A question on this site asked whether a one-component system is at equilibrium when melting or boiling, and the disparate answers were somewhat dependent on the definition of phase equilibrium. Another question asks whether two phases can ever exist in equilibrium at the boiling point. The term "phase equilibrium" seems to imply that under certain conditions, a one-component system with two phases is indeed at equilibrium.

Searching for the definition of phase equilibrium, I found these two:

Definition 1

Phase equilibrium is the study of the equilibrium which exists between or within different states of matter namely solid, liquid and gas. Equilibrium is defined as a stage when chemical potential of any component present in the system stays steady with time.

Source: https://www.wattco.com/casestudy/phase-equilibrium

Definition 2

Phase equilibrium is the state of thermodynamic system, in which the different phases of the substance having common boundary surfaces do not vary quantitatively.

Source: https://www.elsevier.com/books/computational-materials-science/ovrutsky/978-0-12-420143-9

Implications of the distinct definitions

According to definition 1, water present as two phases at the melting point would be at equilibrium even if slowly melting isothermally. According to definition 2, the same situation would not be called equilibrium because the amount of ice is decreasing over time (and thermodynamic parameters such as entropy and inner energy are changing over time).

Phase rule

For the mentioned system, the phase rule specifies one independent degree of freedom (you can change one intensive parameter, say temperature, and all other intensive parameters, in this case just pressure, will be determined, given that the system is supposed to be at equilibrium). Because it is a one-component system, the concentrations (or activities) are constant and are not a degree of freedom. If you are used applying the equilibrium concept to solution chemistry, this is somewhat unusual.

Question: What is the official definition of phase equilibrium, and how does it apply to a one-component system where the chemical potentials are independent of the mole ratios?


The second definition probably refers to the intensive properties of the phases, not the extent of the phases, being invariant, making the two definitions you present equal. The confusion within the first question you link to has to do with the definition of "boiling" and its application to open systems.

Phase transitions of the sort being discussed here are reversible. For instance, quoting a chem libre text:

There are six ways a substance can change between these three phases [solid, liquid, gas]; melting, freezing, evaporating, condensing, sublimination, and deposition(2). These processes are reversible [...]

A reversible process is possible only if all intervening points in the process represent equilibrium states.

The phase rule ($f=c-p+2$) is upheld: here c=1, p=2, so that f=1. You can vary the pressure and then the equilibrium temperature is set (or vice-versa), as described by the Clapeyron equation. This puts no limits on the relative extents of the phases at any particular coexistence point, provided T, p and the chemical potential are not altered. I would restate the condition of equilibrium (which amounts to a definition) as follows$^\ast$:

Two different phases of a substance in contact with each other in a closed system at some uniform temperature and pressure (thermal and mechanical equilibrium) will be in equilibrium if the chemical potential of the substance is the same in both phases.

Note that a process ("boiling", "melting") cannot be an equilibrium state. However, points along the process of melting and boiling are mutually in equilibrium. The proof is that mixing two such states (two states at the MP or two states at the BP) should not alter either of their physical properties.

$^\ast$ Having slightly tweaked my answer to the linked question.

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