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Modified McKay–Perring Equation and Its Two Particular Solutions

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Abstract

Zdanovskii’s rule is the simplest isopiestic molality relation of mixed electrolyte aqueous solutions and the McKay–Perring equation is a differentio-integral equation particularly suitable for calculating solute activity coefficients from isopiestic measurements. However, they have two unsolved problems, which have puzzled solution chemists for several decades: (1) Zdanovskii’s rule has been verified by precise isopiestic measurements. But, several scientists suggested that the rule contradicts the Debye–Hückel limiting law for extremely dilute unsymmetrical mixtures. (2) In the McKay–Perring equation, a solute activity coefficient is multiplied by a solute composition variable. Different scientists have suggested that the composition variable may be the total ionic strength, common ion concentration, total ionic concentration, or an additive function with arbitrary proportionality constants. But, the different choices of the composition variable may lead to different activity coefficient results. Here, I derive a modified McKay–Perring equation in which the composition variable has the exclusive physical meaning of total ionic concentration for mixed electrolyte solutions (or of total solute particle concentration for the mixed solutions containing nonelectrolyte solutes). I also demonstrate that Zdanovskii’s rule is consistent with the Debye–Hückel limiting law for extremely dilute unsymmetrical mixtures. I derive two particular solutions of the modified McKay–Perring equation: one for the systems obeying Zdanovskii’s rule and another for the systems obeying a limiting linear concentration rule. These theoretical results have been verified with literature experiments and model calculations.

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Notes

  1. One of the Reviewers has presented an alternative derivation of Eq. 10. First, without the restriction of \( \ln a_{W} \to 0, \) an equation \( \left( {{{\nu_{1} m_{1}^{0} } \mathord{\left/ {\vphantom {{\nu_{1} m_{1}^{0} } {\nu_{2} m_{2}^{0} }}} \right. \kern-0pt} {\nu_{2} m_{2}^{0} }}} \right)_{{a_{\text{W}} }} = \left( {{{\phi^{0(2)} } \mathord{\left/ {\vphantom {{\phi^{0(2)} } \phi }} \right. \kern-0pt} \phi }^{0(1)} } \right)_{{a_{\text{W}} }} \) for single aqueous solutions is derived from the definition \( \phi^{0(i)} = - {{1000\ln a_{\text{W}}^{0(i)} } \mathord{\left/ {\vphantom {{1000\ln a_{\text{W}}^{0(i)} } {M\nu_{i} m_{i}^{0} }}} \right. \kern-0pt} {M\nu_{i} m_{i}^{0} }} . \) When applied to the restricted concentration region where the Debye–Hückel limiting law is valid and when using the Debye–Hückel limiting law expression for the osmotic coefficient, it leads to \( \left( {{{\nu_{1} m_{1}^{0} } \mathord{\left/ {\vphantom {{\nu_{1} m_{1}^{0} } {\nu_{2} m_{2}^{0} }}} \right. \kern-0pt} {\nu_{2} m_{2}^{0} }}} \right)_{{a_{\text{W}} }} = \left\{ {{{\left[ {1 - \left( {{{\left| {z_{a}^{0(2)} z_{c}^{0(2)} } \right|A} \mathord{\left/ {\vphantom {{\left| {z_{a}^{0(2)} z_{c}^{0(2)} } \right|A} 3}} \right. \kern-0pt} 3}} \right)I^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} } \right]} \mathord{\left/ {\vphantom {{\left[ {1 - \left( {{{\left| {z_{a}^{0(2)} z_{c}^{0(2)} } \right|A} \mathord{\left/ {\vphantom {{\left| {z_{a}^{0(2)} z_{c}^{0(2)} } \right|A} 3}} \right. \kern-0pt} 3}} \right)I^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} } \right]} {\left[ {1 - \left( {{{\left| {z_{a}^{0(1)} z_{c}^{0(1)} } \right|A} \mathord{\left/ {\vphantom {{\left| {z_{a}^{0(1)} z_{c}^{0(1)} } \right|A} 3}} \right. \kern-0pt} 3}} \right)I^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} } \right]}}} \right. \kern-0pt} {\left[ {1 - \left( {{{\left| {z_{a}^{0(1)} z_{c}^{0(1)} } \right|A} \mathord{\left/ {\vphantom {{\left| {z_{a}^{0(1)} z_{c}^{0(1)} } \right|A} 3}} \right. \kern-0pt} 3}} \right)I^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} } \right]}}} \right\}_{{a_{\text{W}} }} . \) A similar derivation can be used for the mixed solution. They will further lead to \( \mathop {lim}_{{\ln a_{\text{W}} \to 0}} \left( {{{\nu_{1} m_{1}^{0} } \mathord{\left/ {\vphantom {{\nu_{1} m_{1}^{0} } {\nu_{2} m_{2}^{0} }}} \right. \kern-0pt} {\nu_{2} m_{2}^{0} }}} \right)_{{a_{\text{W}} }} = 1 \) and \( \mathop {lim}_{{\ln a_{\text{W}} \to 0}} \left\{ {{{(\nu_{1} m_{1} + \nu_{2} m_{2} )} \mathord{\left/ {\vphantom {{(\nu_{1} m_{1} + \nu_{2} m_{2} )} {\nu_{i} m_{i}^{0} }}} \right. \kern-0pt} {\nu_{i} m_{i}^{0} }}} \right\}_{{a_{\text{W}} }} = 1, \) resulting in Eq. 10.

References

  1. Rard, J.A., Platford, R.F.: Experimental methods: isopiestic. In: Pitzer, K.S. (ed.) Activity Coefficients in Electrolyte Solutions, 2nd edn. CRC Press, Boca Raton (1991)

    Google Scholar 

  2. Zdanovskii, A.B.: Regularities in the property variations of mixed solutions. Tr. Solyanoi Lab. Akad. Nauk SSSR, No. 6, 5 (1936)

  3. Stokes, R.H., Robinson, R.A.: Interactions in aqueous nonelectrolyte solutions. I. Solute–solvent equilibria. J. Phys. Chem. 70, 2126–2130 (1966)

    Article  CAS  Google Scholar 

  4. Mikhailov, V.A.: Thermodynamics of mixed electrolyte solutions. Russ. J. Phys. Chem. 42, 1414–1416 (1968)

    Google Scholar 

  5. Clegg, S.L., Seinfeld, J.H.: Improvement of the Zdanovskii–Stokes–Robinson model for mixtures containing solutes of different charge types. J. Phys. Chem. A 108, 1008–1017 (2004)

    Article  CAS  Google Scholar 

  6. McKay, H.A.C., Perring, J.K.: Calculation of the activity coefficients of mixed electrolytes from vapour pressures. Trans. Faraday Soc. 49, 163–165 (1953)

    Article  CAS  Google Scholar 

  7. Robinson, R.A.: A numerical illustration of the McKay–Perring equation. Trans. Faraday Soc. 49, 1411–1412 (1953)

    Article  CAS  Google Scholar 

  8. Bonner, O.D., Holland, V.F.: Activity coefficients of p-toluenesulfonic acid and sodium p-toluenesulfonate in mixed solutions. J. Am. Chem. Soc. 77, 5828–5829 (1955)

    Article  CAS  Google Scholar 

  9. Morgan, R.S.: A thermodynamic study of the system sodium sulfite–sodium bisulfate–water at 25 °C. Tappi 43, 357–364 (1960)

    CAS  Google Scholar 

  10. Shul’ts, M.M., Makarov, L.L., Yu-Jeng, S.: Activity coefficients of NiCl2 and NH4Cl in binary and ternary solutions at 25 °C. Russ. J. Phys. Chem. 36, 2194–2198 (1962)

    Google Scholar 

  11. Fu, H.-T., Chen, I., Su, C.-H., Chung, F.-Y.: Thermodynamic properties of the ternary system MeCl2–Me’Cl–H2O. I. Activity coefficients of electrolytes in MgCl2–NaCl–H2O. In: The First National Electrochemical Conference, Changchun, China, October 1963

  12. Wang, Z.-C.: On the McKay–Perring equation (unpublished work). Year-end Thesis, Nanjing University, Nanjing, China (1963)

  13. Fu, H.-T., Chen, I., Su, C.-H., Chung, F.-Y.: Thermodynamic properties of the ternary system MeCl2–Me’Cl–H2O. I. Activity coefficients of electrolytes in MgCl2–NaCl–H2O. J. Nanjing Univ. (Nat. Sci.) 8, 96–108 (1964)

    Google Scholar 

  14. Robinson, R.A., Bower, V.E.: Thermodynamics of the ternary system: water–sodium chloride–barium chloride at 25 °C. J. Res. Nat. Bur. Stand. (Phys. Chem.) 69A, 19–27 (1965)

    Article  CAS  Google Scholar 

  15. Vdovenko, V.M., Ryazanov, M.A.: Activity coefficients in polycomponent systems. I. Radiokhimiya (Engl. Transl.) 7, 39–44 (1965)

    CAS  Google Scholar 

  16. Wang, Z.-C.: Novel general linear concentration rules and their theoretical aspects for multicomponent systems at constant chemical potentials—ideal-like solution model and dilute-like solution model. Ber. Bunsen-Ges. Phys. Chem. 102, 1045–1058 (1998)

    Article  CAS  Google Scholar 

  17. Bower, V.E., Robinson, R.A.: Thermodynamics of the ternary system: water–glycine–potassium chloride at 25 °C from vapor pressure measurements. J. Res. Nat. Bur. Stand. (Phys. Chem.) 69A, 131–135 (1965)

    Article  CAS  Google Scholar 

  18. Schonhorn, H., Gregor, H.P.: Multilayer membrane electrodes. III. Activity of alkaline earth salts in mixed electrolytes. J. Am. Chem. Soc. 83, 3576–3579 (1961)

    Article  CAS  Google Scholar 

  19. Lanier, R.D.: Activity coefficients of sodium chloride in aqueous three-component solutions by cation-sensitive glass electrodes. J. Phys. Chem. 69, 3992–3998 (1965)

    Article  CAS  Google Scholar 

  20. Harned, H.S., Owen, B.B.: The Physical Chemistry of Electrolyte Solutions, 3rd edn. Reinhold, New York (1958)

    Google Scholar 

  21. Robinson, R.A., Stokes, R.H.: Activity coefficients of mannitol and potassium chloride in mixed aqueous solutions at 25 °C. J. Phys. Chem. 66, 506–507 (1962)

    Article  CAS  Google Scholar 

  22. Guggenheim, E.A.: Mixtures. Oxford University Press, London (1952)

    Google Scholar 

  23. Temkin, M.: Mixtures of fused salts as ionic solutions. Russ. J. Phys. Chem. 20, 411–420 (1945)

    CAS  Google Scholar 

  24. Fφrland, T.: Thermodynamic properties of simple ionic mixtures. Discuss. Faraday Soc. 32, 122–127 (1961)

    Article  Google Scholar 

  25. Wang, Z.-C., Zhang, X.-H., He, T.-Z., Bao, Y.-H.: High-temperature isopiestic studies on {(1 − y)Hg + y(1 − t)Bi + ytSn}(l) at 600 K. Comparison with the partial ideal-solution model. J. Chem. Thermodyn. 21, 653–665 (1989)

    Article  CAS  Google Scholar 

  26. Wang, Z.-C.: The linear concentration rules at constant partial molar quantity \( \bar{\Uppsi }_{0} \)—extension of Turkdogan’s rule and Zdanovskii’s rule. Acta Metall. Sin. 16, 195–206 (1980)

    CAS  Google Scholar 

  27. Wang, Z.-C.: Thermodynamics of multicomponent systems at constant partial molar quantity \( \bar{\Uppsi }_{0} \)—thermodynamical aspect of iso-\( \bar{\Uppsi }_{0} \) rules of Zdanovskii-type and Turkdogan-type. Acta Metall. Sin. 17, 168–176 (1981)

    CAS  Google Scholar 

  28. Wang, Z.-C.: The theory of partial simple solutions for multicomponent systems. In First China–USA Bilateral Metallurgical Conference, pp. 121–136. The Metall. Ind. Press, Beijing, China (1981)

  29. Wang, Z.-C.: The theory of partial simple solutions for multicomponent systems. Acta Metall. Sin. 18, 141–152 (1982)

    CAS  Google Scholar 

  30. Wang, Z.-C., Lück, R., Predel, B.: Pure component A + classically ideal solution (B + C + …) = ? J. Chem. Soc. Faraday Trans. 86, 3641–3646 (1990)

    Article  CAS  Google Scholar 

  31. Wang, Z.-C.: Relationship among the Raoult law, Zdanovskii–Stokes–Robinson rule, and two extended Zdanovskii–Stokes–Robinson rules of Wang. J. Chem. Eng. Data 54, 187–192 (2009)

    Article  CAS  Google Scholar 

  32. Wang, Z.-C.: Relationship among the Henry law, Turkdogan rule, and two extended Turkdogan rules of Wang. J. Chem. Eng. Data 55, 1821–1827 (2010)

    Article  CAS  Google Scholar 

  33. Morachevskii, A.G., Butukhanova, T.V.: Application of the Zdanovskii rule to liquid metal systems with strong interaction of components. Russ. J. Appl. Chem. 80, 1300–1303 (2007)

    Article  CAS  Google Scholar 

  34. Kolosova, E.Y., Morachevskii, A.G., Tsymbulov, L.B.: Applicability of the Zdanovskii rule to sulfide melts. Russ. J. Appl. Chem. 81, 1287–1289 (2008)

    Article  CAS  Google Scholar 

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Acknowledgements

The author thanks Dr. Joseph A. Rard for helpful comments and for supplying Ref. [15].

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  1. Department of Chemistry, Northeastern University, Shenyang, 110004, Liaoning, China

    Zhi-Chang Wang

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Wang, ZC. Modified McKay–Perring Equation and Its Two Particular Solutions. J Solution Chem 47, 484–497 (2018). https://doi.org/10.1007/s10953-018-0723-2

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