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 (Helgeson et al., 1981) depends on , so that the standard state of a solute is at the pressure of the system. A record that declares a molar volume independent of temperature and pressure (mv_constant in the ThermoFun databases: the solids and the solutes described by a heat-capacity polynomial) has its standard state at the pressure of the system as well, its standard Gibbs energy and enthalpy carrying

which vanishes at   bar (P_STANDARD). For portlandite, with  , it amounts to   at 10 bar and to   at 1 kbar: negligible for a laboratory sample or a structure, not for a deep reservoir. The solvent follows the equation of state of water at every temperature and pressure, from the values its record tabulates at and (Thermochemistry §3). A gas is referred to the pure ideal gas at , and the pressure enters its activity (§2). A species built by hand keeps the functions it is given: a heat-capacity polynomial alone carries no pressure term.

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. Here is expressed in kg/mol, so is in mol/kg of solvent, not mol/g or mol/kg of solution. Molarity instead uses solution volume, in mol/L. At fixed composition, thermal expansion changes molarity but not molality; reaction or water exchange can change either.

The state is not equally hypothetical in every property. Its partial molar enthalpy, volume and heat capacity are those of the solute at infinite dilution; its entropy and its standard potential are not, because of the term in (Richet, 2001; Sec. 11.2e, p. 254). The temperature dependence of therefore measures how far the partial molar enthalpy of the solute has moved from its value at infinite dilution, :

(Richet, 2001; Sec. 11.2e, Eq. (11.38), p. 255), and the standard properties of a solute, those the HKF equations give, are properties of infinite dilution.

HKFActivityModel, DaviesActivityModel and the Pitzer model (Pitzer, 1975) 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 models obtain it from the osmotic coefficient , which is one for the ideal dilute model,

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.

This is a consistency requirement for a thermodynamic model. Integrating a chosen path does not by itself prove that arbitrary solute activity laws come from one potential; the integrability checks in Activity models establish the limits of that claim.

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 the ratio of its fugacity to that pressure,

so that its chemical potential grows with pressure as  , the molar volume of an ideal gas. That is also the volume the package gives a gas, whether its record declares the ideal gas (mv_pvnrt) or it is built without a molar volume, so that the two are consistent at every pressure. A real gas keeps this standard state and gains a fugacity coefficient in its activity,  , from an equation of state (Real gases).

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 free sites are then counted as part of the host, whose database formula already contains the surface groups they are made of, so that a grain whose sites are all free has the composition of that formula; it has the database energy only if the free site carries no energy of its own. The reference is therefore fixed at zero, following Kulik (2002), and it is no longer a convention: any other value moves the host's solubility by  , which host_coupling_bias evaluates. Giving the free site XsOH the energy of the oxygen and hydrogen it holds, ``\mu^\circ(\mathrm{H_2O})

• \mu^\circ(\mathrm{H^+}) = -237.2`` kJ/mol, would count that energy twice, and

at the weak-site density of Dzombak and Morel (1990),  , it is worth 8.3 log units; a coupled family whose free site is more than 0.05 log units away from zero is refused at construction.

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.