Redox in Magmatic
Systems
Activities in Non-Ideal
Solutions
Lecture 10
Line 5 has a slope of -1 and an
intercept of log K.
We can also use pe-pH diagrams to
illustrate stability of solid phases in
presence of solution. In this case, we
must choose concentration.
More about pe-pH
diagrams
• pe-pH diagrams are a
kind of stability or
predominance diagram.
• They differ from phase
diagrams because lines
indicate not phase
boundaries, but equal
concentrations.
o There is only 1 phase in this this
diagram – an aqueous solution.
• Regions are regions of
predominance.
o The aqueous species continue to
exist beyond their fields, but their
concentrations drop off
exponentially.
Environmental
Interpretation of pe-pH
Redox in Magmas
Oxygen Fugacity
• Igneous geochemists use oxygen fugacity ƒO2 to
represent the redox state of the system. Hence, the
oxidation of ferrous to ferric iron would be written
as:
2FeO + O2(g) = Fe2O3
• For example, oxidation of magnetite to hematite:
2Fe3O4+ ½O2(g) = 3Fe2O3
• (Actually, there isn’t much O2 gas in magmas.
Reaction more likely mediated by water and
hydrogen).
Redox in Magmatic
Systems
• For magnetite-hematite
• 2Fe3O4+ ½O2(g) = 3Fe2O3
• assuming the two are
pure solids
K MH =
1
ƒ1/2
O
2
K=e
-∆ G of /RT
• At a temperature such as
1000K
æ 6∆ G of Fe O ,1000 - 4∆ G of Fe O ,1000 ö
1
( 2 3
)
( 3 4
)
÷
-log K = log ƒO2 = çç
÷
2
2.303RT
è
ø
Oxygen Fugacity Buffers
• The log ƒO2 – T diagram is a
phase diagram illustrating
boundaries of phase
stability. The two phases
coexist only at the line.
• Reactions such as
magnetite-hematite (or ironwüstite or fayalitemagnetite-quartz) are
buffers.
• For example, if we bleed O2
into a magma containing
magnetite, the ƒO2 cannot
rise above the line until all
magnetite is converted to
hematite (assuming
equilibrium!)
Real Solutions:
Minerals and Magmas
Chapter 4
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
or
Vex = BX2 - BX22
• 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
X 1 = 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.