Topic2270 Ion Association the Term ' Strong Electrolyte' Has a Long And

Topic2270 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 Na Cl , Et4N 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 r from this ion the electric potential equals ψj whereby the potential energy ⋅ ⋅ ψ of ion i with charge number zi equals z i e j . The solvent is a structureless continuum and each ion is a hard non-polarisable sphere characterised by its charge, ⋅ z j e , and radius rj. If the bulk number concentration of i ions is pi , the average local concentration of ' i ions pi is given by equation (a) [3]. ' = ⋅ − ⋅ ⋅ ψ ⋅ pi n i exp( zi e j / k T) (a) The number of i-ions, dni in a shell thickness dr distance r from the reference j ion is given by equation (b) [4]. = ⋅ ()− ⋅ ⋅ ψ ⋅ ⋅ ⋅ π ⋅ 2 ⋅ p j n i exp zi e j / k T 4 r dr (b) At small r, the electric potential arising from the j ion is dominant. z ⋅ e Hence [5], ψ = j (c) j ⋅ π ⋅ ε ⋅ ε ⋅ 4 0 r r z ⋅ e z ⋅ e Hence, dn = p ⋅ exp− i ⋅ j ⋅ 4 ⋅ π ⋅ r 2 ⋅ dr (d) i i ⋅ ⋅ π ⋅ ε ⋅ ε ⋅ k T 4 0 r r z ⋅ z ⋅ e2 Or [6], dn = p ⋅ exp− i j ⋅ 4 ⋅ π ⋅ r 2 ⋅ dr (e) i i ⋅ π ⋅ ⋅ ⋅ ε ⋅ ε ⋅ 4 k T 0 r r Using equation (e), the number of ions in a shell , thickness dr and distance r from the j ion, at temperature T in a solvent having relative permittivity εr is obtained for ions with charge numbers zi and zj. For two ions having the same sign dni increases with increase in r, a pattern intuitively predicted. However for ions of opposite sign an interesting pattern emerges in which dni decreases with increase in r , passes through a minimum and then increases. In other words there exists a distance q at which there is a minimum in the probability of finding a counterion. ⋅ ⋅ 2 zi z j e Thus [7] q = (f) ⋅ π ⋅ ε ⋅ ε ⋅ ⋅ 8 0 r k T For a given salt, q increases with decrease in εr at fixed T. Bjerrum suggested that the term ‘ion pair’ describes two counter ions where their distance apart is less than q [8]. In other words the proportion of a given salt in solution in the form of ions pairs increases with decrease in εr . The interplay between solvent permittivity and ion size aj as determined by the sum of cation and anion radii is important. For a fixed aj, the fraction of ions present as ion pairs increases with decrease in relative permittivity of the solvent. Thus high εr 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 εr decreases the extent of ion pair formation increases with decrease in ion size [9]. Ion Association The fraction of salt in solution θ in the form of ion pairs is given by the integral of equation (e) within the limits a and q where a is the distance of closest approach of cation and anion. q z ⋅ z ⋅ e2 Thus θ = 4⋅ π ⋅ p ⋅ exp− + − ⋅ r 2 ⋅ dr (g) i ∫ ⋅ π ⋅ ε ⋅ ε ⋅ ⋅ ⋅ a 4 0 r k T r Hence [10], for a solution where the concentration of salt cj expressed using the unit, mol dm-3 , θ is given by equation (h). 3 ⋅ ⋅ 2 4 ⋅ π ⋅ N z + z − e θ = ⋅ ⋅ Q(b) (h) 103 4 ⋅ π ⋅ ε ⋅ ε ⋅ k ⋅ T 0 r b where Q(b) = ∫ x −4 ⋅ e x ⋅ dx (i) 2 2 z + ⋅ z − ⋅ e with b = (j) ⋅ π ⋅ ε ⋅ ε ⋅ ⋅ ⋅ 4 0 r k T a z ⋅ z ⋅ e2 and x = − + − (k) ⋅ π ⋅ ε ⋅ ε ⋅ ⋅ ⋅ 4 0 r k T r The integral Q(b) has been tabulated as a function of b [1,10]. According to equation (h), θ increases with increase in b; i.e. with increase in a and decrease in εr . 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 Na+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 µ j (sln) is given by equation (l) . µ = µ 0 + ⋅ ⋅ ⋅ ⋅ j (sln) j (s ln) 2 R T ln(c j y ± / c r ) (l) The mean ionic activity coefficient (concentration scale) is defined by equation (m) → = limit(c j 0)y ± 1.0 at all T and p. (m) The thermodynamic properties of the neutral (dipolar) ion pair are treated as ideal. µ = µ0 + ⋅ ⋅ Then, ip (s ln) ip (s ln) R T ln(cip / c r ) (n) The equilibrium between ‘free’ ions (i.e. salt j) and ion pairs is described by the following equation. M + X − (s ln) ⇐⇒ M + (sln) + X − (sln) (o) µ = µ Then, ip (s ln) j (sln) (p) Hence the ion pair dissociation constants KD is given by equation (q). ∆ 0 = − ⋅ ⋅ diss G R T ln(K D ) (q) ∆ 0 = µ0 − µ0 where diss G j (sln) ip (s ln) (r) (c ⋅ y / c ) 2 = j ± r Hence, K D (s) (cip / cr ) = θ⋅ = − θ ⋅ + - But c j cS and cip (1 ) cs where cs is the total concentration of salt M X . θ2 ⋅ y 2 ⋅ c Then, K = ± s (t) D − θ ⋅ (1 ) c r KD is dimensionless. The long-established convention in this subject defines a ' quantity KD . θ2 ⋅ y 2 ⋅ c Thus K ' = ± s (u) D (1− θ) For very dilute solutions, the assumption is made that θ = 1 and y ± = 1. Hence using equation (h), − θ 1 ≅ 1 ' (v) K D cS 3 ⋅ ⋅ 2 4 ⋅ π ⋅ N z + z − e where θ = ⋅ ⋅ Q(b) 3 ⋅ π ⋅ ε ⋅ ε ⋅ ⋅ 10 4 0 r k T The relative permittivity of the solvent is present explicitly in equation (v) and implicitly in Q(b). Conductivities of Salt Solutions The molar conductances of salt solutions at fixed T and 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 ≤ ε ≤ mixtures [11] at 298.15 K over the range 2.2 r 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 pemittivities 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].

Load more