The Standard Approximation

Consider the general case:

$$\input{equations/25-3A}$$ (33-4)

Assuming the system is closed, the condition for equilibrium is just:

$$\input{equations/25-3B}$$ (33-5)

which becomes, if reference is made to the pure states

$$\input{equations/25-3C}$$ (33-6)

so the condition for equilibrium becomes

$$\input{equations/25-3D}$$ (33-7)

where the activity of a component is $a_A = \gamma_A X_A$ which is to be determined empirically.

However, there are standard approximations in which the solid phases in the reaction can be considered to be pure. In this approximation, and through use the additional thermodynamic approximations for material behavior:

Approximate equilibrium conditions can be obtained by practical means.

Consider the oxidation (or, the reverse, the reduction) of a pure metal:

$$\input{equations/metaloxide-rx}$$ (33-8)

The chemical potentials of each each component in each solid phase is in equilibrium with the gaseous phase.

$$\input{equations/25-5A}$$ (33-9)

Therefore, it is appropriate to consider equilibrium in the gas phase.

Considering an ideal gas mixture

$$\input{equations/25-5B}$$ (33-10)

What is remarkable about Equation 33-10 is that it is true!

$P_M$ for typical metals is $10^{-8}$- $10^{-12}$atm. $P_{MO}$ for typical oxides is $10^{-18}$- $10^{-24}$atm. Such tiny numbers which would be very, very difficult to measure.

Also $\ensuremath{\overline{G}}_{M(gas)}(P=1,T)$ and $\ensuremath{\overline{G}}_{MO(gas)}(P=1,T)$ represent molar free energies that are highly unstable with respect to forming a solid or a liquid at sub-solar temperatures.

Expressions for $\ensuremath{\overline{G}}_{M(gas)}$ and $\ensuremath{\overline{G}}_{MO(gas)}$ can be obtained by integrating the pressures for the gas phase and the condensed phase:

$$\input{equations/25-6A}$$ (33-11)

For almost every condensed phase the last term in Equation 33-11 is always very small compared to the others, so to very good approximation:

$$\input{equations/25-6B}$$ (33-12)

Putting this into Equation 33-10, the following approximation is obtained:

$$\input{equations/26-7A}$$ (33-13)

In other words,