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Using the Carnot cycle to determine changes of the phase transition temperature

The Clausius-Clapeyron relation and its analogs in other first-order phase transitions, such as type-I superconductors, are derived using very elementary methods, without appealing to the more advanced concepts of entropy or Gibbs free energy. The reasoning is based on Kelvin's formulation of the second law of thermodynamics, and should be accessible to high school students. After recalling some basic facts about the Carnot cycle, we present two very different systems that undergo discontinuous phase transitions (ice/water and normal/superconductor), and construct engines that exploit the properties of these systems to produce work. In each case, we show that if the transition temperature $T_tr$ were independent of other parameters, such as pressure or magnetic field, it would be possible to violate Kelvin's principle, i.e., to construct a perpetuum mobile of the second kind. Since the proposed cyclic processes can be realized reversibly in the limit of infinitesimal changes in temperature, their efficiencies must be equal to that of an ordinary Carnot cycle. We immediately obtain an equation of the form $dT /dX = f(T, X)$, which governs how the transition temperature changes with the parameter $X$.