1.16.1: Ion Association

The term ‘ strong electrolyte’ has a long and honourable history in the development of an understanding of the properties of salt solutions. This term describes salt solutions where each ion contributes to the properties almost independently of all other ions in a given solution. The word ‘almost’ signals that the properties of a given salt solution are determined in part by charge –charge interactions between ions through the solvent separating ions in solution. Otherwise the ions can be regarded as free. Such is the case for aqueous salt solutions at ambient temperatures and pressures prepared using 1:1 salts such as \(\mathrm{Na}^{+} \mathrm{Cl}^{-}\), \(\mathrm{Et}_{4}\mathrm{N}^{+} \mathrm{Br}^{-}\) …

However with decrease in relative permittivity of the solvent, the properties of salt solutions indicate that not all the ions can be regarded as free; a fraction of the ions are associated. For dilute salts solutions in apolar solvents such as propanone a fraction of the salt is described as being present as ion pairs formed by association of cations and anions. With further decrease in the permittivity of the solvent higher clusters are envisaged; e.g. triple ions, quadruple ions…. Here we concentrate attention on ion pair formation building on the model proposed by N. Bjerrum [1,2].

The analysis identifies a given \(j\) ion in a salt solution as the reference ion such that at distant \(\mathrm{r}\) from this ion the electric potential equals \(\psi_{j}\) whereby the potential energy of ion \(\mathrm{i}\) with charge number \(\mathrm{z}_{\mathrm{i}}\) equals \(\mathrm{z}_{i} \, e \, \psi_{j}. The solvent is a structureless continuum and each ion is a hard non-polarisable sphere characterised by its charge, \(\mathrm{z}_{j} \, e\), and radius \(\mathrm{r}_{j}\).

If the bulk number concentration of \(\mathrm{i}\) ions is \(\mathrm{p}_{\mathrm{i}}\), the average local concentration of \(\mathrm{i}\) ions' \(\mathrm{p}_{\mathrm{i}}\) is given by equation (a) [3].

\[\mathrm{p}_{\mathrm{i}}^{\prime}=\mathrm{n}_{\mathrm{i}} \, \exp \left(-\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}} / \mathrm{k} \, \mathrm{T}\right)\]

The number of \(\mathrm{i}\)-ions, \(\mathrm{dn}_{\mathrm{i}}\) in a shell thickness \(\mathrm{dr}\) distance \(\mathrm{r}\) from the reference \(j\) ion is given by equation (b) [4].

\[\mathrm{p}_{\mathrm{j}}=\mathrm{n}_{\mathrm{i}} \, \exp \left(-\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}} / \mathrm{k} \, \mathrm{T}\right) \, 4 \, \pi \, \mathrm{r}^{2} \, \mathrm{dr}\]

At small \(\mathrm{r}\), the electric potential arising from the \(j\) ion is dominant. Hence [5],

\[\psi_{\mathrm{j}}=\frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\]

Hence,

\[\mathrm{dn}_{\mathrm{i}}=\mathrm{p}_{\mathrm{i}} \, \exp \left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e}}{\mathrm{k} \, \mathrm{T}} \, \frac{\mathrm{z}_{\mathrm{j}} \, \mathrm{e}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\right) \, 4 \, \pi \, \mathrm{r}^{2} \, \mathrm{dr}\]

Or [6],

\[\mathrm{dn}_{\mathrm{i}}=\mathrm{p}_{\mathrm{i}} \, \exp \left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}} \, \mathrm{e}^{2}}{4 \, \pi \, \mathrm{k} \, \mathrm{T} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\right) \, 4 \, \pi \, \mathrm{r}^{2} \, \mathrm{dr}\]

Using equation (e), the number of ions in a shell, thickness \(\mathrm{dr}\) and distance \(\mathrm{r}\) from the \(j\) ion, at temperature \(\mathrm{T}\) in a solvent having relative permittivity \(\varepsilon_{\mathrm{r}}\) is obtained for ions with charge numbers \(\mathrm{z}_{\mathrm{i}}\) and \(\mathrm{z}_{j}\).

For two ions having the same sign \(\mathrm{dn}_{\mathrm{i}}\) increases with increase in \(\mathrm{r}\), a pattern intuitively predicted. However for ions of opposite sign an interesting pattern emerges in which \(\mathrm{dn}_{\mathrm{i}}\) decreases with increase in \(\mathrm{r}\), passes through a minimum and then increases. In other words there exists a distance \(\mathrm{q}\) at which there is a minimum in the probability of finding a counterion. Thus [7]

\[\mathrm{q}=\frac{\left|\mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}}\right| \, \mathrm{e}^{2}}{8 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}\]

For a given salt, \(\mathrm{q}\) increases with decrease in \(\varepsilon_{\mathrm{r}}\) at fixed \(\mathrm{T}\). Bjerrum suggested that the term ‘ion pair’ describes two counter ions where their distance apart is less than \(\mathrm{q}\) [8]. In other words the proportion of a given salt in solution in the form of ions pairs increases with decrease in \(\varepsilon_{\mathrm{r}}\). The interplay between solvent permittivity and ion size \(\mathrm{a}_{j}\) as determined by the sum of cation and anion radii is important. For a fixed \(\mathrm{a}_{j}\), the fraction of ions present as ion pairs increases with decrease in relative permittivity of the solvent. Thus high \(\varepsilon_{\mathrm{r}}\) favours description of a salt as present as only ‘free’ cations and anions. The properties of such a real solutions might therefore be described using the Debye-Huckel Limiting Law. By way of contrast as \(\varepsilon_{\mathrm{r}}\) decreases the extent of ion pair formation increases with decrease in ion size [9].

Ion Association

The fraction of salt in solution \(\theta\) in the form of ion pairs is given by the integral of equation (e) within the limits \(\mathrm{a}\) and \(\mathrm{q}\) where \(\mathrm{a}\) is the distance of closest approach of cation and anion. Thus

\[\theta=4 \, \pi \, p_{i} \, \int_{a}^{q} \exp \left(-\frac{z_{+} \, z_{-} \, e^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{r} \, k \, T \, r}\right) \, r^{2} \, d r\]

Hence [10], for a solution where the concentration of salt \(\mathrm{c}_{j}\) expressed using the unit, \(\mathrm{mol dm}^{-3}\), \(\theta\) is given by equation (h).

\[\theta=\frac{4 \, \pi \, N}{10^{3}} \,\left(\frac{\left|\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}\right|}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}\right)^{3} \, \mathrm{Q}(\mathrm{b})\]

where

\[Q(b)=\int_{2}^{b} x^{-4} \, e^{x} \, d x\]

with

\[\mathrm{b}=\frac{\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T} \, \mathrm{a}}\]

and

\[\mathrm{x}=-\frac{\mathrm{z}_{+} \, \mathrm{z}_{-} \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T} \, \mathrm{r}}\]

The integral \(\mathrm{Q}(\mathrm{b})\) has been tabulated as a function of \(\mathrm{b}\) [1,10]. According to equation (h), \(\theta\) increases with increase in \(\mathrm{b}\); i.e. with increase in \(\mathrm{a}\) and decrease in \(\varepsilon_{\mathrm{r}}\).

Ion Pair Association Constants

The analysis leading to equation (h) is based on concentrations of salts in solution. Therefore the equilibrium between ions and ion pairs is described using concentration units. Here we consider the case of a 1:1 salt (e.g \(\mathrm{Na}^{+} \mathrm{Cl}^{-}\)) in the form of the following equilibrium describing the dissociation of ion pairs. [A common convention in this subject is to consider ‘dissociation’.] For a 1:1 salt \(j\) in solution the chemical potential \(\mu_{j}(s \ln )\) is given by equation (l).

\[\mu_{\mathrm{j}}(\mathrm{s} \ln )=\mu_{\mathrm{j}}^{0}(\mathrm{~s} \ln )+2 \, \mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{j}} \, \mathrm{y}_{\pm} / \mathrm{c}_{\mathrm{r}}\right)\]

The mean ionic activity coefficient (concentration scale) is defined by equation (m)

\[\operatorname{limit}\left(c_{j} \rightarrow 0\right) y_{\pm}=1.0 \text { at all T and } p\]

The thermodynamic properties of the neutral (dipolar) ion pair are treated as ideal. Then,

\[\mu_{\mathrm{ip}}(\mathrm{s} \ln )=\mu_{\mathrm{ip}}^{0}(\mathrm{~s} \ln )+\mathrm{R} \, \mathrm{T} \, \ln \left(\mathrm{c}_{\mathrm{ip}} / \mathrm{c}_{\mathrm{r}}\right)\]

The equilibrium between ‘free’ ions (i.e. salt \(j\)) and ion pairs is described by the following equation.

\[\mathrm{M}^{+} \mathrm{X}^{-}(\mathrm{s} \ln ) \Leftarrow \Rightarrow \mathrm{M}^{+}(\mathrm{s} \ln )+\mathrm{X}^{-}(\mathrm{s} \ln )\]

Then,

\[\mu_{i p}(s \ln )=\mu_{j}(s \ln )\]

Hence the ion pair dissociation constants \(\mathrm{K}_{\mathrm{D}}\) is given by equation (q).

\[\Delta_{\text {diss }} G^{0}=-R \, T \, \ln \left(K_{D}\right)\]

where

\[\Delta_{\text {diss }} G^{0}=\mu_{j}^{0}(s \ln )-\mu_{\mathrm{ip}}^{0}(\mathrm{~s} \ln )\]

Hence,

\[K_{D}=\frac{\left(c_{j} \, y_{\pm} / c_{r}\right)^{2}}{\left(c_{i p} / c_{r}\right)}\]

But \(\mathrm{c}_{\mathrm{j}}=\theta \, \mathrm{c}_{\mathrm{s}}\) and \(\mathrm{c}_{\mathrm{ip}}=(1-\theta) \, \mathrm{c}_{\mathrm{s}}\) where \(\mathrm{c}_{\mathrm{s}}\) is the total concentration of salt \(\mathrm{M}^{+} \mathrm{X}^{-}\). Then,

\[\mathrm{K}_{\mathrm{D}}=\frac{\theta^{2} \, \mathrm{y}_{\pm}^{2} \, \mathrm{c}_{\mathrm{s}}}{(1-\theta) \, \mathrm{c}_{\mathrm{r}}}\]

\(\mathrm{K}_{\mathrm{D}}\) is dimensionless. The long-established convention in this subject defines a quantity \(\mathrm{K}_{\mathrm{D}}^{\prime}\). Thus

\[\mathrm{K}_{\mathrm{D}}^{\prime}=\frac{\theta^{2} \, \mathrm{y}_{\pm}^{2} \, \mathrm{c}_{\mathrm{s}}}{(1-\theta)}\]

For very dilute solutions, the assumption is made that \(\theta = 1\) and \(\mathrm{y}_{\pm} = 1\). Hence using equation (h),

\[\begin{aligned}
&\frac{1}{\mathrm{~K}_{\mathrm{D}}^{\prime}} \cong \frac{1-\theta}{\mathrm{C}_{\mathrm{S}}} \\
&\theta=\frac{4 \, \pi \, N}{10^{3}} \,\left(\frac{\left|\mathrm{z}_{+} \, \mathrm{z}_{-}\right| \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{k} \, \mathrm{T}}\right)^{3} \, \mathrm{Q}(\mathrm{b})
\end{aligned}\]

Conductivities of Salt Solutions

The molar conductances of salt solutions at fixed \(\mathrm{T}\) and \(\mathrm{p}\) can be precisely measured. To a first approximation the molar conductance of a given solution offers a method of counting the number of free ions. For salt solutions in solvents of low permittivity the molar conductance offers a direct method for assessing the fraction of salt present as free ions and hence the fraction present as ion pairs. Hence electrical conductivities of salt solutions in solvents of low relative permittivity have been extensively studied in order to probe the phenomenon of ion pair formation.

The classic study was reported [10] by Fuoss and Kraus in 1933 who studied the electrical conductivities of tetra-iso-amylammonium nitrate in dioxan + water mixtures [11] at \(298.15 \mathrm{~K}\) over the range \(2.2 \leq \varepsilon_{\mathrm{r}} \leq 78.6\). The dependence of measured dissociation constants followed the pattern required by Bjerrum’s theory. Following the publication of the study by Fuoss and Kraus [10], many papers were published confirming the general validity of the Bjerrum ion-pair model. We note below a few examples of these studies which lead in turn to developments of the theory. For example in solvents of very low relative pemitivities triple ions are formed of the ++- and +-- type [12,13]. Many experimental techniques have been used to support the Bjerrum model; e.g. cryoscopic studies [14], electric permittivities of solutions [15,16] and Wien effects [17].

Following the Bjerrum model, other models were suggested and developed. Denison and Ramsey [18] suggested that the term ‘ion pair’ describes ions in contact, all other ions being free. Sadek and Fuoss [19] proposed that association of free ions to form contact ion ion pairs involved formation of solvent separated ion pairs, although they later withdrew the proposal[20]. Gilkerson [21] modified equations describing ion-pair formation to include parameters describing ion-solvent interaction. In 1957 Fuoss [22] restricted the definition of the term ‘ion pair’ to ions in contact. The dipolar nature of an ion pair was confirmed by dielectric relaxation studies [23,24]. In the development of theories of ion pair formation Hammett notes the models of ion pair formation which involve charged spheres in a continuous dielectric may only be relevant under especially favourable circumstances [25].

General Comments

The initial proposal by Bjerrum concerning ion pair formation has had an enormous impact in many branches of chemistry including mechanistic organic chemistry [26,27]. Spectroscopic studies identified ion-pairs in solution using charge transfer to solvent spectra [28]. Electron spin resonance identified the presence of ion pairs in solution. Particularly interesting are those solutions where the counterion hops between two sites in an organic radical anion [29].

Returning to the context of thermodynamics, the Bjerrum model of ion association has been extended to descriptions of partial molar volumes [30], apparent molar heat capacities and compressibilities of salts in non-aqueous solutions including cyanomethane [31].

Nevertheless the debate concerning ion association in solution has continued particularly with the development of statistical thermodynamic treatments of salt solutions. Grunwald [32] comments on the debate. To some extent the question arises as to the extent to which formation of ions pairs is either assumed from the outset or emerges from a given theoretical model for a salt solution.

Footnotes

[1] N. Bjerrum, K. Danske Vidensk Selskab, 1926,7, No. 9.

[2] For more recent accounts see—

1. R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London, 2nd ed. Revised,1965, chapter 14.
2. H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolytic Solutions, Reinhold, New York, 2nd. edition, revised and enlarged, 1950, section 3-7.
3. S. Glasstone, An Introduction to Electrochemistry, D. Van Nostrand, New York, 1942.
4. J. O’M. Bockris and A. J K. N. Reddy, Modern Electrochemistry: Ionics, Plenum Press, New York, 2nd. edn.,1998,chapter 3.

[3]

\[\begin{gathered}
\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \psi_{\mathrm{j}}}{\mathrm{k} \, \mathrm{T}}=\frac{[1] \,[\mathrm{C}] \,[\mathrm{V}]}{\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{[\mathrm{A} \mathrm{s}] \,\left[\mathrm{J} \mathrm{A} \mathrm{s}^{-1}\right]}{[\mathrm{J}]}=[1] \\
\mathrm{p}_{\mathrm{i}}=\left[\mathrm{m}^{-3}\right] \quad \mathrm{p}_{\mathrm{i}}^{\prime}=\left[\mathrm{m}^{-3}\right]
\end{gathered}\]

[4]

\[\mathrm{p}_{\mathrm{i}} \, \exp \left(-\mathrm{z}_{\mathrm{i}} \, \mathrm{e} \, \Psi_{\mathrm{j}} / \mathrm{k} \, \mathrm{T}\right) \, 4 \, \pi \, \mathrm{r}^{2} \, \mathrm{dr}=\frac{1}{\left[\mathrm{~m}^{3}\right]} \,[1] \,[1] \,\left[\mathrm{m}^{3}\right]=[1]\]

[5]

\[\psi_{\mathrm{j}}=\frac{[1] \,[\mathrm{C}]}{[1] \,[1] \,\left[\mathrm{F} \mathrm{m}^{-1}\right] \,[1] \,[\mathrm{m}]}=\frac{[\mathrm{A} \mathrm{s}]}{\left[\mathrm{As} \mathrm{V}^{-1}\right]}=[\mathrm{V}]\]

[6]

\[\begin{aligned}
&\left(-\frac{\mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}} \, \mathrm{e}^{2}}{4 \, \pi \, \mathrm{k} \, \mathrm{T} \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{r}}\right) \\
&=\frac{[1] \,[1] \,[\mathrm{C}]^{2}}{[1] \,[1] \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}] \,\left[\mathrm{Fm}^{-1}\right] \,[1] \,[\mathrm{m}]}=\frac{[\mathrm{As}]^{2}}{[\mathrm{~J}] \,[\mathrm{F}]} \\
&=\frac{[\mathrm{As}]^{2}}{[\mathrm{~J}] \,\left[\mathrm{As} \mathrm{A} \mathrm{s} \mathrm{J}^{-1}\right]}=[1]
\end{aligned}\]

Hence, \(\mathrm{p}_{\mathrm{j}}=\left[\mathrm{mol} \mathrm{m}^{-3}\right] \,[1] \,[1] \,[1] \,\left[\mathrm{m}^{2}\right] \,[\mathrm{m}]=[1]\)

[7]

\[\mathrm{q}=\frac{[1] \,[1] \,[\mathrm{C}]^{2}}{[1] \,[1] \,\left[\mathrm{Fm}^{-1}\right] \,[1] \,\left[\mathrm{J} \mathrm{K}^{-1}\right] \,[\mathrm{K}]}=\frac{[\mathrm{A} \, \mathrm{s}]^{2}}{\left[\mathrm{~A}^{2} \mathrm{~s}^{2} \mathrm{~J}^{-1} \mathrm{~m}^{-1}\right] \,[\mathrm{J}]}=[\mathrm{m}]\]

[8] Distance \(\mathrm{q}\) corresponds to the distance where \(\frac{\left|\mathrm{z}_{\mathrm{i}} \, \mathrm{z}_{\mathrm{j}}\right| \, \mathrm{e}^{2}}{4 \, \pi \, \varepsilon_{0} \, \varepsilon_{\mathrm{r}} \, \mathrm{q}}=2 \, \mathrm{k} \, \mathrm{T}\)

[9] We stress the distinction between association of cation \(\mathrm{M}^{+}\) and anion \(\mathrm{X}^{-}\) to form an ion pair and association in solution of \(\mathrm{H}^{+}\) and \(\mathrm{CH}_{3}\mathrm{COO}^{-}\) ions to form undissociated ethanoic acid. In the later case the cohesion is discussed in quantum mechanical terms.

[10] R. M. Fuoss and C. A. Kraus, J. Am. Chem. Soc.,1933,55,1019.

[11] The liquid mixture dioxan + water is notable for being completely miscible and ambient \(\mathrm{T}\) and \(\mathrm{p}\), the relative permitivities having a remarkable range. No other water + organic liquid offers such a range.

[12] R. M. Fuoss and C. A. Kraus, J. Am. Chem. Soc.,1933,55,2387.

[13] R. M. Fuoss, Chem. Rev.,1935,17,227.

[14] F. M. Batson and C. A. Kraus, J. Am. Chem. Soc.,1934,56,2017.

[15] G. S. Hooper and C. A. Kraus, J. Am. Chem. Soc.,1934,56,2265.

[16] R. M. Fuoss, J.Am.Chem.Soc.,1934,56,1031.

[17] D. J. Mead and R. M. Fuoss, J. Am. Chem.Soc.,1939,61,2047.

[18] J. T. Denison and J. B. Ramsey, J.Chem.Phys.,1950,18,770.

[19] H. Sadek and R. M. Fuoss, J. Am. Chem.Soc.,1954,76,5897,5902,5905.

[20] H. Sadek and R. M. Fuoss, J. Am. Chem. Soc.,1959,81,4511.

[21] W. Gilkerson, J. Chem. Phys.,1956,25,1199.

[22] R. M. Fuoss, J. Am. Chem. Soc.,1957,79,3304.

[23] A. H. Sharbaugh, H. C. Eckstrom and C. A. Kraus, J. Chem, Phys.,1947.15,54.

[24] G. Williams, J. Phys.Chem.,1959,63,532.

[25] L. P. Hammett, Physical Organic Chemistry, McGraw-Hill, New York, 2nd. edition, 1970.

[26] See for example, S. Winstein, P. E.Klinedinst and E. Clippinger, J Am. Chem. Soc., 1961,83,4986.

[27] E. A. Moelwyn-Hughes, Chemical Statics and Kinetics of Solutions, Academic Press, London, 1971,p.405.

[28] For further references see M. J. Blandamer, and M. F. Fox., Chem. Revs., 1970.70, 59.

[29] T. A. Claxton, J. Oakes and M. C. R. Symons, Trans. Faraday Soc., 1968, 64, 596.

[30] J.-F. Cote, J. E. Desnoyers and J-C. Justice, J Solution Chem.,1996,25,113.

[31] J.-F. Cote and J. E. Desnoyers, J. Solution Chem..,1999,28,395.

[32] E. Grunwald, Thermodynamics of Molecular Species, Wiley, New York, 1997,chapter 12.