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In this paper, we give a succinct derivation of the fundamental equation of classical equilibrium thermodynamics, namely the Gibbs equation. This derivation builds on our equilibrium relaxation theorem for systems in contact with a heat reservoir. We reinforce the comments made over a century ago, pointing out that Clausius’ strict inequality for a system of interest is within Clausius’ set of definitions, logically undefined. Using a specific definition of temperature that we have recently introduced and which is valid for both reversible and irreversible processes, we can define a property that we call the change in calorimetric entropy for these processes. We then demonstrate the instantaneous equivalence of the change in calorimetric entropy, which is defined using heat transfer and our definition of temperature, and the change in Gibbs entropy, which is defined in terms of the full N-particle phase space distribution function. The result shows that the change in Gibbs entropy can be expressed in terms of physical quantities.

A Derivation of the Gibbs Equation and the Determination of Change in Gibbs Entropy from Calorimetry