9458
J. Phys. Chem. 1995,99, 9458-9465
Thermodynamics of Ion Exchange Selectivity at Interfaces John D. Morgan,"" Donald H. Napper, and Gregory G. Warr Division of Physical and Theoretical Chemistry, University of Sydney, Sydney, New South Wales 2006, Australia Received: October 27, 1994; In Final Form: February 14, 1995@
The thermodynamic theory of ion exchange selectivity at an airholution interface is rigorously derived. The free energy change driving ion exchange is shown to be a sum of terms arising from counterion interactions with the solvent, chemical interactions with the surfactant, and mechanical effects at the surface. Each of these terms can be explicitly calculated and the solvation interaction is shown to dominate the exchange free energy. These results may be extended to provide a qualitative description of ion exchange in other systems, such as resins or micelles. A partial dehydration model of ion exchange is presented and is shown to provide falsifiable, quantitative predictions conceming ion exchange selectivity. Most available ion exchange data for anion and cation exchange, both within a valence group and between ions of different valences, are unified in this model.
Thomas3 in the particular case of ion exchange resins. They showed that
1. Introduction Ion exchange has been observed to occur at many surfaces, including ion exchange resins, micelles, and other surfactant aggregates,' and at the airtsolution interface.* The selectivity of an ion exchanger for any two competing counterions may be described by two parameters, which it is the goal of theory to explain. They are the selectivity constant K, defined in terms of thermodynamic activities, and the conventional selectivity coefficient K , defined in terms of concentrations. For the equilibrium of two counterions A and B between the aqueous state and the adsorbed state. written as
where z is the ion valence, K and K are defined as
and
where ai is the aqueous activity of component i, Zi is the activity in the adsorbed state, and Ti is the surface excess of i. K is a parameter that is commonly measured, but it is purely a phenomenological coefficient and so provides little information. The selectivity constant on the other hand may be used to determine the free energy of ion exchange and so provides information conceming the microscopic mechanism of selectivity. Unfortunately, K cannot be directly measured. In order for theory to be joined to experiment a way must be found to relate K to K . Such a relation has been given by Gaines and Present address: Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716. @Abstractpublished in Advance ACS Abstracts, May 15, 1995. +
0022-3654/95/2099-9458$09.00/0
where ZB is the mole fraction of counterion B in the adsorbed state,$ is the activity coefficient of counterion i in the adsorbed state, n, is the number of moles of adsorbed solvent per exchange equivalent of the resin, and a, is the activity of the resin-adsorbed solvent. (The limits a and b of the integrals are explained below.) Unfortunately the quantities n, and a, are not experimentally measurable, and thefi have apparently only once been determined in a real ~ y s t e m .In ~ practice only the integral over Kt is calculated in the application of eq 3, which leaves unresolved the magnitude of the contribution of the remaining terms. In contrast to the case of resins, the airtwater interface of a solution containing an ionic surfactant and two or more counterions affords an ideal system in which to study the thermodynamics of ion exchange. A similar expression to eq 3 may be derived for the airtsurfactant solution interface which makes reference only to experimentally accessible quantities. This result is developed in section 2 and its application to a real system illustrated in section 3. In section 4 we use the temperature dependence of the selectivity coefficient to resolve the exchange free energy into enthalpic and entropic contributions. One of the goals of the theoretical investigation of ion exchange selectivity is to be able to predict the selectivity constant of two ions from independently measurable properties of the individual ions. In section 5 we develop a quantitative form of Eisenman's theory of ion exchange? the partial dehydration model, which successfully describes with few exceptions ion exchange for both anions and cations, for exchanges within a valence group and for exchanges of ions of different valencies. It is shown that, of the inorganic counterions whose ion exchange behavior has been studied, most fall into a small number of hydration classes characterized by an experimentally measurable partial dehydration parameter. This microscopic model fits naturally into the general thermodynamic framework and unifies a large body of ion exchange data. 0 1995 American Chemical Society
Ion Exchange Selectivity at Interfaces
J. Phys. Chem., Vol. 99, No. 23, 1995 9459
2. Thermodynamic Framework In this section we derive a result corresponding to eq 3 for ion exchange at the free interface of a solution containing a surfactant ion S and two competing counterions A and B. At the outset we adopt the assumption that there are no coions to the surfactant present. This is situation is realized in a number of measurements of K reported elsewhere.* After stating the main result, we will then illustrate its extension to coion systems. The derivation follows that of Gaines and Thomas’ original r e ~ u l t with , ~ some important differences. We begin with the Gibbs adsorption equation at constant temperature,
-A d n
+ TiSA WSA+ Ti,
WSB= 0
(4)
Now, recalling the definitions of the exchange constant (eq 1) and coefficient (eq 2), and the fugacity coefficients, we may write
In K = In K
,iisA =
+ RT In as,
(5)
- lnf,,
(10)
Since K is strictly constant, d In K
+ d lnf,,
= d Inf,, = 0
(1 1)
The combination of eq 9 with eq 11 leads to the elimination of the two activity coefficients in turn. Eliminatingfsa first gives d lnf,, =
II is the surface pressure, and ,&A is the chemical potential of the electrically neutral aggregate of surfactant and counterion. This does not imply association of the surfactant and counterion in any chemical sense. Quantities with bars refer to the surface phase. A is the area occupied by 1 mol of adsorbed surfactant ion S, and E, is the number of moles of species i adsorbed in such an area. The following analysis is therefore based on a constant amount of adsorbed matter and variable interfacial area, rather than the other way around. This is unconventional but entirely consistent. This approach is adopted because it is the natural framework in which to treat the question “What is the change in the free energy of the system when 1 mol of counterion B is substituted for 1 mol of counterion A at the interface?’ Using the Gibbs convention for the surface, the solvent surface excess is zero. Consequently the problematic solvent term in eq 3 does not appear in our calculations, which considerably simplifies the results. The chemical potential of the surface adsorbed species is
+ lnf,,
dII iA d In K - d In r + TRT
(12)
Eliminating the other activity coefficient gives d lnf,,
= -i, d In K - d In r
+dn TRT
(13)
Integration of these last two results will afford expressions that may be substituted into eq 10 to determine In K . Let the state of the system be designated ‘a’ when the only counterion present is A, and ‘b’ when only B is present, and let ‘Q’ designate the state of the system at some intermediate composition. We carry out the following integrations at constant temperature, constant in the interface (but variable A and r),and constant ionic composition in the bulk solution, save for the counterions undergoing substitution in the transition from a to b. Beginning with our expression for f s ~ ,
Therefore, at Q
&,
is the standard state chemical potential. This statement does not imply association of the surfactant and counterion in any chemical sense. A similar expression may be written for B. A definition of the standard state of adsorbed SA is now needed. We choose to refer all surface states to a hypothetical standard state of a surface excess of 1 mol per square meter behaving as an ideal two-dimensional gas, the equation of state of which is
n = rRT
(7) wherej is the surface fugacity coefficient and r”= 1 mol m-*. Substituting for the chemical potential in the Gibbs equation and dividing by the molar area, we find
=0
(8)
From here we follow Gaines and Thomas’ d e r i ~ a t i o n . On ~ expanding the activities and dividing by r = r S A r S B , eq 8 becomes
+
--dn
rRT
lnfsA = lnfsA(a) - iB In K -
(6)
This definition is analogous to the conventional standard state for solutions, defined as a concentration of 1 mol per liter behaving as an ideal solution. The activities of the species in the interface are then defined as
-dn + TSAd In as, + TSBd In z,i RT
Similarly, integrating eq 13 from a to Q, we have
+ d In r + i, d lnfsa + i, d lnf,,
=0
(9)
kQIn K d i B +
sQe a
TRT
- k Q dIn r (16)
Substituting eqs 15 and 16 into eq 10, we get, after some manipulation,
which finally becomes In K =
S,I In K d i , + ln-as,(b) + sbdn TRT a
(18)
This result relates the thermodynamic selectivity coefficient to K, ZsA(a), and ZsB(b), which are experimentally accessible. It is analogous to Gaines and Thomas’ result for ion exchange resins. However, there is no solvent term, and all the terms contributing to K are experimentally measurable and may be determined from appropriate adsorption isotherm data. The physical significance of each of the terms appearing in eq 18 is not immediately obvious, so we shall briefly discuss the underlying chemistry of this result. We begin by observing
Morgan et al.
9460 J. Phys. Chem., Vol. 99, No. 23, 1995
that the last term is just the mechanical work involved in changing the area and surface tension of the system as the counterion composition is altered. This term is specific to ion exchange at the surfactant solution interface. A different accounting for the mechanical work of ion exchange would be required to describe ion exchange at a micelle surface or in an ion exchange resin, for instance. The remaining terms, however, should be applicable to any ion exchange system. Next, we consider the integral over the selectivity coefficient, which can be written as
The integrand of the first term on the right hand side is antisymmetric with respect to an origin of jiB = I/*. Therefore the integral over the stated limits vanishes and
The terms representing the surface composition have canceled out. Therefore this term is best interpreted as accounting for the free energy change in the aqueous subphase in going from state a to state b. Similarly the term zS,A.(a)
In accounts for the change in the free energy due to the change in the chemical nature of the surface phase in going from state a to state b. One now sees that eq 18 expresses the equilibrium constant as the sum of chemical terms corresponding to energy balances in the aqueous and surface phases and a surface work term. Grouping the aqueous terms, and the surface terms we may write eq 18 as -RT In K = g
+g
(21)
where g is the free energy change in the solution phase and g is the free energy change in the surface phase upon substitution of A for B. The integral over In K, g, is usually the only contribution that is considered in calculating the selectivity constant K , as the other contributions have not previously been made explicit the way they have been here. We will therefore calculate the term g for the Br/Cl ion exchange at a surfactant/ solution interface in the following section. The foregoing discussion has assumed that there are no coions to the surfactant present. This assumption is justified when the coion surface excess is a great deal less than the surfactant excess. It is not the case when, for instance, the coion is itself a surfactant. Coions can be incorporated into the analysis in a manner akin to the constructionof mean pair activity coefficients in standard solution thermodynamics. We will illustrate this for a single coion, M, with the extension to any number of coions being trivial. The Gibbs equation for such a system is -A d n
+ rsd In fsrs+ rMd In fMrM + rAd In fArA + rBd In fBrB =0
(22)
This may be written in terms of properties associated with neutral aggregates of the ionic species SMA and SMB. Let r s M A and rsMB be the surface excesses of these species. These
quantities satisfy the following mass balances:
r ~ ~ *S M B= r~ rsMA + rsMB = rs+ rM =r ~ S M A
We also define the mean triplet fugacities fSMA
=fsfnf,$fSMB
=f%fMfB
and the mean triplet surface activities
When the appropriate substitutions are made, eq 22 can be written as
The previous manipulations now carry through as before using the mean triplet activities and activity coefficients instead of the mean pair quantities, ensuring that the integrations are carried out under constant Tis EM. For the case of a coion of low surface activity, the surface fugacity fM 1 and r M
