1. Activities in Non-Ideal Solutions
Chapter 4
Lecture 10

2. We will cover Chapter 4 only through section 4.6 (page 144).
You will not be responsible for the remaining material in the chapter.

3. Power of Solutions
Our thermodynamics would be so much simpler if all solutions behaved ideally! Even simpler if there were no solutions at all!
But, once we learn how to handle them, we’ll see that we can use solution behavior to do some real geochemistry and learn things about the Earth.
Because the distribution of Mg and Fe between olivine and a magma depends on temperature, we can use the observed distribution to determine magma temperatures.
We can predict the temperature at which K- and Na-feldspar will exsolve and use this to determine metamorphic temperatures.
We can actually predict the plagioclase-spinel peridotite phase transition
We can determine equilibration temperatures of garnet peridotites.

4. In This Chapter We’ll learn how to handle non-ideal solutions
Learn how to construct phase diagrams from thermodynamic data
Learn how thermodynamics is used for geothermometry and geobarometry
See how thermodynamics can be used to predict the sequence of minerals precipitating from magma.

5. Non-Ideality
We found a useful approach to non-ideality in electrolyte solutions (Debye-Hückel), but there are many other kinds of non-ideal solutions, including solids, liquid silicates, etc.
Some substances undergo spontaneous exsolution (oil and water; K- and Na-feldspar; clino- and ortho-pyroxene; silicate and sulfide magmas). When that happens, we know the solution is quite non-ideal (ideal solutions should always be more stable than corresponding physical mixtures).

6. Margules Equations
When you need to fit an empirical function to an observation and don’t fully understand the underlying phenomena, a power series is a good approach because it of its versatility.
So, for example, we can express the excess volume of a solution (e.g., alcohol and water) as:
where X2 is the mole fraction of component 2 and A, B, C, … are constants, Margules parameters, to be determined empirically (e.g., by curve fitting).

7. Symmetric Solutions
Let’s first look at some simple cases. One such case would be where we need only the first and second order terms.
The excess volume (and other excesses) should be entirely functions of mole fraction, so for a binary solution where X1 = 1 - X2, A= 0 and our equation should reduce to:
The simplest such case would be symmetric about the midpoint X1 = X2.
In that case,
Substituting X1 = 1 – X2, we find that
B = -C

8. Interaction Parameters
Let WV = B
W is known as an interaction parameter (volume interaction parameter in this case) because non-ideal behavior results from interactions between dissimilar species.
The interaction parameter is a function of T, P, and the nature of the solution, but not of X.
We can define similar interaction parameters for free energy, enthalpy, and entropy.

9. Regular Solutions Since ∆G = ∆H-T∆S
The free energy interaction parameter is:
WG = WH – TWS
Regular solutions are the special case where WS = 0 and therefore WG = WH and WG is independent of T.
what does this imply about ∆Sexcess?

10. Interaction Parameters and Activity Coefficients
For a binary symmetric solution, λ1 must equal λ2 at X1 = X2.
From this we can derive:
For X2 ≈ 1 (very dilute solution of X1), then
and WG = RT ln h
What happens when X1 ≈ 1?

11. Asymmetric Solutions
More complex situation where expansion carried out to third order.
Excess free energy given by:
Activity coefficient:
Calculated Free Energy at 600˚C in the Orthoclase-Albite System as a function of Albite mole fraction.

12. Free Energy at various T
Albite-Orthoclase
Free Energy at various T
A 3-D version

13. G-bar–X and Exsolution
We can use G-bar–X diagrams to predict when exsolution will occur.
Our rule is the stable configuration is the one with the lowest free energy.
A solution is stable so long as its free energy is lower than that of a physical mixture.
Gets tricky because the phases in the mixture can be solutions themselves.

14. Inflection Points
At 800˚C, ∆Greal defines a continuously concave upward path, while at lower temperatures, such as 600˚C (Figure 4.1), inflections occur and there is a region where ∆Greal is concave downward. All this suggests we can use calculus to predict exsolution.
Inflection points occur when curves go from convex to concave (and visa versa).
What property does a function have at these points?
Second derivative is 0.
Albite-Orthoclase

15. Inflection Points Second derivative is:
First term on r.h.s. is always positive (concave up).
Inflection will occur when

16. Spinodal
Actual solubility gap can be less than predicted because an increase is free energy is required to begin the exsolution process.

17. Phase Diagrams
Phase diagrams illustrate stability of phases or assemblages of phases as a function of two or more thermodynamic variables (such as P, T, X, V).
Lines mark boundaries where one assemblage reacts to form the other (∆Gr=0).

18. Thermodynamics of Melting
Melting occurs when free energy of melting, ∆Gm, is 0 (and only when it is 0).
This occurs when:
∆Gm = ∆Hm –T∆Sm
Hence:
Assuming ∆S and ∆H are independent of T:
where Ti,m is the freezing point of pure i, T is the freezing point of the solution, and the activity is the activity of i in the liquid phase.
T-X phase diagram for the system anorthite-diopside.

19. Computing an Approximate Phase Diagram
We assume the liquid is an ideal solution (ai = Xi)
and compute
over the range of Xi