First-order phase transitions are commonly associated with a discontinuous behavior of some of the thermodynamic variables and the presence of a latent heat. In the present study it is shown that this is not necessarily the case. Using standard thermodynamics, the general characteristics of phase transitions are investigated, considering an arbitrary number of conserved particle species and coexisting phases, and an arbitrary set of state variables. It is found that there exist two different possible types of realizations of a phase transition. In the first type, one has the immediate replacement of a single phase with another one. As a consequence, some of the global extensive variables indeed behave discontinuously. In the second type, one has instead the gradual (dis-) appearance of a single phase over a range of the state variables. This leads to a continuous behavior of the (global) basic thermodynamic variables. Furthermore, in this case it is not possible to define a latent heat in a trivial manner. It is derived that the latter (former) case happens if the number of extensive state variables used is larger or equal (lower) than the number of coexisting phases. The choice of the state variables thus place a crucial role for the qualitative properties of the phase transition.
First-order phase transitions play an important role in various different areas of physics and chemistry, but also in every-day life. The most prominent example, which every human encounters almost every day, is the boiling of water. Some other first-order phase transitions are the subject of present-day fundamental research. For example in relativistic heavy-ion collision experiments, e.g., done at CERN or at the future FAIR facility at GSI, physicists are searching for signs of an anticipated phase transition in which hadrons are dissolved into Email address: matthias.hempel@freenet.de (Matthias Hempel) quarks, their elementary building blocks. This so-called QCD phase transition can also occur in neutron stars, if the so-far unknown phase transition density is low enough. The phase transition of water takes place at about 10 orders of magnitude lower temperatures and 14 orders of magnitude lower densities than the QCD phase transition. Nevertheless, both of these phase transitions are described with the same thermodynamic framework for the coexistence of phases.
The standard definition of a first-order phase transition, given, e.g., in Ref. [1] is that one has spatially separated, coexisting phases which can be distinguished from each other by some of their (local) thermodynamic quantities, e.g., by their densities. Another definition is that the first derivatives of the thermodynamic potential behave discontinuously, corresponding to the Ehrenfest classification. In its original formulation [2] (see also Ref. [3]), the Gibbs free energy is used as the thermodynamic potential. In the present article it will be shown that the choice of the thermodynamic potential, corresponding to a particular set of natural state variables, plays a crucial role in this definition. 1 Another common definition is that a first-order phase transition involves a ’latent heat’, which is only applicable for a phase transition at constant temperature. An example for a first-order phase transition at non-constant temperature is the boiling of water in a pressure cooker. Here the boiling takes place over a range of temperatures, in contrast to the constant boiling temperature of water in an open pot. As the temperature does not stay constant, it is not possible to define a latent heat in a simple manner.
This example illustrates the importance of the used state variables: for boiling water in an open pot, the pressure is kept constant, in a pressure cooker it is the volume, and this leads to very different characteristics of the phase transition. In the present work these two variants are referred to as continuous and discontinuous realizations of a first-order phase transition. The term ‘realization’ is meant to refer to a particular thermodynamic process, corresponding to a particular set of state variables which are changed in a continuous way.
Phase transitions can also change their characteristics if one goes to a multicomponent system, i.e., with several conserved particle species or quantum numbers. One example is the aforementioned QCD phase transition in neutron stars. Neutron stars are described as being at zero temperature and in hydrostatic equilibrium, so that the pressure is decreasing strictly monotonically with radius. The conserved quantum numbers are baryon number and the net electric charge, which has to be zero to maintain charge neutrality. Usually local charge neutrality is considered, and then the phase transition happens at a single radius inside the star. If instead the electric charge is treated as a second globally conserved quantum number (ignoring any Coulomb interactions, which is a commonly used simplification in this research field), a spatially extended phase-coexistence region occurs, as the phase transition takes place over a range of pressures [4]. In the field of neutron star physics, these two different realizations are known as ‘Maxwell’ and ‘Gibbs’ phase transitions, respectively. Effects of additionally conserved quantum numbers are also discussed in the context of heavy-ion collisions, see, e.g., Ref. [5]. In chemistry and chemical physics, a phase transition with phase coexistence of two (or more) macroscopic phases with different chemical compositions is also known as a ’non-congruent’ phase transition, see the IUPAC definition [6] and Refs. [7, 8, 9].
In the present paper, the most general case of a phase transition of a multicomponent substance with any number of conserved particle species and any number of coexisting phases (respecting Gibbs phase rule) is considered. It will be shown that only the number of extensive state variables used, E, and the number of involved phases, K, determine whether the realization of the phase transition is continuous (E ≥ K), where all basic thermodynamic variables behave continuously, or discontinuous (E < K) where some of the extensive thermodynamic variables behave discontinuously. Furthermore, in the former case, the intensive variables such
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