Conditions of Multiphase Equilibrium

Below it will be shown, for a multiphase system, that the chemical potential in each phase must be uniform and equal.

Consider the following simple multiphase system:

Figure 25-1: An example of a multiphase system. $P$ and $T$ are constant and equilibrium with a reservoir.

Application of the conditions of internal equilibrium to the entire system considering that it is composed of $f$ phases:

$$\input{equations/19-1A}$$ (25-10)

$\mu^j_i$ is the chemical potential of chemical species $i$ in phase $j$.

Write this out for a three ( $f=3$) phase system composed of two ( $C=2$) species $W$ and $B$:

For a closed system,

$$\input{equations/19-2A}$$ (25-11)

this follows for each possible species $i$, therefore:

$$\input{equations/19-2Abis}$$ (25-12)

In other words, the chemical potentials of any chemical species is equal in all the present phases.

Or if we number the species $i=1,2,\ldots,C$ and the number of phases $j=I,II,\ldots,f$:

$$\input{equations/19-4A}$$ (25-13)

Each row has $f-1$ equal signs; i.e. $f-1$ equations. So in the above there are $C(f-1)$ equations.

In addition we have, via the Gibbs-Duhem equation for each phase, another relation between the variables:

$$\input{equations/19-3A}$$ (25-14)

that gives us another $f$ equations. Therefore,

Let the number of free variables be $D$ (degrees of freedom). Then,

$$\input{equations/19-3B}$$ (25-15)

or:

$$\input{equations/gibbs-phase-rule}$$ (25-16)

This is a relation between the number of degrees of freedom in a system and the number of components. Commonly, one can think of the number of degrees of freedom in a system as the number of phases that can co-exist, or $D = P$.

And this brings to mind the following..... limerick:

There was a recent graduate from MIT
Who was forced to send back her course three degree
she couldn't make a phase plot
Because she had simply forgot
that P + F = 2 + C