Activities in NonIdeal Solutions
Chapter 4
Lecture 10
We will cover Chapter 4
only through section 4.6
(page 144).
You will not be responsible for the remaining material in the
chapter.
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.
o 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.
o We can predict the temperature at which K- and Na-feldspar will exsolve
and use this to determine metamorphic temperatures.
o We can actually predict the plagioclase-spinel peridotite phase transition
o We can determine equilibration temperatures of garnet peridotites.
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.
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 orthopyroxene; 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).
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:
Vex = A + BX2 + CX22 + DX23 +…
• 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).
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:
Vex = BX2 + CX22
• The simplest such case would be symmetric about the
midpoint X1 = X2.
• In that case,
BX + CX 2 = BX + CX 2
2
2
1
• Substituting X1 = 1 – X2, we find that
B = -C
1
Interaction Parameters
Vex = BX2 + CX22
• Let WV = B
Vex = WV X2 -WV X22 = WV X2 (1- X2 ) = X1 X2Wv
• W is known as an interaction parameter (volume
interaction parameter in this case) because nonideal 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.
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.
o what does this imply about ∆Sexcess?
Interaction Parameters
and Activity Coefficients
• For a binary symmetric solution, λ1 must equal λ2 at
X1 = X2 .
• From this we can derive:
RT ln l1 = X 22WG
µ1 = µ1o + RT ln X1 + X22WG
• For X2 ≈ 1 (very dilute solution of X1), then
µ1 = µ1o + RT ln X1 + WG
• and
WG = RT ln h
• What happens when X1 ≈ 1?
Asymmetric Solutions
• More complex situation
where expansion
carried out to third
order.
• Excess free energy
given by:
Gex = (WG1 X2 + WG2 X1 )X1 X2
• Activity coefficient:
Calculated Free Energy at 600˚C
in the Orthoclase-Albite System as
a function of Albite mole fraction.
ln li =
(2WG j - WGi )X 2j + 2(WGi - WG j )X j3
n i RT
Albite-Orthoclase
Free Energy at various T
A 3-D version
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.
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
Inflection Points
• Second derivative is:
æ ¶2 G ö
RT æ ¶2 Gex ö
çè ¶X 2 ÷ø = X X + çè ¶X 2 ÷ø
2
1 2
2
• First term on r.h.s. is
always positive
(concave up).
• Inflection will occur
when
æ ¶2 Gex ö
RT
£
çè ¶X 2 ÷ø
XX
2
1
2
Spinodal
Actual solubility gap can be less than
predicted because an increase is free
energy is required to begin the exsolution
process.
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).
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:
DH m
TM =
∆ Sm
Assuming ∆S and ∆H are
independent of T:
Ti,m
T Rln ai,
= 1- i,m
T
∆ H i,m
•
T-X phase diagram for the
system anorthite-diopside.
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.
Computing an Approximate Phase
Diagram
We assume the liquid is an ideal solution (ai = Xi)
and compute
over the range of Xi
T=
∆ H i,m
∆ H i,m
- R ln Xi
Ti,m