F I G . 12-9. Activity coefficient plots for various electrolytes at 25°C.
of the same z+z_ product appear to be due to several causes. Extensive hydration of the ions ties up the solvent water and makes the true mole fraction higher than the apparent one; a similar explanation was offered for the behavior of aqueous sucrose solutions (note Table 10-3). This may be a major reason for the rather high activity coefficients of electrolytes such as HCl in concentrated solutions.
A second factor is that many electrolytes are not fully dissociated. Although a chemical bond is not ordinarily expected to form in the case of electrolytes involving a rare gas type of ion such as K+, ions may associate strongly as an ion pair. Salts such as C u S 04, however, may actually form a coordinate bond in their association. Acids such as H2S 04 are relatively weak, and are not fully dissociated except in dilute solution. If association occurs for any of these reasons, the result is generally to lower the activity coefficient of the electrolyte.
A less chemically specific effect is that the actual size of the ion becomes impor-tant in concentrated solutions. The result is an increase in activity coefficient over what it would otherwise be.
These various explanations are important in the sense that they provide some rationale for an otherwise bewildering variety of behavior. They are discussed in somewhat more detail in the Commentary and Notes section. On the other hand, an activity coefficient is a phenomenological quantity; when multiplied by the mean molality, it gives the mean activity of the electrolyte and therefore its chemical potential, regardless of explanation.
12-8 The Debye-Huckel Theory
Repeated remarks have been made throughout this chapter about the effect of the long-range Coulomb interactions between ions on both their transport behavior or mobilities and their activity coefficients. The theoretical approach to the former effect is mentioned further in the Special Topics section. The same interactions that affect ion mobilities also affect their activity coefficients, and the treatment of this second effect is important enough to cover here.
The theory of Debye and Huckel (1923) constituted a major breakthrough.
Earlier observations, especially by G. Lewis, had shown that the activity coefficients of electrolytes were, in dilute solutions, determined more by ionic strength than by any specific chemical property. The implication that nonideality effects are due mainly to long-range Coulomb interactions between ions was accepted, but the problem of treating a collection of ions seemed insurmountable. The normal statistical mechanical approach would require one somehow to list the energy states of each ion while recognizing that its energy was affected by the location of all other ions.
The path that Debye and Huckel found was through the assumption that there exists some average potential around each ion and that each potential field is independent, that is, one does not perturb another. Further assumptions were that the electrolyte is completely dissociated and, at first, that the ions are of negligible size. It was also assumed that electrical interactions are solely responsible for
•deviations from nonideality. The treatment is therefore limited to rather dilute solutions.
We proceed as follows, taking for simplicity the electrolyte to be of the uni-univalent type. It is assumed that there is some average potential φ which is a function of distance r from any particular ion. The potential energy of an ion in this potential is εφ, where e is the electronic charge. Since we are dealing with Coulomb's law, φ will be in esu units and likewise e. The probability of finding a positive ion in a region of potential φ around a particular ion of like charge is given by the Boltzmann principle:
n+ = ne~e*/kT9 (12-74)
where η is the average concentration in molecules per cubic centimeter. Similarly
n_ = ne**/*r. (12-75) The net charge density ρ is then
Ρ = ( n+ — n_)e = ne(e~e^kT - ee^kT). (12-76) The next major assumption, without which further progress would have stopped,
was that a theorem from electrostatics, known as the Poisson equation, could be used. This states that the rate of change or the divergence of the gradient of the electrostatic potential at a given point is proportional to the charge density at that point. The equation is valid for a continuous medium of uniform dielectric constant D and the equation for spherical coordinates is
^-hii-Z)—^- <
12-
77>
This implies that the random motion of ions gives a smeared-out charge density to which Eq. (12-77) can be applied.
We now combine Eqs. (12-76) and (12-77) to obtain what has come to be known as the Poisson-Boltzmann equation:
V*^ = _ i ^ 5 i (e-'*lkT — e**lkT). (12-78) The assumptions implicit up to this point (such as the independence of the potential
12-8 THE DEBYE-HUCKEL THEORY 463 fields around each ion) require that the effect not be a large one, that is, that eifjjkT be a small number. We therefore proceed to expand the exponentials, keeping only the first term, to obtain
"•-(τκτ)*· »
2-
79>
The collection of quantities multiplying φ on the right-hand side of Eq. (12-79) is assembled into a single parameter κ, defined as
K * = W f % z*U i • ( 1 2-8 0 ) i
Equation (12-80) applies to the general case of a collection of ions of charges zt and reduces to κ2 = %nne2/DkT for a uni-univalent electrolyte. Equation (12-79) then becomes
ν2φ = κ2φ. (12-81)
The solution to Eq. (12-81) is
# 0 = (12-82)
This may be verified by insertion of the expression for Φ(Γ) back into Eq. (12-81). Thus
dr Dr2 " Dr~ 9
Λ άφ ze zeKr
r2 1 _ — 0-KR Ο —Kr
dr D D '
d / α*ψ\ zeK ζβκ zeK2r
— r2 — = e~Kr e~Kr Η e~l
dr\ dr) D D D
zeK2r
D 1 d I M\ ζβκ*
We again expand the exponential, keeping only the first terms, to get
φϊ) = £ - ^ Γ κ . (12-83)
The first term on the right is just the potential due to the charge on the ion itself, and we are interested only in the second term, —ζβκ/Ό, which gives the alteration in the potential due to the distribution of other ions around the given ion. Notice that this second term corresponds to the potential of a charge — ze as observed at a distance l/#c. The quantity Ι/κ has the dimensions of distance (centimeters in the cgs system) and has come to be known as the effective or equivalent radius of the ion atmosphere. The simple physical picture, illustrated in Fig. 12-10, is thus one of an ion of charge ze having a statistical excess of ions of opposite charge around it, the excess amounting to just — ze and behaving as though it were located on a spherical shell of radius l/κ.
It is next necessary to find the free energy associated with this extra potential
Θ © Θ
Q 0 0 ° Ο
0© Θ
Θ ©
F I G . 12-10. Illustration of the ion atmosphere effect.originating from the ion atmosphere. This is done by calculating the reversible work needed to form the atmosphere: We integrate the product of potential times charge as the ion is allowed to build up its charge from zero to the full value:
Cze Cze / Z€K \
electrical work = J </fatmd(ze) = J y jy) d(ze) or
This electrical work then contributes to the chemical potential μί of the ith ion:
μί = μ{° + kT In di + Wei .
Since the whole derivation is for a very dilute solution, it seems safe to assume a% = Wi > that is, that the ion obeys Henry's law apart from the electrical contribu
tion. This last is the source of the observed deviation from ideality, reported experimentally in terms of an activity coefficient. The conclusion is then
wei = kT In j i = zfe'K
~2D~
or
In Ύ ί = Zi2e2K
2Dkf' (12-84)
We next need to find the mean activity coefficient for the case of an electrolyte
12-8 THE DEBYE-HUCKEL THEORY 465
having just two kinds of ions. From Eq. (12-64) we have
In y± = ^ In y+ + ^ In y_ . (12-85)
Algebraic manipulation of Eqs. (12-85) and (12-61) gives
l n y±= " I * * * -I 2 Z S ^ - (1 2"8 6>
Finally, /c can be related to the ionic strength / [Eq. (12-70)] since nt = (C(/l000) JVQ and in dilute solution C? = , where />0 is the solvent density. Equation (12-80) becomes
8irgW0Vo
1000DRT1 (U * / }
and so Eq. (12-86) becomes
1η
^ = - ' ^ - ΐ 5 ^ ( » τ Γ
/ 1 / 2 <1 2-8 8>= - Λ I z+z _ I
Insertion of the values for the general constants {e in esu and R in erg m o l e- 1 K~x) gives
In y± = - 4 . 1 9 8 X 10· | z+z _ | [ ^ r ^ /1 / 2. (12-89) For water at 25°C, D = 78.54 and Eq. (12-89) reduces t of
In y± = - 1 . 1 7 2 I z+z _ 111 1 2. (12-90) The Debye-Huckel treatment leads, first of all, to a theoretical explanation
of the empirical observation that the activity coefficient of an electrolyte is deter
mined by the ionic strength / of the medium. The quantitative predictions, as for example, from Eq. (12-90), have been well verified for uni-univalent electrolytes.
The theoretical slopes are included in Figs. 12-7 and 12-8, and the experimental points approach agreement at low concentrations. One may make a more direct check by plotting the activity coefficient data of Table 12-7 according to Eq. (12-90).
This is done in Fig. 12-11, and it is apparent that the uni-univalent electrolytes approach agreement with theory. It is less clear whether accurate agreement occurs in the case of the higher charge types. Equation (12-90) is a limiting law in that various approximations restrict its validity to dilute solutions, and it has been assumed that activity coefficients for the more highly charged electrolytes will approach the theoretical predictions at sufficiently low concentrations. This aspect is discussed further in the Commentary and Notes section.
+ T h e preceding derivation h a s b e e n m a d e in t h e c g s s y s t e m , a s the m o r e natural o n e t o use.
T h e e q u a t i o n s are the s a m e in the SI s y s t e m except that D is everywhere multiplied b y 4ne0
(8.854 χ 1 0 "i a— s e e Section 3 - C N - l ) a n d the factor o f 1 0 0 0 disappears f r o m the d e n o m i n a t o r s o f E q s . (12-87) a n d (12-88). I o n i c strength remains in M ( m o l e k g- 1 solvent) but P0 is n o w in k g m-8 a n d 1/*, t h e i o n a t m o s p h e r e radius, in meters.
0 0.1 0 . 2 0 . 3 0 . 4 0.5
F I G . 1 2 - 1 1 . Plot of activity coefficient data at 25°C so as to test the Debye-HUckel theory.
Dashed lines give the various limiting law slopes.