## 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*) = *e*^{2}*z*_{+}*z*_{−}/4π*εr* 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}/∂*x*^{2} + ∂^{2}/∂*y*^{2} + ∂^{2}/∂*z*^{2} 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*is given by

_{i}*ez*. This is also the work required in charging an ion up to charge

_{i}Ψ_{j}*ez*in the potential

_{i}*Ψ*. We assume that the number density of ions

_{j}*n*of ions

_{ij}*i*around a central ion

*j*is given by the correlation function

*g*(

_{ij}*r*) and the density at zero potential

^{6)}*n*. For low ionic density the potential of mean force in

_{i}*g*(

_{ij}*r*) is the potential energy

*ez*(see

_{i}Ψ_{j}__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* + *x*^{2}/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*are constants to be determined from the boundary conditions. Since for

_{j}*r →*∞,

*Ψ*→ 0, the constant

*A*must be zero and the general solution reads .

_{j}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 ions

__, that is, the RPM model). Hence, for__

^{7)}*r*<

*a*no other ions are present. The potential

*Ψ*near an ion, that is, at

_{j}*r*=

*a*, of charge

*z*is then given by

_{j}__(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*and the electric field

_{j}*E*= −∂

*Ψ*/∂

_{j}*r*should be continuous. Since obviously and

*Ψ*should represent the same specific solution, we can from these two conditions evaluate both

_{j}*B*and

_{j}*C*and if we do so, we obtain

_{j}__(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*. For each increment the work at constant

_{j}*P*and

*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*defined by, respectively,

_{c}(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*, which is important because this leads to extra terms upon evaluating the enthalpy, heat capacity, and so on. The general relationship between

_{i}*c*and

_{i}*m*is (see Section 2.1)

_{i}where the subscript 1 refers to the solvent. Hence (remembering that *n _{i}* =

*N*

_{A}

*c*with

_{i}*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 × 10^{9} 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*< 0.01 m. The limiting Debye–Hückel expression corresponds with using

_{m}(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 ln

*γ*

_{±}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 reads

__(see Ref. [16], p. 39).__

^{8)}(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 H_{2}O 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.