# Gibbs phase rule

Until now we have not considered any constraint on the number of phases that can coexist in a system at given values of ${\displaystyle T}$ and ${\displaystyle P}$. However such constraints exist and are given by the Gibbs phase rule, which we now derive.

Let us consider a system composed of ${\displaystyle c}$ different components and with ${\displaystyle \varphi }$ coexisting phases (which we will label with roman numbers ${\displaystyle I}$, ${\displaystyle II}$ etc.); from what we have previously stated ${\displaystyle T}$ and ${\displaystyle P}$ must be common to all phases, and all their chemical potentials must be equal. However, in order to specify the composition of the system we need only ${\displaystyle c-1}$ variables, namely the ratios of the concentrations of the various components with respect to the concentration of a previously chosen component. Therefore the chemical potentials will depend on ${\displaystyle T}$, ${\displaystyle P}$ and ${\displaystyle c-1}$ relative concentrations ${\displaystyle (x^{1},\dots ,x^{c-1})}$, not necessarily common to all phases. We therefore have:

${\displaystyle \mu _{I}^{1}(T,P,x_{I}^{1},\dots ,x_{I}^{c-1})=\mu _{II}^{1}(T,P,x_{II}^{1},\dots ,x_{II}^{c-1})=\cdots =\mu _{\varphi }^{1}(T,P,x_{\varphi }^{1},\dots ,x_{\varphi }^{c-1})}$
${\displaystyle \vdots }$
${\displaystyle \mu _{I}^{j}(T,P,x_{I}^{1},\dots ,x_{I}^{c-1})=\mu _{II}^{j}(T,P,x_{II}^{1},\dots ,x_{II}^{c-1})=\cdots =\mu _{\varphi }^{j}(T,P,x_{\varphi }^{1},\dots ,x_{\varphi }^{c-1})}$
${\displaystyle \vdots }$
${\displaystyle \mu _{I}^{c}(T,P,x_{I}^{1},\dots ,x_{I}^{c-1})=\mu _{II}^{c}(T,P,x_{II}^{1},\dots ,x_{II}^{c-1})=\cdots =\mu _{\varphi }^{c}(T,P,x_{\varphi }^{1},\dots ,x_{\varphi }^{c-1})}$

We therefore have ${\displaystyle 2+\varphi (c-1)}$ variables and ${\displaystyle c(\varphi -1)}$ equations; if we want a solution to exist we must have at least as many variables as equations, namely ${\displaystyle 2+\varphi (c-1)\geq c(\varphi -1)}$. This means that:

${\displaystyle \varphi \leq c+2}$
which means that the maximum number of coexisting phases in a system composed of ${\displaystyle c}$ different substances at fixed temperature and pressure is ${\displaystyle c+2}$. In particular, we see that for a single-component ${\displaystyle PVT}$ system we can have at most three coexisting phases for a particular choice of ${\displaystyle T}$ and ${\displaystyle P}$: this is the triple point that we have already encountered.

Equivalently, we can say that in general the number of independent variables with which we can choose to describe the system at the coexistence of phases is:

${\displaystyle f=2+\varphi (c-1)-c(\varphi -1)=c+2-\varphi }$
called thermodynamic degrees of freedom.