FUGACITY (and activity).

First, some symbols:

f(i) = fugacity of component i in a mixture.

P = total pressure
Pi = partial pressure of component i
e.g., P(CO_{2}) = partial pressure of CO_{2}

j = activity coefficient of i (subscripts left off for simplicity) J = fugacity coefficient of pure i "

a(i) = activity of component i in a A SINGLE PHASE mixture X(i) = mole fraction " "

µ(i) = chemical potential of component i

ss = standard state K = equilbrium constant

**Fugacity**

Formal definition: whenever we are dealing with the chemical potential of a component in a gas phase, or a component that *MAY* be in a gas phase (see footnote 1), then we can use the fugacity to account for the difference between the chemical potential of interest µ(P,T), and the chemical potential of the pure substance at T,1 bar:

f(i) = exp{[µ(T,P) - µ(T,1 bar, pure)]/RT}

qualitative concept: Note that at a given T,there are only two variables in the above equation: f(i) and µ(T,P). Therefore, as, fi increases, so does µ. A high fugacity of water or oxygen means a high chemical potential of water or oxygen, respectively. A high chemical potential of water or oxygen indicates a "wet" or "oxidized" system, respectively.

BUT what does it MEAN??? well, look at the most general equation for fugacity:

f(i) = j X(i) J P.

If a gas is ideal AND the mixture is ideal, then, f(i) = X(i) P, and fi is also equal to Pi .

So, if you ask "what is the fugacity, and how do I think about it", just think of it as a partial pressure: it is a strong function of the mole fraction of the component in the gas phase, and of the total pressure of the gas phase, just like a partial pressure; more precicely, remembering that chemical potential is a quantitative measure of the reactivity of a component in a phase, we can think of fiugacity as a measure of how much the chemical potential of the component in the gas deviates from the chemical potential of some reference, namely, the standard state, due to changes in P and/or the mole fraction of the component i.

How is it different then, from partial pressure or total pressure? Well, taking a pure gas first, we know that fi deviates from P by:

f(i) = J P.

WHY is this so, you ask? Well, IT GOES BACK TO the fact that the rate of change of the chemical potential of a pure gas (i) with respect to changes in P (at a given T) is equal to the volume of the gas, V. For an ideal gas, V = RT/P; dµ = VdP, then becomes dµ = RT(dP/P), and integration yields µ(P) = µ(1 bar) + RtlnP, where the RtlnP term corrects the one-bar chemical potential up to the pressure of interest, P. BUT, for a non-ideal gas, V has a different value from RT/P, so something other than RTlnP must be used as the "pressure correction" term. The solution to this problem, is, on the surface, a rather "unsatisfying" one: we use Rtlnf(i), instead of RTlnP. Now the pressure has been multiplied by the fugacity coefficient, which are tabulated, or are available as output from computer models, but which ultimately are based on experimental determinations made by by hard working experimental geochemists and others scientists by integrating the *difference* between the observed molar volume of the gas and the ideal molar volume (RT/P), with respect to pressure (dP) from one bar to the pressure of interest (see footnote 2).

So, pressure is still pressure, and fugacity is not a "corrected pressure". Fugacity has meaning only when related to chemical potential, and the correction factor term RtlnF(i) corrects the chemical potential of gas for the fact that tabulated free energies of the gas are at a stated total pressure (usually one bar), and for the pure gas; because the chemical potentail is central to thermodynamics, fugacity becomes so also.

As µ(T,P, pure) = µ(T,1,pure) +Rtlnf(pure )i, we can, at a given T, use f(i)i as a proxy for the chemical potential, when we are thinking about how pressure will affect a reaction where we have a gas phase present (e.g., "ah yes, at T = 200 C, as we increase the fugacity of water in the gyp-anh-gas assemblage, we will definitely favor the stability of gypsum relative to anhydrite" ).

When we have a mixture, fi does double duty, because

fi = j X(i) J P,

which is of course equal to: fi = a(i) f (pure i). Then, for a gas,

mu(T,P, mix) = mu (T,1,pure) +Rtlnf(i),

and we may leave it in this form in our "condition for equilibrium" statement (e.g., when subbing for the FIRST TERM in the equilibrium condition:

mu-CO_{2}(T,P,
vapor,mix) =
mu-CO_{2}(T,P, liquid,mix)

so that it ends up in the equilibrium constant as f(i). IN THIS CASE, the "delta mu - zero, of Rxn" HAD BETTER contain mu (T,1,pure). Remember, K is calculated from "delta mu - zero, of Rxn" = -RTlnK.

NOW, *if you want* you CAN have the activity of (i) in the gas in the equilibrium constant. then,

mu(T,P, mix) = mu (T,1,pure) + Rtlnf (pure i) + RTlna(i) , GAS EQ 1.

AND, mu(T,P, mix) = mu (T,1,pure) +Rtlnf (pure i) is put into the "delta mu - zero, of Rxn"; because "delta mu - zero, of Rxn" = -RTlnK, and the form of K (whether you will use fi or a(i) )for gases HAS TO BE CONSISTENT with what you put into the "delta mu - zero, of Rxn"!!!!!!

EQ 1 is DIRECTLY analogous to what we do for non-gases (liquids and solids, see footnote 3), where

µ(T,P, mix) = µ(T,1,pure) + (P-1)delta V + Rtlna(i) LIQ/SOLID EQ 2

PLEASE COMPARE EQ 1 AND EQ 2.

When you use a(i) for liquid or solid in an equil constant, K, and all you have from a table, etc, is µ(T,1,pure), then you MUST include µ(T,1,pure) + (P-1)delta V into the "delta µ - zero, of Rxn" = -RTlnK.

footnote 1. like when we are dealing with:

1. the vapor pressure of liquid water at 25 C, and **1 bar total pressure without a gas present**; or

2. a system where oxygen or sulfur is a system component, but there is no gas phase (see below).

footnote 2. (These result from many hundreds of measurements of actual molar volumes of gasses, and in the case of gas mixtures, measurements of the molar volumes of the gas mixture,so that once we have the fugacity coefficient, we can also estimate the activity coefficient).

footnote 3. we call liquids and solids "condensed" phases, because of their relative incompressibility. THIS DOES NOT MEAN THAT THEIR CHEMICAL POTENTIAL's DO NOT HAVE TO BE CORRECTED FOR PRESSURE!!!! IT ONLY MEANS WE CAN HOPE TO INTEGRATE VdP to (P-1)delta V, without V changing too much with pressure. We cannot EVEN PRETEND to integrate VdP for gases with V constant, HENCE, fugacity!!!!!!!!!!!

Special Note on oxygen fugacity.

This is a master variable in
geology. f_{O2} is important to you whether you are
working with swamps or stones. Reducing conditions are "low
f_{O2}", and oxiding conditions are "high
f_{O2}". Obviously, these terms are relative. An
"oxidized granite" still formed at an f_{O2} MUCH LOWER than the fO_{O2} = x(i) P
= 0.21 x (1 bar) , or f_{O2} = 0.21 bar of the present atmosphere. (note that
both J and j are assumed to be one in this equation, because the
atmosphere is approximately an ideal gas, AND the gases mix approx.
ideally, respectively).

Also, thermo is quantitative, and f_{O2} MUST be well defined. When both
magnetite (mt) and hematite (hm) are present, the equilibrium constant for the
condition of chemical equilibrium:

6hm = 4mt + O_{2}

is K = f_{O2}. The value of K obviously changes with T, because all the mu
(T,1bar, pure) terms in K=exp("delta mu-zero of Rxn"/RT), along with the
T in "RT", will change as temperature changes. ALSO, K is a function of P
because FOR EACH SOLID, there is a (P-1)delta V term in addition to mu
(T,1,pure) in "delta µ-zero of Rxn". BUT, at any P, T, if mt and hm are
present and are pure, the value of K gives us the oxygen fugacity.

For subsurface environments (just about anything from the water table on down), this f is considered "oxidizing". hematite is not found crystallizing from granites or basalts, but it can occur as a product of the high (250 - 600(?) C) or low (250 - surface) temperature hydrothermal alteration of these rocks. hm or its surficial precursors (Fe(OH)3, goethite, etc.) are stable in oxygenated hydrological or sedimentary environments, and may persist in their diagenetic or metamorphic progeny. Note, however, that as diagenesis/metamorphism progresses, reactions like

hm + FeSiO_{3} (rock frags)--> Fe_{3}O_{4} +
SiO_{2}
(greatly oversimplified) remove
hm from assemblages up-grade.

Just having pyrite present, FeS_{2}, says nothing by itself
about f_{O2}, because hematite and pyrite can coexist,
and do so in what we call the epithermal (hot, near-surface) environment
that obtains in the shallow parts of active volcanic environments where
gold deposits are forming today; BOTH f_{O2} and
f_{S2} are high! If pyrite AND organic matter
(alkanes, fatty acids, proteins/amino acids, saccharides, polysaccharides,
kerogen, cadaverine, putricine, scatoles, indoles, urine, etc.) are
present, THEN f_{O2} is probably pretty low (Note,
however, that at low temperatures { a room T up to about 250 C??} that
"oxidative disequilibrium" is RAMPANT - in those cases, there is NOT a
unique oxygen fugacity. This is one reason why Pt electrde measurements
in most surficial geology environments are meaningless. In a swampy
environment, methane, hydrogen and H2S are prominent, and the equilibria:

CH4 + O_{2} --> CO_{2} + H2O, H2S + O_{2} --> H2SO4 2H2 + O_{2}
--> 2H2O

are
shifted the left; note that the three gases I mentioned will achieve high
fugacities at low f_{O2} when we apply Le Chatelier's
principle. To be more quantitative, however, write the equilibrium
constants for these equilibria, (using fugacity for O_{2},
and all gases; use
activity for H2O and H2SO4, because, at least at low T, they are more
prevalent in the liquid phase). (HOWEVER, WRITING THEM AS ACTIVITIES IS
*NOT* THERMODYNAMICALLY REQUIRED). Then, you will find that the
f_{O2}
is related
to the RATIOS of oxidized to reduced species fugacities/activities.

An additional problem arises at all but the highest
f_{O2}, especially in higher T (>250 C) subsurface
systems; namely, there is NO O_{2} present in ANY of the phases;
the gases
are basically mixtures of water and hydrogen gas plus other gases, but
with no O_{2}. In this case, we say that the
f_{O2} is simply a measure of the
chemical potential of oxygen in the system, and the f_{O2} is "realized" in
the gas by:

f_{O2} = K
{f_{H2O}^{2}}/{f_{H2}^{2}}
. where K is the equilibrium constant for the third
reaction, above. So, in a mineral assemblage, WITHOUT a gas present, if
we have an equilibrium like:

6hm-->4mt+O_{2}, then there is an oxygen fugacity. period.

In an aqueous solution:

Fe2+ + 0.5 H+ +0.25 O_{2} --> Fe3+ + H2O

Unfortunately, you will commonly see this written in terms of
Fe^{2+} --> Fe3+ + e-, and the measure of oxidation is then the
Eh. That is a fine formalism when there are REAL electrochemical
reactions occurring (e.g., in batteries, or when electrons are conducted
thru nuts and bolts on ships in seawater, or even through interconnected
sufides in a weathering ore deposit), but Eh IS A WASTED concept in most
of geochemistry where it leads to many pit-falls. Students then think that
the activity, or "concentration" of electrons is somehow an important
variable, and unfortunately, I have seen many researchers say something
like, "oh, do not worry, there are plenty of electrons floating around in
the solution, etc." ... this is pure fiction, and the reaction HAS NOTHING
TO DO WITH **ANY** electrons floating around anywhere. There IS an oxygen
fugacity, gas phase or not, because the chemical potential of
O_{2} is FULLY defined in ANY thermodynamic system with elements
of variable oxidation state (usually, S,C,H,O,N, Fe, Mn etc). Just write
the equilibrium constant, use the free energy of formation of
O_{2} in the "delta mu-zero of Rxn", and calc the
f_{O2}, which is a measure of the chemical
potential of
O_{2}. This is one of the most powerful concepts in geochemistry.
Even in a silicate melt, the dissolved FeO and Fe2O3 define the
f_{O2}. So we can define f_{O2} for
GROUNDWATERS, MINERAL ASSEMBLAGES, GASES, MELTS, METAMORPHIC ROCKS, ETC.
You can't do that with Eh, or, for that matter, with H2S/SO3, or
H2/H2O ratios, etc.

THE END