# 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?

• Your question intrigues me, so I made some research. This is the only verifiable information I've found so far: doitpoms.ac.uk/tlplib/solid-solutions/index.php – CHM Jun 30 '12 at 2:52
• Looks OK to me (not sure though, not too familiar with this area). Note that "no answer" does not mean off topic-- just that the right people may not have seen it. Your error propagation one is probably off topic though, so Ill leave it deleted for now. Next time, flag the post, though its good you are willing to clean up your own posts :) – ManishEarth Aug 25 '12 at 18:17

## 1 Answer

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!