The concept of entropy in classical thermodynamics has its roots in attempts to improve the design and efficiency of steam engines; through Sadi Carnot's work, scientists eventually came to realize the significance of the notion of "reversible process" for estimation of the theoretical performance of heat engines (with the most efficient thermodynamic cycle being the "Carnot cycle"). This in turn led to the conception of the "second law of thermodynamics" and to the most important "state function" connected with it -- the entropy. If i understand its history correctly, classically the entropy was an artificial measure of the tendency of thermodynamics events (flow of heat from hot to cold bodies, spontaneous expansion when gas is allowed to make free expansion, etc.) to occur spontaneously; if $\Delta S >0$ for a process that it should naturally occur, and, with the introduction of Helmholtz and Gibbs free energies, the entropy enabled to express "available energy" for work in different conditions.
Later, Boltzmann gave the statistical interpretation of entropy, $S = k_B \mathbb{log \Omega}$ , as a measure of the number of microstates corresponding to the same macro-state. This gave a new way of looking into the second law of thermodynamics (which states the entropy of the universe always rises): since macrostates that correspond to greater amount of micro-states are more probable, the second law states that the universe tends to advance from the less probable to the more probable state, which sounds quite logical.
However, as with all great scientific theories, the final form of the theory somehow hides the evolution of ideas behind it, and renders it more difficult to gain insight into the thought process behind it. As a case example for which I lack both the statistical understanding and the historical understanding, I'd like to ask about the Clausius-Clapeyron relation for characterising the coexistence curve between two phases of matter.
- I always found it quite tricky to gain intuition for Clausius-Clapeyron relation, $\frac{dP}{dT} = \frac {L}{T\Delta v}$ , and although I understand its modern derivation (using entropy and chemical potential), I still feel like I don't really understand it. Therefore, I'd like to know how those authors (Emile Clapeyron and Rudolf Clausius) originally thought and derived their relation. This question is interesting since Clapeyron derived this relation before the notion of entropy was introduced.
- In addition, and this question is more physical than historical, I'd like to know how one can give a mechanical statistics interpretation of the change of entropy between two phases of matter, especially between liquid and gas; it's clear to me why there is a difference in the "level of disorder" between solid and liquid (since solid tends to be crystalline), but less clear in the case of liquid-gas phase transition.
