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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 Physical chemistry is the study of macroscopic and microscopic phenomena in chemical systems in terms of the principles, practices, and concepts of physics such as motion, energy, force, time, thermodynamics, quantum chemistry, statistical mech ...
Kenneth Pitzer Kenneth Sanborn Pitzer (January 6, 1914 – December 26, 1997) was an American physical and theoretical chemist, educator, and university president. He was described as "one of the most influential physical chemists of his era" whose work "sp ...
. The parameters of the Pitzer equations are linear combinations of parameters, of a virial expansion of the excess
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work (physics), work that may be performed by a closed system, thermodynamically closed system a ...
, 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 An osmotic coefficient \phi is a quantity which characterises the deviation of a solvent from ideal behaviour, referenced to Raoult's law. It can be also applied to solutes. Its definition depends on the ways of expressing chemical composition o ...
, mixed ion activity coefficients, and salt solubility. They can be used to calculate mixed ion
activity coefficient In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same (o ...
s 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 In theoretical chemistry, Specific ion Interaction Theory (SIT theory) is a theory used to estimate single- ion activity coefficients in electrolyte solutions at relatively high concentrations. It does so by taking into consideration ''interacti ...
(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 In mechanics, the virial theorem provides a general equation that relates the average over time of the total kinetic energy of a stable system of discrete particles, bound by potential forces, with that of the total potential energy of the system. ...
equation of state for a gas. : PV= R T + B P + C P^2 + D P^3 \dots where P is the pressure, V is the volume, T is the temperature and B, C, D ... are known as virial coefficients. The first term on the right-hand side is for an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
. The remaining terms quantify the departure from the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first st ...
with changing pressure, 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 Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for am ...
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 coefficient In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same (o ...
s as a function of
ionic strength The ionic strength of a solution is a measure of the concentration of ions in that solution. Ionic compounds, when dissolved in water, dissociate into ions. The total electrolyte concentration in solution will affect important properties such ...
. 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 = - \alpha m^ - 2 \beta m : 1-\varphi = (\alpha/3) m^ + \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 In theoretical chemistry, Specific ion Interaction Theory (SIT theory) is a theory used to estimate single-ion activity coefficients in electrolyte solutions at relatively high concentrations. It does so by taking into consideration ''interaction ...
. 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 An osmotic coefficient \phi is a quantity which characterises the deviation of a solvent from ideal behaviour, referenced to Raoult's law. It can be also applied to solutes. Its definition depends on the ways of expressing chemical composition o ...
. 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 In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work (physics), work that may be performed by a closed system, thermodynamically closed system a ...
:\frac = f(I) +\sum_i \sum_j b_ib_j\lambda_(I)+\sum_i \sum_j \sum_kb_ib_jb_k\mu_+\cdots ''Ww'' is the mass of the water in kilograms,'' bi, bj'' ... are the
molal Molality is a measure of the number of moles of solute in a solution corresponding to 1 kg or 1000 g of solvent. This contrasts with the definition of molarity which is based on a specified volume of solution. A commonly used unit for molali ...
ities of the ions and ''I'' is the ionic strength. The first term, ''f(I)'' represents the Debye-Hückel limiting law. 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. The quantities ''μ''''ijk'' represent the interactions between three particles. Higher terms may also be included in the virial expansion. Next, the free energy is expressed as the sum of
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a species ...
s, or partial molal free energy, :G= \sum_i \mu_i\cdot N_i = \sum_i \left (\mu^0_i +RT \ln b_i\gamma_i \right )\cdot N_i and an expression for the activity coefficient is obtained by differentiating the virial expansion with respect to a molality b. :\ln \gamma_i = \frac =\fracf' +2\sum_j \lambda_b_j +\frac\sum_j\sum_k \lambda'_ b_jb_k + 3\sum_j\sum_k \mu_ b_jb_k+ \cdots \phi-1=\left(\sum_ib_i\right)^\left f'-f + \sum_i\sum_j\left(\lambda_+I\lambda'_ \right)b_ib_j +2\sum_i\sum_j\sum_k \mu_ b_ib_jb_k + \cdots\right/math> For a simple electrolyte ''M''''p''''X''''q'', at a concentration ''m'', made up of ions ''M''''z''+ and ''X''''z'', the parameters f^\phi, B^\phi_ and C^\phi_ are defined as :f^\phi=\frac :B^\phi_=\lambda_+I\lambda'_ +\left(\frac\right)\left(\lambda_+I\lambda'_\right)+\left(\frac\right)\left(\lambda_+I\lambda'_\right) :C^\phi_ =\left frac\right\left(p\mu_+q\mu_\right). The term ''f''''φ'' is essentially the Debye-Hückel term. Terms involving \mu_ and \mu_ are not included as interactions between three ions of the same charge are unlikely to occur except in very concentrated solutions. The ''B'' parameter was found empirically to show an ionic strength dependence (in the absence of
ion-pairing In chemistry, ion association is a chemical reaction whereby ions of opposite electric charge come together in solution to form a distinct chemical entity. Ion associates are classified, according to the number of ions that associate with each ot ...
) which could be expressed as :B^\phi_=\beta^_ +\beta^_ e^. With these definitions, the expression for the osmotic coefficient becomes :\phi-1=, z^+z^-, f^\phi+b\left(\frac\right)B^\phi_ +m^2\left \frac\right^\phi_. A similar expression is obtained for the mean activity coefficient. :\ln \gamma_\pm=\frac :\ln \gamma_\pm =, z^+z^-, f^\gamma+m\left(\frac\right)B^\gamma_ +m^2\left \frac\right^\gamma_ 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. The treatment can be extended to mixed electrolytes and to include association equilibria. Values for the parameters ''β''(0), ''β''(1) and ''C'' for inorganic and organic acids, bases and salts have been tabulated. Temperature and pressure variation is also discussed. One area of application of Pitzer parameters is to describe the ionic strength variation of
equilibrium constant The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency ...
s measured as concentration quotients. Both SIT and Pitzer parameters have been used in this context, For example, both sets of parameters were calculated for some
uranium Uranium is a chemical element with the symbol U and atomic number 92. It is a silvery-grey metal in the actinide series of the periodic table. A uranium atom has 92 protons and 92 electrons, of which 6 are valence electrons. Uranium is weakly ...
complexes and were found to account equally well for the ionic strength dependence of the stability constants. Pitzer parameters and SIT theory have been extensively compared. There are more parameters in the Pitzer equations than in the SIT equations. Because of this the Pitzer equations provide for more precise modelling of mean activity coefficient data and equilibrium constants. However, the determination of the greater number of Pitzer parameters means that they are more difficult to determine. Chapter 9, Estimation of medium effects on thermodynamic data


Compilation of Pitzer parameters

Besides the set of parameters obtained by Pitzer et al. in the 1970s mentioned in the previous section. Kim and Frederick 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. For some complex electrolytes, Ge et al. obtained the new set of Pitzer parameters using up-to-date measured or critically reviewed osmotic coefficient or activity coefficient data.


A comparable TCPC model

Besides the well-known Pitzer-like equations, there is a simple and easy-to-use semi-empirical model, which is called the three-characteristic-parameter correlation (TCPC) model. It was first proposed by Lin et al. It is a combination of the Pitzer long-range interaction and short-range solvation effect: :ln ''γ'' = ln ''γ''PDH + ln ''γ''SV Ge et al. 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. Temperature dependent parameters for a number of common single salts were also compiled, available at. The performance of the TCPC model in correlation with the measured activity coefficient or osmotic coefficients is found to be comparable with Pitzer-like models.


See also

*
Bromley equation The Bromley equation was developed in 1973 by Leroy A. Bromley with the objective of calculating activity coefficients for aqueous electrolyte solutions whose concentrations are above the range of validity of the Debye–Hückel equation. This eq ...
* Davies equation *
Osmotic coefficient An osmotic coefficient \phi is a quantity which characterises the deviation of a solvent from ideal behaviour, referenced to Raoult's law. It can be also applied to solutes. Its definition depends on the ways of expressing chemical composition o ...


References

* Chapter 3. *Pitzer, K.S. ''Ion interaction approach: theory and data correlation'', pp. 75–153. {{Refl