Final Remarks - Lesson 2 - Mixing Liquids: Ionic Solutions - Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)

12. Mixing Liquids: Ionic Solutions

12.9. Final Remarks

In this chapter we have discussed only a few basic aspects of ionic solutions. The field of ionic solutions is extensive yet, despite its old age, its practical importance means that much research is still being conducted on electrolytes and monographs written on the subject, for example by Barthel et al. [19] and Wright [29]. In spite of all these developments, the Debye–Hückel–Onsager theory remains a cornerstone of the topic. Whilst the approximations involved have been discussed [30], perhaps the most appealing approach for higher concentrations is a combined use of the Debye–Hückel–Onsager model and Bjerrum's ion-association concepts. The relationship between the ion-pair concept and solution of the nonlinear Poisson–Boltzmann equation has been discussed [31]. For the many other effects that occur, for example by using strong or time-dependent electric fields, we refer to the literature.


1) The value of ΔhydH(H+) obtained is greater in magnitude than most older values. The ± sign indicates random errors only.

2) To be consistent with Pauling's radii, Shannon also used a value of rion(O2−) = 140 pm; data using that value are referred to as “effective” ionic radii. Shannon states that “it is felt that crystal radii correspond more closely to the physical size of ions in a solid”.

3) Hydrogen atoms have a large so-called “incoherent” scattering cross-section for neutrons as compared to the coherent cross-section, which troubles the interpretation of these experimental data significantly. For deuterium, this ratio is much more favourable, rendering the interpretation much easier.

4) On the one hand, X-ray diffraction (XRD) provides information on distances between all elements, while EXAFS (extended X-ray absorption fine structure) yields information specifically on the element whose absorption edge is selected. Unfortunately, EXAFS is limited to relatively nearby distances, while diffraction is informative to, say, 2 nm away.

5) In Chapter 2, we used fx,X, fm,X and fc,X for component X for systematics. Here, we use γXfm,X for brevity. The coefficient fm,X is related to fc,X by fm,X = cX fc,X/ρmX with ρ the mass density of the solvent. Moreover, to avoid confusion with the degree of dissociation α, we use for the remainder of this chapter the label i, j (or +, −) for components instead of α, β.

6) We use n for number density in order to avoid confusion with the charge density ρ.

7) As said, typically one takes a = (σ+ + σ)/2 (with σi the ionic diameter) because the probability for having pairs +/+ and −/− pairs is much smaller than for +/− pairs.

8) Note the change from ln to log.

9) Unfortunately the conventional symbol for conductivity κ is the same as used for the Debye length κ−1.

10) We use here a linear relationship between the “generalized force” (here, the field E) and the “generalized displacement” (here, the current density i). Irreversible thermodynamics shows that this linear relationship is often applicable. Anyway, nonlinear response is beyond the scope of this book.

11) Water has been intensively studied: See (31-08-2012).

12) It does not, however, provide a value for Λ°, since this parameter is related to ion–solvent interaction. Equating Stokes' frictional force to the electrical driving force provides an estimate, of course, but an estimate for the radius a must still be made. This value can be estimated by employing a concentration series and fitting Eq. (12.81), using a as a parameter.


1 Ketelaar, J.A.A. (1958) Chemical Constitution, Elsevier, Amsterdam.

2 Tissandier, M.D.J., et al. (1998) Phys. Chem., A102, 7787.

3 Pauling, L. (1960) The Nature of the Chemical Bond, 3rd edn, Cornell University Press, Ithaca.

4 (a) Shannon, R.D. and Prewitt, C.T. (1969) Acta Crystallogr., B25, 925; (b) Shannon, R.D. (1976) Acta Crystallogr. A32, 751.

5 Marcus, Y. (1988) Chem. Rev., 88, 1475.

6 Born, M. (1920) Z. Phys., 1, 45.

7 Latimer, W.M., Pitzer, K.S., and Slansky, C.M. (1939) J. Chem. Phys., 7, 108.

8 Stiles, P.J. (1980) Aust. J. Chem., 33, 1389.

9 Blum, L. and Fawcett, W.R. (1992) J. Phys. Chem., 96, 408 and references cited therein.

10 Marcus, Y. (1985) Ion Solvation, Wiley-Interscience, New York.

11 See Volkov et al. (1998).

12 Marcus, Y. (1996) Pure Appl. Chem., 89, 1495.

13 (a) Frank, H.S. and Evans, M.W. (1945) J. Chem. Phys., 13, 507; (b) Frank, H.S. and Wen, W.Y. (1957) Disc. Faraday Soc. 24, 133.

14 Bjerrum, N. (1909) Proceedings, 7th International Congress of Pure and Applied Chemistry, London, section X, p. 58; See also Bjerrum, N. (1909) Z. Elektrochem., 23, 321.

15 Güntelberg, E., page 155 in Bjerrum, N. (1926) Z. Phys. Chem., 119, 145.

16 See Davies (1962).

17 Pitzer, K.S. (1995) Thermodynamics, 3rd edn, McGraw-Hill, New York and references cited therein.

18 See Lee (2008).

19 See Barthel et al. (1998).

20 (a) Pitzer, K.S. (1977) Acc. Chem. Res., 10, 371; (b) Pitzer, K.S. (1973) J. Phys. Chem., 77, 268.

21 See Robinson and Stokes (1970).

22 (a) de Grotthuss, C.J.T. (1806) Ann. Chim., LVIII, 54; (b) Cukierman, S.L. (2006) Et tu, Grotthuss! and other unfinished stories. Biochim. Biophys. Acta Bioenerg., 1757, 876.

23 Pashley, R.M., Francis, M.J., and Rzechowicz, M. (2008) Curr. Opin. Coll. Interface Sci., 13, 236.

24 Fowler, R.H. and Guggenheim, E.A. (1939) Statistical Thermodynamics, Cambridge University Press, London.

25 (a) Bjerrum, N. (1926) Kgl. Danske Vid. Selsk., Math.-fys. Medd., 7, 9; (b) Fuoss, R.M. (1934) Trans. Faraday Soc. 30, 967. See also Davies (1962), Chapter 15.

26 Fuoss, R.M. (1958) J. Am. Chem. Soc., 80, 5059.

27 Fuoss, R.M. (1934) Trans. Far. Soc., 30, 967.

28 Fuoss, R.M. and Kraus, C.A. (1933) J. Am. Chem. Soc., 55, 1019 and 2387.

29 See Wright (2007).

30 Kirkwood, J.G. (1934) J. Chem. Phys., 2, 767.

31 (a) Guggenheim, E.A. (1957) Disc. Faraday Soc., 24, 53; (b) Skinner, J.F. and Fuoss, R.M. (1964) J. Am. Chem. Soc. 86, 3423.

32 Fawcett, W.R. (1999) J. Phys. Chem., B103, 11181.

33 Burgess, J. (1988) Ions in solution. Ellis Horwood, Chichester.

34 Enderby, J.E. and Neilson, G.W. (1979) Water (ed. F. Franks), Plenum, New York, Vol. 7, Ch. 1.

35 Jenkins, H.D.B. and Thakur, K.P. (1979) J. Chem. Ed., 56, 576.

36 Kohlrausch, F. (1885) Ann. Phys. Chem. 26, 161.

37 Kohlrausch, F. and Grotrian, F. (1875) Ann. Phys. 154, 215.

Further Reading

Barthel, J.M.G., Krienke, H., and Kunz, W. (1998) Physical Chemistry of Electrolyte Solutions, Springer, Berlin.

Davies, C.W. (1962) Ion Association, Butterworths, London.

Falkenhagen, H. (1971) Theorie der Elektrolyte, Hirzel, Stuttgart.

Fowler, R.H. and Guggenheim, E.A. (1939) Statistical Thermodynamics, Cambridge University Press, London.

Friedman, H.L. (1962) Ionic Solution Theory, Interscience, London.

Harned, H.S. and Owens, B.B. (1950) The Physical Chemistry of Electrolyte Solutions, Reinhold, New York.

Lee, L.L. (2008) Molecular Thermodynamics of Electrolyte Solutions, World Scientific, Singapore.

McInnes, D.A. (1939) The Principles of Electrochemistry, Reinhold, New York.

Robinson, R.A. and Stokes, R.H. (1970) Electrolyte Solutions, 2nd rev. ed, Butterworth, London. See also Dover Publishers reprint, 2002.

Volkov, A.G., Deamer, D.W., Tanelian, D.L., and Markin, V.S. (1998) Liquid Interfaces in Chemistry and Biology, John Wiley & Sons, Inc., New York.

Wright, M.R. (2007) An Introduction to Aqueous Electrolyte Solutions, John Wiley & Sons, Ltd, Chichester.