Anal. Chem. 1986,58,2939-2943
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Further Studies of the Hydration Model for Ionic Activities in Unassociated Electrolytes Roger G. Bates Department of Chemistry, University of Florida, Gainesville, Florida 32611
At ionic strengths less than 0.1 mol kg-’, where long-range ion-ion interactions are dominant, several reasonable estlmates of the indivlduai ion actlvlties are In substantlai agreement. I n the past, conventlons based on the separation of mean activity coefficients into their ionic components in accord with suggestions of MacInnes and Guggenheim or with the Debye-Huckel theory have found use. At elevated ionic strengths, however, the activity coefflclents of single Ionk species become dependent not only on the Ionic strength but also on the specific nature of the counterlon. A lwo-parameter convention based on the Stokes-Robinson-Glueckauf modification of the Bjerrum hydratlon theory can relate the activity of a given ion species at high Ionic strengths to the difference of the hydration Indexes h, h , of the cation and anlon in the unassociated electrolytes MX, MX,, and M2X. The hydration parameters for 53 electrolytes have been determined for a modification of the model that allows the hydratlon index lo change with the molality of the electrolyte. Evidence supporting the convention h,, = h,, = h , = 0 is given. On this basis, ion activities in solutions of LICI, NaCI, KCI, NH,CI, KF, and CaCI,, useful for the standardization of ion-selective electrodes, have been derived.
-
The appearance of potentiometric sensors responsive to individual ionic species is one of the most dramatic developments of the past two decades. Under certain conditions, such as at constant ionic strength, the response of these electrodes may be directly related to ionic concentrations. In the majority of instances, however, the derivation of analytically useful information demands some assumption regarding the magnitude of single ionic activities or activity coefficients. Data of this sort are needed, for example, to standardize the potentiometric assembly and to convert a voltage response from a function of activities to ionic concentrations. As is well-known, activities of individual ionic species are not thermodynamically accessible. It is therefore necessary to resort to conventional scales, which, despite their arbitrary basis, may nonetheless be internally consistent and reasonable in the light of electrolyte solution theory. Activities of single ionic species are the key, as well, to evaluating single electrode potentials and liquid-junction potentials. Furthermore, Frank ( I ) has emphasized that a complete account of the thermodynamics of aqueous electrolytes awaits a knowledge of the thermodynamic properties of individual ionic species.
PROPOSALS FOR DILUTE SOLUTIONS In the past, the problem of the Gibbs energies of individual ionic species has been addressed primarily in two ways, namely through attempts to estimate the potential of a single reference electrode (2-4) and by procedures for separating mean ionic activity coefficients (y+)or energies of transfer of electrolytes from water to a second solvent (both of which are thermodynamically defined) into the contributions of the component ions (5-8). Both approaches involve an arbitrary step not subject to proof, and at the present time the second appears to be the more promising.
In solutions of symmetrical electrolytes MX, which are (1) not subject to ion association and (2) so dilute that only long-range electrostatic interactions are significant, the Debye-Huckel theory suggests that (1) ?’M = ?’X = y* Furthermore, for unsymmetrical electrolytes under these same conditions the theory embodies the following effects of ionic charge: y+4
= 7-4= y2+ =
72-
(2)
and
(3) = y-9 = y3+ = 7 3 Faced with the problem of evaluating liquid-junction potentials, MacInnes ( 5 ) proposed that the relationship of eq 1 be applied to KCl at all ionic strengths. The way was thus paved for calculating the activity coefficients of other ions, provided mean activity coefficients were known. Thus, in CaC1, (0.2 mol kg-’), y+9
y+3 (CaCl,, m = 0.2) YCa
=
yb2 (KC1, m = 0.6)
(4)
This procedure, sometimes called the “mean salt method”, has been widely used, particularly in marine chemistry. In the ionic strength range 0.01-0.1 mol kg-’, the MacInnes convention is consistent with the convention chosen (7) for establishing the standard reference solutions on which the National Bureau of Standards pH scale is fixed. The pH convention log ycl = -A11/2/(l
+ 1.511/’)
(5)
where A is the Debye-Huckel slope, was intended to apply only to specific buffer/chloride mixtures with ionic strength (1) no higher than 0.1 mol kg-’, and its extension to higher ionic strengths has proved impractical (9). In order to minimize residual liquid-junction potentials and, hence, to realize the full capabilities of ion-selective electrodes (some of which behave well in concentrated electrolyte solutions), standards of ionic activity at Z = 1 and above are needed. From the point of view of modern theories of electrolyte mixtures (e.g., ref 1@-12), it is unreasonable to expect ycl, for example, to have the same value in 1 m KC1 as in 1 m LiCl or 1m CsC1, where the short-range interactions between the chloride ion and the cations will be different. Thus, a “single-parameter’’ convention is inadequate (23). Despite their demonstrated usefulness under conditions of restricted ionic strength, the conventions mentioned above do not meet the requirements of reasonableness when applied at elevated ionic strengths.
ELABORATION OF THE HYDRATION MODEL FOR MEAN ACTIVITY COEFFICIENTS Bjerrum (14) pointed out that specific differences among the activity coefficients of strong electrolytes of the same charge type at a given ionic strength can be attributed to differences in the hydration of the ions. Bjerrum’s approach offered a basis for a “two-parameter” model for the activity coefficients of unassociated electrolytes at high concentrations.
0003-2700/86/0358-2939$01.50/0 0 1986 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 14, DECEMBER 1986
Stokes and Robinson (15) and Glueckauf (16) elaborated the treatment by separating the electrostatic contribution fel (described by the “extended” Debye-Huckel equation with ion-size parameter a) from the contribution fhyd resulting from reduction of the solvent activity (described by a hydration number, h ) : log Yi = f d
+ fhyd
= f(Ict
a) + f(m,h )
(6)
where I, is the ionic strength on the concentration (mol dm-3) basis and m is the molality (mol kg-l). Their equations were remarkably successful, fitting the mean activity coefficients of many unassociated electrolytes up to ionic strengths of 6 and above. The value of h was considered to be a constant for a given electrolyte as long as the moles of unbound water exceeded the amount (hm)of bound water by a considerable margin. Stokes and Robinson set this limit at hm = 12 (15). Assuming further that ionic hydration numbers are additive, and following Frank ( I ) , we have proposed (17,18) that the separation of yi into y M and yx could be governed by the relative magnitudes of hM and hx. The values of h for the alkali chlorides suggested that hcl must be small, and it was proposed that ionic hydration numbers be based on the convention hc, = 0. It was then possible to use the GibbsDuhem equation to separate yi for unassociated chlorides MCl and MC1, into their cationic and anionic parts, with the result 0.018 log r+(MX) = log y+ + -(h+ - k ) m @ (7) In 10 and, for a 2:l chloride, 0.018 log y+(MX,) = 2 log y+ + -(h+- hJm@ In 10 log [l 0.018(3 - h M + hcl)m] (8)
+
+
where cp is the osmotic coefficient. The activity coefficient of the anion of the electrolyte M,+X,. is then obtained from the mean activity coefficient y+ by y. = (y+U/y+”+)l’U-
(9)
where the number of ions is given by u = u+ + u-. For a symmetrical electrolyte u+ = v _ = 1 and one has, corresponding to eq 7, 0.018 log y- = log y* - -(h+ - h.)m@ (10) In 10 Equations 7 and 8 suggest ( I ) that y+ = y- = y+ when both ions of a symmetrical electrolyte are hydrated to the same degree and ( 2 ) that y+/yt = y+/y- when they are not. There is evidence that the Glueckauf modification (19) of the Debye-Huckel equation, relating to a disordered lattice model (20,21), is more successful at moderate and high ionic strengths than the ”extended” equation. Glueckauf has indeed demonstrated that this formula is superior to the DebyeHuckel model for estimating fel up to KC^ = 2.8, or to an ionic strength of about 5 mol kg-’. For this reason, the hydration equation preferred by Bates and Robinson (22) utilizes the Glueckauf (1969) equation for j,, and combines it with fhyd derived by “volume-fraction statistics”, as in the 1955 treatment of Glueckauf (16): log y* = --A(z+.z_~I,~’’
1
I+
+ 0.5BuIc’/2
1 + BaIC1l2 0.018mr(r + h - v ) + u(ln 10)(1+ 0.018mr) [ ( h- u ) / u ] log (1 0.018mr) - ( h / u ) log (1 - 0.018mh) (11) In eq 11,A and B are Debye-Huckel constants dependent on dielectric constant and temperature, u is the number of ions of charges z+ and z- from one molecule of the electrolyte, and
+
r is the ratio of the apparent molar volume of the electrolyte to that of water. The calculation of molar concentrations and apparent molar volume ratios will be discussed in a later section.
DERIVATION OF HYDRATION PARAMETERS FOR ELECTROLYTES FROM MEAN ACTIVITY COEFFICIENTS The first term on the right of eq 11 is expected to yield values of fel (eq 6) that vary linearly with m when the proper value of the ion-size parameter, a, is used (23,24). Similarly, there is evidence (16) that the hydration contribution, fhyd, is a closely linear function of m as well. In our earlier work (22),a graphical method was used to obtain “best” values of (1 and h. By trial, the value of 6 required to produce the best straight-line plot of log y+ - fel was found, and the slope PI of this line was determined. Next, values of h were chosen by trial until the slope Pz of fhyd vs. m matched PI. The present study was devoted to refinements of this approach (method A) of deriving the parameters and to examining a more sophisticated model in which allowance was made for a possible variation of h with molality (method B). Preliminary results have been described elsewhere (25). Glueckauf (19) has pointed out that the value of fhyd doubtless reflects factors other than ion hydration, notably hard-sphere effects and the like, but that these are likely also to vary linearly with the molality. For this reason, we prefer to denote h the “hydration index” rather than the “hydration number”. Nevertheless, the values of h derived from activity coefficients agree satisfactorily with primary hydration numbers estimated for ionic species. The mean activity coefficients, in general, were taken from the compilations of Robinson and Stokes (26). For 1:l electrolytes, the data for this study covered the molality range 0.1-3.0 mol kg-l (N = 17 points) or to a maximum of 4.0-5.0 mol kg-’ (N = 19 or 21). For the halides of the bivalent alkaline earth metals, the maximum molality was either 2 or 3 mol kg-’ ( I = 6 or 9 and N = 15 or 17, respectively). Method A. For further studies by method A, a computer program was used instead of the graphical procedure to match p2 with PI. The salient steps of the program were as follows: (1)input first estimates of u , h, and increments .ki and Ah; ( 2 ) calculate log y+ - fel and perform linear regression vs. m to obtain first estimates of fll and the variance of regression; (3) change ci by the assigned increment Ad and repeat until variance is at a minimum (giving best estimate of &); (4) calculate fhyd from the first estimate of h and perform linear regression vs. m to obtain Pz; (5) change h by the increment Ah and repeat until & matches the best estimate of PI; and (6) print best values of (1 and h. Consistent values of these parameters were obtained to fO.O1 unit by this procedure. Method B. When the hydration index of a uni-univalent electrolyte was small ( h < 4), eq 11 have a good fit of the activity coefficient data; standard deviations of fit, u, were less than 0.002 in log y+. When h was greater than 5, u rose to 0.007 (LiC1) and 0.008 (HC1). The f i t was still worse for the highly hydrated (h from 8 to 12) alkaline earth chlorides, where u exceeded 0.02 in some instances. Figure 1is a plot of the data for CaC12 Differences between the “known” log y+ and that calculated with the values of a and h provided by method A are plotted as a function of m. It is clear that fhy< is, in this case, not strictly a linear function of m, as assumed in method A. In a refinement of this method (not shown), the best values of d and h were determined simultaneously by a nonlinear least-squares fitting. In this way, standard deviations of fit for the alkaline-earth chlorides were reduced to less than 0.01. Nevertheless, the assumption of a hydration index that does not change with the molality of the electrolyte is obviously
ANALYTICAL CHEMISTRY, VOL. 58,NO. 14, DECEMBER 1986
2941
, I
CaCiz
Method A hzI049
0 02
4-
-
h
h i I / 0 4 - 0 462m
/
I \
I I
/
I
q
0.117 0.089 0.021 0.019 0.016 -0.045 0.044 0.061 0.042 0.041 0.034 0.020 -0.058 0.102 0.001 0.039 0.048 -0.012 0.001 0.042 0.094 -0.051 0.044 0.030 0.025 0.147 -6.882 1.152 0.098 0.213
(log Y * P
6 9 9 5.4 6 9 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5.4 5.4 6
8.5 0.9 6.6 1.1 1.5 0.4 1.o
0.2 1.1 0.3 2.1 1.3 1.5 0.3 9.4 0.7 7.2 1.7 4.2 1.0 1.3 0.3 1.5 0.3 3.1 1.3 7.8 1.4 5.7 2.4 2.6 1.3 3.3 0.4 3.2 0.5 1.7 1.0 5.4 0.6 3.7 0.5 3.0 1.6 1.7 1.0 1.6 0.8 5.3 0.2 4.6 1.0 0.5 2.5 3.3 3.4 9.7 08
u = standard deviation of fit.
dexes for the cations listed in the first column. The mean deviations given in the last column show that the uncertainty in evaluating h, is of the order of 0.4 or less. In a later section we show that an error of this magnitude in h, - h- does not alter significantly the single-ion activities derived by the hydration convention. It is of some interest to compare the hydration indexes assigned in this way with primary-sphere hydration numbers estimated from various physical and thermodynamic data. From compressibilities, entropies, apparent molar volumes,
5.48 5.44 5.41 5.05 5.47 5.50 5.55 5.25 6.29 6.00 6.10 5.55 5.23 4.67 4.69 5.01 5.59 6.22 4.41 4.03 4.21 4.24 3.03
11.97 11.04 10.45 9.88 12.83 11.55 10.73 10.04 12.68 11.77 11.18 10.91 11.17 8.22 6.27 10.55 10.99 13.47 7.67 3.97 3.87 4.08 4.26
0.523 0.462 0.433 0.686 0.681 0.485 0.402 0.563 0.512 0.491 0.399 0.540 0.712 0.584 0.553 0.699 0.660 0.582 0.307 -0.207 -0.113 0 0.184
1.8 1.7 2.9 1.6 1.4 1.6 1.6 1.5 1.3 1.6 2.0 3.0 2.2 2.3 2.2 1.3 1.4 1.4 0.9 0.7 0.9 1.2 1.9
u = standard deviation of fit. ~
~~~~
Table 111. Comparison of Hydration Indexes ( h ) for Halides Derived by Methods A and B (Equations 1: and 12) at a Molality of 1 mol kg-'" h for given halide
H+ Li+ Na+
K+ Rb+
Cs+ Mg2+ Ca2+ SrZ+ Ba'+
chloride
bromide
iodide
mean
5.53 5.17 3.59 2.48 2.23 1.80 11.45 10.58 10.02 9.19
5.71 5.25 3.89 2.42 1.89 1.36 12.15 11.06 10.33 9.48
5.73 5.10 3.99 2.50 1.60 0.80 12.17 11.28 10.78 10.37
5.66 f 0.08 5.18 f 0.05 3.82 f 0.16 2.47 f 0.03 1.91 f 0.22 1.32 f 0.35 11.92 f 0.32 10.97 f 0.26 10.38 f 0.26 9.68 f 0.46
"Values from method A are included only when the standard deviation of fit was less than 0.0025.
and mobilities, Bockris and Reddy (29) estimate the most probable integral values as follows (our hydration indexes in parentheses): Li', 5 (5.2); Na+, 4 (3.8);K', 3 (2.5); F-, 4 (1.5); C1-, 2 (0); Br-, 2 (0); I-, 1 (0). The agreement seems quite acceptable. In a recent paper, however, Pan (30) has elected to identify the ion-size parameter ci in the Stokes-Robinson hydration theory (15)with the sum of the Pauling crystal radii of the cation and anion and has then derived values of h for the lithium, sodium, and potassium halides. This procedure yielded values considerably larger than those given i n Table 111: K halides, about 7; Na halides, about 13; Li halides, about 22.
INDIVIDUAL ION ACTIVITIES With the aid of values of h, from the last column of Table I11 ionic activity coefficients y+ and y-in solutions of unassociated halides at elevated ionic strengths can be calculated. The mean activity coefficients and osmotic coefficients (or water activities) are needed to utilize eq 7, 8, and 10. From these, ionic activities useful in the standardization of ionselective electrodes can be derived, as shown in Table 1V. It is now important to ascertain the effect on the calculated ion activity resulting from the uncertainties in ionic hydration
ANALYTICAL CHEMISTRY, VOL. 58, NO. 14, DECEMBER 1986
Table IV. Recommended Values for Ion Activities in Solutions of Strong Electrolytes at 298.15 K ion electrolyte LiCl NaCl
KC1 NHICl KF CaC1,
(i)
m = 0.1
5.2 3.8
Li+ Na+ C1K+ NH4+
1.100 1.106 1.112 1.112 1.112 1.111 1.570
2.5 2.3 1.0 11.0
FCaZ+
aia t m = 1.0 m = 2.0
-log
h+- h_
0.070 0.155 0.210 0.201 0.204 0.198 0.580
-0.358 -0.184 -0.067 -0.095 -0.090 -0.104 -0.188
Table V. Sensitivity of the Calculated Ion Activity in Solutions of Electrolytes MX and MX2 to Uncertainties in the Hydration Index ( h ) electrolyte
m=1
uncertainty in h a t m=2 m=3
m=4
S(pM)/Gh
LiCl NaCl KC1 CSCl MgClz CaC1, BaClz
-0.008 -0.007 -0.007 -0.007 0.000 0.001 0.000
-0.017 -0.016 -0.014 -0.014 -0.002 0.000 -0.004"
-0.030 -0.025 -0.022 -0.021
-0.045 -0.035 -0.030 -0.029
0.025
0.035
S(pX)/%
KF
0.008
0.015
'At m = 1.8 mol ke-I.
values indicated in the last column of Table 111. The data in Table V show than an error of 1 unit in h+or h- alters the value of -log aiof the cation and anion by less than 0.03 at a molality of 3 mol kg-'. This lack of sensitivity to the value of h+- h- is fortunate, enhancing the validity of a conventional set of ionic hydration indexes such as that proposed here. In many instances, ion-selective electrodes can be standardized usefully in terms of ionic concentrations, as in measurements a t low or constant ionic strengths. In this connection, further studies of metal-ion buffersfor the standardization a t very low ion concentrations are needed. For concentrated solutions, however, it is often necessary to match the ionic strength of the "unknown" with that of a standard of known ionic activity, in order to minimize the residual liquid-junction potential error and to measure accurately a difference of ion activity. It is axiomatic that no convention can be proved "right" or "wrong". The virtues of one reasonable procedure over
2943
another depend almost entirely on wide acceptance. A plurality of arbitrary definitions of the single-ion activity would lead rapidly to chaos. The hydration theory described here appears to offer a reasonable, albeit conventional, approach to scales of single-ion activity, applicable not only to potentiometry but also to other areas of solution chemistry. The first step toward international endorsement has been taken (31), and it is hoped that this procedure will eventually be accorded the universal acceptance needed to bring order to this important area. Registry No. LiC1,7447-41-8 NaC1,7647-14-5;KCl, 7447-40-7; NH,Cl, 12125-02-9; KF, 7789-23-3; CaCl,, 10043-52-4.
LITERATURE CITED Frank, H. S. J . Phys. Chem. 1963, 67,1554. Randles, J. E. B. Trans. Faraday SOC.1956, 52, 1573. Goldberg, R. N.; Frank, H. S. J. Phys. Chem. 1972, 76, 1758. Leckey, J. H.; Horne, F. H. J. Phys. Chem. 1981, 85. 2504. (5) MacInnes, D. J. Am. Chem. SOC. 1919, 4 1 , 1086. (6) Guggenheim, E. A. J. Phys. Chem. 1930, 34, 1758. (7) Bates, R. G.; Guggenhelm, E. A. Pure Appi. Chem. 1980, 1 , 163. (8) Popovych, 0.; Tomkins, R. P. T. Nonaqueous Solution Chemistry: Wiley: New York, 1981: Chapter 5. (9) Bates, R. G.: Alfenaar, M. I n Ion-Selective Electrodes; Durst, R. A,, Ed.: National Bureau of Standards: Washington, DC, 1969; NBS Spec. Publ. 314, p 191. (IO) Friedman, H. L. Ionic Solution Theory; Interscience: New York, 1962. (11) Reilly, P. J.; Wood, R. H. J. Phys. Chem. 1969, 73,4292. (12) Pitzer, K. S . J. Phys. Chem. 1973, 77,268. (13) Bates, R. G. Pure Appi. Chem. 1973, 36,407. (14) Bjerrum, N. 2.Anorg. Chem. 1920, 109, 275. (15) Stokes, R. H.; Robinson, R. A. J. Am. Chem. SOC. 1948, 70, 1870. (16) Glueckauf, E. Trans. Faraday Soc. 1955, 5 1 , 1235. (17) Bates, R. G.; Staples, B. R.; Robinson, R. A. Anal. Chem. 1970, 4 2 , 067. (18) Robinson, R. A.: Duer, W. C.; Bates, R. G. Anal. Chem. 1971, 4 3 , 1862. (19) Glueckauf, E. Proc. R . Soc. London. A 1969, 370,449. (20) Kirkwood, J. G. Chem. Rev. 1936, 19, 275. (21) Frank, H. S.; Thompson, P. T. I n The Structure of Eiectroiyfe Soiutions: Hamer, W. J., Ed.; Wiley: New York, 1959. (22) Bates, R. G.: Robinson, R. A. I n Ion-Selective Electrodes; Pungor, E., Ed.; Akademlai KladB: Budapest, 1978: p 3. (23) Huckel, E. Phys. Z . 1925, 2 6 , 93. (24) Davies, C. W. J. Chem. SOC. 1938, 2093. (25) Bates, R. G. I n Ion-Selective Nectrodes, 4; Pungor, E., Ed.: AkadOmiai Kladb: Budapest, 1985; p 725. (26) Robinson, R. A.; Stokes, R. H. Eiectroiyte Solutions, 2nd ed. revised; Butterworths: London, 1970; app. 8.10. (27) Loewenthal, R. E.; Marals, G. v. R. Carbonate Chemistry of High Saiinity Waters; Res. Rpt. W46, Department of Civil Engineering, University of Cape Town, 1983. (28) Harned, H. S . , Owen, B. B. The Physical Chemistry of Eiectroiytic Solutions, 3rd ed.; Reinhold: New York, 1958; Chapter 8. (29) Bockris, J. O'M.; Reddy, A. K. N. Modern Electrochemistry; Plenum: New York, 1970; Vol. I.p 131. (30) Pan, C.-F. J. Phys. Chem. 1985, 6 9 , 2777. (31) Bates, R. G.; Robinson, R. A. Pure Appi. Chem. 1974, 37,575 (1) (2) (3) (4)
RECEIVED for review May 30, 1986. Accepted July 21, 1986. This work was supported by the National Science Foundation under Grant CHE81 20592.
