# Gibbs phase rule

The phase rule (after Josiah Willard Gibbs in 1876, simply phase rule ) states that in thermodynamic equilibrium many not arbitrary phases can coexist side by side. In addition, it can be used to determine the maximum possible degrees of freedom at a certain point in the phase diagram . For a physically homogeneous thermodynamic system , two state variables are sufficient to determine the state of equilibrium.

Gibbs' phase rule is derived from the Gibbs-Duhem equation , which shows that in a thermodynamic system not all intensive variables can be changed independently of one another.

## Phase rule for gases and liquids

For fluids , i.e. gases and liquids , it reads:

${\ displaystyle f = N-P + 2}$ • ${\ displaystyle N}$ - Number of independent types of particles in the system (e.g. H 2 O, CO 2 )
• ${\ displaystyle P}$ - number of stages (different states of aggregation of one or more components or coexisting liquid phases (such as water and oil are immiscible and forming two phases).)
• ${\ displaystyle f}$ - Number of degrees of freedom (here: number of state variables that can be changed without changing the number of phases of the system)

The number of existing phases P depending on N and f is shown in the following table:

${\ displaystyle \ Leftrightarrow P = N-f + 2}$ N = 1 N = 2 N = 3
f = 0 3 4th 5
f = 1 2 3 4th
f = 2 1 2 3
f = 3 0 1 2

For a one-component system ( ), for example , which exists in two different aggregate states ( ), it follows that there is exactly one remaining degree of freedom, that is, a coexistence line in the phase diagram. If three phases exist at the same time (e.g. liquid, gaseous and solid water), exactly one point remains in the phase diagram, the triple point , since no degree of freedom remains (f = 0). ${\ displaystyle N = 1}$ ${\ displaystyle P = 2}$ There must be, it follows from Gibbs' phase rule that in a one-component system there can be a maximum of three coexisting phases at one point (see triple point in the graphic on the right); in a two-component four phases, etc. ${\ displaystyle f \ geq 0}$ ## Phase rule for solids

Since with solids a (small) change in pressure due to the low gas pressure has no or only very little effect, Gibbs' phase rule in this case is:

${\ displaystyle f = N-P + 1}$ However, this simplification does not apply to higher pressures, such as those already in the earth's crust . For the determination of pressure and temperature in geothermal barometry , this is of great importance for the correct recording of the respective equilibrium paragenesis .