Standard states

Before this page

Thermochemistry, whose notation is used throughout, in particular the split    .

The chemical potential of a species is split in Thermochemistry into a standard potential, read from a database, and an activity, computed by a model. The split is not unique: it rests on a reference state of the species, its standard state, which the database and the activity model must share and which is fixed by convention rather than by physics. This page states the convention adopted by the package for each class of species, the way to pass from one convention to another, and the one case in which the reference energy ceases to be a convention and becomes a statement about matter, namely a surface site whose budget follows a dissolving solid.

1. One potential, two ways of writing it

Once is chosen, the relation

defines the activity, and conversely. The chemical potential itself is a property of the system that does not depend on any convention, and at equilibrium it takes the same value in every phase between which the species can be exchanged; neither the activity nor the standard potential shares that property, only their combination does. Carbon dioxide distributed between a gas phase and water thus has one chemical potential and two activities, a fraction of the pressure in the gas and a molality in the solution. The ratio of the two activities is the Henry constant , and equating the two expressions of the potential shows that    is nothing but the difference between the two standard potentials (Anderson and Crerar, 1993) (§12.6).

A standard state is a state of the pure substance, or of a reference solution, completely specified by its temperature, its pressure, its composition and its state of aggregation. The phrase "25 °C and 1 bar", often met in place of a standard state, fixes two of these four attributes and leaves the other two open (Anderson and Crerar, 1993) (§12.2). Nothing requires the state to be realizable either: the standard state of a solute used below is a hypothetical solution, whose interest lies in the fact that its properties are known, not in the possibility of preparing it.

The temperature of the standard state is always that of the system. The relation above results from integrating    at constant temperature, from the standard state up to the state considered, so that both must be taken at the same ; this is why is stored as a function of temperature and evaluated at the temperature of each state (see Apparent and formation Gibbs energies) rather than as a number.

Pressure is treated according to the model attached to the species. The Helgeson-Kirkham-Flowers equation of state of aqueous solutes depends on , so that the standard state of a solute is at the pressure of the system. The heat-capacity polynomial used for solids and gases carries no pressure term, and their standard state is at   bar whatever the pressure of the state. This is exact at 1 bar. Above it, a condensed phase misses the contribution     and an ideal gas the contribution . For portlandite, with  , the first amounts to   at 10 bar and to   at 1 kbar: negligible for a laboratory sample or a structure, not for a deep reservoir.

2. The conventions in use, class by class

Solutes: the hypothetical ideal one-molal solution

The standard state of a solute is a solution at the molality   mol/kg in which the solute would behave as it does at infinite dilution, that is a solution obeying Henry's law up to one molal. Such a solution does not exist, the interactions between ions being significant well below that concentration, and its properties are obtained by extrapolating measurements made on dilute solutions. The two parts of the choice have separate reasons (Anderson and Crerar, 1993) (§12.4.4 and §12.4.6). Referring the ideal behavior to infinite dilution makes the activity coefficient tend to one in the very limit where the measurements are made, so that every departure from ideality is carried by alone; the value of one molal makes the activity numerically equal to the molality in that limit, whereas any other value would add a constant to every standard potential without any benefit. The activity then reads

being the amount of water and its molar mass. HKFActivityModel, DaviesActivityModel and the Pitzer model compute from the composition of the solution (Activity models). The dilute model takes   and forms the same ratio , which it declares to the aqueous accessors as a molarity, on the ground that a dilute solution has a density close to 1 kg/L; concentration_scale says which reading a model adopts.

The solvent: pure water

Water is referred to the pure liquid at the temperature of the system, with Raoult's law as the ideal limit, so that   as the solution tends to pure water. The dilute model takes  , the mole fraction of water in the aqueous phase. The other models obtain it from the osmotic coefficient ,

which is the form imposed by the Gibbs-Duhem relation once the activity coefficients of the solutes are given, as argued in Activity models §3.

Pure solids

A pure solid is its own standard state, its activity is one and its chemical potential reduces to . It follows that a pure phase contributes to the Gibbs energy a term linear in its amount, and that the minimization decides whether it is present by comparing with the combination of component potentials its formula implies (Thermochemistry §4 and §5).

Gases

A gas is referred to the pure ideal gas at   bar, and its activity in an ideal mixture is then  . The package retains the mole fraction alone, which coincides with it at 1 bar, and the remark on pressure made in §1 applies.

End-members of a solid solution

An end-member is referred to its own pure phase, as a pure solid is, and its activity in an ideal solution is its mole fraction in the phase,  ; the excess models add , with   as the phase becomes pure (Solid solutions). The mole fraction depends on the way the formula of each end-member is written. Doubling a formula halves the number of moles for a given mass, changes every mole fraction of the phase and doubles the standard potential, so that database values cannot be carried over to another formula unit. The mixing implemented here counts one site per formula unit, which ties the size of the mole to the formula as written in the database (Anderson and Crerar, 1993) (§12.7).

Species bound to a surface

A species occupying a site is referred to a surface entirely covered by it, and its activity is its site fraction  , being the budget of the site family (Chemistry that happens on a surface §3). The reference density fixing the zero of that scale is distinct from the capacity fixing how many sites exist (Kulik, 2002): the former is a convention of the kind discussed on this page, the latter a physical property of the sorbent.

3. Changing from one convention to another

Since does not depend on the convention, passing from an old standard state to a new one amounts to transferring a constant from one term to the other,

the right-hand side being evaluated in any state, and most conveniently in a limit where both activity coefficients are known (Anderson and Crerar, 1993) (§12.6.1). The most frequent instance in aqueous chemistry is the passage between the mole-fraction scale of a solute, used by formulations that treat the aqueous phase as a mixture like any other, and the molality scale used here. With    and  , the ratio equals and tends to one at infinite dilution, where both activity coefficients do, which yields

The constant is not small, as its evaluation with the molar mass of water computed from the atomic masses shows:

using ChemistryLab, DynamicQuantities
Mw = Species("H2O")[:M]
m° = 1.0u"mol/kg"
T = 298.15u"K"
uconvert(us"kJ/mol", R_GAS_Q * T * log(ustrip(m° * Mw)))

-9.956856183807943 mol⁻¹ kJ

A standard potential transcribed from a mole-fraction database without this correction shifts the solubility of every phase formed from that solute by   per unit of stoichiometric coefficient.

4. When the reference energy stops being a convention

A convention can be changed at no cost as long as the constant it introduces cancels from every computed quantity. For a surface site with a fixed budget it does: every surface reaction carries one site on each side, so that adding the same constant to the of all the members of a family changes no reaction energy. The test suite checks it by shifting a whole family by as much as 20 kJ/mol, after which the amount of the sorbent is unchanged to in relative terms, which is the tolerance of the solve itself.

The cancellation fails as soon as the budget follows the amount of the solid that carries the sites, the case treated in Chemistry that happens on a surface §10. The host then carries of the site component, the reference energy of the free site enters its chemical potential, and that energy becomes a statement about matter. A free site XsOH holds an oxygen and a hydrogen, and assigning it   states that a surface hydroxyl forms from its elements at no cost. The error is the energy of that matter, which host_coupling_bias evaluates from the conservation matrix. At the weak-site density of (Dzombak and Morel, 1990) for hydrous ferric oxide, where  , it amounts to 8.3 log units on the solubility of the host, enough to dissolve it completely where the same system with a fixed budget keeps its solid. The value consistent with the rest of the database is the standard energy of the matter the site carries,     kJ/mol for a hydroxylated oxide, and a coupled family left at zero is refused at construction with that value in the error message. (Kulik, 2002) avoids the question altogether by keeping the free site out of the balance, as a surface solvent of fixed activity; the surface page compares the two routes.

Where to go next

Proving that an answer is the answer is the next page of the chapter: it uses the potentials defined here to state what makes a computed equilibrium provably the minimum. The two places where the activity departs from its ideal form are treated in Activity models and Solid solutions, and the surface conventions in Chemistry that happens on a surface.