TdS, is increasing in a way that is incompatible with energy conservation
Summary
The Carnot cycle is the foundation upon which modern thermodynamics has been built, but we owe our modern understanding not to Carnot but to Clausius. More recent developments in irreversible thermodynamics have attempted to treat reversible and irreversible processes alike by mapping a space of non-equilibrium states on to the space of equilibrium states and by defining non-equilibrium quantities such as a contact temperature, which determines the entropy exchange with the environment arising from heat flow [6] These are complex mathematical ideas that have the potential to treat complex irreversible phenomena, but still they depend on the notion that entropy is a property of matter and that it is produced in an irreversible process. It was invoked by Clausius to account for internal work against what he called, “the separative force of heat” These three quantities, H, T and Z, either defined, or were defined by, the thermodynamic state and Clausius regarded entropy as a property of a body. The connection between entropy changes in such a cycle and the mathematical notion of entropy as a state function are discussed
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