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Thermodynamical relations at characteristic points

We define the characteristic temperature T C of a material by using the behaviour of the specific heat in the vicinity of this temperature: C p,N (T ) = f(1 − (T/T C ),p,N ) for T < T C and C p,N (T ) = g((T/T C ) − 1,p,N ) for T > T C where p is the external pressure and N is the number of particles. The characteristic temperature T C = T C ( p, N ) can e.g. be identified with the transition temperature in the case of continuous phase transitions or with one of the two characteristic temperatures (on heating) and (on cooling) in the case of discontinuous phase transitions. Using the idea of the characteristic temperature, we present a new method on how to calculate the Gibbs potential G = G(T, p, N) in the vicinity of the characteristic temperature, without it being a necessity to know the explicite expressions for f and g. This general and uniform approach allows to derive many thermodynamical relations connecting measurable quantities at the characterisic point as e.g. Clausius–Clapeyron equation and its analogs, valid for discontinuous phase transitions and 15 relations valid for both continuous and discontinuous phase transitions, 6 of them are new. As a special case of these latter relations, we find six relations valid for continuous phase transitions (Ehrenfest equation and its analogs). The derived relations result exclusively from the definition of the characteristic point and are independent from the underlying microscopical mechanism leading to phase transitions (as e.g. solid–liquid, liquid–vapour, magnetic, ferroelectric, structural and superconducting transitions) and therefore, they are generally valid. These relations can be very important for material science. Using some of them it is possible to predict (without performing additional experiments) whether the transition or characteristic temperature T C of a given material sample will increase or decrease under doping (injection) or applied pressure. Similar 24 thermodynamical relations can also be found with ease for systems described by the Helmholtz potential F = F(T,V, N ).