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

12.2. The Born Model and Some Extensions

The first question that comes to the mind when modeling hydration is how to represent an ion. The conventional solution is to consider it as a sphere with a certain radius and charge. There are two size scales for ions in regular use:

· The Pauling²⁾ scale [3], in which the effective nuclear charge is used to proportion the distance between ions in crystals into anionic and cationic radii. This analysis is based on r_ion(O²⁻) = 140 pm.

· The Shannon and Prewitt scale [4], which is based on a major review of crystallographic data. Different radii for different coordination numbers and for high and low spin states of the ions are given. The data are based on r_ion(O²⁻) = 126 pm, and are referred to as “crystal” ionic radii. Essentially, the cation radii are about 20 pm larger and the anion radii about 15 pm smaller than those of Pauling. Some data are given in Appendix E. Ionic radii in aqueous solutions have been reviewed by Marcus [5].

From electrostatics we know that the potential of a particle with charge q and radius a in a medium with relative permittivity ε_r is given by

(12.2)

If we assume that we may take a hard sphere for the ion and replace the solvent by a (continuous) dielectric, we can calculate from this expression the Gibbs energy difference Δ_hydG necessary to transfer the ion with charge ze(with z the valency and e the unit charge) from the gas phase (ε_r = 1) to a hydrated state (ε_r = ε_r). In this model, due to Born [6], the process is achieved in three steps. First, we decharge the ion in the gas phase; second, we transfer the ion to the solution; and, third, we recharge the ions in the solvent.

For calculating Δ_hydG we use the work w of charging given by

(12.3)

The total work done for a mole of ions is thus, assuming a zero energy change for step 2,

(12.4)

and we see that it is dependent, apart from the permittivity ε_r, on the charge ze and the radius a of the ions. Since the work W is done at constant tempera­ture and pressure, it represents the change in Gibbs energy Δ_hydG° for the process.

As an example, take the Na⁺ ion for which the Shannon–Prewitt (Pauling) radius is 116 (95) pm. We calculate that Δ_hydG° = −591 (−720) kJ mol⁻¹. Experimental hydration data for some ions are given in Table 12.1 (note that the entropy contributions are small), from which it can be seen that the experimental value for Na⁺ is −424 kJ mol⁻¹. In this case – and this appears to be generally true – the Born equation overestimates Δ_hydG°. Moreover, the Born equation predicts equal values for Δ_hydG for positive and negative ions of the same radius. Experimentally, this is not true when using the Pauling scale, as can be seen by comparing the values for the K⁺ ion (a = 133 pm) and the F⁻ion (a = 136 pm) in Table 12.1, although the estimates are much closer to each other when using the Shannon–Prewitt scale.

Table 12.1 Hydration Gibbs energy and entropy for various ions at 25 °C.^a)

a) Data from Ref. [32].

This type of discrepancy has led research groups to modify the Born expression, the simplest empirical modification being an adaptation of the effective ionic radius, recognizing that water coordinates differently with positive and negative ions. By adding 10 pm to the radius of a negative ion and 85 pm to the radius of a positive ion using the Pauling scale, a good correspondence with the experimental data [7] can be obtained.

Another consideration is that the value of ε_r close to the ion will be lower than the bulk value, in view of the high electric field strength close to the ion. As an example, we quote a simple empirical expression for this dependence due to Stiles [8] that reads

(12.5)

In this way the permittivity varies from ε_a at r = a to ε_b at r = b. Since ε is now position-dependent, we must use ∇·ε(r)E(r) = 0 (instead of the usual ε∇·E(r) = 0), leading to with . The electrostatic energy, to be evaluated numerically, becomes

(12.6)

It appears that R = (Δ_hydG)_Stiles/(Δ_hydG)_Born is not particularly sensitive to the precise choice of the profile, but also that saturation effect is incapable of explaining the overestimate of the Born model completely. For example, using ε_a = 2 and ε_b = 80, R = 0.85, 0.92 and 0.96 for a/b = 0.2, 0.4 and 0.6, respectively.

While the above-described adaptations are based on continuum considerations, the mean spherical approximation (MSA) is based on the discrete nature of the solvent, as described by the integral equation approach. The ion is represented by a sphere with radius a and charge ze, while for the solvent a sphere with radius a and dipole moment μ is taken. Hydrogen bonding is neglected. We only quote here the results with some background and refer to the literature [9] for details. For ionic radii, the Shannon–Prewitt scale is used.

The MSA consists of the core condition g(r) = 0 for r < 2a and the approximation c(r) = −βϕ(r) for r > 2a, together with the OZ equation (see Chapter 6). For hard-spheres with ϕ(r) = ∞ for r < 2a, this approach is identical to the PY approach and an analytical solution is possible (see Section 7.3). In the case of hydration, ϕ(r) represents either the ion–ion potential or the ion–dipole potential, as given by Eq. (3.7). The dipole–dipole interactions between the solvent molecules are taken into account by the factor f, which we take as f = 1 initially. The Gibbs energy of hydration in the limit of infinite dilution reads

(12.7)

where the polarization parameter λ for the solvent is given by

(12.8)

If we take for ε_r the bulk value of the solvent, the effect of polarizability, softness and asymmetry of the solvent is taken effectively into account. For water at 25 °C, ε_r = 78.5, resulting in λ = 2.65. The parameter δ ≡ b/λ depends only on the nature of the solvent. For δ = 0 we obtain Δ_hydG = (Δ_hydG)_Born, and hence δ acts as a radius correction. A reasonable hard-sphere radius for H₂O is b = 142 pm [10], so that δ = 54 pm, which leads to a significant change in Δ_hydG. The entropy Δ_hydS is obtained by differentiating Δ_hydG with respect to T, which leads (as δ is temperature-dependent) to

(12.9)

and where ∂λ/∂T is obtained from differentiating Eq. (12.8). Using the same data as before, we obtain for Na⁺ in H₂O, Δ_hydG° = −403 kJ mol⁻¹, which is considerably closer to the experimental value of −424 kJ mol⁻¹ than the Born estimate of −591 kJ mol⁻¹. The value obtained obviously still depends on the choice of radius made. As the Shannon–Prewitt radii for positive ions are typically 20 pm larger than the Pauling radii, the agreement with experiment for these ions is quite good. However, for negative ions the size correction is overestimated. Although dipole–dipole interactions could be taken into account, it is felt that the formal treatment overestimates the effect. Hence, Eq. (12.7)can be rewritten as −z²/Δ_hydG = k(a + δ) with k = 8πε₀ε_r/N_Ae²f(ε_r – 1), where now the experimental data for Δ_hydG are used and f and δ are determined separately for cations and anions. For alkali metal and earth alkali metal ions, it appears that the slope k obtained corresponds to f = 1 and δ⁺ = 49 pm. This suggests that for cations the dipole–dipole interactions are not important in hydration, while the value of δ⁺ is close to the theoretical value δ = 54 pm. For the anions used (halides and sulfide), the slope k obtained corresponds to f = 0.74 and δ⁻ = 0 pm, showing that dipole–dipole interactions are important but also that the anions have a less disruptive effect on the water network than cations. It should be noted that the results depend heavily on the ionic radius used. A radius corresponding to coordination number six is used, whilst for smaller-sized ions (Li⁺, Mg²⁺) this may not be appropriate. Also, if Pauling radii are used, the agreement is not satisfactory. Nevertheless, overall a rather satisfying description is obtained.

For more complete reviews on hydration, we refer to Marcus [10] and Volkov et al. [11]. In principle, the above-described approach can also be used to calculate Δ_traG for the transfer of an ion from one solvent to another. Several reviews on ion transfer are available in the literature (e.g., Ref. [12]).

Problem 12.3

Determine TΔ_hydS using the Born model. Show that TΔ_hydS is small as compared to Δ_hydG, and indicate the reason why. Estimate the numerical values of TΔ_hydS and Δ_hydG for Li⁺ in water, given that ε_r = 302.6 exp(−T/220.5).

Problem 12.4

As with any model, the Born model and its extensions contains assumptions. Indicate at least two aspects that deserve criticism, and include the reason(s) why.

Problem 12.5

A simple model for dielectric saturation is ε_r = ε_sat ≅ 4 for r < 0.2 nm, and ε_r = ε_bulk ≅ 78 for r > 0.4 nm with a linear transition region in between. Using this model, calculate Δ_hydG° for Na⁺ and compare the answer with the Born result.

Problem 12.6

Calculate Δ_hydG for Li⁺ in H₂O using the MSA. Compare the result with the experimental value.

Problem 12.7

Indicate why a difference in effective radius for positive and negative ions, as determined from hydration energies, is expected.