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

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

(12.25)

For a solution we need Ψ_j(r) representing the electrical potential due to the potential of ion j itself and the potential of its atmosphere. The charge density ρ_j(r) at distance r from ion j is

(12.26)

where n_ij denotes the number density of ion i at distance r from ion j. Charge density ρ_j(r) and potential Ψ_j(r) are self-consistently connected by a basic equation of electrostatics, Poisson's equation (see Appendix D), given by

(12.27)

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

(12.28)

Combining Eq. (12.27) and (12.28) yields for the Poisson equation

(12.29)

With the electrical potential at a certain point given by Ψ_j, the potential energy of an ion of charge ez_i is given by ez_iΨ_j. This is also the work required in charging an ion up to charge ez_i in the potential Ψ_j. We assume that the number density of ions n_ij of ions i around a central ion j is given by the correlation function g_ij(r) and the density at zero potential⁶⁾ n_i. For low ionic density the potential of mean force in g_ij(r) is the potential energy ez_iΨ_j (see Chapter 5) or, equivalently, given by Boltzmann's law, so that

(12.30)

The charge density ρ_j is obtained by summing over all ions i and reads

(12.31)

Expanding the exponentials in Eq. (12.31) via exp(x) = 1 + x + x²/2 + …, we obtain

(12.32)

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

(12.33)

Using Ψ_j = Y(r)/r, Eq. (12.33) is transformed to

(12.34)

for which the solution is

(12.35)

where A_j and B_j are constants to be determined from the boundary conditions. Since for r → ∞, Ψ → 0, the constant A_j must be zero and the general solution reads .

Before, we assumed that Ψ_j 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 Ψ_j near an ion, that is, at r = a, of charge z_j is then given by

(12.36)

where C_j as defined is a constant due to the presence of other ions for r > a. At r = a, both the potential Ψ_j and the electric field E = −∂Ψ_j/∂r should be continuous. Since obviously and Ψ_j should represent the same specific solution, we can from these two conditions evaluate both B_j and C_j and if we do so, we obtain

(12.37)

so that the total potential Ψ_j at a distance r ≥ a from a finite size ion j becomes

(12.38)

The potential of the atmosphere is the difference between the total potential Ψ_j and the potential of the single ion and thus

(12.39)

The value C_j ≡ , 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 ez_j. Substituting Eq. (12.36) and (12.37) in Eq. (12.32) we find the result

(12.40)

so that by integration over all space with r > a the final result obtained is

(12.41)

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, G^E is calculated as the work required to introduce one extra ion to the solution containing a density n_j of each type of ion j, multiplied by N_A. 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 ez_j. For each increment the work at constant Pand T is equal to the charge multiplied by the potential due to the other ions . Hence [15], (using x as dummy variable)

(12.42)

For an ionic solution of ν₊ ions of charge z₊ and ν₋ ions of charge z₋, the mean activity coefficient γ_± is defined by

(12.43)

Moreover, we have charge neutrality expressed by

(12.44)

Substituting Eq. (12.42) in Eq. (12.43) meanwhile using Eq. (12.44) results in

(12.45)

The next concept we need is the ionic strength I_m and I_c defined by, respectively,

(12.46)

for solutions of molality m and of concentration c, respectively. For a 1–1 electrolyte solution I_m is just the molality, while for other electrolytes it represents a type of “effective” molality. As is customary, we convert for κ from molarity c_i to molality m_i, which is important because this leads to extra terms upon evaluating the enthalpy, heat capacity, and so on. The general relationship between c_i and m_i is (see Section 2.1)

where the subscript 1 refers to the solvent. Hence (remembering that n_i = N_Ac_i with N_A as Avogadro's number)

(12.47)

with ρ′ the mass density of the solution. For small m_i, ρ′ ≅ M/V_m, where M is the molar mass and V_m is the molar volume of the solvent. Introducing all of this into the expression for κ, we obtain

(12.48)

so that the final Debye–Hückel expression for the activity coefficient becomes

(12.49)

(12.50)

For water at 25 °C using ρ′ = 1.00 g cm⁻³ and ε_r = 78.5, A = 1.172 kg^1/2 mol^−1/2 and B = 3.29 × 10⁹ m⁻¹ kg^1/2 mol^−1/2. The expression is valid up to about I_m ≅ 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 I_m < 0.01 m. The limiting Debye–Hückel expression corresponds with using

(12.51)

The latter model is usually denoted as the extended Debye–Hückel model.

12.5.2 Extensions

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 I_m, so that the expression for lnγ_± at high molality becomes

(12.52)

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 < I_m < 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. [16], p. 39).

(12.53)

The typical error is 5% at I_m ∼ 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 I_m ≅ 6, is probably due to Pitzer [17]. Generalization to multisolvent electrolytes is difficult, however. Another approach, based on the MSA using the (restrictive) primitive model, has been described by Lee [18] and Barthel et al. [19].

Problem 12.12

Verify Eq. (12.37).

Problem 12.13

Calculate κ for a 2–1 electrolyte in H₂O at 25 °C for c = 0.01 mol l⁻¹.

Problem 12.14

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.