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Activity coefficients of MgCOo3 and CaSOo4 ion pairs as a function of ionic strength
Geochunica
et Cosmochimica
PressPrintedin GreatBritain Acta.1976. Vol.40.pp 54910554 Pergamon
Activity coefficients of MgCO”, and CaSOz ion pairs as a function of ionic strength ERIC J. REARDON* and DONALD LANCMUIR Department
of Geosciences, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A.
(Received
12 August 1975; accepted
in rel;ised form
13 November
1975)
Abstract-Based on potentiometric data and gypsum solubility in mixed salt solutions, respectively, the activity coefficients of MgCO; and CaSOi ion pairs decrease with ionic strength (I) at 25°C. Computed y’s for the ion pairs fit the empirical equation log yi = -BI. B coefficients of 0.63 k 0.10 for MgCO”, and 0.45 ) 0.15 for CaSOF, are obtained from linear regression of log yL values vs I between 0.04 and 0.6 molal. Assumptions that the activity coefficients of these neutral ion pairs equal
unity or are approximated by the Setchenow expression (log yt = kl) are therefore invalid at moderate ionic strengths. Log yi = -BI is the same general form as the equation of KIRKWOOD (Chem. Rea. 24, 233-251,
1939) for neutral
dipoles.
INTRODUCTION
MOST WORKERS concerned with ionic equilibria assume that activity coefficients (y’s) for neutral ion pairs equal unity or are approximated by the Setchenow expression as modified by HARNED and OWEN (1958). These assumptions are based on the extensive compilations of activity coefficient data for gases in water, and the well-established salting-out effect of neutral solutes (RANDALL and FAILEY, 1927a, b, c; HARNED and OWEN, 1958). In carbonate equilibria calculations, workers generally presume that y for neutral SO:-, HCO;, and CO:- ion pairs equals yH,cO;, arguing that the structure of H&O: is similar to that of the neutral ion pairs. However, yH,COZis actually the net activity coefficient of ycoZtaq) and Y&Co; since the Gibbs free energy change for the reaction: H,O
+ CO,(aq) 4 H&O”,
(1)
is taken by convention to be zero. ELLIS (1959) estimates that only 0.259% of the total dissolved CO2 in pure water is in the form of molecular H,CO”,. The remainder is present as hydrated COZ gas molecules. Therefore, on structural grounds, the assumption y neutral ion pair = YH,co; is not well based. The dipole nature of ion pairs is correlative with established theories of ion pair formation. The constituent hydrated ions may occur with a fixed number of separating water molecules (FUOSS, 1959) or within a specified distance of each other separated by a varying number of water molecules (BJERRUM, 1926). In either case the ion pair can be viewed as two point charges at a distance r, and having a dipole moment P. * Present address: Department of Earth Sciences, University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl.
Many workers have recognized the importance of this dipole character as it explains the solution behavior of ion pairs. (See, for example, YEATTSand MARSHALL,1969; KESTERand PYTKOWICZ, 1970; PYTKOWICZ and KESTER, 1971.) KESTER (1969) found that the activity coefficient of MgSOi in sea water was approximately 0.8 instead of the usually accepted value of 1.1 based on the Setchenow expression. YEATTSand MARSHALL (1969) noted an inverse dependence of yCasO, with ionic strength (I) from gypsum solubility studies. RIDDELLet al. (1972) studied laser Raman and i.r. vibrational spectra of NaNO, in solvent D20. They computed yNaNO, = 0.311 for the most dilute solution studied (I = 1.23 molal). The experimental evidence suggests, then, that at moderate ionic strength it is incorrect to assume that y-neutral ion pair values equal unity or can be computed from the Setchenow expression. In this study we have determined the ionic strength variation of yCasOi from gypsum-salt solubility data from pH measurements in and of ?MgCO, MgC03-MgC12 solutions. Throughout the paper. unless otherwise defined (1) ionic strengths are effective molal values corrected for the presence of ion pairs, and (2) molalities and activity coefficients are of the “free” solute species. EXPERIMENTAL
RESULTS
yCaS0: There are numerous excellent studies of gypsum solubility in pure water and in a variety of aqueous salt solutions (CAMERON and SEIDELL, 1901; CAMERON and BREAZEALE, 1904; SULLIVAN, 1905; HILL and YANICK, 1935; HILL and WILLS, 1938; MARSHALL and SLUSHER, 1966). Based on gypsum solubility in pure water and on measurements of the dissociation of the CaSOi ion pair (MONEY and DAVIES, 1932; NAKAYAMA and RASNICK, 1967; BELL and GEORGE, 1953), there is general agreement that pK (gypsum) = 4.59 + 0.03 and pK(CaS0:) = 2.31 + 0.04,
E. J. REARLWNand D. LANGMUIR
550
where pK is the negative logarithm of the dissociation constant (K) for the solid or the solute indicated. Equilibria in the system CaS04.2H2@HzO-MiAj, where MiAj is a monovalent cation salt which dissociates to form i moles of M+ and j moles of A’-, can be expressed by the following six linearly independent equations. From charge balance relations: mCa2’ = 0.5(2mSO:-
+ mMS0; - mM’ + jMA’-).
(2)
From mass balance of total calcium (XmCa): mCaS0:
= CmCa - mCa’+.
(3)
From mass balance of total sulfate (ZmSO,): mMS0;
= EmSO, - mSO:-
- mCaSO2.
(4)
From dissociation reaction expressions: = K sypsuml(YCa~+.YsO:~.mCaii)CH2032 (5) Fig. 1. Yso;- vs I. Curve fitted through Yso;- values computed using the Debye-Hiickel expression (circles) for (YM’.YSO;-.mSO:-) (6) I < 0.05, and yso:- values from REARD~N(1975) (boxes) ycaso; = (Y~~~+.Yso:~.mCa~+.rnSO:-)/ for I > 0.24. The vertical bar denotes two standard devi(Kc,so:)mCaSOZ (7) ations of the estimated uncertainty of yso;-, Brackets denote activities of the enclosed species. YMso, values were modelled after ya- in the following manner. Solving the appropriate set of above equations for a A solution of M2S04, where the ion pair MSO; is formed. series of gypsum-salt solutions, one can determine the varican be regarded as a mixture of two completely dissociated ation of Ycaso: with I. The published empirical data used electrolytes, M2S04 and M(MS04). The hypothetical 1: I in these calculations are listed in Table 2. Results of the electrolyte M(MSOL) has a mean activity coefficient calculations are plotted in Fig. 2. The regression line, Y+M(M~o,)which is assumed equal to Y+MCI.Justification which is forced through the origin (corresponding to for this last assumption and possible err& associated with y~soz = 1.0 at infinite dilution), has a slope of it are considered in detail elsewhere (REARWN, 1975). Since -0.45 * 0.15. ;:M(Mso,) = Y~+.Y~~o;,then YMSO;/YM*.?'SO:~ inequation In the computation of ycasot we adopted pK(gyp(6) can be expressed as Y:~(~so,)lYi$+.Yso;-. Values sum) = 4.59 and stated in the introduction that y neutral of KM~o;. dissociation constants of MSO; ion pairs ion pair = 1.00 was often assumed in ionic equilibria calused in this study, are listed in Table 1. YQ~+and YM+are culations. To illustrate internal consistency, pK(gypsum) computed by the mean salt method assuming must be recomputed using the new expression for yoaso,. Yk4 = 7~1~= yk~,.-j,using chloride salt activity coefficient The average of 0.0154 + 0.0002 moles/l for the solubility data from ROBINX~Nand STOKES(1970). A plot of yso: of gypsum in pure water at 25°C based on 14 measurevalues vs I, based on REARD~N(1975) for I > 0.24, and ments reported in SEIDELL (1958). combined with Yso;- values computed from the Debye-Hiickel expression pK(CaS0:) = 2.31 (BELL and GEORGE, 1953) and for I < 0.05 is given in Fig. 1. The curve ignores the two Y~so; = 0.959 from the expression log y~so, = -0.451, lowest Debye-Htickel yso:- values because these are ceryields pK(gypsum) = 4.59. This value is identical to tainly less reliable than the other Debye-Htickel values pK(gypsum) adopted in the development of our arguments. at lower ionic strengths, or than yso:- based on the mean The agreement reflects the fact that the accepted pK(gypsalt approach. sum) value is also based on gypsum solubility in pure Computation of Ycaswkfrom gypsum solubility in CaCl, water, and that at these relatively low ionic strength\ solutions involves an analogous but simpler approach. (I = 0.0406), ycaso; is only slightly less than unity. Charge balance dictates: mSO:-
mM+ = (KMso;)(YMso;.~MSO;)/
mCa’+ = 0.5 (2mSO:Equations becomes
+ mCl_).
(3) (5) and (7) are unchanged,
mCaS0, and (6) is eliminated.
@a) however, (4)
= ZmSO, - mSO:-
(4a)
yMgC0; In a previous publication on MgCO”, and CaCO; ion pairs (REARWNand LANGMUIR,1974), we noted significant trends in computed pK(MgCO”,) values for I > 0.05 which
Table 1. Ion pair dissociation constants at 25°C and their sources
nsHeoa+
0.9
+ 0.1
"oetetler (1963)
c&03+
1.1
+ 0.1
Jacobson and Langmuir (1974)
CaCO3o
3.15 + 0.08
Reardon end Langmuir (1974)
N*C03o
2e88 + 0.05
Reardon and Langmuir (1974)
cas040
2.31 + 0.04
Honey end Davies (1932) Naksyama and Rasnick (1967)
MgOH+
2.20 + 0.05
McGee and Hostetler (1975)
caotz+
1.38 + 0.05
Bell and George (1953)
NaS04-
0.82 + 0.1
Reardon (1975)
KS04-
0.85 + 0.02
Chlebek and Meter (1967) Truesdell and Hostecler (1968)
551
Activity coefficients of MgCO”, and CaSOP, ion pairs Table 2. Gypsum solubility data at 25°C in a variety of electrolyte solutions used to calculate ~,-~sw; by equations (2H7). Original published concentrations have been converted to molalities where necessary
(1947)
Sveshnikova
0.0166 0.0101 0.0177 0.0202 0.0228 0.0611 0.0238 0.0818 0.0280 0.123
ELatehall and Slusher(1966) -KaCl
- KC1
0.052 0,065 0.113 0.136 0.182
cameron and Setdell
(1301)
0.0091 0.0665 0.216 0.0087 0.108 0.336
-0.058 -0.076 -0.234 -0.231 -0.333
0.005
0.0126 0.010 0.0118 0.015 0.0112 0.020 0.0108 0.025 0.0104 0.030 0.0097 0.0402 0.0093 0.0503 ca7neron
and
0.047
0.057 0.069 0.082 0.095 0.109 0.137 0.166
Braaleale
0.054 0.071 0.101 0.173 0.258 0.301
-0.015 -0.039 -0.066 -0.113 -0.138 -0.150
0.0180 0.0214 0.0252 0.0285 0.0320
0.0432 0.0854 0.173 0.257 0.342
0.092 o.l.40 0.238 0.329 0.420
iO.006 -0.087 -0.091 -0.123 -0.185
Sveshnikova (1947)- CsCl2 0.0118 0.0145 0.0678 0.0103 0.0244 0.0934 0.0097 0.0399 0.136
-0.033 -0.038 -0.041 -0.040 -0.043 -0.042 -0.029 -0.026
(1904)
0.0121 0.0169 0.071 0.0108 0.0674 0.188 0.0103 0.100 0.261 0.0109 0.173 0.417 0.0117 0.264 0.604
0.0117 0.0257 0.0513 0.115 0.192 0.232
- Cam2
-0.041 -0.052
van “eldhulzen (1929). Eiredin SeideLl(1958) - CaC12 0.0137
0.0162 0.0175 0.0194 0.0231 0.0266 0.0281
-0.035 -0.004 -0.030
Hilland Wills11938)-Na2SO4
-ca2S04
0.0109 o.cl422 0.130 0.0102 0.115 0.293 0.0196 0.233 0.541
-0.057 -0.120 -0.114 -0.182 -0.241
-0.084 -0.119 -0.180
Cameronand Breazeale (1904)- K2S04
"an "eldhulzen (1929)- R2S04
0.0118 0.028 0.0115 0.029 0.0107 0.057 0.0110 0.113 0.0115 0.164 0.0118 0.178
0.0139 0.0128 0.0122 0.0115
0.096 0.099
-0.094 -0.080
0.162
-0.095
0.284 0.387 0.416
-0.160 -0.205 -0.223
Sullivan(1905)- (Nn4)2s04 0.0146 0.00195 0.0‘2 0.0133 0.00621 ii49 0.0122 0.0129 0.061 0.0113 0.0249 0.0890 0.0106 0.0499 O&43 0.0108 0.0999 0.249
Hill
0.0469 0.0564 0.067 0.089
we attributed to possible errors in YMsCO;.At I < 0.05, little or no trend was observed. Table 3 records data for several additional runs designed to obtain YMsCo; vs I using plC(MgC0;) = 2.88 f 0.05 based on 31 determinations at I < 0.05. An outline of the experimental procedure will be given here, but for a full description, the reader is referred to the earlier paper. A solution of K2C03 prepared with distilled, de-ionized, C02-free water was titrated with HCI to bring the pH to approximately 10.0. Increasing amounts of MgC1,.6H20 were then added and each pH change recorded. The change in pH reflected Mg’+ ion pairing with CO;- ion, disrupting the carbonate equilibria, resulting in HCO; dissociation to H+ and CO:-. The eight linearly independent equations below, which describe the system, were solved by the Newton-Raphson method iGROvE,1966) and recycled to a constant ionic strength. mCO:- = [HCO;]IY,/(IO-‘“)yco:~ (8) mMgOH + = ~Mg’+]mOH-/K~soH* (9)
-0.038 -0.034 -0.045 -0.064
andYanick(1935)- (NkZ4)2S04
0.0127 0.0122 0.0607 0.0107 0.0859 0.219 0.0119 0.261 0.553
-0.020 -0.012 -0.011 -0.044 -0.070 -0.131
mOH- = [H~O]~~/(~~~-)lO~pH = CT - (mCO:- + mMgCOj + mMgHC0:) mMgC0: = ZmMg - (mMg’+ + mMgOH+ + mMgHC0;)
0.005 0.010 0.015 0.0251
-0.043 -0.120 -0.238
mMg2” = O.S(mOH- + 2mCO:- + mCl- + mHC0; -mK+ - mMgOH+ - mMgHC0:) (13) mMgHC0: = [Mg2+]mHCO;j~MsuCo; (14) YMV~~CO; = CW?‘l Ico:-J/(rc~,o;)(mMgCO;), (15) where R, is the ionization constant of water, and Cr the total solution carbonate. To define and solve the above relations, it was assumed that 1/~suco; = 7~~0, and y~sou+ = ~oH-. Values for K~sou+, K~sco;, and K~sn~o; are recorded in Table 1. ~~sz+ was computed by the mean salt method using MgClz mean activity coe& cient data from ROBINSON and STOKES(I970). ~~~~~ and yct~- were taken from WALKER et ni. (1927). A plot of log Y&CO; vs I from the data in Table 3 is given in Fig. 3. The regression line has a slope of -0.63 + 0.10 and intercept of 0.02 + 0.025. At the 95% confidence level, this intercept is not significantly different from its expected theoretical value of zero at infinite dilution.
(10)
mHC0;
(1 I) (12)
DISCUSSION
OF RESULTS
The computed values of yMgco; and yCasO,in Figs. 2 and 3 clearly decrease with I, and there is general
E. J. REARDON and D. LANGMUIR
552
0
I 0.30
-0.3 0
0.15
I 0.45
1
060
r
Fig. 2. Variation of log ycaso, with I at 25°C based on experimental data in Table 2. agreement using data from different workers. In the and computation of yCaso;, values of pK(CaS0:)
pK(MS0,) when varied within the limits of their uncertainty had little effect on the trend of yCaSO;with I. Based on the regression lines in Figs. 2 and 3, at I = 0.5 log yEasO;= -0.23 and log y&Co; = -0.32. To ascribe these non-zero values to uncertainties in yMg’+and yco:- used to calculate yMgco; or to uncertamtles m yCa2+and yso:- used to derive yCaSOX, would require the activity coefficient products (yMMg’+. yro; ) or (yea” .yso:-) to be higher by 100 and SO%,respectively. At this ionic strength such uncertainties are improbable, and so real significance is attached to the trends of log yMeco, and log l+-,sO, with 1.
I Fig. 3. Variation of log yMg&o, with I at 25’C based on experimental data in Table 3.
These results are in essential agreement with those of YEATT~ and MARSHALL (1969) who observed an inverse relation between log ycasp, and I from measurements of gypsum solublhty in mixed NaNO,-Na,S04 solutions. Their computed yCaSO, values are larger than ours; however, they ignored
CaNO: and NaSO, ion pairs. This may account for some or all of the discrepancy between our results. RJZARD~Nand LANGMUIR(1974) observed a strong trend with I in pK(MgCO”,) values recomputed from th& data of CARRELS et al. (1961). The recomputation
was made using the expression log yMgco; = 0.061
Table 3. Results of experimental runs at 25°C to determine the variation of y&Co, with ionic strength. Concentrations are in molalities. Reproducibility of pH measurements was kO.01 units
1.0327 2.2756 4.2501 6.2523 7.9579
9.650 9.345 9.150 8.960 8.820
0.0153 0.0431 0.0891 0.1356 0.1750
0.0087 0.0080 0.0073 0.0066 0.0061
0.00362 0.00195 0.00131 0.00083 0.00059
0.0092 0.0109 0.0115 0.0119 0.0121
Run X 2: 20Oml of 0.02918 ti2C03. 2.4 ml of 1.Oh HClwere brought the p" to 10.180 0.3609 1.2506 2.4321 2.9511 4.8351 7.2083
10.006 9.630 9.380 9.286 9.052 8.880
0.0030 0.0177 0.0435 0.0552 0.0983 0.1529
0.0121 0.0115 0.0107 0.0104 0.0095 0.0085
0.01107 0.00482 0.00288 0.00236 0.00141 0.00093
0.3622 1.3679 2.4837 3.4135 4.5400 6.7005
9.840 9.480 9.279 9.140 9.008 8.820
0.0063 0.0297 0.0563 0.0785 0.1053 0.1564
0.0033 0.0030 0.0029 0.0027 0.0026 0.0023
0.00175 0.00088 0.00060 0.00045 0.00034 0.00022
-0.015 -0.082 -0.123 -0.178 -0.200 -0.299 -0.384
added which
0.0055 0.011* 0.0137 0.0142 0.0151 0.0155
R"" u 3: 200 ml of 0.00747 ti2c03. 0.6 ml of l.Olm HClvere brovght the p" f0 10.230
0.094 0.176 0.314 0.453 0.572
0.078 0.116 0.192 0.227 0.356 0.520
-0.075 -0.094 -0.126 -0.169 -0.271 -0.341
added which
0.0024 0.0033 0.0036 0.0037 0.0038 0.0039
0.036 0.105 0.185 0.251 0.332 0.485
-0.013 -0.025 -0.098 -0.176 -0.249 -0.355
R"" K 4: 200 ml of 0.02558 mK2c03. 2.4 ml of l.OnI "Cl were added Which brought the pH to 10.130 0.3959 0.7559 1.1944 1.7320 2.4297 3.2360 4.0773 5.0834 6.4496
9.897 9.708 9.538 9.396 9.264 9.154 9.065 8.984 8.880
0.0040 0.0098 0.0184 0.0301 0.0459 0.0644 0.0838 0.1070 0.1386
0.0115 0.0112 0.0109 0.0105 0.0101 0.0097 0.0093 0.0088 0.0084
0.00798 0.00526 0.00365 0.00273 0.00209 0.00166 0.00137 0.00109 0.00089
0.0054 0.0081 0.0097 0.0106 0.0112 0.0116 0.0119 0.0122 0.0123
0.071 0.085 0.110 0.145 0.192 0.247 0.305 0.375 0.470
-0.057 -0.079 -0.098 -0.122 -0.15, -0.190 -0.216 -0.229 -0.286
Activity coefficients of MgCO; and CaSOi ion pairs
553
Expression (16) and a somewhat similar equation of BATEMAN et al. (1940) reduce at constant temperature and infinite dilution to a limiting law form of:
log& = -BI,
(17)
which predicts a linear decrease in log A with ionic strength. It is well established that the constituent ions in ion pairs such as CaSOi and MgCO”, are water separated, and that bonding between the ions results chiefly from long range electrostatic forces. (See, for example, NANCOLLAS, 1966; DALY et al., 1972.) Based on the Fuoss (1959) model for ion pairs which assumes a fixed distance between constituent ions, and using the Kirkwood expression, we may compute I B values for CaSO$ and MgCOS. With average hydFig. 4. Variation of pK(MgCO”,) with I at 25°C based on recomputed data from GARRELS et al. (1961) at 25°C. Cirrated radii for Mg2+, Ca2+, SO:-. and CO:- of 7.5, cles are pK(MgCO”,) values computed using the Setchenow 6.0. 4.5. and 5.0 A from KIELLAND(1937). and an averexpression log YM~CO,= 0.06 I. Triangles are based on the empirical expression log YMgCwj= - BI, with B = 0.63 age of 5.0A for all ions in solution, we obtain a obtained by linear regression of points plotted in Fig. 3. and h values of 7.7 and 5.3 for CaSO> and 9.0 and 6.5 for MgCos dipoles, respectively. Substitution into from HARWED and OWEN (1958) based on the activity equations (16) and (t 7) yields B values of 1.5 and coefficient of COz gas in aqueous BaCl, solutions. 1.9 for CaSOz and MgCO”,, compared with 0.45 and Figure 4 depicts the effect on pK(MgCO$) of using 0.63 derived experimentally. Although theory predicts an empirical B value of 0.63 in the expression log the correct relative magnitude and the trend of log yMpc@;= -BI. As can be seen, use of this B value fi with I, it yields values three times larger than comlargely eliminates the trend in pl<(MgCO”,) with I. puted from the empirical data. This disparity eviThe resultant mean pl( value of 2.84 i 0.02 is in dently reflects a fallacy among the several assumpgood agreement with 2.88 + 0.05 determined by tions implicit in Kirkwood’s equation. These assumpREARDONand LANGMUJR (1974) from the data at ionic tions are that: (a) the dipole is a rigid molecule of strengths below 0.05. spherical shape; (b) only long-range electrostatic contributions to the chemical potential are important; THEORETICAL TREATMENT (c) the rotational entropy of the dipole, and dipoleA theoretical exp~nation for the variation of y-neusolvent interactions may be ignored; and (d) the tral ion pair with ionic strength is needed. As advodipole concentration is negligible, since long range cated by YEATTSand MARSHALL(1969) and KESTER dipole-dipole interactions are not considered. As a and PYTKOWICZ(1970), it is appropriate to consider further limitation, Kirkwood’s equation is strictly MgCO”, and CaSOz as dipoles which interact with applicable only in very dilute solutions where short a solution to varying degrees depending on the interrange ion-dipole interactions are negligible. Whatever atomic distance and charge of the constituent ions and the reason, Kirkwood’s equation is clearly inadequate structure of the dipole. Several workers have to explain the empirical data for di-divalent ion pairs attempted to predict the chemical potential of dipolar at 25°C. species in solution (SCATCHARD and KIRKWOOD,1932; Based solely on the empirical data for CaSO; and KIRKWOOD,1934, 1939; BATEMANet al., 1940). For MgCO& the activity coefficients of other di-divalent the simplified treatment by KIRKWOOD(1939), the ion pairs at 25°C can probably be estimated with a molar activity coeflicient g) of neutral dipoles of B value of 0.5 in equation (17). The Kirkwood equaspherical shape is computed from: tion suggests that B values for mono-monovalent neutral ion pairs may be roughly one-fourth as large as B’s for the di-divalent pairs. Further research is needed: (1) to determine the where N is Avogadro’s number: e, the electronic activity coefficients of other di-divalent and of monocharge; k, the Boltzman constant; D, the dielectric conmonovalent neutral ion pairs as well as charged ion stant of water; I, the ionic strength; b, the distance pairs with ionic strength and at ditterent temperabetween the centers of charge of the dipole; a. the tures; and (2) to develop an a priori method of comsum of b and the radius of ions of charge Z present in puting y ion pair values. the solution (all ions present assumed to have identical CONCLUSIONS radii); p (equal to bZe), the dipole moment of the ion pair expressed in esu’s; and E(P),a structural parameter There is growing evidence to suggest that activity which is a function of b/a_ Values of a(p) for particular coefficients for neutral ion pairs decrease with ionic h/a values are tabulated by KIRKWOOD(1939). strength (YEATTSand MARSHALL,1969; KESTER,1969;
554
E. J. REARDONand D. LANGMUIR
RIDELL et trl.. 1972). In this paper, ycaso; was computed from gypsum solubility data in mixed salt solu-
tions, and YMgCO, from potentiometric measurements in MgC03-MgC12 solutions. Log y values for the neutral ion pairs decrease linearly with ionic strength, in qualitative accord with the theoretical equation of KIRKWOOD (1939) for neutral dipoles. However, specific y’s computed using Kirkwood’s equation are in
marked disagreement
with the experimental
values.
Acknowledgements-We are indebted to DANA KESTERfor a constructive critical review of an earlier version of this paper. Support for the research has been provided by the Mineral Conservation Section of The Pennsylvania State University. REFERENCES BATEMANL. C., CHURCH M. G., HUGHESE. D., INGOLD C. K. and TAHERN. A. (1940) Substitution at a saturated carbon atom: Part XXIII. J. Chem. Sot. 979-1011. BELL R. P. and GEORGEJ. H. B. (1953) Dissociation of thallous and Ca sulfate salts at different temperatures. Trans. Faraday Sot. 49, 619-627.
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Fuoss R. M. (1959) Ionic association. III. The equilibria between ion pairs and free ions. J. Amer. Chem. Sot. 80, 5059-5061. GARRELSR. M., THOMPSONM. E. and SIEVERR. (1961) Control of carbonate solubility by carbonate complexes. Amer. J. Sci. 259, 24-45.
GROVE W. E. (1966) Brief Numerical Methods, 117 pp. Prentice-Hall. HARNEDH. S. and OWEN B. B. (1958) The Physical Chemistry of’Electro/yfic Solutions, 803 pp. ReinGold. HILL A. E. and WILLSJ. H. (1938) Ternarv svstems. XXIV. Calcium sulphate, sodium’ sulphate and water. J. Amer. Chem. Sot. 60, 1647-1655. HILL A. E. and YANICKN. S. (1935) Ternary systems. XX. Calcium sulfate, ammonium sulfate and water. J. Amer. Chem. Sot. 57, 645-651. HOSTETLERP. B. (1963) Complexing of magnesium with bicarbonate. J. Phvs. Chem. 67. 72@721. JACOH~~N R. L. and ~ANGMLJIR D.’(1974) Dissociation constants of calcite and CaHCO: from 0 to 50°C. Geochim. Cosmochim. Acta 38, 301-318.
KESTERD. R. (1969) Ion association of sodium, magnesium, and calcium with sulfate in aqueous solution. %npublished Ph.D. thesis, Oregon State University, 116 pp. KESTERD. R. and PY~KOWICZR. M. (1970) Effect of temperature and pressure on sulfate ion association in seawater. Geochim. Cosmochim. Acta 34. 1039-1051. K IELLANDJ. (1937) Activity coefficients of ions in aqueous solutions. J. Amer. Chem. Sot. 59, 1675-1678.
KIRKWO~DJ. G. (1934) Theory of solutions of molecules containing widely separated charges with special applications to zwitterions. J. Chem. Phys. 2, 351-361. KIRKWO~DJ. G. (1939) Theoretical studies upon dipolar ions. Chem. Rev. 24, 233-251. MARSHALLW. L. and SLUSHERR. (1966) Thermodynamics of calcium sulfate dihydrate in aqueous sodium chloride solutions O-110. J. Phys. Chem. 70, 4015-4027. MCGEE K. A. and HOSTETLERP. B. (1975) Studies in the system MgGSi02&02-H,0 (IV): the stability of MgOH’ from IO” to 90°C. Amer. J. Sci. 275, 304-3 17. MONEYR. W. and DAVIES,C. W. (1932) The extent of dissociation of salts in water. Part IV. Bi-bivalent salts. Trans. Furaday Sot. 28, 609.
NAKAYAMAF. S. and RASNICKB. A. (1967) Ca-electrode method for measuring dissociation and solubility of calcium sulfate dihydrate. Anal. Chem. 39, 1022-1023. NANCOLLASG. H. (1966) Interactions in Electrolyte Solutions, 214 pp. Elsevier. PYTKOWICZR. M. and KESTERD. R. (1971) The physical chemistry of seawater. Oceanogr. Marine Biol. Ann. Rev. 9, 11-60. RANDALLM. and FAILEYC. F. (l927a) The activity coefficients of gases in aqueous salt solutions. Chem. Rev. 4, 271-284.
RANDALLM. and FAILEYC. F. (1927b) The activity coefficient of non-electrolytes in aqueous salt solutions from solubility measurements. The salting-out order of the ions. Chem. Rev. 4, 285-290. RANDALLM. and FAILEYC. F. (1927~) The activity coefficient of the undissociated part of weak electrolytes. Chem. Rev. 4, 291-318.
REARWN E. J. (1975) Dissociation constants of some monovalent sulfate ion pairs at 25°C from stoichiometric activity coefficients. J. Phys. Chem. 79, 422425. REARPONE. J. and LANGMUIRD. (1974) Thermodynamic properties of the ion oairs MgCO”, and CaCOt from i0
to 50°C. Amer. J. S’ci. 274, 599-612.
RIDDELLJ. D.. LOCKWMD D. J. and IRISH D. E. (1972) Ion pair formation in NaN03/D20 solutions: Raman and infrared spectra, parital molal volumes, conductance and viscosity. Can. J. Chem. 50, 2951-2962. ROBINSON R. A. and STOKESR. H. (1970) Electrolyte So/utions, 571 pp. Butterworths. SCATCHARD G. and KIRKWOODJ. G. (1932) Das Verhalten von Zwitterion und von mehrwertigen Ionen mit weit entfernten Ladungen in Electrolytlosungen. Phys. 2. 33, 397-301. SEIDELLA. (1958) Solubilities of Inorganic and Metal Organic Compounds, Vol. 1, 4th ed., 1487 pp. Am. Chem. Sot. SHTERNINAE. P. and FROLOVAE. V. (1957) Solubility in the system CaC03-NaCl-COZ-H,O. Zh. Neorg. Khim. 2, 1655-1657.
SULLIVANA. L. (1905) Calcium sulphate in ammonium sulphate solutions. J. Chem. Sot. 27, 529. SVESHNIKOVA V. N. (1947) Solubility isotherms of the systems CaS04-CaC12-H,O and CaS04-KCI-H,O. Izv. Sektora Fiz-Khim. An&a., Inst. Ohshch. Khim. Akad. Nauk. SSSR 79, 447450.
Neorgan.
TRUESDELL A. H. and HOSTETLER P. B. (1968) Dissociation constants of KSO, from 10 to 50°C. Geochim. Cosmochim. Acta 32, 1019-1022.
WALKERA. C., BRAYV. B. and JOHNSTON J. (1927) Equilibrium in solutions of alkali carbonates. J. Amer. Chem. Sot. 49, 1235-1256. YEATTSL. B. and MARSHALLW. L. (1969) Apparent invariance of activity coefficients of calcium sulfate at constant ionic strength and temperature in the system CaSO,-Na,SO,-NaNO,-H,O to the critical temperature of water. Association equilibria. J. Phys. Chem. 73. 8 l-90.
et Cosmochimica
PressPrintedin GreatBritain Acta.1976. Vol.40.pp 54910554 Pergamon
Activity coefficients of MgCO”, and CaSOz ion pairs as a function of ionic strength ERIC J. REARDON* and DONALD LANCMUIR Department
of Geosciences, The Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A.
(Received
12 August 1975; accepted
in rel;ised form
13 November
1975)
Abstract-Based on potentiometric data and gypsum solubility in mixed salt solutions, respectively, the activity coefficients of MgCO; and CaSOi ion pairs decrease with ionic strength (I) at 25°C. Computed y’s for the ion pairs fit the empirical equation log yi = -BI. B coefficients of 0.63 k 0.10 for MgCO”, and 0.45 ) 0.15 for CaSOF, are obtained from linear regression of log yL values vs I between 0.04 and 0.6 molal. Assumptions that the activity coefficients of these neutral ion pairs equal
unity or are approximated by the Setchenow expression (log yt = kl) are therefore invalid at moderate ionic strengths. Log yi = -BI is the same general form as the equation of KIRKWOOD (Chem. Rea. 24, 233-251,
1939) for neutral
dipoles.
INTRODUCTION
MOST WORKERS concerned with ionic equilibria assume that activity coefficients (y’s) for neutral ion pairs equal unity or are approximated by the Setchenow expression as modified by HARNED and OWEN (1958). These assumptions are based on the extensive compilations of activity coefficient data for gases in water, and the well-established salting-out effect of neutral solutes (RANDALL and FAILEY, 1927a, b, c; HARNED and OWEN, 1958). In carbonate equilibria calculations, workers generally presume that y for neutral SO:-, HCO;, and CO:- ion pairs equals yH,cO;, arguing that the structure of H&O: is similar to that of the neutral ion pairs. However, yH,COZis actually the net activity coefficient of ycoZtaq) and Y&Co; since the Gibbs free energy change for the reaction: H,O
+ CO,(aq) 4 H&O”,
(1)
is taken by convention to be zero. ELLIS (1959) estimates that only 0.259% of the total dissolved CO2 in pure water is in the form of molecular H,CO”,. The remainder is present as hydrated COZ gas molecules. Therefore, on structural grounds, the assumption y neutral ion pair = YH,co; is not well based. The dipole nature of ion pairs is correlative with established theories of ion pair formation. The constituent hydrated ions may occur with a fixed number of separating water molecules (FUOSS, 1959) or within a specified distance of each other separated by a varying number of water molecules (BJERRUM, 1926). In either case the ion pair can be viewed as two point charges at a distance r, and having a dipole moment P. * Present address: Department of Earth Sciences, University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl.
Many workers have recognized the importance of this dipole character as it explains the solution behavior of ion pairs. (See, for example, YEATTSand MARSHALL,1969; KESTERand PYTKOWICZ, 1970; PYTKOWICZ and KESTER, 1971.) KESTER (1969) found that the activity coefficient of MgSOi in sea water was approximately 0.8 instead of the usually accepted value of 1.1 based on the Setchenow expression. YEATTSand MARSHALL (1969) noted an inverse dependence of yCasO, with ionic strength (I) from gypsum solubility studies. RIDDELLet al. (1972) studied laser Raman and i.r. vibrational spectra of NaNO, in solvent D20. They computed yNaNO, = 0.311 for the most dilute solution studied (I = 1.23 molal). The experimental evidence suggests, then, that at moderate ionic strength it is incorrect to assume that y-neutral ion pair values equal unity or can be computed from the Setchenow expression. In this study we have determined the ionic strength variation of yCasOi from gypsum-salt solubility data from pH measurements in and of ?MgCO, MgC03-MgC12 solutions. Throughout the paper. unless otherwise defined (1) ionic strengths are effective molal values corrected for the presence of ion pairs, and (2) molalities and activity coefficients are of the “free” solute species. EXPERIMENTAL
RESULTS
yCaS0: There are numerous excellent studies of gypsum solubility in pure water and in a variety of aqueous salt solutions (CAMERON and SEIDELL, 1901; CAMERON and BREAZEALE, 1904; SULLIVAN, 1905; HILL and YANICK, 1935; HILL and WILLS, 1938; MARSHALL and SLUSHER, 1966). Based on gypsum solubility in pure water and on measurements of the dissociation of the CaSOi ion pair (MONEY and DAVIES, 1932; NAKAYAMA and RASNICK, 1967; BELL and GEORGE, 1953), there is general agreement that pK (gypsum) = 4.59 + 0.03 and pK(CaS0:) = 2.31 + 0.04,
E. J. REARLWNand D. LANGMUIR
550
where pK is the negative logarithm of the dissociation constant (K) for the solid or the solute indicated. Equilibria in the system CaS04.2H2@HzO-MiAj, where MiAj is a monovalent cation salt which dissociates to form i moles of M+ and j moles of A’-, can be expressed by the following six linearly independent equations. From charge balance relations: mCa2’ = 0.5(2mSO:-
+ mMS0; - mM’ + jMA’-).
(2)
From mass balance of total calcium (XmCa): mCaS0:
= CmCa - mCa’+.
(3)
From mass balance of total sulfate (ZmSO,): mMS0;
= EmSO, - mSO:-
- mCaSO2.
(4)
From dissociation reaction expressions: = K sypsuml(YCa~+.YsO:~.mCaii)CH2032 (5) Fig. 1. Yso;- vs I. Curve fitted through Yso;- values computed using the Debye-Hiickel expression (circles) for (YM’.YSO;-.mSO:-) (6) I < 0.05, and yso:- values from REARD~N(1975) (boxes) ycaso; = (Y~~~+.Yso:~.mCa~+.rnSO:-)/ for I > 0.24. The vertical bar denotes two standard devi(Kc,so:)mCaSOZ (7) ations of the estimated uncertainty of yso;-, Brackets denote activities of the enclosed species. YMso, values were modelled after ya- in the following manner. Solving the appropriate set of above equations for a A solution of M2S04, where the ion pair MSO; is formed. series of gypsum-salt solutions, one can determine the varican be regarded as a mixture of two completely dissociated ation of Ycaso: with I. The published empirical data used electrolytes, M2S04 and M(MS04). The hypothetical 1: I in these calculations are listed in Table 2. Results of the electrolyte M(MSOL) has a mean activity coefficient calculations are plotted in Fig. 2. The regression line, Y+M(M~o,)which is assumed equal to Y+MCI.Justification which is forced through the origin (corresponding to for this last assumption and possible err& associated with y~soz = 1.0 at infinite dilution), has a slope of it are considered in detail elsewhere (REARWN, 1975). Since -0.45 * 0.15. ;:M(Mso,) = Y~+.Y~~o;,then YMSO;/YM*.?'SO:~ inequation In the computation of ycasot we adopted pK(gyp(6) can be expressed as Y:~(~so,)lYi$+.Yso;-. Values sum) = 4.59 and stated in the introduction that y neutral of KM~o;. dissociation constants of MSO; ion pairs ion pair = 1.00 was often assumed in ionic equilibria calused in this study, are listed in Table 1. YQ~+and YM+are culations. To illustrate internal consistency, pK(gypsum) computed by the mean salt method assuming must be recomputed using the new expression for yoaso,. Yk4 = 7~1~= yk~,.-j,using chloride salt activity coefficient The average of 0.0154 + 0.0002 moles/l for the solubility data from ROBINX~Nand STOKES(1970). A plot of yso: of gypsum in pure water at 25°C based on 14 measurevalues vs I, based on REARD~N(1975) for I > 0.24, and ments reported in SEIDELL (1958). combined with Yso;- values computed from the Debye-Hiickel expression pK(CaS0:) = 2.31 (BELL and GEORGE, 1953) and for I < 0.05 is given in Fig. 1. The curve ignores the two Y~so; = 0.959 from the expression log y~so, = -0.451, lowest Debye-Htickel yso:- values because these are ceryields pK(gypsum) = 4.59. This value is identical to tainly less reliable than the other Debye-Htickel values pK(gypsum) adopted in the development of our arguments. at lower ionic strengths, or than yso:- based on the mean The agreement reflects the fact that the accepted pK(gypsalt approach. sum) value is also based on gypsum solubility in pure Computation of Ycaswkfrom gypsum solubility in CaCl, water, and that at these relatively low ionic strength\ solutions involves an analogous but simpler approach. (I = 0.0406), ycaso; is only slightly less than unity. Charge balance dictates: mSO:-
mM+ = (KMso;)(YMso;.~MSO;)/
mCa’+ = 0.5 (2mSO:Equations becomes
+ mCl_).
(3) (5) and (7) are unchanged,
mCaS0, and (6) is eliminated.
@a) however, (4)
= ZmSO, - mSO:-
(4a)
yMgC0; In a previous publication on MgCO”, and CaCO; ion pairs (REARWNand LANGMUIR,1974), we noted significant trends in computed pK(MgCO”,) values for I > 0.05 which
Table 1. Ion pair dissociation constants at 25°C and their sources
nsHeoa+
0.9
+ 0.1
"oetetler (1963)
c&03+
1.1
+ 0.1
Jacobson and Langmuir (1974)
CaCO3o
3.15 + 0.08
Reardon end Langmuir (1974)
N*C03o
2e88 + 0.05
Reardon and Langmuir (1974)
cas040
2.31 + 0.04
Honey end Davies (1932) Naksyama and Rasnick (1967)
MgOH+
2.20 + 0.05
McGee and Hostetler (1975)
caotz+
1.38 + 0.05
Bell and George (1953)
NaS04-
0.82 + 0.1
Reardon (1975)
KS04-
0.85 + 0.02
Chlebek and Meter (1967) Truesdell and Hostecler (1968)
551
Activity coefficients of MgCO”, and CaSOP, ion pairs Table 2. Gypsum solubility data at 25°C in a variety of electrolyte solutions used to calculate ~,-~sw; by equations (2H7). Original published concentrations have been converted to molalities where necessary
(1947)
Sveshnikova
0.0166 0.0101 0.0177 0.0202 0.0228 0.0611 0.0238 0.0818 0.0280 0.123
ELatehall and Slusher(1966) -KaCl
- KC1
0.052 0,065 0.113 0.136 0.182
cameron and Setdell
(1301)
0.0091 0.0665 0.216 0.0087 0.108 0.336
-0.058 -0.076 -0.234 -0.231 -0.333
0.005
0.0126 0.010 0.0118 0.015 0.0112 0.020 0.0108 0.025 0.0104 0.030 0.0097 0.0402 0.0093 0.0503 ca7neron
and
0.047
0.057 0.069 0.082 0.095 0.109 0.137 0.166
Braaleale
0.054 0.071 0.101 0.173 0.258 0.301
-0.015 -0.039 -0.066 -0.113 -0.138 -0.150
0.0180 0.0214 0.0252 0.0285 0.0320
0.0432 0.0854 0.173 0.257 0.342
0.092 o.l.40 0.238 0.329 0.420
iO.006 -0.087 -0.091 -0.123 -0.185
Sveshnikova (1947)- CsCl2 0.0118 0.0145 0.0678 0.0103 0.0244 0.0934 0.0097 0.0399 0.136
-0.033 -0.038 -0.041 -0.040 -0.043 -0.042 -0.029 -0.026
(1904)
0.0121 0.0169 0.071 0.0108 0.0674 0.188 0.0103 0.100 0.261 0.0109 0.173 0.417 0.0117 0.264 0.604
0.0117 0.0257 0.0513 0.115 0.192 0.232
- Cam2
-0.041 -0.052
van “eldhulzen (1929). Eiredin SeideLl(1958) - CaC12 0.0137
0.0162 0.0175 0.0194 0.0231 0.0266 0.0281
-0.035 -0.004 -0.030
Hilland Wills11938)-Na2SO4
-ca2S04
0.0109 o.cl422 0.130 0.0102 0.115 0.293 0.0196 0.233 0.541
-0.057 -0.120 -0.114 -0.182 -0.241
-0.084 -0.119 -0.180
Cameronand Breazeale (1904)- K2S04
"an "eldhulzen (1929)- R2S04
0.0118 0.028 0.0115 0.029 0.0107 0.057 0.0110 0.113 0.0115 0.164 0.0118 0.178
0.0139 0.0128 0.0122 0.0115
0.096 0.099
-0.094 -0.080
0.162
-0.095
0.284 0.387 0.416
-0.160 -0.205 -0.223
Sullivan(1905)- (Nn4)2s04 0.0146 0.00195 0.0‘2 0.0133 0.00621 ii49 0.0122 0.0129 0.061 0.0113 0.0249 0.0890 0.0106 0.0499 O&43 0.0108 0.0999 0.249
Hill
0.0469 0.0564 0.067 0.089
we attributed to possible errors in YMsCO;.At I < 0.05, little or no trend was observed. Table 3 records data for several additional runs designed to obtain YMsCo; vs I using plC(MgC0;) = 2.88 f 0.05 based on 31 determinations at I < 0.05. An outline of the experimental procedure will be given here, but for a full description, the reader is referred to the earlier paper. A solution of K2C03 prepared with distilled, de-ionized, C02-free water was titrated with HCI to bring the pH to approximately 10.0. Increasing amounts of MgC1,.6H20 were then added and each pH change recorded. The change in pH reflected Mg’+ ion pairing with CO;- ion, disrupting the carbonate equilibria, resulting in HCO; dissociation to H+ and CO:-. The eight linearly independent equations below, which describe the system, were solved by the Newton-Raphson method iGROvE,1966) and recycled to a constant ionic strength. mCO:- = [HCO;]IY,/(IO-‘“)yco:~ (8) mMgOH + = ~Mg’+]mOH-/K~soH* (9)
-0.038 -0.034 -0.045 -0.064
andYanick(1935)- (NkZ4)2S04
0.0127 0.0122 0.0607 0.0107 0.0859 0.219 0.0119 0.261 0.553
-0.020 -0.012 -0.011 -0.044 -0.070 -0.131
mOH- = [H~O]~~/(~~~-)lO~pH = CT - (mCO:- + mMgCOj + mMgHC0:) mMgC0: = ZmMg - (mMg’+ + mMgOH+ + mMgHC0;)
0.005 0.010 0.015 0.0251
-0.043 -0.120 -0.238
mMg2” = O.S(mOH- + 2mCO:- + mCl- + mHC0; -mK+ - mMgOH+ - mMgHC0:) (13) mMgHC0: = [Mg2+]mHCO;j~MsuCo; (14) YMV~~CO; = CW?‘l Ico:-J/(rc~,o;)(mMgCO;), (15) where R, is the ionization constant of water, and Cr the total solution carbonate. To define and solve the above relations, it was assumed that 1/~suco; = 7~~0, and y~sou+ = ~oH-. Values for K~sou+, K~sco;, and K~sn~o; are recorded in Table 1. ~~sz+ was computed by the mean salt method using MgClz mean activity coe& cient data from ROBINSON and STOKES(I970). ~~~~~ and yct~- were taken from WALKER et ni. (1927). A plot of log Y&CO; vs I from the data in Table 3 is given in Fig. 3. The regression line has a slope of -0.63 + 0.10 and intercept of 0.02 + 0.025. At the 95% confidence level, this intercept is not significantly different from its expected theoretical value of zero at infinite dilution.
(10)
mHC0;
(1 I) (12)
DISCUSSION
OF RESULTS
The computed values of yMgco; and yCasO,in Figs. 2 and 3 clearly decrease with I, and there is general
E. J. REARDON and D. LANGMUIR
552
0
I 0.30
-0.3 0
0.15
I 0.45
1
060
r
Fig. 2. Variation of log ycaso, with I at 25°C based on experimental data in Table 2. agreement using data from different workers. In the and computation of yCaso;, values of pK(CaS0:)
pK(MS0,) when varied within the limits of their uncertainty had little effect on the trend of yCaSO;with I. Based on the regression lines in Figs. 2 and 3, at I = 0.5 log yEasO;= -0.23 and log y&Co; = -0.32. To ascribe these non-zero values to uncertainties in yMg’+and yco:- used to calculate yMgco; or to uncertamtles m yCa2+and yso:- used to derive yCaSOX, would require the activity coefficient products (yMMg’+. yro; ) or (yea” .yso:-) to be higher by 100 and SO%,respectively. At this ionic strength such uncertainties are improbable, and so real significance is attached to the trends of log yMeco, and log l+-,sO, with 1.
I Fig. 3. Variation of log yMg&o, with I at 25’C based on experimental data in Table 3.
These results are in essential agreement with those of YEATT~ and MARSHALL (1969) who observed an inverse relation between log ycasp, and I from measurements of gypsum solublhty in mixed NaNO,-Na,S04 solutions. Their computed yCaSO, values are larger than ours; however, they ignored
CaNO: and NaSO, ion pairs. This may account for some or all of the discrepancy between our results. RJZARD~Nand LANGMUIR(1974) observed a strong trend with I in pK(MgCO”,) values recomputed from th& data of CARRELS et al. (1961). The recomputation
was made using the expression log yMgco; = 0.061
Table 3. Results of experimental runs at 25°C to determine the variation of y&Co, with ionic strength. Concentrations are in molalities. Reproducibility of pH measurements was kO.01 units
1.0327 2.2756 4.2501 6.2523 7.9579
9.650 9.345 9.150 8.960 8.820
0.0153 0.0431 0.0891 0.1356 0.1750
0.0087 0.0080 0.0073 0.0066 0.0061
0.00362 0.00195 0.00131 0.00083 0.00059
0.0092 0.0109 0.0115 0.0119 0.0121
Run X 2: 20Oml of 0.02918 ti2C03. 2.4 ml of 1.Oh HClwere brought the p" to 10.180 0.3609 1.2506 2.4321 2.9511 4.8351 7.2083
10.006 9.630 9.380 9.286 9.052 8.880
0.0030 0.0177 0.0435 0.0552 0.0983 0.1529
0.0121 0.0115 0.0107 0.0104 0.0095 0.0085
0.01107 0.00482 0.00288 0.00236 0.00141 0.00093
0.3622 1.3679 2.4837 3.4135 4.5400 6.7005
9.840 9.480 9.279 9.140 9.008 8.820
0.0063 0.0297 0.0563 0.0785 0.1053 0.1564
0.0033 0.0030 0.0029 0.0027 0.0026 0.0023
0.00175 0.00088 0.00060 0.00045 0.00034 0.00022
-0.015 -0.082 -0.123 -0.178 -0.200 -0.299 -0.384
added which
0.0055 0.011* 0.0137 0.0142 0.0151 0.0155
R"" u 3: 200 ml of 0.00747 ti2c03. 0.6 ml of l.Olm HClvere brovght the p" f0 10.230
0.094 0.176 0.314 0.453 0.572
0.078 0.116 0.192 0.227 0.356 0.520
-0.075 -0.094 -0.126 -0.169 -0.271 -0.341
added which
0.0024 0.0033 0.0036 0.0037 0.0038 0.0039
0.036 0.105 0.185 0.251 0.332 0.485
-0.013 -0.025 -0.098 -0.176 -0.249 -0.355
R"" K 4: 200 ml of 0.02558 mK2c03. 2.4 ml of l.OnI "Cl were added Which brought the pH to 10.130 0.3959 0.7559 1.1944 1.7320 2.4297 3.2360 4.0773 5.0834 6.4496
9.897 9.708 9.538 9.396 9.264 9.154 9.065 8.984 8.880
0.0040 0.0098 0.0184 0.0301 0.0459 0.0644 0.0838 0.1070 0.1386
0.0115 0.0112 0.0109 0.0105 0.0101 0.0097 0.0093 0.0088 0.0084
0.00798 0.00526 0.00365 0.00273 0.00209 0.00166 0.00137 0.00109 0.00089
0.0054 0.0081 0.0097 0.0106 0.0112 0.0116 0.0119 0.0122 0.0123
0.071 0.085 0.110 0.145 0.192 0.247 0.305 0.375 0.470
-0.057 -0.079 -0.098 -0.122 -0.15, -0.190 -0.216 -0.229 -0.286
Activity coefficients of MgCO; and CaSOi ion pairs
553
Expression (16) and a somewhat similar equation of BATEMAN et al. (1940) reduce at constant temperature and infinite dilution to a limiting law form of:
log& = -BI,
(17)
which predicts a linear decrease in log A with ionic strength. It is well established that the constituent ions in ion pairs such as CaSOi and MgCO”, are water separated, and that bonding between the ions results chiefly from long range electrostatic forces. (See, for example, NANCOLLAS, 1966; DALY et al., 1972.) Based on the Fuoss (1959) model for ion pairs which assumes a fixed distance between constituent ions, and using the Kirkwood expression, we may compute I B values for CaSO$ and MgCOS. With average hydFig. 4. Variation of pK(MgCO”,) with I at 25°C based on recomputed data from GARRELS et al. (1961) at 25°C. Cirrated radii for Mg2+, Ca2+, SO:-. and CO:- of 7.5, cles are pK(MgCO”,) values computed using the Setchenow 6.0. 4.5. and 5.0 A from KIELLAND(1937). and an averexpression log YM~CO,= 0.06 I. Triangles are based on the empirical expression log YMgCwj= - BI, with B = 0.63 age of 5.0A for all ions in solution, we obtain a obtained by linear regression of points plotted in Fig. 3. and h values of 7.7 and 5.3 for CaSO> and 9.0 and 6.5 for MgCos dipoles, respectively. Substitution into from HARWED and OWEN (1958) based on the activity equations (16) and (t 7) yields B values of 1.5 and coefficient of COz gas in aqueous BaCl, solutions. 1.9 for CaSOz and MgCO”,, compared with 0.45 and Figure 4 depicts the effect on pK(MgCO$) of using 0.63 derived experimentally. Although theory predicts an empirical B value of 0.63 in the expression log the correct relative magnitude and the trend of log yMpc@;= -BI. As can be seen, use of this B value fi with I, it yields values three times larger than comlargely eliminates the trend in pl<(MgCO”,) with I. puted from the empirical data. This disparity eviThe resultant mean pl( value of 2.84 i 0.02 is in dently reflects a fallacy among the several assumpgood agreement with 2.88 + 0.05 determined by tions implicit in Kirkwood’s equation. These assumpREARDONand LANGMUJR (1974) from the data at ionic tions are that: (a) the dipole is a rigid molecule of strengths below 0.05. spherical shape; (b) only long-range electrostatic contributions to the chemical potential are important; THEORETICAL TREATMENT (c) the rotational entropy of the dipole, and dipoleA theoretical exp~nation for the variation of y-neusolvent interactions may be ignored; and (d) the tral ion pair with ionic strength is needed. As advodipole concentration is negligible, since long range cated by YEATTSand MARSHALL(1969) and KESTER dipole-dipole interactions are not considered. As a and PYTKOWICZ(1970), it is appropriate to consider further limitation, Kirkwood’s equation is strictly MgCO”, and CaSOz as dipoles which interact with applicable only in very dilute solutions where short a solution to varying degrees depending on the interrange ion-dipole interactions are negligible. Whatever atomic distance and charge of the constituent ions and the reason, Kirkwood’s equation is clearly inadequate structure of the dipole. Several workers have to explain the empirical data for di-divalent ion pairs attempted to predict the chemical potential of dipolar at 25°C. species in solution (SCATCHARD and KIRKWOOD,1932; Based solely on the empirical data for CaSO; and KIRKWOOD,1934, 1939; BATEMANet al., 1940). For MgCO& the activity coefficients of other di-divalent the simplified treatment by KIRKWOOD(1939), the ion pairs at 25°C can probably be estimated with a molar activity coeflicient g) of neutral dipoles of B value of 0.5 in equation (17). The Kirkwood equaspherical shape is computed from: tion suggests that B values for mono-monovalent neutral ion pairs may be roughly one-fourth as large as B’s for the di-divalent pairs. Further research is needed: (1) to determine the where N is Avogadro’s number: e, the electronic activity coefficients of other di-divalent and of monocharge; k, the Boltzman constant; D, the dielectric conmonovalent neutral ion pairs as well as charged ion stant of water; I, the ionic strength; b, the distance pairs with ionic strength and at ditterent temperabetween the centers of charge of the dipole; a. the tures; and (2) to develop an a priori method of comsum of b and the radius of ions of charge Z present in puting y ion pair values. the solution (all ions present assumed to have identical CONCLUSIONS radii); p (equal to bZe), the dipole moment of the ion pair expressed in esu’s; and E(P),a structural parameter There is growing evidence to suggest that activity which is a function of b/a_ Values of a(p) for particular coefficients for neutral ion pairs decrease with ionic h/a values are tabulated by KIRKWOOD(1939). strength (YEATTSand MARSHALL,1969; KESTER,1969;
554
E. J. REARDONand D. LANGMUIR
RIDELL et trl.. 1972). In this paper, ycaso; was computed from gypsum solubility data in mixed salt solu-
tions, and YMgCO, from potentiometric measurements in MgC03-MgC12 solutions. Log y values for the neutral ion pairs decrease linearly with ionic strength, in qualitative accord with the theoretical equation of KIRKWOOD (1939) for neutral dipoles. However, specific y’s computed using Kirkwood’s equation are in
marked disagreement
with the experimental
values.
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