What is the difference between an ionic and molecular activity model? (Example: Diopside-Hedenbergite) Original question:

When looking at a Diopside-Hedenbergite solid solution, what is the difference between an ionic and molecular activity model? Why does the moleclar model yield higher values?

Post-Edit question:

The two activities calculated below seem to me like I'm just comparing some random values. What is the true activity of the Fe- or Mg-mineral?

So far I got:

Dipside and Hedenbergite are Pyroxenes. Pyroxenes have to metal sites (M1 and M2):

\begin{align}
M^1M^2[Si_2O_6]
\end{align}

The difference between Hedenbergite and Diopside is:

\begin{align}
Hedenbergit (Fe^{[M1]}Ca^{[M2]}[Si_2O_6]) \longleftrightarrow Diopside (Mg^{[M1]}Ca^{[M1]}[Si_2O_6])
\end{align}

The activity of an element is assumed to be equal to the concentration:

\begin{align}
a=\color{Red}{\gamma}x \\
a=x
\end{align}

In the next step one has to calculate the concentrations on the M1 site because it's the site where the ion exchange happens:

\begin{align}
x[Fe]=\frac{[Fe]}{[Fe]+[Mg]}
\end{align}

\begin{align}
x[Mg]=\frac{[Mg]}{[Fe]+[Mg]}
\end{align}

The molecular mixing model apparently compares the activity of the Fe- and Mg-mineral on the M1 site:

\begin{align}
a[Mg]=x[Mg]
\end{align}

\begin{align}
a[Fe]=\frac{1-x[Mg]}{x[Fe]}
\end{align}

Whereas the ionic mixing model compares the activities of all the cations of the minerals:

\begin{align}
a[Mg]=x[Mg]^{[M1]}x[Ca]^{[M2]}
\end{align}

\begin{align}
a[Fe]=x[Fe]^{[M1]}x[Ca]^{[M2]}
\end{align}

Why are there two activities? Shouldn't there be one true activity? What does the one activity mean and what does the other mean?
The activity of a chemical species is a key concept in thermodynamics of solutions. While the thermodynamic potential is the “true thermodynamic variable”, i.e. the one that plays a primary role in statistical mechanics, chemists do tend to like activity (which is univocally related to chemical potential) because it is close to another quantity which has clear physical meaning: concentration.

Now, relating the activity (or chemical potential) of a species to its concentration in a mixture is impossible to do in a general way. Models have been developed to provide approximations, however, the best known being the ideal solution, where activity equals molarity (concentration). Your “molecular mixing model” appears to be a model of ideal solid solution. However, one cannot expect this approximation to hold in case of strongly non-ideal solutions, and ionic solutions are prominent in that category. The “ionic mixing model” you describe thus appears to be another such approximation, more suited to these ions in a solid solution. I have never encountered it before, but I found a series of lecture slides where it seems to be very neatly introduced (see around slide 31).

While I cannot fully address the question, I hope this helps!