Pitzer equations

Last updated January 19, 2024

Pitzer equations^[1] are important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water.^[2]^[3]^[4] They were first described by physical chemist Kenneth Pitzer.^[5] 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.

PV=RT+BP+CP^{2}+DP^{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. The remaining terms quantify the departure from the ideal gas law 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.^[6]

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 ^[7] 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,^[8]

\ln {\gamma }=-\alpha m^{1/2}-2\beta m

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.^[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. 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 ^[11]

{\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

W_w is the mass of the water in kilograms, b_i, b_j ... are the molalities of the ions and I is the ionic strength. The first term, f(I) represents the Debye–Hückel limiting law. The quantities λ_ij(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 potentials, or partial molal free energy,

G=\sum _{i}\mu _{i}\cdot N_{i}=\sum _{i}\left(\mu _{i}^{0}+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 {\partial ({\frac {G^{ex}}{W_{w}RT}})}{\partial b_{i}}}={\frac {z_{i}^{2}}{2}}f'+2\sum _{j}\lambda _{ij}b_{j}+{\frac {z_{i}^{2}}{2}}\sum _{j}\sum _{k}\lambda '_{jk}b_{j}b_{k}+3\sum _{j}\sum _{k}\mu _{ijk}b_{j}b_{k}+\cdots

$\phi -1=\left(\sum _{i}b_{i}\right)^{-1}\left[If'-f+\sum _{i}\sum _{j}\left(\lambda _{ij}+I\lambda '_{ij}\right)b_{i}b_{j}+2\sum _{i}\sum _{j}\sum _{k}\mu _{ijk}b_{i}b_{j}b_{k}+\cdots \right]$

For a simple electrolyte M_pX_q, at a concentration m, made up of ions M^z⁺ and X^z⁻, the parameters $f^{\phi }$ , $B_{MX}^{\phi }$ and $C_{MX}^{\phi }$ are defined as

f^{\phi }={\frac {f'-{\frac {f}{I}}}{2}}

B_{MX}^{\phi }=\lambda _{MX}+I\lambda '_{MX}+\left({\frac {p}{2q}}\right)\left(\lambda _{MM}+I\lambda '_{MM}\right)+\left({\frac {q}{2p}}\right)\left(\lambda _{XX}+I\lambda '_{XX}\right)

C_{MX}^{\phi }=\left[{\frac {3}{\sqrt {pq}}}\right]\left(p\mu _{MMX}+q\mu _{MXX}\right).

The term f^φ is essentially the Debye–Hückel term. Terms involving $\mu _{MMM}$ and $\mu _{XXX}$ 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) which could be expressed as

B_{MX}^{\phi }=\beta _{MX}^{(0)}+\beta _{MX}^{(1)}e^{-\alpha {\sqrt {I}}}.

With these definitions, the expression for the osmotic coefficient becomes

\phi -1=|z^{+}z^{-}|f^{\phi }+b\left({\frac {2pq}{p+q}}\right)B_{MX}^{\phi }+m^{2}\left[2{\frac {(pq)^{3/2}}{p+q}}\right]C_{MX}^{\phi }.

A similar expression is obtained for the mean activity coefficient.

\ln \gamma _{\pm }={\frac {p\ln \gamma _{M}+q\ln \gamma _{X}}{p+q}}

\ln \gamma _{\pm }=|z^{+}z^{-}|f^{\gamma }+m\left({\frac {2pq}{p+q}}\right)B_{MX}^{\gamma }+m^{2}\left[2{\frac {(pq)^{3/2}}{p+q}}\right]C_{MX}^{\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.^[12]^[13] The treatment can be extended to mixed electrolytes^[14] and to include association equilibria.^[15] Values for the parameters β⁽⁰⁾, β⁽¹⁾ and C for inorganic and organic acids, bases and salts have been tabulated.^[16] Temperature and pressure variation is also discussed.

One area of application of Pitzer parameters is to describe the ionic strength variation of equilibrium constants 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 complexes and were found to account equally well for the ionic strength dependence of the stability constants.^[17]

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.^[18]

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^[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.

For some complex electrolytes, Ge et al.^[21] 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.^[22] 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.^[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.^[25]

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.

Related Research Articles

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In chemical thermodynamics, activity is a measure of the "effective concentration" of a species in a mixture, in the sense that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution. The term "activity" in this sense was coined by the American chemist Gilbert N. Lewis in 1907.

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<span class="mw-page-title-main">UNIFAC</span> Liquid equilibrium model in statistical thermodynamics

In statistical thermodynamics, the UNIFAC method is a semi-empirical system for the prediction of non-electrolyte activity in non-ideal mixtures. UNIFAC uses the functional groups present on the molecules that make up the liquid mixture to calculate activity coefficients. By using interactions for each of the functional groups present on the molecules, as well as some binary interaction coefficients, the activity of each of the solutions can be calculated. This information can be used to obtain information on liquid equilibria, which is useful in many thermodynamic calculations, such as chemical reactor design, and distillation calculations.

<span class="mw-page-title-main">UNIQUAC</span> Model of phase equilibrium in statistical thermodynamics

In statistical thermodynamics, UNIQUAC is an activity coefficient model used in description of phase equilibria. The model is a so-called lattice model and has been derived from a first order approximation of interacting molecule surfaces. The model is, however, not fully thermodynamically consistent due to its two-liquid mixture approach. In this approach the local concentration around one central molecule is assumed to be independent from the local composition around another type of molecule.

An osmotic coefficient $is a quantity which characterises the deviation of a solvent from ideal behaviour, referenced to Raoult's law. It can be also applied to solutes. Its definition depends on the ways of expressing chemical composition of mixtures.$

The Davies equation is an empirical extension of Debye–Hückel theory which can be used to calculate activity coefficients of electrolyte solutions at relatively high concentrations at 25 °C. The equation, originally published in 1938, was refined by fitting to experimental data. The final form of the equation gives the mean molal activity coefficient $f \pm$ of an electrolyte that dissociates into ions having charges $z 1$ and $z 2$ as a function of ionic strength $I$ :

In theoretical chemistry, Specific ion Interaction 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 coefficients between the various ions present in solution. Interaction coefficients are determined from equilibrium constant values obtained with solutions at various ionic strengths. The determination of SIT interaction coefficients also yields the value of the equilibrium constant at infinite dilution.

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 equation, together with Specific ion interaction theory (SIT) and Pitzer equations is important for the understanding of the behaviour of ions dissolved in natural waters such as rivers, lakes and sea-water.

MOSCED is a thermodynamic model for the estimation of limiting activity coefficients. From a historical point of view MOSCED can be regarded as an improved modification of the Hansen method and the Hildebrand solubility model by adding higher interaction term such as polarity, induction and separation of hydrogen bonding terms. This allows the prediction of polar and associative compounds, which most solubility parameter models have been found to do poorly. In addition to making quantitative prediction, MOSCED can be used to understand fundamental molecular level interaction for intuitive solvent selection and formulation.

References

Pitzer, K.S., ed. (1991). Activity coefficients in electrolyte solutions (2nd ed.). C.R.C. Press. ISBN 0-8493-5415-3. Chapter 3. *Pitzer, K.S. Ion interaction approach: theory and data correlation, pp. 75–153.

↑ Pitzer, Kenneth S. (1991). Activity coefficients in electrolyte solutions (2nd ed.). Boca Raton: CRC Press. ISBN 0849354153.
↑ Stumm, W.; Morgan, J.J. (1996). Water Chemistry . New York: Wiley. ISBN 0-471-05196-9.
↑ Snoeyink, V.L.; Jenkins, D. (1980). Aquatic Chemistry: Chemical Equilibria and Rates in Natural Waters. New York: Wiley. ISBN 0-471-51185-4.
↑ Millero, F.J. (2006). Chemical Oceanography (3rd ed.). London: Taylor and Francis. ISBN 0-8493-2280-4.
↑ E. Connick, Robert E. Connick (December 2000). "Kenneth Pitzer, 6 January 1914 · 26 December 1997". Proceedings of the American Philosophical Society. 14 (4): 479–483. JSTOR 1515624.
↑ McMillan, W.G.; Mayer, J.E. (1945). "The statistical thermodynamics of multicomponent systems". J. Chem. Phys. 13 (7): 276. Bibcode:1945JChPh..13..276M. doi:10.1063/1.1724036.
↑ Debye, P.; Hückel, E. (1923). "Zur Theorie der Electrolyte". Phys. Z. 24: 185.
↑ Brønsted, J.N. (1922). "Studies on solubility IV. The principle of the specific interaction of ions". J. Am. Chem. Soc. 44 (5): 877–898. doi:10.1021/ja01426a001.
↑ Guggenheim, E.A.; Turgeon, J.C. (1955). "Specific interaction of ions". Trans. Faraday Soc. 51: 747–761. doi:10.1039/TF9555100747.
↑ Scatchard, G. (1936). "Concentrated solutions of strong electrolytes". Chem. Rev. 19 (3): 309–327. doi:10.1021/cr60064a008.
↑ Pitzer, Kenneth S. (1991). Activity coefficients in electrolyte solutions (2nd ed.). Boca Raton: CRC Press. p. 84. ISBN 0849354153.
↑ Pitzer, K.S.; Mayorga, G. (1973). "Thermodynamics of Electrolytes, II. Activity and osmotic coefficients with one or both ions univalent". J. Phys. Chem. 77 (19): 2300–2308. doi:10.1021/j100638a009.
↑ Pitzer, K.S.; Mayorga, G. (1974). "Thermodynamics of Electrolytes. III. Activity and osmotic coefficients for 2–2 electrolytes". J. Solution. Chem. 3 (7): 539–546. doi:10.1007/BF00648138.
↑ Pitzer, K.S.; Kim, J.J. (1974). "Thermodynamics of electrolytes. IV. Activity and osmotic coefficients for mixed electrolytes". J. Am. Chem. Soc. 96 (18): 5701–5707. doi:10.1021/ja00825a004.
↑ Pitzer, Kenneth S. (1991). Activity coefficients in electrolyte solutions (2nd ed.). Boca Raton: CRC Press. p. 93. ISBN 0849354153.
↑ Pitzer (1991), Tables 2-11
↑ Crea, F.; Foti, C.; Sammartano, S. (2008). "Sequestering ability of polycarboxylic acids towards dioxouranium(V)". Talanta. 28 (3): 775–778. doi:10.1016/j.talanta.2007.12.009.
↑ Grenthe, I.; Puigdomenech, I. (1997). Modelling in aquatic chemistry. Nuclear Energy Agency, O.E.C.D. ISBN 92-64-15569-4. Chapter 9, Estimation of medium effects on thermodynamic data
↑ Kim, Hee Taik; Frederick, William J. (April 1988). "Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25.degree.C. 1. Single salt parameters". Journal of Chemical & Engineering Data. 33 (2): 177–184. doi:10.1021/je00052a035.
↑ Kim, Hee Taik; Frederick, William J. (July 1988). "Evaluation of Pitzer ion interaction parameters of aqueous mixed electrolyte solutions at 25.degree.C. 2. Ternary mixing parameters". Journal of Chemical & Engineering Data. 33 (3): 278–283. doi:10.1021/je00053a017.
↑ Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (April 2008). "Correlation and Prediction of Thermodynamic Properties of Some Complex Aqueous Electrolytes by the Modified Three-Characteristic-Parameter Correlation Model". Journal of Chemical & Engineering Data. 53 (4): 950–958. doi:10.1021/je7006499.
↑ Lin, Cheng-Long; Lee, Liang-Sun; Tseng, Hsieng-Cheng (September 1993). "Thermodynamic behavior of electrolyte solutions". Fluid Phase Equilibria. 90 (1): 57–79. doi:10.1016/0378-3812(93)85004-6.
↑ Ge, Xinlei; Wang, Xidong; Zhang, Mei; Seetharaman, Seshadri (March 2007). "Correlation and Prediction of Activity and Osmotic Coefficients of Aqueous Electrolytes at 298.15 K by the Modified TCPC Model". Journal of Chemical & Engineering Data. 52 (2): 538–547. doi:10.1021/je060451k.
↑ Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (January 2008). "Correlation and Prediction of Thermodynamic Properties of Nonaqueous Electrolytes by the Modified TCPC Model". Journal of Chemical & Engineering Data. 53 (1): 149–159. doi:10.1021/je700446q.
↑ Ge, Xinlei; Wang, Xidong (12 February 2009). "A Simple Two-Parameter Correlation Model for Aqueous Electrolyte Solutions across a Wide Range of Temperatures". Journal of Chemical & Engineering Data. 54 (2): 179–186. doi:10.1021/je800483q.

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[1] Pitzer, Kenneth S. (1991). Activity coefficients in electrolyte solutions (2nd ed.). Boca Raton: CRC Press. ISBN 0849354153.

[stumm_morgan-2] Stumm, W.; Morgan, J.J. (1996). Water Chemistry . New York: Wiley. ISBN 0-471-05196-9.

[aquatic-3] Snoeyink, V.L.; Jenkins, D. (1980). Aquatic Chemistry: Chemical Equilibria and Rates in Natural Waters. New York: Wiley. ISBN 0-471-51185-4.

[4] Millero, F.J. (2006). Chemical Oceanography (3rd ed.). London: Taylor and Francis. ISBN 0-8493-2280-4.

[Connick-5] E. Connick, Robert E. Connick (December 2000). "Kenneth Pitzer, 6 January 1914 · 26 December 1997". Proceedings of the American Philosophical Society. 14 (4): 479–483. JSTOR 1515624.

[6] McMillan, W.G.; Mayer, J.E. (1945). "The statistical thermodynamics of multicomponent systems". J. Chem. Phys. 13 (7): 276. Bibcode:1945JChPh..13..276M. doi:10.1063/1.1724036.

[7] Debye, P.; Hückel, E. (1923). "Zur Theorie der Electrolyte". Phys. Z. 24: 185.

[8] Brønsted, J.N. (1922). "Studies on solubility IV. The principle of the specific interaction of ions". J. Am. Chem. Soc. 44 (5): 877–898. doi:10.1021/ja01426a001.

[G-9] Guggenheim, E.A.; Turgeon, J.C. (1955). "Specific interaction of ions". Trans. Faraday Soc. 51: 747–761. doi:10.1039/TF9555100747.

[10] Scatchard, G. (1936). "Concentrated solutions of strong electrolytes". Chem. Rev. 19 (3): 309–327. doi:10.1021/cr60064a008.

[11] Pitzer, Kenneth S. (1991). Activity coefficients in electrolyte solutions (2nd ed.). Boca Raton: CRC Press. p. 84. ISBN 0849354153.

[12] Pitzer, K.S.; Mayorga, G. (1973). "Thermodynamics of Electrolytes, II. Activity and osmotic coefficients with one or both ions univalent". J. Phys. Chem. 77 (19): 2300–2308. doi:10.1021/j100638a009.

[13] Pitzer, K.S.; Mayorga, G. (1974). "Thermodynamics of Electrolytes. III. Activity and osmotic coefficients for 2–2 electrolytes". J. Solution. Chem. 3 (7): 539–546. doi:10.1007/BF00648138.

[14] Pitzer, K.S.; Kim, J.J. (1974). "Thermodynamics of electrolytes. IV. Activity and osmotic coefficients for mixed electrolytes". J. Am. Chem. Soc. 96 (18): 5701–5707. doi:10.1021/ja00825a004.

[15] Pitzer, Kenneth S. (1991). Activity coefficients in electrolyte solutions (2nd ed.). Boca Raton: CRC Press. p. 93. ISBN 0849354153.

[16] Pitzer (1991), Tables 2-11

[17] Crea, F.; Foti, C.; Sammartano, S. (2008). "Sequestering ability of polycarboxylic acids towards dioxouranium(V)". Talanta. 28 (3): 775–778. doi:10.1016/j.talanta.2007.12.009.

[grenthe-18] Grenthe, I.; Puigdomenech, I. (1997). Modelling in aquatic chemistry. Nuclear Energy Agency, O.E.C.D. ISBN 92-64-15569-4. Chapter 9, Estimation of medium effects on thermodynamic data

[19] Kim, Hee Taik; Frederick, William J. (April 1988). "Evaluation of Pitzer ion interaction parameters of aqueous electrolytes at 25.degree.C. 1. Single salt parameters". Journal of Chemical & Engineering Data. 33 (2): 177–184. doi:10.1021/je00052a035.

[20] Kim, Hee Taik; Frederick, William J. (July 1988). "Evaluation of Pitzer ion interaction parameters of aqueous mixed electrolyte solutions at 25.degree.C. 2. Ternary mixing parameters". Journal of Chemical & Engineering Data. 33 (3): 278–283. doi:10.1021/je00053a017.

[21] Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (April 2008). "Correlation and Prediction of Thermodynamic Properties of Some Complex Aqueous Electrolytes by the Modified Three-Characteristic-Parameter Correlation Model". Journal of Chemical & Engineering Data. 53 (4): 950–958. doi:10.1021/je7006499.

[22] Lin, Cheng-Long; Lee, Liang-Sun; Tseng, Hsieng-Cheng (September 1993). "Thermodynamic behavior of electrolyte solutions". Fluid Phase Equilibria. 90 (1): 57–79. doi:10.1016/0378-3812(93)85004-6.

[23] Ge, Xinlei; Wang, Xidong; Zhang, Mei; Seetharaman, Seshadri (March 2007). "Correlation and Prediction of Activity and Osmotic Coefficients of Aqueous Electrolytes at 298.15 K by the Modified TCPC Model". Journal of Chemical & Engineering Data. 52 (2): 538–547. doi:10.1021/je060451k.

[24] Ge, Xinlei; Zhang, Mei; Guo, Min; Wang, Xidong (January 2008). "Correlation and Prediction of Thermodynamic Properties of Nonaqueous Electrolytes by the Modified TCPC Model". Journal of Chemical & Engineering Data. 53 (1): 149–159. doi:10.1021/je700446q.

[25] Ge, Xinlei; Wang, Xidong (12 February 2009). "A Simple Two-Parameter Correlation Model for Aqueous Electrolyte Solutions across a Wide Range of Temperatures". Journal of Chemical & Engineering Data. 54 (2): 179–186. doi:10.1021/je800483q.

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