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> At present I am looking at the simple problem of calculating the
equilibrium solution of pure water in equilibrium with the minerals pyrite
and goethite.  The simulation has been performed successfully, but I am now
looking at the equations that PHREEQC is using behind the simulation.  I
have read through the User's Guide and would like to clarify that the
equations I am saying are being used are in actual fact correct.

> Master unknowns:      lnaH2O, lnae-, Waq, lnaH+, lnaFe (master species
for Fe), lnaS (master species for S), npyrite, ngoethite

Also, ionic strength, so total of nine unknowns.

>Equations:
>fH2O
This is the activity of water equation.

>fO ? mole balances on oxygen
>fH ? mole balance on hydrogen
>fz ? charge balance
>fFe ? mole balance on iron
>fS ? mole balance on sulphur
>fpyrite ? mass action expression for pyrite
>fgoethite ? mass action expression for goethite

Plus ionic strength equation, for 9 equations. Sounds right.

> In the output in the description of solution it states next to the
calculated pH 'charge balance' and next to the pe 'adjusted to redox
equilibrium'.  Does this mean that an additional equation is being used
which adjusts the solution to redox equilibrium and calculates the pe in
the same way that the charge balance calculated the pH?  I have not used
?redox in the solution input card and therefore not indicated that any
particular redox couple be used to calculate the pe.

A redox couple or input pe is only used in the initial speciation of a
SOLUTION. It is used primarily if the total concentration of a redox
element is defined. In the initial solution case the redox information is
used to distribute the element among its redox states. In the initial
solution, redox disequilibria is allowed, for instance you can define S-2,
and O2, which under equilibrium would not coexist, and then choose either
couple to distribute total iron.

However, during the "reaction" calculations--any time you bring together a
solution (SOLUTION, MIX, or USE) and any one of EQUILIBRIUM_PHASES,
SURFACE, EXCHANGE, REACTION, REACTION_TEMPERATURE, KINETICS,
SOLID_SOLUTION, or GAS_PHASE--all redox elements react to redox
equilibrium. You can think of it as all of the redox couples reacting to
produce the same pe for each couple. Redox equilibrium is always enforced
in reaction calculations and it is tied up in the charge-, oxygen-, and
hydrogen-balance equations; I usually associate the activity of electron
unknown with the hydrogen-balance equation, but it is really a set of
simultaneous equations and a corresponding number of unknowns, one of which
is the activity of the electron.

> In PHREEQE, the conservation of electrons equation was used to calculate
pe.  Am I correct in saying that the equation is not used in PHREEQC for
forward modelling, but rather the O and H  element balances are included as
equations and the mass of solvent water, (Waq) is included as an additional
unknown?

Yes. Charge balance gets into it too.

> In checking to see if the equations I think PHREEQC is using are the
right ones, I had a look at the equations used to calculate pure water in
equilibrium with oxygen (at fixed pressure).  The unknowns and associated
equations that I think are being used are:

Unknowns  lnaH2O, , lnae-, Waq, lnaH+,  noxygen
Equations
fH2O
f
fO ? mole balances on oxygen
fH ? mole balance on hydrogen
fz ? charge balance
fg - mass action expression for oxygen gas    PO2 = K  [H2O]2 / [H+]4 [e-]4

Let's see, the master equations are faH2O, fmu, fH, fO, and fcharge
balance, plus mass action equations for H2, O2, OH-; unknowns are aH2O, mu,
mass of water, e-, and H+, and from mass action H2, O2, and OH-
(activities). This is the minimum set of equations and unknowns for a
reaction calculation. For an initial solution the mass of water is fixed at
1 kg (usually) and the mole balance equation for O is not used for this
calculation. Charge balance is not included if "charge" is not specified;
and mole balance on H is not included if pe is fixed. However, for reaction
calculations, these equations are always included. In addition, there are
activity coefficients that are based on the ionic strength, that let the
molalities be calculated from the activities, but for conceptualizing, it
is simpler to equate activity and molality, or simply assume the activity
coefficients can always be calculated to make the conversions.

> aOH- is then calculated by mass action equation [OH-][H+] = 10-14

Yes. H2 and O2 are also calculated from mass-action expressions.

> Firstly, I was wondering whether these equations are in actual fact the
correct equations being used.

I think you have done pretty well.

> Secondly, if these equations are correct, how can the charge balance be
included when it is just a linear combination of all the other balances
(OT and HT)?

The charge balance is not redundant, but is another piece of information.
You need all three to balance an equation, 2H2O = 4H+ + O2, balances in
oxygen and hydrogen, but not charge. Same with a solution, if you define
the pH of the solution and do not specify "charge" for any constituent, you
will calculate a charge imbalance for the solution. Another way to look at
it is by varying the charge imbalance for the solution, you get a different
pH. So charge balance is a separate equation with additional information.

David

David Parkhurst (dlpark@xxxxxxxx)
U.S. Geological Survey
Box 25046, MS 413
Denver Federal Center
Denver, CO 80225

Project web page: https://wwwbrr.cr.usgs.gov/projects/GWC_coupled

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Complete Water Resources Division Software