# Phase transitions and phase diagrams

Experimentally, any element or compound can be found, depending on the thermodynamic conditions in which it is, in different *phases*. When we say that a system is in a particular *phase* we mean that its physical properties (like density or magnetization) are uniform.
We have also seen that the states of equilibrium for a thermodynamic system are minima of appropriate thermodynamic potentials.
Let us consider for example a system and its Gibbs free energy ; the molar Gibbs free energy will be:

*phase diagram*is a representation of these regions in the appropriate parameter space (for the system we are considering, space). If we now plot the values of as a function of and in space for every phase of the system, we can determine the regions where the two phases will be the stable ones, namely we can determine the phase diagram of the system, as shown in the following figure:

The very interesting region of this space (and the one on which we will focus our attention in this section) is the line where the surfaces of the two phases intersect: along this the two phases coexist, and when the system crosses it we say that it undergoes a *phase transition*.

Obviously, what we have stated can be generalized to systems which exhibit an arbitrary number of phases^{[1]}.

For a system the phase diagram can be represented by a surface in space described by a state equation like (with a function depending on the particular system considered). In this case it is more useful to consider the projection of this surface on and planes. Generally these projections look like those shown in the following figures:

(there are however notable exceptions, like water for example). In particular, systems are generally characterized by the existence of a *triple point* (see Gibbs phase rule) and a *critical point* in the phase diagram (in the first figure these are determined, respectively, by the temperatures and ).
The existence of a critical point has a very intriguing consequence: since the liquid-gas coexistence line ends in a point, this means that a liquid can continuously be transformed in a gas (or viceversa), and in such a way that the coexistence of liquid and gaseous phases is *never* encountered^{[2]}.

Obviously, our considerations are valid also for different thermodynamic systems.

For example, if we consider a magnetic system and its phase diagram (where is the external magnetic field^{[3]}), then we find another critical point as shown in the following figure:

In this case when the system is paramagnetic, namely it has no spontaneous magnetization when ; when however, the system is magnetic and its magnetization can be positive (the total magnetic dipole moment points upwards) or negative (it points downwards), and depends on the sign of .

The same system in space can be represented for different values of the temperature as in the following figure: In particular, we can see that when the magnetization has a jump discontinuity at .- ↑ There are however constraints on how many phases can coexist at a given temperature and pressure, see Gibbs phase rule.
- ↑ It can be legitimately asked if such a point exists also for the solid-liquid transition. As far as we know this doesn't happen, and a reasonable explanation for this has been given by Landau: critical points can exist only between phases that differ quantitatively and not qualitatively. In the case of liquid and gas, in fact, the two phases have the same internal symmetry (both are invariant under
*continuous*spatial translations) and differ only for the mean distance between the particles, while the solid and liquid phases have qualitatively different internal symmetries (solids are invariant only under*discrete*spatial translations). - ↑ For simplicity, we are supposing that the real magnetic field , which is a vector, is directed along an axis, for example the vertical axis, so that we can consider only its magnitude and thus treat it as a scalar.