5518
J. Phys. Chem. 1983, 8 7 , 5516-5521
provide definitive quantitative details about these interactions.
for providing computer time for the analysis of the data. Registry No. Cyclobutanone, 1191-95-3.
Acknowledgment. The authors gratefully acknowledge the financial assistance of The National Science Foundation, Grant No. CHE-8111739. We are also grateful to the computer center of The North Dakota State University
Supplementary Material Available: Tables of experimental intensity data .and correlation and error matrices are available ( 5 pages). Ordering information is given on any current masthead page.
Silicon-29 Nuclear Magnetic Resonance Study of Hydroxyl Sites on Dehydrated Silica Gel Surfaces, Using Silylation as a Probe Dean W. Slndorf and Gary E. Maclel" Department of Chemlstty, Colorado State Universlty, Fort Colllns, Colorado 80523 (Received: August 18, 1982)
A detailed 29SiNMR analysis is presented on the silylation of a series of heat-treated silica gels by hexamethyldisilizane (HMDS)with the aim of elucidating details of the surface hydroxyl population of the unsilylated samples. It is found that the complete removal of all silanol groups from silica is probably approached only at temperatures higher than those normally employed in silica treatments (i.e., higher than 1000 "C)and that some molecular water may remain on the surfaces of samples normally considered "dried" (e.g., evacuated at 200 "C). The population and reactivity patterns of surface hydroxyls are considered in terms of reasonable surface and dehydration models.
Introduction Solid-state 29Si NMR experiments, employing cross polarization (CP) and magic-angle spinning (MAS) techniques, have been shown previously to provide a promising and valuable approach for the characterization of silica gel surfaces and surface-silylatedsilicas.l* It has been shown that the pertinent NMR relaxation parameters (THsi for CP and TlpH)are suitable for quantitative 29Si NMR analysis of the surface silanol silicon atoms1 and the attached silane-silicons of silylated samples: and render ?Si CP/MAS essentially a surface technique. Previous reports have described the analysis of 2gSi NMR intensities of silica samples for characterizing the nature of the silica surface (e.g., hydroxyl densities, concentrations of geminal, and lone-hydroxyl silanol sites) and how it is changed by heat treatmenL2v5 Other 29SiCP/ MAS studies3* have addressed the details of silylation by simple surface silylating agents, focussing on characterizing the 29SiNMR method of determining the nature and extent of the silylation process and its relation to surface structure. In the present paper we reexamine earlier 29Si NMR results on the silylation of silica gel by hexamethyldisilazane (HMDS) to yield trimethylsilane (TRMS) derivatives, and present additional Y3i CP/MAS data aimed at using the 2eSiNMR analysis of the silylation process as a probe of the surface hydroxyl population. Previous results are confirmed and new insights are made possible by this approach, especially as regards the nature of dehydrated (dehydroxylated) silica samples. (1)G.E.Maciel and Dean W. Sindorf, J.Am. Chem. SOC.,102,7607 (1980). (2)D. W.Sindorf and G. E. Maciel, J. Am. Chem. SOC.,105, 1487 (1983). (3)G.E.Maciel, D. W. Sindorf, and V. J. Bartuska, J. Chromatogr., 205,438(1981!. (4)D. W.Sindorf and G. E. Maciel, J. Am. Chem. Soc., 103, 4263 (1981). (6)D. W.Sindorf and G. E. Maciel, J.Phys. Chem., 86,5208 (1982). (6) D. W. Sindorf and G. E. Maciel, J. Am. Chem. Soc., 105, 3767 (1983).
Experimental Section Silylated samples were prepared from l-g portions of each of the heat-treated Fisher S-157 silica gels, dehydrated as described earlier, by reaction with toluene solutions of HMDS at room temperature for 24 h.2v5 Partially reacted samples were prepared by using 1 mL of 9% HMDS/ toluene (v/v), and maximally reacted samples were prepared by using 1 mL (excess) of 50% HMDS/toluene (v/v). TRMS loading levels were determined from the weight gain observed following subsequent evacuation of the silylated sample at 110 "C. ?Si CP/MAS NMR spectra were obtained at 39.75 MHz on a Nicolet NT-200 spectrometer, modified for the observation of solids. The cross polarization contact time employed in this study was 10 ms, a value that previous relaxation s t ~ d i e s ~ have 9 ~ 1 ~shown to be sufficiently long to permit essentially full cross polarization for the single and geminal surface silanol silicons, as well as silicons in attached TRMS groups. The longer CP relaxation times of the surface silicon atoms with no attached hydroxyl groups prevents full polarization of these silicon nuclei in a 10-ms CP time. Hence, the 29Siintensities of the single and geminal silanols and the attached silane groups are considered quantitatively valid in this study. The signals from other surface silicons were not used in the analysis. Results and Discussion Surface OH Content. The 29SiCP/MAS NMR spectra of partially and maximally silylated samples of dehydrated silica gel are shown in Figures 1and 2, respectively. From these spectra, using peak areas obtained from integration and deconvolution techniques described earlier,2,5we can obtain the surface hydroxyl concentrations of the unsilylated samples using eq 1 (eq 8 of ref 5 ) (OH)' = S -j(l + f + fi) (1) where f I i / ( I i + I { ) , f 3 I i / ( I i + I{). I:, I:, and I{ are 29Sipeak areas measured at 15 (attached silane), -90
0 1983 American Chemical Society 0022-3654/03/2007-5516~01.50/0
The Journal of Physical Chemistw, Vol. 87,
29SiNMR Study of Dehydrated Silica Gel Surfaces
1
.
.
*
'
"
0
'
*
J
l
-100
'
PPM
.
*
l
.
0
.
.
,
No. 26, 1983
5517
A
PPM
-100
Figure 1. 39.75-MHz 29Si CP/MAS NMR spectra of dehydrated S-157 silicas that have been parttally reacted with HMDS (using experimental method 3 of ref 5). Temperatures shown are the temperatures (OC) used in dehydrating the silica gels before siiylation.
I
,
0
1
.
I
I
.
I
O
-100
,
PPM
l
s
0
-
*
B
.
a
-100
*
n
,
PPM
Figure 2. 39.75-MHz 2sSiCP/MAS NMR spectra of dehydrated S-157 silicas that have been maximally reacted with HMDS (using experimental method 3 of ref 5), showing the initial dehydration temperatures ("C).
(unreacted geminal sites), and -100 ppm (unreacted lone-hydroxyl sites plus reacted geminal-hydroxyl sites), respectively. f is the ratio of attached silane silicons to residual surface silanol silicons in the reacted sample. f ; is the fraction of residual surface silanols in the reacted sample that are of the geminal-hydroxyl type. S is defined as the surface silane concentration, i.e., the number of bound TRMS groups per 100 A2 of silica surface. As was shown earlier: S can be determined from the surface area
of the silica gel, and the observed weight gain of the silylation reaction or carbon elemental analysis of the silylated product. Equation 1 assumes that all of the intensity in the silanol resonances can be identified with surface hydroxyl species. For a more general formulation which includes the possibility of internal SiOH groups the alternate approach discussed below will be more appropriate. This approach is based on the bulk hydroxyl density, pOH,defined as the number of mmole of OH per gram of
5518
The Journal of Physical Chemistty, Vol. 87, No. 26, 1983 '
SURFACE HYDROXYL CONTENT FISHER S.157
7:FROM S i - 2 9 NMR
0 :FROM WT. LOSS UPON HEATING A T 10-5 TORR
6-
5-
t
4-
3-
\
2-
1
-'
T
100
ZOO
300
400
500
660 700
860
900 1000 ,100 ' C
Figure 3. Bulk hydroxyl density determined by the NMR approach (poHNMR) and by the weight-change approach (poHdas a function of dehydration temperatures. These values can be converted to the (OH)' scale shown by assuming a constant surface area.
sample. This quantity can be estimated from TRMS uptake and 29Siintensity data (eq 2 ) , or from a direct measurement of weight loss following dehydration (eq 3). The
(3)
NMR-derived pOHvalue (poHNm) is expressed in units of mmole per gram, and is proportional to the measured silane weight gain per gram of silica ( A w ) divided by the effective molecular weight of the attached TRMS group (72 g/mol); the expression (1+ f + f l ) /isfa proportionality factor that relates the TRMS group concentration to the concentration of surface silanol groups by use of the formalism developed previ~usly.~ The dehydration-derived POH parameter (poHd) is obtained from the difference between the weight per gram of hydroxyls remaining on the surface following dehydration at a given temperature and that at a some maximum temperature (AW-.) corresponding to complete dehydroxylation,divided by the effective molecular weight, 9 g/mol, of a dehydration-liberated hydroxyl group, with a correction factor, (1- A V X 10-3)-1,that takes into account the reduction in substrate weight that results from the dehyration process. A comparison of hydroxyl density values determined by both of these methods is shown graphically in Figure 3 which presenh plots of pOH vs. pretreatment (dehydration) temperature. Open circles give the results obtained with the direct weight loss method (poHd), using dehydration data presented earlier,2including an estimated maximum weight loss of 74 mg/g. Solid circles are average values obtained with the %SiNMR method (poHNm) for the two loading levels examined. Error bars show the actual deviation observed in the value of p0Hm calculated for the
(Am
Sindorf and Maciel
two loading levels. Also given in this figure is a (OH)' scale, ~ to (OH)' values, which permits the conversion of p 0 values if one assumes that the surface area of each dehydrated sample is the same as that of the unheated sample. Figure 3 shows that pOH results obtained by these two methods are reasonably consistent (within experimental error) over most of the range of temperatures examined. At high dehydration temperatures, where the possibility of adsorbed or occluded bulk molecular H,O can be considered highly unlikely, changes in pOH as T i s increased can be interpreted as reflecting the direct elimination of water by condensation reactions at the surface. The good agreement between the results obtained from 29SiNMR and those obtained on the basis of dehydration weight-loss data confirms that in this high-temperature region (600-1100 "C) a significant density of hydroxyl groups remain. This substantiates earlier results and assumptions made that at 1000 "C only about 90% of the original hydroxyl groups have been removed by condensation.2 At low pretreatment temperatures a slight but significant difference is evident between results obtained via 29Si NMR and those obtained by the weight-loss method. This discrepancy, which amounts to about 5 mg of H20/g of silica, corresponds to a difference of about 0.5 OH groups per 100 A2 in the estimated population of silanol groups on the surface a t 150 "C (i.e., 5.2 vs. 5.7). A possible explanation for this difference is that, even on the surface of silica samples that have been evacuated at 100-300 "C, a certain amount of tightly bound, physically adsorbed molecular water remains. While not affecting the 29Si NMR measurements, this physically bound water would result in erroneously high values of (OH)' and poH determined by the dehydration weight-loss method. The simultaneous occurrence of both silanol condensation reactions and water-desorption processes on the surface of silica gels heated at low-to-moderate temperatures may offer one explanation for the general inconsistency of surface hydroxyl concentrations reported in the literature over the years. The results of the present study show that solid-state 29SiNMR methods can be employed successfully to measure surface hydroxyl concentrations or bulk OH densities with good precision and accuracy for silica gel samples preheated over a wide temperature range. In addition, such measurements are not complicated by the presence of molecular water, and therefore do not require the preparation of completely anhydrous samples. Surface Reactivity. Differences between the loading levels and surface coverages of samples reacted with an excess or limited amount of silylating agent reflect both the total number and overall accessibility of residual SiOH groups on the silica surface. For steric reasons, only a certain fraction of the original hydroxyls can react as long as their surface concentration is greater than about 2.8 per 100 A2,the maximum possible density of the TRMS phase on any surface predicted on the basis of molecular geometry and packing considerations.6 Even for surfaces with OH concentrations considerably less than 2.8/ 100 A2, bonding can be less than 100% efficient if the spacial distribution of sites is such that pairs or multiples of surface hydroxyls are physically located in regions of surface too small to attach more than one TRMS group. With the reaction conditions employed in these studies it was found that only 44% of the surface hydroxyl sites present on fully hydrated Fisher 5-157 actually reacted with excess HMDS.5 (In contrast, for a silica surface with 4.5 sites/100 A2 one predicts a theoretical maximum coverage of 2.814.5 = 0.62.) This 44% coverage corresponds
The Journal of Physical Chemistv, Vol. 87, No. 26, 1983 5519
*?Si NMR Study of Dehydrated Silica Gel Surfaces
TABLE I: Surface Coverages of TRMS Groups on Dehydrated Silica Gels 9% HMDSa
dehydration temp, OC 160 20 9 300 353 417 507 567 650 706 800 a
psites,
PTRMS,
mmol/g 5.7 5.5 4.7 4.0 3.2 3.3 1.7 1.3 1.0 0.8
mmol/g
Percent HMDS in toluene.
0.79 0.82 0.83 0.84 0.84 0.79 0.69 0.63 0.17
PTRMS,
81
0.14 0.17 0.21 0.26 0.38 0.46 0.58 0.64 0.21
8 2
mmol/g
81
2.47 2.32 2.15 2.07 1.90 1.40 1.24 0.98 0.74 0.18
0.43 0.42 0.46 0.51 0.59 0.64 0.73 0.8.2 0.74 0.23
0.14 0.19 0.19 0.24 0.44 0.48 0.64 0.58 0.13
8 2
0.44 0.44 0.46 0.54 0.61 0.65 0.68 0.77 0.73 0.39
No reaction observed for samples preheated at or above 1000 OC.
to a surface loading of about 2.47 mmol/g, a loading value that can also be considered a practical upper limit for dehydrated samples reacted under similar conditions, since condensation reactions are not likely to bring residual surface sites into more favorable bonding positions. Qualitatively these considerations suggest that, as the average density of hydroxyl groups on the silica surface is decreased, the maximum loading density obtainable for the TRMS phase will decrease monotonically from an initial (maximum)value of 2.47 mmol/g to essentially zero. However, if we assume random condensation (dehydration) processes, the surface coverage (e) should increase from 0.44 to 1.0, a value of 1.0 implying that all of the residual silanol sites are located far enough apart so that the reaction of any given group is unhindered by previous reactions on neighboring groups. For heat-treated silica samples that have been reacted with a limited amount of HMDS, a somewhat different behavior can be expected. In the reaction conditions used here, about 0.85 mmol of potential TRMS was added to 1-g portions of silica. As long as the density of hydroxyl groups greatly exceeds this level, the silylation process will be essentially 100% efficient (i.e., 0.85 mmol/g of TRMS will be attached to the surface). One can expect in this case to observe a constant degree of loading of the silane phase on all but the most completely dehydroxylated samples. When the total OH density is decreased below about 0.85 mmol/g, silane uptake for silica samples reacted with either concentration of HMDS should be the same, since in both cases reagent is in excess of surface sites, and should decrease monotonically with further OH elimation. Thus, 8 is expected to increase from some initial value (-0.13 in this case) to a maximum value of 1.0. To investigate predictions based upon the considerations given above, we analyzed the experimental NMR and gravimetric data in terms of the following parameters:
(in mmol/g) is the bulk density of the attached TRMS phase, obtained as the ratio of the measured silylation weight increase per gram of silica (AW) to the molecular weight increment accompanying TRMS silylation. psitesis the bulk density (in mmol/g) of silanol sites, expressed in terms of a "normalized" hydroxyl density. (The factor 1 + fi adjusts for the fact that each geminal site contributes two OH groups to the hydroxyl population contributing to pOH.) When psiteawas calculated, values of POH were obtained from the solid curve shown in Figure 3 (averaging both 29Siand dehydration-method results) PTRMS
50% HMDSa
I
,i I
1 .od
/J
1'
2
3
4
5
6'
PSllES
Figure 4. Plot of measured bulk density of attached sllane phase bTRMS) vs. the estimated bulk density of the surface silanol sites for silica gels dehydrated at the Indicated temperatures ("C):0 , data obtained for maxlmally reacted samples (using 50% HMDS); 0, data obtained for parthlly reacted samples (using 9 % HMDS). Dashed lines showing loading that would be predlcted on the basis of assuming 100 % reaction.
and values of f g i were obtained from previously reported data on the same samples.2 The calculation of the surface-coverage parameter, 01, is based upon the definition of 6 (i.e., reacted silanol sites per initial silanol sites)5and assumes that only one hydroxyl groups in each geminalhydroxyl site can participate in bonding. e2 is the NMRderived surface coverage, its formulation based entirely on analysis of spectral intensities, using the methods described in detail el~ewhere.~ Values of these parameters for silica gels dehydrated at several temperatures are summarized in Table I. Plots of pm vs. paitesfor samples reacted with both 9 and 50% solutions of HMDS are shown in Figure 4. For maximally reacted samples the surface coverage (average of O1 and e,) is plotted as a function of dehydration temperature in Figure 5. For silica samples heated to temperatures less than or equal to about 550 O C the data in Table I and Figure 4 exhibit behavior in reasonable agreement with what is expected on the basis of the discussion above. As can be seen from Figure 4, bonding levels (pTRMS) remain essentially constant over this temperature range for samples reacted with the 9% HMDS solution, and are within experimental error of what one would expect assuming 100%
5520
The Journal of Physical Chemistty, Vola87, No. 26, 1983
03
0.1
Figure 5. Plot of data for maximum total surface coverage (6) vs. dehydration temperature. Curve A shows behavior predicted assumlng random reactions and moderate crowding effects. Curve 6 shows constraint imposed by the formation of “viclnal pairs”. Curve C incorporates both the mechanisms of curve A and curve 6.
reaction. This suggests that surface OH groups in this series of samples are distributed sufficiently far apart so that at this degree of coverage crowding effects are completely avoided. As can be seen from an examination of Table I and Figure 5, maximally reacted samples of this dehydrationtemperature range also exhibit behavior qualitatively consistent with considerations presented above. For example, 6 increases substantially with increasing dehydration temperatures, going from an initial value of 0.44 at 200 “C to 0.80 at 560 “C.However, for higher dehydration temperatures 6 decreases. Modeling studies similar to those described previously in studies of TRMS bonding with nondehydrated silica samples5suggest that surface coverages should follow (only qualitatively)the dashed curve (A) shown in Figure 5. This curve was constructed with the data on the fraction of original sites remaining, following dehydration, on each of two hypothetical surface types.5 By assuming that hydroxyl groups are eliminated randomly from the surface by condensation reactions between pairs of neighboring silanols, site-depleted analogues of the surface models described previously5 were constructed. For random TRMS reactions on such surfaces (with moderate crowding) one predicts the loading behavior indicated by line A in Figure 5. The observed deviation of the experimental data from this line can have a number of possible interpretations. For example, if hydroxyl pairs are eliminated autocatalytically (i.e., the resulting condensation products stimulate the dehydration of neighboring hydroxyls), rather than randomly (uncooperative dehydration), then expanding regions of hydroxyl-depleted surface will be formed. However, residual silanols in noneffected “islands” will be packed as tightly together as usual, so that surface coverages of TRMS will be limited in these areas by crowding to values little changed from those observed for unheated silica gels. This view might explain the generally lower experimental 6 values from curve A of Figure 5 for lowto-moderate temperatures. For higher dehydration temperatures 6 does appear to increase significantly, indicating that to a certain extent dehydration probably proceeds as a result of random or semirandom condensation mechanisms. Beyond a dehydration temperature of about 550 “C a more substantial deviation from predicted behavior (curve A) is evident. Figures 4 and 5 show that, not only does the total bonding capacity of the silica surface reduce to a very low value at 800 “C,but low values of 6 indicate that
Sindorf and Macle1
the relative reactivity of residual surface OH groups also decreases abruptly. Thus, hydroxyl groups appear to be present, but for some reason cannot react with the silylating agent. Several possibilities for this kind of behavior can be suggested. One interpretation is that the kinetics of the silylation reaction become unfavorable for highly dehydroxylated surfaces. HMDS is a polar molecule, and might reasonably be expected to interact unfavorably with the relatively hydrophobic surfaces formed as a result of dehydroxylation. For residual hydroxyls located deep in the highly porous silica structure, with pore walls composed of such surfaces, the rate-limiting step in the reaction pight easily involve the initial adsorption of HMDS. This contention may be supported by the observation that a slight dependence on the concentration of silylating agent is exhibited in Figure 4 for reactions involving samples heated from 650 to 800 “C (i.e,, the more concentrated solution yields higher loading levels than the dilute solution, even though in both cases HMDS is in excess of surface silanol sites). Another possible explanation is that the surface accessibility of residual hydroxyls is decreased by the condensation process. Such a situation might arise, for example, if the entrances to pores or cavities containing unreacted OH groups are blocked by surface-to-surface condensation reactions. Such processes might be expected to accompany the “sintering” and loss of surface area often reported for silica samples heated to excessively high temperatures (Le., >700 0C).7 An alternative interpretation of the data in Figure 5 relies upon a detailed consideration of the dehydration mechanism suggested earlier for the 100 face of p-cristobalite (as a model of the silica ~ u r f a c e ) .In ~ this model initial condensation reactions between adjacent geminal hydroxyls were predicted to result in the formation of “vicinal” (adjacent) hydroxyl pairs of two lone-hydroxyl sites interconnected by a siloxane bond. Unlike the single-silanol sites of a 111-face ,@-cristobalitemodel, where even on fully hydrated surfaces neighboring sites are located no closer than about 5 A, on a 100-type surface hydroxyl sites that are associated with such vicinal pairs would be only about 3.3 A apart. As in the case of geminal sites, steric considerations imposed by the size of the TRMS groups (with an effective diameter of 6.4 A) may prohibit the reaction of more than one OH group on a given vicinal pair. In addition, previous results5 have indicated that, at least for samples heated to temperatures less than about 550 “C,these vicinal pairs may be highly resistant to further condensation processes, since such processes would necessarily involve unfavorable transformations, requiring, for example, the formation of highly strained edge-linked structure^.^ The significance of this possibility is that on dehydrated silica surfaces a substantial fraction of the residual OH content may be associated with condensation-resistant vicinal pairs, instead of the widely separated, isolated hydroxyls that one would expect on dehydrated 111-type surfaces. In this situation surface silane coverages will always be less than unity, since half of the sites associated with these vicinal-pair structures will be unable to react with HMDS. The actual bulk density of vicinal sites present at a given dehydration temperature, defined as pW,can be estimated ~
(7) W. A.Patrick, J. C. Foyer, and R. I. Rush,J. Phys. Chem., 31,511 (1927). (8) R.K.Iler, “The Chemistry of Silica, Solubility,Polymerization and Surface Properties and Biochemistry”, Wiley, New York, 1979. (9) J. B. Peri and A. L. Hensley, Jr., J.Phys. Chem., 72, 2926 (1968).
J. Phys. Chem. lS03. 87. 5521-5522
by using previously reported dehydration data2 on the fraction of geminal sites remaining following dehydration (F,) and assuming that all of the geminal sites that have been eliminated through condensation are converted into single hydroxyls that are associated with vicinal pairs. If half of this vicinal-site density is subtracted from the total silanol site density (psitestgiven in Table I), maximum coverages (emc)permitted by this model can be estimated as follows: Psites -
=
PVS/Z
(4) Psitas
with Pvs
= PgiFg
The initial geminal site density, pgi, will be equal to (fgi)(pBites),where psites is the total silanol site density determined for a completely hydrated sample. With fgi = 0.15 and psites = 5.6 (obtained from the data for the 160 “C sample in Table I), p i is about 0.85 mmol/g for Fisher S-157 silica. For the present study, use of eq 4 and the data of Table I results in the maximum coverages shown by curve B of Figure 5.
5521
A more integrated treatment that incorporates both of the schemes associated with curves A and B in Figure 5 yields the predicted qualitative behavior indicated by the solid curve C.’O Considering the speculative nature of the arguments that have been presented, the agreement between curve C and the experimental data exhibited in Figure 5 is surprisingly good, particularly at low-to-moderately high dehydration temperatures (up to 550 OC). The deviations observed for very high dehydration temperatures may be attributable to the condensation of vicinal hydroxyl pairs through mechanisms such as those suggested earlier: and/or a combination of other considerations suggested here. Acknowledgment. The authors gratefully acknowledge partial support of this research by the U.S. Department of Energy (Contract No. DE-AT20-81LC0652 from the Laramie Energy Technology Center) and the assistance of the Colorado State University Regional NMR Center, funded by National Science Foundation Grant CHE 7818581. Registry No. 29Si,14304-87-1; HMDS, 999-97-3. (10) D. W. Sindorf, Ph.D. Thesis, Colorado State University, 1982.
COMMENTS Comment on “Theoretical Single-Ion Activity of Calcium and Magnesium Ions in Aqueous Electrolyte Mixtures”
Sir: A recent publication reported the single-ion activity coefficients of calcium and magnesium ions in aqueous calcium chloride-sodium chloride and magnesium chloride-sodium chloride mixtures, as calculated by the mean spherical approximation (MSA).’ Mean ionic activity coefficients for calcium chloride in mixtures with sodium chloride were also reported, and showed reasonable agreement with literature values2 for concentrations of sodium chloride up to 0.15 M and calcium chloride to 10 mM. Although mean ionic activity coefficients for single electrolytes are readily evaluated from the MSA without the need to utilize single-ion values,3v4the only feasible way to extend the treatment to mixtures of electrolytes is through prior evaluation of single-ion activity coefficients. It is therefore unfortunate that the treatment of Vericat and Grigera’ is incomplete; their numerical values quoted in Tables 11,111, and IV are incorrect, as they incorporate only that contribution arising from the charges on the ions. The values for the mean activity coefficients of CaC1, quoted in Table 11, for example, correspond to exp(A In 7 4 of Blum and H ~ y eor , ~to exp(g(I’)]of Sorensen.6 It is necessary to include also the contribution from the uncharged reference fluid (hard-sphere contributions).I (1) Vericat, F.; Grigera, J. R. J. Phys. Chem. 1982,86, 1030. (2) Butler, J. N. Biophys. J. 1968, 8, 1486. (3) Triolo, R.; Blum, L.; Floriano, M. A. J.Phys. Chem. 1977,67,5956. (4) Triolo, R.; Blum, L.; Floriano, M. A. J.Phys. Chem. 1978,82, 1368. (5) Blum, L.; Hoye, J. S . J.Phys. Chem. 1977, 81, 1311. (6) Sorensen, T. S.Acta Chem. Scand. 1978, A32,571 (Appendix 111). (7) Triolo, R.; Floriano, M. A.; Ruffo, I.; Blum, L. Ann. Chim. 1977, 67, 433.
For single electrolytes, at low concentrations, the hardsphere contribution ( y H Sto ) the mean activity coefficient is most readily evaluated, as outlined by Sorensen,6 by analytical integration of the Gibbs-Duhem equation, using the simplified expression of Triolo et aL8 for (PO, the hard-sphere contribution to the osmotic coefficient. Evaluation of single-ion activity coefficients, however, requires the calculation of the hard-sphere contribution to the single-ion activity coefficient, i.e., yiHS. This can be done in two ways: (i) by a modification of the method of Robinson and stoke^,^ involving integration from the cross-differentiation relation,1° or (ii) by differentiation of the expression for the free energy of a mixture of hard spheres.’l For single electrolytes (two kinds of ions only), both methods give identical results;1° the second method is somewhat more readily applied to a mixture of three kinds of ions and was employed (Appendix) to calculate the hard-sphere contributions to the single-ion activity coefficients for calcium or magnesium and chloride ions in mixtures of sodium and calcium or magnesium chlorides, using the parameters listed by Vericat and Grigera.l Representative results for 0.05 m CaC1, + m molal NaCl are given in Chart I. Results for In yaHS are very similar; for 0.05 m MgC12 + m molal NaC1, In yMgHS ranges from 0.038 at m = 0.01 to 0.115 at m = 1.0. Representative values of mean activity coefficients for CaCl,, calculated with incorporation of the hard-sphere contributions, are listed in Table I, which should replace Table I1 of ref 1. Since the hard-sphere contributions are always positive, the general effect is to increase the calculated mean activity coefficient, and hence to improve the agreement with lit(8) Triolo, R.; Grigera, J. R.; Blum, L. J. Phys. Chem. 1976,80, 1858. (9) Robinson, R. A.; Stokes, R. H. J. Phys. Chem. 1961, 65, 1954. (10)Humffray, A. A., submitted for publication. (11) Ebeling, W.; Scherwinski, K. 2.Phys. Chem. (Leipzig) 1983, 264.
0022-365418312087-5521$01.50/0 0 1983 American Chemical Society
