Pitzer equations[1] are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water.
The remaining terms quantify the departure from the ideal gas law with changing pressure,
Expressions were derived for the variation of single-ion activity coefficients as a function of ionic strength.
This theory was very successful for dilute solutions of 1:1 electrolytes and, as discussed below, the Debye–Hückel expressions are still valid at sufficiently low concentrations.
Moreover, Debye–Hückel theory takes no account of the specific properties of ions such as size or shape.
Brønsted had independently proposed an empirical equation,[8] in which the activity coefficient depended not only on ionic strength, but also on the concentration, m, of the specific ion through the parameter β.
[9] Scatchard[10] extended the theory to allow the interaction coefficients to vary with ionic strength.
Note that the second form of Brønsted's equation is an expression for the osmotic coefficient.
The exposition begins with a virial expansion of the excess Gibbs free energy[11] Ww is the mass of the water in kilograms, bi, bj ... are the molalities of the ions and I is the ionic strength.
The quantities λij(I) represent the short-range interactions in the presence of solvent between solute particles i and j.
This binary interaction parameter or second virial coefficient depends on ionic strength, on the particular species i and j and the temperature and pressure.
Next, the free energy is expressed as the sum of chemical potentials, or partial molal free energy, and an expression for the activity coefficient is obtained by differentiating the virial expansion with respect to a molality b.
For a simple electrolyte MpXq, at a concentration m, made up of ions Mz+ and Xz−, the parameters
are not included as interactions between three ions of the same charge are unlikely to occur except in very concentrated solutions.
These equations were applied to an extensive range of experimental data at 25 °C with excellent agreement to about 6 mol kg−1 for various types of electrolyte.
One area of application of Pitzer parameters is to describe the ionic strength variation of equilibrium constants measured as concentration quotients.
Because of this the Pitzer equations provide for more precise modelling of mean activity coefficient data and equilibrium constants.
[18] Besides the set of parameters obtained by Pitzer et al. in the 1970s mentioned in the previous section.
Kim and Frederick[19][20] published the Pitzer parameters for 304 single salts in aqueous solutions at 298.15 K, extended the model to the concentration range up to the saturation point.
Those parameters are widely used, however, many complex electrolytes including ones with organic anions or cations, which are very significant in some related fields, were not summarized in their paper.
It was first proposed by Lin et al.[22] It is a combination of the Pitzer long-range interaction and short-range solvation effect: Ge et al.[23] modified this model, and obtained the TCPC parameters for a larger number of single salt aqueous solutions.
This model was also extended for a number of electrolytes dissolved in methanol, ethanol, 2-propanol, and so on.
[24] Temperature dependent parameters for a number of common single salts were also compiled, available at.
Due to its empirical aspects, the Pitzer modelling framework has a number of well-known limitations.
[26] Most importantly, to improve the fits to experimental data, different variations of the equations have been described.
One alternative modelling approach[27] has been specifically designed to address this extrapolation issue by reducing the number of equation parameters while maintaining similar predictive precision and accuracy.