Oxidation state, and the potential conjugate to it

A Portland cement can be modeled without ever mentioning oxidation states. Its sulfur is sulfate, its iron is ferric, and nothing in the paste changes either. A cement made with blast-furnace slag cannot: the slag brings sulfur as sulfide, S(-II), into a pore solution whose sulfur is sulfate, S(+VI), and a calculation that cannot hold both has already decided the answer.

This page is about what has to be added to a Gibbs minimization for it to hold both, and — as importantly — about what that addition does not buy.

Charge is conserved separately from the elements, but not always

The constraint of an equilibrium calculation is A n = b: the amounts must reproduce the component totals. The question is what the components are.

Take a system whose every element has a single valence — sodium always Na(I), chlorine always Cl(-I). Then the charge of a species is a fixed linear function of its composition: one sodium contributes +1, one chloride −1, always. The charge row of the stoichiometric matrix is a combination of the element rows, it adds no information, and carrying it would impose the same constraint twice.

Now add sulfur at two valences, say SO4-2 and HS-. No fixed multiplier turns a sulfur atom into a charge any more: it depends on which sulfur. The charge row is then independent, and dropping it would let the calculation move electrons around for free — turning sulfide into sulfate at no cost, which is precisely the wrong answer.

So the test is a rank test, and it is the one the package performs:

When it holds, Zz — the unit-charge pseudo-component — is kept as a row of the conservation matrix, and the oxidation state of the system is conserved alongside its elements. When it does not, the row is dropped as redundant.

Both ranks are computed exactly

The matrices here hold element counts and charges, so their rank is an integer fact about the chemistry and the test above is a comparison of two integers. Computing them by counting singular values above a tolerance would answer that question with a floating-point comparison, and a singular value near the threshold then decides differently on different machines — which is not a rounding difference but a different conservation law: with the charge row dropped, the electron's column is identically zero and a half-reaction "balances" without it. Both ranks come from exact rational row reduction, so this system either has redox freedom or it does not, everywhere.

This happens more often than a reader expects

Species lists are usually built with speciation, which derives species from the atoms of what it is given. Ask for SO4-2 and it brings HS-, SO3-2 and S2O3-2 along, because they are made of the same atoms. Any system containing sulfate therefore has redox freedom whether or not its author wanted any — which is correct, and worth knowing before reading a result.

The potential conjugate to it

Conserving a quantity is half of a thermodynamic description; the other half is the intensive variable conjugate to it. For the elements those are the chemical potentials. For charge it is the electron activity, written

with the Faraday constant — 0.05916 V per pe unit at 25 °C. pe and are the same statement in different units, as pH and the proton chemical potential are.

There is no electron to read it from

Here the analogy with pH breaks, and the break is structural rather than an inconvenience of implementation. pH is read off a species: H+ is in the system, it has an activity, take its logarithm. No database tabulates a free electron in solution, because there is no such thing to tabulate — the electron in a redox reaction is an accounting device, not a dissolved substance.

So the electron activity is inferred, from a couple that is in the system. Any redox couple obeys a half-reaction, which the package balances from the element and charge balance alone:

Its equilibrium constant relates the activities of the members to the activity of the electron, so with the members' activities known the electron's follows:

The is computed from the same standard Gibbs energies as everything else, with the electron at the conventional standard state   — the same convention that puts H+ at zero, and equally a convention rather than a measurement. That it is the usual convention is what makes the numbers comparable with published half-reaction constants:

half-reaction	computed here	published
	33.69	33.66
	13.02	13.03

Computed from CEMDATA18's own data, so the agreement also checks that CEMDATA18 and the sources of those published constants share a reference state.

What this does not buy: one potential per system

A single pe describes a system only if every couple in it is at mutual equilibrium. That is a strong assumption, and in a cement it is false.

The demonstration costs nothing. On one solution carrying sulfate, sulfide and both irons at comparable amounts, the two couples report

couple	pe
	+13.0	+0.77 V
	−3.7	−0.22 V

almost a volt apart. Neither is wrong. They are the potentials of those couples, and they differ because the solution is not at redox equilibrium with itself: sulfate reduction is kinetically frozen on any time scale a cement cares about, so the sulfur couple retains whatever state the slag gave it while the iron couple relaxes.

This is why pe takes the couple as an argument and reports which one it used. Computing it from two couples and comparing them is a measurement of how far the single-potential assumption is from holding, and it is worth making before trusting either number.

Two ways to pose a redox calculation

Conserve it. The default, and what a sealed paste does: the charge row is in b, the oxidation state is whatever the reactants brought, and the potential comes out as a result. Nothing needs to be said; it happens because the component is there.

Prescribe it. FixedpE and FixedEh impose the potential and let the oxidation state follow, for a system genuinely open to an outside buffer — a controlled atmosphere, an electrode, a measured one wants to reproduce. The mechanism is the implicit titrant of FixedpH, with the difference forced by the absence of an electron species: what is prescribed is not one species' activity but the linear combination the half-reaction gives,

one equation, one unknown (the titrant amount), one extra column in the conservation rows.

Prescribing a potential on a closed paste is a modeling error rather than a modeling choice: it says the system exchanges electrons with something, and a sealed specimen does not. The first form is the one a cement wants.

What a slag cement actually needs

Putting the pieces together, the reason this chapter exists:

1. the slag brings S(-II), the clinker brings S(+VI), so the species list spans two valences and the charge row survives — the conservation is automatic;

2. the equilibrium then distributes sulfur between the two, and how far it goes is a thermodynamic result rather than an input;

3. the resulting is strongly reducing, which is why a slag cement protects embedded steel differently from a Portland one, and why its pore solution can carry sulfide;

4. but the answer is an equilibrium answer, and sulfate reduction is slow. A real slag paste is somewhere between the sulfur its slag brought and the sulfur equilibrium would give it, and nothing in a Gibbs minimization knows where.

Point 4 is the honest limit of this chapter. The calculation is now able to pose the question; a kinetic description of sulfate reduction, which this package does not have, is what would answer it.