Environ. Sci. Technol. 1992, 26, 1780- 1786
Thermodynamically Derived Relationships between the Modified Langmuir Isotherm and Experimental Parameters Wilfred L. Polzer”
Earth and Environmental Science Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 M. Gopala Rao
Department of Chemical Engineering, Howard University, Washington, DC
20059
Hector R. Fuentes
Departments of Civil Engineering and Geological Sciences, University of Texas at El Paso, El Paso, Texas 79968 Rlchard J. Beckman
Analysis and Assessment Dlvision, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 B The modified Langmuir isotherm parameters and the
thermodynamic ion-exchange properties are developed from theoretical considerations leading to the derivation of a relationship between the two. For the binary ionexchange system, the relationship indicates that a unique set of isotherm parameters can be defined by a minimum set of experimental parameters. Those parameters are the solid-liquid ratio and the initial concentration of the competing solute in the liquid and exchanger phases. In principle, unique sets of isotherm parameters can be predicted for different conditions from a known set of experimentally determined parameters. The effect of experimental parameters (e.g., solid-liquid ratio and competing solute concentration) on isotherm parameters (e.g., linear Kd)have been documented in the literature. Experimental data are used to illustrate the derived relationships.
Introduction In the past, retardation caused by sorption has been represented by a single term (e.g., &) that relates the concentration of the solute in the solution phase to that in the solid phase (I,2). This simplistic representation is used because sorption processes are generally complex and not well understood, and thus effective coupling of these processes to transport models is difficult to achieve. However, adsorption of solutes on solid materials can be described in terms of isotherms. These isotherms range from simple (e.g., linear &) to complex types (3-6). Use of isotherm equations allows coupling of adsorption parameters to transport equations to account for retardation of a solute under flow conditions (7). Sophisticated mechanistic models, such as the triple-layer model, are also being developed to evaluate sorption (8). However, the information necessary to describe such sophisticated models for natural environmental conditions is usually not available (9). Therefore, conventional isotherms are expected to be used in the future to describe sorption and to couple sorption models to transport models. Adsorption isotherm equations are mathematical data fitting devices, i.e., empirical fitting equations. Their mathematical form has, for the most part, been developed empirically and subsequently rationalized by various theories [e.g., the Gibbs (IO),Langmuir ( l l ) ,BET (12), and IAS (13)theories, among others]. Parameters derived from theoretical considerations should be independent of experimental or environmental conditions. However, many of the relatively simple conventional models have been 1780 Envlron. Sci. Technol., Vol. 26, No. 9, 1992
extended beyond the limits of theory. For example, Langmuir theory assumes homogeneity of adsorption sites, whereas most natural adsorbents exhibit heterogeneity of adsorption sites. Distribution functions that describe the heterogeneity of adsorption sites have been derived by rigorous mathematical methods (14,15) and from kinetic theory (16) for sorption in monosolute systems. However, only semiempirical or multicomponent analogues (5,17, 18) have been used to extend the theory to multisolute sorption systems. Thus, for competitive multisolute systems, isotherm parameters are still considered empirical (619). Empirical parameters depend on the conditions for which they are determined. Thus, if the effects of sorption on retardation need to be characterized for radionuclides that may leak from a geologic repository, the number of isotherm parameter determinations could be overwhelming because of the numerous variables encountered along flow paths from the repository to the accessible environment. Investigators of ion-exchange reactions have attempted to minimize the number of variables needed for predictive purposes by relating isotherm parameters to fundamental thermodynamic constants using phase equilibrium models (20-22). However, these attempts have been restrictive in their application. For example, Elprince et al. (20) and Serne and Relyea (21)described the Kd parameter of the linear isotherm in terms of a thermodynamic parameter for ion-exchange adsorption. However, the linearity of the model was applicable over a very limited adsorption range. Elprince and Sposito (22) also employed classical thermodynamics to derive Langmuir-type isotherms. This relation was restricted to low concentrations of the ion in an electrolyte. Other investigators (5, 23, 24) derived isotherm equations for univalent-univalent ion exchange. The overall purpose of this study is to derive relationships between the parameters (constants) of 8 generalized isotherm (modified Langmuir) and experimental parameters (conditions) for binary ion exchange through the thermodynamic model of Gaines and Thomas (25)and to demonstrate those relationships using experimental data. Specifically, the experimental conditions necessary for the prediction of isotherm Constants of a specified sorbent and specified solutes will be identified. The experimental data will be used to (i) estimate constants of the modified Langmuir isotherm, (ii) predict rational selectivity coefficients and exchanger-phase activity coefficients from the isotherm parameters, (iii) calculat,e the thermodynamic equilibrium constants, and (iv) demonstrate the dependency of isotherm parameters on experimental variables.
0013-936X/92/0926-1780$03.00/0
0 1992 American Chemical Society
Mathematical Relationships In the theoretical development of the relationships between the modified Langmuir isotherm and experimental parameters, a binary ion-exchange reaction will be assumed. The stoichiometric ion-exchange reaction for phase equilibrium can be written as vBA + v A B + v B A + vAB (1) where A, B are solutes A and B, respectively, in the liquid phase; A, B are solutes A and B, respectively, in the solid phase; and vA, V B are stoichiometric coefficients of solutes A and B, respectively. Modified Langmuir Isotherm. The following equation is the modified Langmuir isotherm (26) expressed in the nomenclature of eq 1.
where E A and E B are charge fractions of sorption sites occupied by solutes A and B, respectively; CAis the concentration of solute A in the liquid phase (mol m-?; KD is the isotherm parameter that represents the average sorption energy (m3mol-'); and /3 is the isotherm parameter that represents the heterogeneity of sorption sites. Rational Selectivity Coefficient. The rational selectivity coefficient (27) for eq 1 is
Kr$=
[I"
EB"*
g][51
(3)
where Krj$is the rational selectivity coefficient; YA and YB are single-ion activity coefficients of solutes A and B, respectively, in the liquid p h e ; and CBis the concentration of solute B in the liquid phase (mol rn-?. In the logarithmic form In K,AB = V B In E A - VA In E B + VA In C B + VA In YB V B In C A - V B In YA (4) The rational selectivity coefficients are conditional equilibrium constants that can be related to the thermodynamic equilibrium constant through one variable, either E A or ED In order to express Krj$as a function of E B , CA and CB need to be expressed in terms of E B . The concentration of solute A in eq 2 can be substituted into that in eq 4. An assumption is that the modified Langmuir isotherm parameters are obtained under the condition of a constant initial concentration of the competing solute, CBo. As will be seen later, a constant CBq is a necessary, but insufficient, condition for the modified Langmuir model to be valid. The concentration of solute B will also depend on the initial conditions of the experiment. Under the above conditions C B can be expressed in terms of the amount of solute A gained in the exchanger phase and the amount of solute B gained in the liquid phase: (5)
where Qois the total mole charge of exchanger phase (mol, kg-l), E, is the charge fraction of solute A initially in the exchanger phase, and E B o is the charge fraction of solute B initially in the exchanger phase. Thus substitution of logarithmic forms of eq 2 and eq 6 into eq 3 results in an expression of K A as a function of only one variable, E B .
where Y~ and Y~ are assumed to be constant. As indicated above, Kr$ can be expressed as a function of only E B for a specific exchanger and for solutes A and B. Therefore, if the initial conditions change (e.g., if W / V, EB and CBochange), then the isotherm parameters, KD an$ p, must change in order for Kr$ to be dependent only on E B . Values for KD and /3 may be obtained directly from experimental data using best-fit regression analysis when the experimental design does not maintain the experimental parameters W / V, E , and CBpconstant. However, these KD and /3 values s h o d be considered empirical and, as such, will only be applicable under the very same conditions. On the other hand, the isotherm data obtained under the above variable experimental conditions can be used to determine KD and /3 parameters that can be extended beyond the experimental design through the relationship in eq 7. The selectivity coefficients are dependent only on E B ; therefore, they can be determined under conditions of variable W / V ,EBo, and CFo. For example, selectivity coefficients are determined either at constant ionic strength or at variable ionic strength. Once determined, these selectivity coefficients then can be used to predict the KD and 0parameters for any set of arbitrarily selected WIV, E B o , and CBothrough eq 7. Exchange Equilibrium Constant. According to Gaines and Thomas (25),the exchange equilibrium constant, K,,;, can be expressed as an integral of Krj$. In K~,$= (vB - vA)
+ J0
1
aB
In
(8)
Equation 7 can be substituted into eq 8 and the resulting equation integrated to give the following. r
1
r
where q A is the mole charge of solute A in the exchanger phase (mol, kg-I), q, is the initial mole charge of solute A in the exchanger phase (mol, kg-I), CBois the initial concentration of solute B in the liquid phase (mol m-3), 2, is the valence of solute B, and W / V is the solid-liquid ratio (kg me3). Equation 5 can be rewritten in terms of E B .
or
As can be noted, the isotherm parameter /3 does not appear in eq 9; thus it is not related to Keqj$.That is to say, K , $ is an integration of all selectivity coefficients and is only indicative of the spread of selectivity coefficients about an Envlron. Scl. Technol., Vol. 26, No. 9, 1992
1781
Table I. Experimental Data for the Adsorption of Cobalt, Strontium, and Cesium on Calcium-Saturated Bandelier Tuff in 5 mol m-3Calcium Chloride Solution [Polzer et al. (ZS)]" CCO
Pco
(mol m-9 3.97 x 2.34 x 3.23 x 1.02 x 7.90 X 7.05 X 7.40 x 8.90 x 1.82 x 7.90 x 2.69 X 7.15 X 2.73 x 6.20 X 1.46 3.90
3.80 x 2.34 x 2.95 x 9.77 x 6.92 X 6.76 x 6.76 x 6-76 x 6.61 X 2.77 X 4.63 X 7.87 X 1.21 1.85 1.93 2.78
10-10 10-10 10-10 10-9 lo4 10-7 104 10-4 10-3 10-1 lo-'
"Q,= 3.30 mmol,
CSr
(mol, kg-9
(mol m 9
10-7 10-7
1.37 x 10-3 4.81 X 9.70 x 1.96 X lo-' 5.05 X 10-I 1.04 2.11 5.44
10-7 10-7 lo4 10-5 10-4 10-3
lo-' 10-l lo-'
In fA =
(VB - v A ) E B - E B
In K A
+
s,""
In K A
~ E B
(10) and uA
In f B =
(vA
2.27 x 7.22 X 1.32 X 2.64 X 5.07 X 8.72 X 1.38 2.12
10-3 lo-' lo-' lo-' lo-'
CCS
(mol m+) 6.54 x 6.76 x 1.07 X 5.13 x 4.57 x 1.78 x 1.51 x 1.78 x 3.90 X 9.55 x 2.72 X 5.59 x 1.20 3.30
10-9 10-9 lo-* 10-8 10-7 10-5 10-3 10-2 10-2 10-1 10-1
9cs
(mol, kg-') 4.57 x 4.79 x 7.59 x 3.47 x 3.02 x 2.95 x 2.40 X 7.76 X 1.44 X 2.18 X 4.16 X 7.73 x 1.21 1.86
10-6 104 104 10-5 10-4 10-3
lo-' lo-' lo-' 10-1
kg-'.
average selectivity coefficient. The exchange equilibrium constant permits the evaluation of the thermodynamic properties of adsorption. That is, the Gibbs free energy at a specific temperature can be calculated from the exchange equilibrium constant at that temperature; enthalpy can be calculated from equilibrium constants determined at different temperatures, and entropy from its relationship to the free energy and enthalpy. The Gibbs free energy gives an indication of the preference that an exchanger phase has for the competing solutes. It also provides a fundamental basis for handling multicomponent solutes and exchangers. Exchanger-Phase Activity Coefficients. Also from the activity coefficients of solutes Gaines and Thomas (B), A and B in the exchanger phase are given as follows: Vg
QSl
(mol, kg-l)
- vB)EA
+ E A In K A - S,,In ~~b ~ 1
E B
(11) where fA and f B are the activity coefficients of solutes A and B, respectively, in the exchanger phase. The substitution of eq 7 into eq 10 and eq 11followed by the appropriate integration gives
--- I
WQO.
I
and
The exchanger-phase activity coefficient describes the departure of adsorption from that of the assumed reference state and reflects all those effects that are not otherwise considered in the thermodynamic models (27). The reference state of the Gaines-Thomas model is the state in which all exchanger sites are occupied by the adsorbing solute A in an infinitely dilute solution of solute A (25). The modified Langmuir isotherm parameter describes the distribution of sorption energies. That distribution can be dependent on the nonuniformity of surface sites, interactions among adsorbed species, and the dependence of the enthalpy and entropy of adsorption on exchanger composition (28). Once the experimental parameters are fixed, the value of /3 reflects the distribution of exchanger-phase activity coefficients. It is dependent on the V), the initial concentration of the solid-liquid ratio (W/ competing solute in solution (CB,,), and the initial fraction of exchanger sites occupied by the competing solute (E%).
Materials and Procedures The adsorption data are those obtained by Polzer et al. (29) for the adsorption of strontium, cesium, and cobalt on calcium-saturated Bandelier tuff. The investigators (29) did not present a complete tabulation of the experimental data. In order to aid researchers, the data are presented in Table I of this study. Bandelier tuff is described geologically as nonwelded to welded rhyolite ash flows and pumice (30). Mineralogically the tuff is composed primarily of orthoclase feldspar with quartz and very little or no clays. The cation-exchange capacity of the tuff was determined to be 3.3 f 0.5 mmol, kg-l over the pH range of 7-9. Adsorption of strontium, cesium, and cobalt was measured with the aid of radioactive tracers of the respective solutes. Details of the experimental design are given by Polzer et al. (29). In this study the assumption is made that the hydrogen ion did not participate in the ion-exchange reactions studied. Results and Discussion Modified Langmuir Isotherm. The observed adsorption of strontium, cobalt, and cesium on Bandelier tuff is given in Figure 1 along with the best fits of the data achieved with the modified Langmuir isotherm (eq 2). Note that qAin Figure 1is equal to Optimal fitting parameter values were determined by nonlinear regression. The values of the isotherm parameters, their lower and upper limits at the 95% confidence level, "error" variance
QaA.
1782
Environ. Scl. Technol., Vol. 26, No. 9, 1992
Strontium
Observed
- Best Fit
-I
Table 11. Modified Langmuir Isotherm Parameters with Lower and Upper Limits at the 95% Confidence Level, and the Variance (SSR)and Correlation Coefficients (R) of the Regression Analysis for the Adsorption of Strontium, Cesium, and Cobalt on Calcium-Saturated Bandelier Tuff"
lower limit
KD (m3 mol-') Cesium
L J4
1
Y
Ti
upper limit lower limit
P
upper limit SSRb R
strontium
cesium
cobalt
0.33 0.34 0.35 0.92 0.95 0.98 0.0122 0.82
0.40 0.42 0.44 0.79 0.83 0.86 0.00933 0.89
1.6 1.9 2.3 0.5 0.6 0.7 0.433 0.89
OAdsorption data are from experiments of Polzer et al. (29). *SSR is the square root of the resultant quantity of the sum of squares of the residuals divided by number of residuals minus 2. Cobalt
4t
i
1 1
OO l
3
2
r6
5
4
C, (molc rn-3) Flgure 1. Observed and best-flt concentrations of adsorbed strontlum, ceslum, and cobalt on calclumsaturatsd Bandelier tuff as a function of their concentrations In solutlon. Best-fit concentrations are based on the parameters of the modifled Langmulr Isotherm. Adsorption data are from experiments of Polzer et al. (29). I
I
I
I
I
I
I
I
t1 -
1
1 1
9.0
t
I
m
c
8
0.8
1
= -3.0 l,Ot.---.---J
-< 0.6
-7.0 0.0
0.2
0.4
-
0.6
0.8
1.o
Flgure 3. Rational selectivity coefflclents ( K i ) and exchanger-phase activity coefficients (fA, f,) for the adsorption of cesium (A) on calcium (B) saturated Bandelier tuff related to the charge fraction of solute B on the exchanger phase. Adsorption data are from experlments of Polzer et al. (29). The solld clrcles are observed values, and the solid lines are values based on modified Langmulr isotherm parameters determined from a best fit of the adsorption data.
0.0
0.6 0.8 1 .o E, Flgure 2. Ratlonal selectlvlty coefflclents (K;) and exchanger-phase activity coefflclents ( f A , f,) for the adsorptlon of strontium (A) on calclum- (B) saturated Bandeller tuff related to the charge fractlon of solute B on the exchanger phase. Adsorption data are from experlmen& of Poker et al. (29). The solld clrcles are observed values, and the solld lines are based on modifled Langmuir Isotherm parameters determlned from a best flt of the adsorptlon data. 0.0
0.2
0.4
of the regression, and the correlation between the best-fit values and the experimental data are given in Table 11. Based on the 95% confidence level, approximately 4 % variability is observed in the isotherm parameters for strontium adsorption, approximately 6% variability for cesium adsorption, and 20% variability for cobalt adsorption. The initial slope of the data in Figure 1indicates that strontium is adsorbed least strongly and cobalt most strongly on the calcium-saturated tuff; that is, for a specific concentration of the solute in solution, the amount of strontium adsorbed from solution is least, followed by cesium, and then cobalt. Rational Selectivity Coefficient. The selectivity coefficient (K& was calculated from the experimental data as a function of EB using eq 3 (C, was calculated from eq 5 ) . The results of those calculations are plotted for the Envlron. Scl. Technol., Vol. 26, No. 9, 1992
1783
c I
6.0
I
21
c-
42 4.0 Y
-C 2.0
0 -1 .o
0.0
I
0.2
0.4
0.6
0.8
l
l
1 .o
fB
Flgure 4. Rational selectlvity coefflcients (K,;) and exchanger-phase activity coefficlents ( f A ,te)for the adsorption of cobatt (A) on calcium (B) saturated Bandeiler tuff related to the charge fraction of solute B on the exchanger phase. Adsorption data are from experlments of Polzer et al. (29). The soild clrcles are observed values, and the s o l i lines are values based on modified Langmuir Isotherm parameters determined from a best fit of the adsorption data.
adsorption of strontium, cesium, and cobalt as a function of EB (Eca) in Figures 2-4, respectively. Selectivity coefficients predicted from the modified Langmuir parameters were determined using eq 7. Those results are also plotted in Figures 2-4. The above plots indicate that the isotherm parameters do predict the observed selectivity coefficients very well. However at low values of EB, experimental data are not sufficient to evaluate the predictions by the modified Langmuir isotherm. But experimental data documented in the literature do support the general pattern of change in selectivity coefficients at low values of EB (31, 32). The patterns of change in selectivity coefficients as a function of EB for the adsorption of strontium, cesium, and cobalt show similarities. The selectivity coefficient of strontium adsorption changes from concave to convex at EBe 0.5. The range of values is relatively small with the extremes occurring as EB approaches 0 and 1. The pattern for cobalt adsorption is similar to that for strontium. However, the magnitude of the range is greater for cobalt adsorption. The concave section appears to be a mirror image of the convex section for both strontium and cobalt adsorption. In the case of cesium adsorption, the change from concave to convex occurs at EB 0.8; therefore, the concave section is not a mirror image of the convex section. Apparently this is because VA # vB. It is of interest to note that if 0 = 1 (homogeneous adsorption sites for homovalent systems), vA = vB, and ( w / V ) ( Q o / z ~ )
