### Gibbs Phase Rule for Reactive Systems

It is useful to recap the origins of Gibbs Phase Rule in order
to see how the existence of reactions impacts the overall number of
degrees of freedom available.

Recall that, we get adjustable intensive variables from (that
is, we can independently set values for each of these intensive
variables):

- $T$ and $P$ (the state postulate says we only get two
- $N-1$ compositions (since for a mixture $\sum_ix_i =1$ must
also be satisfied)
**for each phase** ($\pi$), so we
can fix $(N-1)\pi$ compositions
- Total: $2+(N-1)\pi$

We then have Thermodynamic relations that relate these intensive
variables (that is, they reduce our degrees of freedom):

- For multi-phase systems (with $\pi$ phases), we have
($\pi-1$) equilibrium relations $\hat f_i^{\alpha} = \hat
f_i^{\beta}$ for each of the $N$ components $i$ (for example for a
2 component, 2 phase system that satisfies Raoult's Law, we can
write both $x_1P_1^{sat} = y_1P_{tot}$ and $x_2P_2^{sat} =
y_2P_{tot}$)
- Total: $(\pi -1)N$

Combining these two arguments, we get the non-reactive Gibbs
Phase Rule that the number of degrees of freedom, $D$ is:

$D = (2+N\pi - \pi) - (N\pi -N) = 2+N-\pi$

If we now include a number of reactions ($r$), each must satisfy
an equilibrium expression, so

- For each $j$ of the $r$ reactions we can write $K_j = \Pi_i
\left [\frac{\hat f_i}{f_i^o} \right ]^{\nu_{i_j}}$

Adding these "extra" Thermodynamic relations to our existing
rule yields the reactive Gibbs Phase Rule where $D$ is now:

##### Outcome:

Calculate the degrees of freedom using Gibbs Phase Rule (for
reactive systems)