In general, the number of degrees of freedom
in a system of phases is equal to the number of system components + 2
* minus* the number of phases:

F = C + 2 - p

the
greater the number of components C, the **greater** the number of
degrees of freedom; the greater the number of coexisting phases, p,
the** fewer ** the number of degrees of freedom.

Let's explore this further. The aim of chemical thermodynamics is to
define, fully and quantitatively, the potential reactivity of materials.
The fundamental quantity that expresses reactivity is the chemical potential. We
define chemical potentials (reactivities) for components of phases. For
every independently variable constituent of the phases in a system (the
phase components) there is a corresponding chemical potential. The
stoichiometric relationships (the so-called "** balanced chemical
reactions**") among the phase components are R in number, where R is
given by the difference between
the total number of phase components (the sum of the components of each
phase, summed over all the phases) minus the number of system
components.

For example, in the simple case of a two phase binary CO2 - H2O system (e.g., bottled sparkling water, or a naturally carbonated spring near e.g., Saratoga NY, USA) there are four phase components (CO2 and water in the gas, and CO2 and water in the liquid. We can write two conditions of chemical equilibrium for this system:

u{CO2(g)} = u{CO2(liq)}

and

u{H2O(g)} = u{H2O(liq)}

where, e.g., u{CO2(g)} is the chemical potential of CO2 in the gas phase.
The ***components of the system*** are CO2 and H2O, and are
*two* in number. If
we subtract the number of system components (2) from the total number of
phase components(TPC) (4), we get the number of "reactions":

R = 4 - 2 = 2.

We can then re-write the phase rule as:

P+F = TPC +2 - R

When the number of degrees of freedom is ZERO, we have NO freedom to
choose the T and P; (for example, when gas, ice and liquid coexist
in the pure H2O system) nature chooses a unique point in T,P space. We
refer
to such a (F=0) assemblage as
** Invariant**.

In an invariant phase assemblage, the number of degrees of freedom is zero. This is really a linear algebraic realtionship among a set of variables [the sum of the chemical potentials of the mineral and fluid-phase *end-members*, (the phase components) and the temperature and pressure of each phase] and a set of constraining equations [comprising the thermal and mechanical conditions of equilibrium (i.e, T(liquid) = T(gas), P(liquid) = P(gas)), the Gibbs-Duhem equations (which are "P" in number since there is one for each phase), and the number of linearly independent *"reactions"* (actually, the number of independent conditions of chemical equilibrium, expressed in terms of the chemical potentials of phase components).

Note that the "**2**" in the phase rule represents the net difference
between {the number of thermal + mechanical variables}, and {the number of
independent thermal and mechanical conditions of equilibrium}); i.e., it
represents the fact that the number of conditons of thermal and mechanical
equilibrium is one less than the number of phases!

P.A. CANDELA