Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
12. Mixing Liquids: Ionic Solutions
12.5. Debye–Hückel Theory
The presence of charges has a significant influence on the structure and properties of solutions. The conventional theory to deal with these effects is due to Debye and Hückel, formulated in 1923, and attempts – common to all solution theories – to model the excess Gibbs (or Helmholtz) energy.
For the purpose of the model, several assumptions are made:
· It is assumed that, apart from a hard-sphere core, the ionic interactions form the only contribution to the excess Gibbs energy. This is the so-called primitive model (PM) for which we have u(r) = ∞ for r < a with a often taken as the distance of closest approach a = (σ+ + σ−)/2 and u(r) = e2z+z−/4πεr for r ≥ a, where ez+ (ez−) and σ+ (σ−) are the charge and diameter of the positive (negative) ions, respectively. If a is a constant, equal for all ions, the model is denoted as the restricted primitive model (RPM).
· The solvent is considered as a dielectric continuum in which the ions interact according to Coulomb's law. The permittivity ε is assumed to be equal to the permittivity of the pure liquid, thereby neglecting the influence of the ions on the permittivity. In the literature replacing the solvent by a dielectric continuum is referred to as the McMillan–Mayer picture.
· The ions are considered to be spherical, nonpolarizable charges producing spherically symmetric electric fields.
· The solution is sufficiently dilute so that at the average ionic distance the potential energy is small as compared with kT.
· The electrolytes are completely dissociated. In the model the structuring effect of the ionic interaction is counteracted by the thermal motion which is trying to disrupt structures. As a result any ion will surrounded by a group of ions, called ionic atmosphere, of which the net charge is of opposite sign to that of the reference ion.
To model this we recall that the electrical potential of a single ion j is given by
For a solution we need Ψj(r) representing the electrical potential due to the potential of ion j itself and the potential of its atmosphere. The charge density ρj(r) at distance r from ion j is
where nij denotes the number density of ion i at distance r from ion j. Charge density ρj(r) and potential Ψj(r) are self-consistently connected by a basic equation of electrostatics, Poisson's equation (see Appendix D), given by
where ∇2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 is the Laplace operator in Cartesian coordinates. Since the charge distribution around an ion is spherically symmetric, it is convenient to use the Laplace operator in polar coordinates from which the angular coordinates are omitted. In this form it reads
Combining Eq. (12.27) and (12.28) yields for the Poisson equation
With the electrical potential at a certain point given by Ψj, the potential energy of an ion of charge ezi is given by eziΨj. This is also the work required in charging an ion up to charge ezi in the potential Ψj. We assume that the number density of ions nij of ions i around a central ion j is given by the correlation function gij(r) and the density at zero potential6) ni. For low ionic density the potential of mean force in gij(r) is the potential energy eziΨj (see Chapter 5) or, equivalently, given by Boltzmann's law, so that
The charge density ρj is obtained by summing over all ions i and reads
Expanding the exponentials in Eq. (12.31) via exp(x) = 1 + x + x2/2 + …, we obtain
The first order term of the expansion is zero because of the charge neutrality of the solution in total. For a symmetric binary salt the third order term is also zero. In any case we limit the expansion to the second order term only. Substituting Eq. 12.32 so truncated in Eq. (12.29), we obtain the (linearized) Poisson–Boltzmann expression
Using Ψj = Y(r)/r, Eq. (12.33) is transformed to
for which the solution is
where Aj and Bj are constants to be determined from the boundary conditions. Since for r → ∞, Ψ → 0, the constant Aj must be zero and the general solution reads .
Before, we assumed that Ψj represents the electrical potential due to the potential of ion j itself plus the potential of its atmosphere. Let us further assume that there is a distance of closest approach a to the central ion for other ions (we take a the same for all pairs of ions7), that is, the RPM model). Hence, for r < a no other ions are present. The potential Ψj near an ion, that is, at r = a, of charge zj is then given by
where Cj as defined is a constant due to the presence of other ions for r > a. At r = a, both the potential Ψj and the electric field E = −∂Ψj/∂r should be continuous. Since obviously and Ψj should represent the same specific solution, we can from these two conditions evaluate both Bj and Cj and if we do so, we obtain
so that the total potential Ψj at a distance r ≥ a from a finite size ion j becomes
The potential of the atmosphere is the difference between the total potential Ψj and the potential of the single ion and thus
The value Cj ≡ , given by Eq. (12.37), represents the potential at the surface of a sphere with radius a due to the effect of all other ions.
To check the self-consistency, we calculate the charge of the atmosphere associated with the reference ion of charge ezj. Substituting Eq. (12.36) and (12.37) in Eq. (12.32) we find the result
so that by integration over all space with r > a the final result obtained is
As expected, this charge equals the negative of the charge of the reference ion.
12.5.1 The Activity Coefficient and the Limiting Law
In the previous chapter we derived that the activity coefficient is related to the excess Gibbs energy. Here, GE is calculated as the work required to introduce one extra ion to the solution containing a density nj of each type of ion j, multiplied by NA. The ion is introduced as uncharged, and this first step is assumed to require no electrical energy. The second step is to gradually charge the ion in solution from 0 to ezj. For each increment the work at constant Pand T is equal to the charge multiplied by the potential due to the other ions . Hence , (using x as dummy variable)
For an ionic solution of ν+ ions of charge z+ and ν− ions of charge z−, the mean activity coefficient γ± is defined by
Moreover, we have charge neutrality expressed by
Substituting Eq. (12.42) in Eq. (12.43) meanwhile using Eq. (12.44) results in
The next concept we need is the ionic strength Im and Ic defined by, respectively,
for solutions of molality m and of concentration c, respectively. For a 1–1 electrolyte solution Im is just the molality, while for other electrolytes it represents a type of “effective” molality. As is customary, we convert for κ from molarity ci to molality mi, which is important because this leads to extra terms upon evaluating the enthalpy, heat capacity, and so on. The general relationship between ci and mi is (see Section 2.1)
where the subscript 1 refers to the solvent. Hence (remembering that ni = NAci with NA as Avogadro's number)
with ρ′ the mass density of the solution. For small mi, ρ′ ≅ M/Vm, where M is the molar mass and Vm is the molar volume of the solvent. Introducing all of this into the expression for κ, we obtain
so that the final Debye–Hückel expression for the activity coefficient becomes
For water at 25 °C using ρ′ = 1.00 g cm−3 and εr = 78.5, A = 1.172 kg1/2 mol−1/2 and B = 3.29 × 109 m−1 kg1/2 mol−1/2. The expression is valid up to about Im ≅ 0.1 m. In the original derivation of the Debye–Hückel model, point ions were assumed so that a = 0 leading to , the so-called Debye–Hückel limiting law. In this limit the activity coefficient is valid for low molality only, say Im < 0.01 m. The limiting Debye–Hückel expression corresponds with using
The latter model is usually denoted as the extended Debye–Hückel model.
For values of I > 0.1 m, extensions for the extended Debye–Hückel model are required. An extensive literature on this topic exists, but we refer here only to one (semi-)empirical extension. According to Hückel, the alignment of the solvent molecules by the ionic atmosphere leads to a linear term in the ionic strength Im, so that the expression for lnγ± at high molality becomes
In the literature this expression is often addressed as the Davies equation (1962), and considered to be a general expression for activity coefficients of solutions with 0.1 < Im < 1.0. Using a “universal” value for the ion radius of 0.3 nm and evaluating the constants at 25 °C, this reads8) (see Ref. , p. 39).
The typical error is 5% at Im ∼ 0.5 m. Other authors use other values for C ranging from C = 0.1 to C = 0.3. The linear term is often addressed as the “salting-out” term, as it accounts for the lowered solubility of salts at high ionic strength.
Several other schemes exist. The most well-known and rather reliable semi-empirical scheme, which relies on the fitting of data and is valid up to Im ≅ 6, is probably due to Pitzer . Generalization to multisolvent electrolytes is difficult, however. Another approach, based on the MSA using the (restrictive) primitive model, has been described by Lee  and Barthel et al. .
Verify Eq. (12.37).
Calculate κ for a 2–1 electrolyte in H2O at 25 °C for c = 0.01 mol l−1.
For a 0.05 m solution of HCl in water, calculate the activity coefficient according to the limiting and extended Debye–Hückel models, as well as the Davies equation. Comment on the differences.