Pitzer equations are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water. They were first described by physical chemist Kenneth Pitzer. The parameters of the Pitzer equations are linear combinations of parameters, of a virial expansion of the excess Gibbs free energy, which characterise interactions amongst ions and solvent. The derivation is thermodynamically rigorous at a given level of expansion. The parameters may be derived from various experimental data such as the osmotic coefficient, mixed ion activity coefficients, and salt solubility. They can be used to calculate mixed ion activity coefficients and water activities in solutions of high ionic strength for which the Debye–Hückel theory is no longer adequate. They are more rigorous than the equations of specific ion interaction theory (SIT theory), but Pitzer parameters are more difficult to determine experimentally than SIT parameters.
Historical development A starting point for the development can be taken as the virial equation of state for a gas.
P V = R T + B P + C P 2 + D P 3 … {\displaystyle PV=RT+BP+CP^{2}+DP^{3}\dots }
where P {\displaystyle P} is the pressure, V {\displaystyle V} is the volume, T {\displaystyle T} is the temperature and B , C , D {\displaystyle B,C,D} ... are known as virial coefficients. The first term on the right-hand side is for an ideal gas. The remaining terms quantify the departure from the ideal gas law with changing pressure, P {\displaystyle P} . It can be shown by statistical mechanics that the second virial coefficient arises from the intermolecular forces between pairs of molecules, the third virial coefficient involves interactions between three molecules, etc. This theory was developed by McMillan and Mayer. Solutions of uncharged molecules can be treated by a modification of the McMillan-Mayer theory. However, when a solution contains electrolytes, electrostatic interactions must also be taken into account. The Debye–Hückel theory was based on the assumption that each ion was surrounded by a spherical "cloud" or ionic atmosphere made up of ions of the opposite charge. 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. The values calculated with Debye–Hückel theory diverge more and more from observed values as the concentrations and/or ionic charges increases. 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,
ln γ = − α m 1 / 2 − 2 β m {\displaystyle \ln {\gamma }=-\alpha m^{1/2}-2\beta m}
1 − φ = ( α / 3 ) m 1 / 2 + β m {\displaystyle 1-\varphi =(\alpha /3)m^{1/2}+\beta m}
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. This is the basis of SIT theory. It was further developed by Guggenheim. Scatchard 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. Measurement of osmotic coefficients provides one means for determining mean activity coefficients.
The Pitzer parameters The exposition begins with a virial expansion of the excess Gibbs free energy
G e x W w R T = f ( I ) + ∑ i ∑ j b i b j λ i j ( I ) + ∑ i ∑ j ∑ k b i b j b k μ i j k + ⋯ {\displaystyle {\frac {G^{ex}}{W_{w}RT}}=f(I)+\sum _{i}\sum _{j}b_{i}b_{j}\lambda _{ij}(I)+\sum _{i}\sum _{j}\sum _{k}b_{i}b_{j}b_{k}\mu _{ijk}+\cdots }
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