J . Phys. Chem. 1986, 90, 895-901
895
Heat Capacity and Other Thermodynamic Properties of Aqueous Magnesium Sulfate to 473 K Ramesh C. Phutela and Kenneth S. Pitzer* Department of Chemistry and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 (Received: August 6, 1985; In Final Form: October 8, 1985)
Measurements are reported for the heat capacity of aqueous MgS0, from 348 to 473 K. Ion-interaction (Pitzer) equations are fitted to these data with additional guidance for The last quantity is obtained indirectly from literature data for MgC12, Na2S04,and NaCI; we have made a new analysis for MgCl, but used published expressions for the sodium compounds. Literature data on the osmotic coefficient and the enthalpy of MgS0, solutions are used to complete a comprehensive thermodynamic treatment. Since all of the other major components of natural waters and sea water have been treated with the same set of equations and over a wide temperature range, the present results for MgS0, complete the basic data set required for prediction of most thermodynamic properties of these systems. The effectiveness of a special second-vinal-coefficient term in representing the ion pairing in MgS0, is considered in some detail.
cpo2.
Introduction measurements at high pressure by Wood.’4 As is described briefly for 20 bar for use in this paper in the Appendix, values of CPo2 Soon after the 1923 paper of Debye and Huckel,’ it was shown were then interpolated. by Bjerrum, that there is significant electrostatic ion pairing in With these independently determined values for C P . O 2 for aqueous 2:2 electrolytes. For MgS0, a t 298 K this has been MgSO,, the difference from the apparent molar heat capacity T p treated by ion association equilibria3” (often including triple, at finite molality can be obtained quite accurately and used to quadruple, or even sextuple ions). Also it has been shown that study the effect of temperature on ion pairing and to develop this ion pairing is moderate enough at 298 K that a virial series reliable equations for various thermodynamic properties. treatment’ is very satisfactory with a negative second virial coefficient corresponding to the ion pairing. Indeed this virial Experimental Method series expression for 2:2 ion interaction has been very s u c c e s s f ~ l ~ ~ ~ The calorimeter was the same in general design as that used in predicting the properties of MgS0, and CaS0, in a variety by Rogers and Pitzer” who also give a detailed description of the of mixed electrolytes a t 298 K. In this paper we present meaexperimental procedure. However, the calorimeter itself was surements of the heat capacity of aqueous MgSO, to 473 K and rebuilt in the interim with the stainless steel tubing replaced by extend a general ion-interaction (Pitzer-equation) treatment for Hastelloy C-276. various thermodynamic properties over the range 298-473 K. This This flow calorimeter is an adaptation to high temperature of treatment is compared with the recent ion-association treatment the design of Picker et al.Is In brief there are twin flow calorof Archer and Wood.6 imetric units in the same thermostat with auxiliary equipment The thermodynamic properties of aqueous MgSO, are of great which provides for an exactly equal volumetric flow rate (at 298.15 practical importance since Mg2+ and SO4,- are next to Na+ and K) of water to one unit and solution to the other. The electrical C1- in abundance in seawater and in many other natural waters heating is adjusted to provide the same temperature rise in each and industrial brines. unit. Thus the heat losses are expected to be nearly equal and The experimental uncertainty of the apparent molar heat cathe ratio of heat capacity of solution to that of water is given by pacity increases rapidly as the molality becomes very small. Also the ion association changes rapidly in the dilute range. Thus one cannot accurately determine the limiting CPo2for Mg2+ + SO4,by extrapolation to zero molality. Rather we obtain that quantity indirectly from the properties of MgCI,, Na2S0,, and NaCl where where P, and P, are the heater powers for solution and water, there is no significant ion pairing and the extrapolation is much respectively, L is the power loss, and the p’s are densities at 298 more reliable. Values for NaCl are taken from a recent, comK. Since the heat capacity of pure water is accurately known,I6 prehensive equationlo of state valid from 273 to 573 K. Rogers it can be used first to evaluate the heat loss from a detailed analysis and Pitzer” measured the heat capacity of aqueous Na2S04and of the unit with water flowing. Then with L determined, the heat combined these results with other data in a general treatment. capacity of the solution can be calculated from eq 1. For aqueous MgCI, we’, have fitted the same type of equations Calibration and test experiments were carried out with water to the available heat capacity data at low pressure including very in both units, with interchange of the water and solution units, recent measurements of S a 1 ~ j a . I ~There are also very recent and with aqueous NaCl as the solution. The flow is generated by a liquid-chromatography pump (Altex (1) Debye, P.; HUckel, E. Phys. Z . 1923, 24, 185, also 305. Model 100) which is always set to maintain a pressure above the (2) Bjerrum, N. K . Dan. Vidensk. Selsk., Mat.-Fys. Medd. 1926, 7, No. vapor pressure of water. Thus there is no vapor phase. All of 9. the present results are for 20 bar. (3) Gardner, A. W.; Glueckauf, E. Proc. R. SOC.London, Ser. A 1969, A313, 131. (4) Pitzer, K. S. J . Chem. SOC.,Faraday Trans. 2 1972, 68, 101. (5) Woolley, E. M.; Hepler, L. G. Can. J . Chem. 1977, 55, 158. (6) Archer, D. G.; Wood, R. H. J . Solution Chem. 1985, 14, 757. (7) Pitzer, K. S.; Mayorga, G. J. Solution Chem. 1974, 3, 539. (8) Harvie, C. E.; Weare, J. H. Geochim. Cosmochim. Acta 1980, 44,981. (9) Harvie, C. E.; Moeller, N.; Weare, J. H. Geochim. Cosmochim. Acta 1984, 48, 723. (10) Pitzer, K. S.; Peiper, J. C.; Busey, R. H. J. Phys. Chem. Ref Data 1984, 13, 1. ( 1 1) Rogers, P. S. Z.; Pitzer, K. S . J. Phys. Chem. 1981,85,2886. 1982, 86, 21 10. (12) Phutela, R. C.; Pitzer, K. S.; Saluja, P. P. S.,manuscript in preparation. (13) Saluja, P. P. S., private communication.
Experimental Results The experimentally measured heat capacities are given in Table I as heat capacities of the solution in J.g-’.K-’ and as apparent molar heat capacities in J-mo1-l.K-l. One estimate of the ex(14) Wood, R. H., private communication. (15) Picker, P.; Leduc, P. A.; Philip, P.; Desnoyers, J . E. J . Chem. Thermodyn. 1971, 3, 631. (16) Haar, L.; Gallagher, J. S . ; Kell, G. S. “NBS/NRC Steam Tables”, Hemisphere Publishing: Washington, 1985. “Proceeding of the Eighth Symposium on Thermophysical Properties”, Vol. 2, Senger, J. V.,Ed.; American Society of Mechanical Engineers: New York, 1981; p 298.
0022-3654/86/2090-0895$01 .50/0 0 1986 American Chemical Society
Phutela and Pitzer
896 The Journal of Physical Chemistry, Vol. 90, No. 5, 1986 TABLE I: Experimental Heat Capacity of Aqueous MgW, at 20 bar Pressure m/mol.kgl PSIPW cP/J*g-'*K-' Vp/J.mol-'.K-l T = 348.15 K (75.0 "C) -44.4 1.02008 4.0905 0.1679 -42.9 1.02008 4.0908 0.1679 4.0474 4.0477 3.8912 3.8961 3.8901 3.7316 3.3784
TABLE II: Pitzer's Equations for the Thermodynamic Properties of an Aqueous Electrolyte Solution"
-46.3 -45.0 -18.5 -9.7 -20.5 4.2 39.6
0.2498 0.2498 0.5946 0.5946 0.5946 1.0091 2.186
1.029 62 1.029 62 1.068 87 1.068 87 1.068 87 1.11400 1.231 80
0.2430 0.2498 0.5841 0.9906 0.9906 0.9906 1.5043 1.5043 2.080 2.080
T = 373.15 K (100.0 "C) 1.028 84 4.0917 1.029 62 4.0870 1.067 70 3.9456 1.11203 3.7943 1.11203 3.7952 1.11203 3.7946 1.165 43 3.6324 1 .I65 43 3.6311 1.221 83 3.4775 1.221 83 3.4754
-5.8 -11.7 17.6 34.3 35.3 34.6 51.4 50.4 65.1 63.8
0.1679 0.2430 0.5841 0.9906 0.9906 1.5043 2.080 2.080
T = 398.15 K (125.0 "C) 1.020 08 4.1718 1.028 84 4.1354 1.067 70 3.9950 1.11203 3.8560 1.11203 3.8554 1.165 43 3.7004 1.221 83 3.5466 1.221 83 3.5421
17.2 13.2 38.9 63.1 62.4 77.9 87.2 84.5
0.0977 0.1679 0.2430 0.5841 0.5841 0.9906 1.5043 1.5043
T = 423.15 K (150.0 "C) 1.011 79 4.2588 1.020 08 4.2254 1.028 84 4.1920 1.067 70 4.0591 1.067 70 4.0597 1.11203 3.9177 1.16543 3.7655 1.16543 3.7644
19.2 22.5 31.2 64.1 65.3 78.5 93.2 92.4
0.0977 0.1679 0.2430 0.5841
T = 448.15 K (175.0 "C) 1.011 79 4.3338 1.02008 4.3022 1.028 84 4.2719 1.067 70 4.1424
40.9 49.6 66.1 90.4
0.0977 0.1679
T = 473.15 K (200.0 "C) 1.01 179 4.4427 1.02008 4.41 14
88.1 84.2
perimental uncertainty was obtained by recalculating the data on the alternate assumption that the heat loss L was proportional to the power introduced into that unit rather than being the same for the units with water and with solution. This has the effect of removing L from both the numerator and denominator of eq 1. On this basis the apparent molar heat capacities are changed by an amount ranging from 0.5 J.K-l.mol-' for the measurement at 2 mol-kg-I to 2.5 J.K-I.mo1-l for the dilute solutions below 0.5 molskg-' at 348 K. This unc, rtainty increases with temperature until it is as much as 4 J.K-I.mo1-l at 423 K and 9 J.K-'.mol-' at 4 7 3 K. Other uncertainties increase the total possible error somewhat and to a greater extent the more dilute the solution. This may increase +hetotal uncertainty to as much as double the values given abov , i.e., 1 J.mo1-l.K-I at 348 K and 2 mol-kg-' or 8 J.mo1-I-K-l at 423 K and 0.1 mol.kg-I. At any particular temperature the uncertainty in @C,, is roughly proportional to m-l. Calculations Review of Equations. We wish to generate a comprehensive equation for the thermodynamic properties of aqueous M g S 0 4 from these heat capacities and other appropriate data. We use the virial-series-plus-Debye-Hiickelequations as developed by
"Definitions of symbols: b = a general constant with value 1.2 kg'/2.mol'1/2; the p's and 0 are ion-interaction parameters specific to each solute MX; p(2)is zero unless both ions have charge 2 or greater; a i = 2.0 kg'/2.mol-i/2 unless both ions have charge 2 or greater whereupon a l = 1.4 kg'/2.mol-'/2 and a2 = 12 kg'/2mol-1/2;I is the ionic strength (1/2xm,z,2);v is the total number of ions formed from the salt ( u = uM + vx); zM and zx are the charges on the ions in protonic units. is the apparent or partial molar heat capacity of the solute at infinite dilution; D is the dielectric constant or relative permittivity; e is the electronic charge; c,, is the permittivity of free space (in esu the factor 4reo is omitted); m, = kg of water.
cPo2
Pitzer"J8 and Pitzer and Mayorga.' Table I1 summarizes these equations for a single electrolyte including the p(*) term which is required for a 2:2 electrolyte but not needed for 1:1, 1:2, or 2:l types. Equations for mixed electrolytes are available from several source~.~,~J~,~~ The equations for enthalpy and heat capacity in Table I1 assume that the exponent a2is a constant. There are theoretical reasons7 that CY* should be proportional to the Debye-Huckel parameter A,, and we will show later that this variation in a2 improves the agreement with the heat-of-dilution data below 0.1 mobkg-I. But at molalities above 0.1, the contribution of the pC2) term ceases to vary with molality whether a2is constant or varies with temperature. Thus, we can fit all of the data above 0.1 mobkg-' (and the intercept at zero molality) with full accuracy with a constant a2
Since the temperature variation of a2greatly complicates the equations for enthalpy and heat capacity, we first treat the data with the constant cy2 equation and with care to avoid any distortion of the fit from the few data below 0.1 mol-kg-' (and above 298 K). For the enthalpy and heat capacity it is convenient to define additional symbols ~~~~
~
~~
~
~
(17) Pitzer, K. S. J . Phys. Chem. 1973, 77, 268. (18) Pitzer, K. S. In "Activity Coefficients in Electrolyte Solutions", Pytkowicz, R. M., Ed.; CRC Press: Boca Raton, FL, 1979; Chapter 7 . (19) Pitzer, K.S.; Kim, J. J. J . Am. Chem. SOC.1974, 96, 5701.
Thermodynamic Properties of MgS04
The Journal of Physical Chemistry, Vol. 90, No. 5, 1986 897 TABLE 111: Parameters for the Equations for Various Thermodynamic Properties of Aqueous MgSO.,
with similar definitions for j3('), /3(2), and C. Values of the Debye-Hiickel parameters were calculated from the equation of Bradley and Pitzer20 for the dielectric constant (relative permittivity) of water and the equations of Haar et a1.I6 for the density. Our measurements of the heat capacity were all at 20 bar. The effect of pressure differences of.less than 20 bar are small compared to the uncertainties of most of the other data used in the global fit; hence no effort was made to obtain the pressure effect on the ion-interaction parameters p(O), etc. The Debye-Huckel parameter was calculated for the pressure of the experiment in each case, although the pressure effect here is also small. These pressure effects would become large at temperatures substantially higher than 473 K, but there are no data for MgS04 at these higher temperatures where its solubility becomes very small. Heat Capacity at Zero Molality. Since the accuracy of experimental measurements of the total heat capacity is approximately independent of molality, the uncertainty in the apparent molar heat capacity increases as the inverse of molality. Thus, with the rapid change of ion association at low m in MgS04, it is not feasible to obtain an accurate value of the limiting heat capacity of Mg2+ S042-by extrapolation. Rather we obtain that quantity indirectly as follows:
+
CPo2(MgSO4)= CPo2(MgCl2)+ CPo2(Na2SO4)- 2Cpo2(NaC1) (22) where equations for NaCl and Na2S04 are available from the literatureI0.l' and that for MgC12has been developed, as described in the Appendix, from measurements in several laboratories. The values" for N a 2 S 0 4were corrected to 20 bar pressure by use of the volumetric properties.2' Fitting of Parameters. With O 2 fixed from eq 22 and the Debye-Hiickel parameter known,& the other parameters of eq 14 were first evaluated separately at each temperature from the experimental heat capacities in Table I. In addition, published ~-~~ measurements near 298 K from several l a b ~ r a t o r i e s ~were fitted. In each case the molality dependence could be represented satisfactorily; indeed CJ could be omitted unless the data extended above 1 mol-kg-I. Equations were then selected to represent the temperature dependency of the virial parameters and of CPo2as follows:
where the q's are fitting parameters and T is temperature in Kelvin. Next, all of the heat capacity data were fitted in a single least-squares regression. At this stage the high-temperature heat capacities measured by Likke and BromleyZ5were included. While less precise than the present measurements, the data of Likke and Bromley agree very well for MgSO, as was the case for Na2S04. (20) Bradley, D. J.; Pitzer, K. S . J. Pfiys. Cfiem. 1979, 83, 1599. 1983, 87, 3798. (21) Phutela, R. C.; Pitzer, K. S. IUPAC Conference on Chemical Thermodynamics, Hamilton, Canada, 1984, abstract p 87; full paper submitted to J. Chem. Eng. Data. (22) Perron, G.; Desnoyers, J. E.; Millero, F. J. Can. J . Chem. 1975, 53, 1134. (23) Drucker, C. Ark. Kemi, Mineral. Geol. 1934, l l A , 1. (24) D'Ans, J.; Tollert, H. Z . Elektrocfiem. 1937, 43, 81. (25) Likke, S . ; Bromley, L. R. A. J . Chem. Eng. Data 1973, 18, 189.
p(O'(298) = 0.21499' /3(')(298) = 3.3646" /3'2'(298) = -32.743' C(298) = 0.006993" 41 = -6.2543 X lo6 q2 = 6.5277 X lo4 q3 = -2.6044 X lo2 q4 = 4.6930 X 10-1 qs = -3.2656 X lo-, q 6 = -1.0282 q7 = 8.4790 X q8 = -2.3366 X q9 = 2.1575 X lo-*
p(0)L(298)= 6.8402 X lo-, /3(1)L(298)= 1.1028 X @(2)L(298)= -2.1515 X lo-' CL(298) = -8.7899 X 410 = -2.9596 X lo-' q I 1 = 9.4564 X lom4 q12 = -1.3764 X 10' 413 = 1.2121 x lo-' qI4 = -2.7642 X q15= 1.0541 X lo-' q I 6= -8.9316 X qI7= 2.5100 X lod q18= -2.3436 X
'From ref 26. Equations 24-27 can be integrated with respect to temperature (following eq 15 and 16) to yield the parameters for the enthalpy and then integrated again to yield the excess Gibbs energy and the activity and osmotic coefficients. The results are illustrated for $O); those for the other three parameters correspond exactly. Here and elsewhere 298 is an abbreviation of 298.15.
pco)' = [(p- 2982)q6/2 + ( T 3- 2983)q,/3 + (74 2984)q,/4 + ( P - 298')q9/5 + 2982/3(0)'(298)]/T2 (28) p(O) = q 6 ( T / 2 + 2 9 S 2 / 2 T - 298) + q7(T2/6
+ +
29S3/3T2982/2) qg(T3/12 29S4/4T - 2 9 8 3 / 3 ) q 9 ( P / 2 0 298'/5T - 29S4/4) (298 - 29S2/T)p(O"(298) /3(')(298) (29)
+
+ +
+
+
The values at 298 K for p(O), @(I), p(2),and C were taken from Rard and Miller.26 These values, based on their isopiestic measurements and those of earlier workers, are so accurately determined that we adopted them as fixed and list them in Table 111. A regression of data from heat of dilution measurement^^'-^^ at 298 K yielded provisional values for the enthalpy integration constants p(O)'(298), etc. With a full set of parameters now assembled, test calculations were made of the osmotic coefficient and the heat of dilution at various temperatures for comparison with literature values of these quantities.3w32 While there were no large disagreements, it was clear that these enthalpy and osmotic data at higher temperatures could contribute to an improved general equation. Consequently, a comprehensive regression was carried out including all available data, except the osmotic coefficients at or below 298 K, and fitting all parameters and for p(O),@(I), /3(2), and C a t 298 K adopted except those for CPo2 - C'(298). from Rard and Miller, i.e., q6 - q I 8and /3(0)L(298) Appropriate weights were assigned to the various data in view of the experimental uncertainties as stated by the authors or judged by us. Complete agreement was not made possible within these uncertainties but the remaining discrepancies are not large. The resulting parameters are listed in Table 111. Effect of Temperature Variation of c y 2 . We now derive the more complex equations wherein a2 is proportional to A,. The only changes are in the term for p(*) or p(*)' or p(2)Jin eq 3 , 5, 10, 12,and implicitly in 15. Let a2 = kA,, with k = 30.6' so that, at 298 K, a2returns to the standard value 12 kg1/2.mol-1/2.Also designate with an asterisk these separate "p(2)terms" and the (d(3(2)*/d7')p, and (d2/3(2)*/dp)pwhich will differ quantities /3(2)*, from but are related to the corresponding quantities when a2is constant. The following equations are simplified for use for a 2:2 (26) Rard, J. A.; Miller, D. G. J. Cfiem.Eng. Data 1981, 26, 33. (27) Snipes, H. P.; Manley, C.; Ensor, D. D. J. Cfiem.Eng. Data 1975, 20, 287. (28) Lange, E.; Streeck, H. 2.Pfiys. Cfiem. 1931, A157, 1. (29) Poczopko, S.; Orzeszko, W. Rocz. Cfiem. 1972, 46, 259. (30) Leung, W. H.; Millero, F. J. J. Solution Cfiem. 1975, 4, 145. (31) Holmes, H. F.; Mesmer, R. E. J. Cfiem. Tfiermodyn.1983, 15, 709. (32) Mayrath, J. E.; Wood, R. H. J. Cfiem. Eng. Data 1983, 28, 5 6 .
898 The Journal of Physical Chemistry, Vol. 90, No. 5, 1986
Phutela and Pitzer
2 oc
-
0
1-
-,E Y -3
\
0
0
-8
-2oc
,D-H 0
I
I
I
2
I
3
m
I
I
4
5
Figure 2. The osmotic coefficient of MgSO, at 383 and 413 K as measured by Holmes and Mesmer” with the solid and dashed curves showing
D , -H
-4OC
I
the calculations of this research and Archer and Wood,6 respectively.
I I
I
I
c
I
Figure 1. The apparent molar heat capacity at 423 K (solid symbols and upper curve) and at 373 K (open symbols and lower curves). Circles are experimental points from this research, squares are from Likke and B r ~ m l e ysolid ; ~ ~ curves are calculations from this research; the dashed curve is for 373 K from Archer and Wood;6the straight lines marked D-H show the Debye-Huckel slopes.
electrolyte, Le., vM = vx = 1 and I = 4m;also we omit the subscript MX. The term in the Gibbs energy, per mole of solute, differs only by the value kA,I‘I2 for a21‘12and the asterisk on pCz). (GE)*/n2RT = (Z/2)p(2)* g(kA,1112)
(30)
Differentiation to obtain the term for the enthalpy now yields an additional term from the temperature dependence of A,.
(+L)* = -(Z/2) x [RF?(~3fl(~)*/dT) g(kA,I1lZ) + (k11/2AH/4)p(2)*g’(kA,I’/2)] (31) In this expression we use the relationships dA,/dT = A H / ~ R ~ g’(x) = dg/dx = -(4/x3)[1 - (1
+ x + x2/2) exp(-x)]
(18) (32)
Finally, the heat capacity term is found to be
05
(6Cp)* = -(I/2)(R[F(C32p(2)*/dP) + 2 T(~3p(~)*/dT)] g( kA,I’I2) kI’i2[(AH/2)(dp(2)*/dT) AJ3(”*/4] g’(kA,Z’/’) k21fl(2)*(AH2/ 1 6 R P ) g”(kA+Z’/*)) (33)
+
g”(x) =
(12/x4)[l
- (1 + x
+
+ x2/2 + x3/6)
+
exp(-x)]
(34)
One can evaluate the starred quantities, Pc2)* and its derivatives, by noting that the p(*) term for the excess Gibbs energy, eq 30, ceases to depend on I when exp(-kA,I’/2) becomes negligible. Then g(x) 2/x2 and
-
lim [(GE)*/n2RZ7 = p(2)*/k2A,2
I--
(35)
When a2 is taken as a constant, the same term takes the value
p(2)/az.These constant values are approached very closely at the relatively small molality of 0.1 molvkg-’ ( I = 0.4 mol-kg-I). By equating these values for GE,one obtains
PC2)* = P(2)A,2(30.65/ 12 kg1/z-mol-’/2)2
(36)
This expression can be differentiated with respect to T as needed.
I12
IO
15
Figure 3. The relative apparent molar enthalpy calculated in this research compared with experimental heats of dilution; see text for details.
Since all of the other thermodynamic functions can be obtained by differentiation of GE,this same relationship will apply for m > 0.1 mol-kg-’ and the same values will be obtained for constant a 2as for az proportional to A, except for m C 0.1. In particular, we note that the limiting value for m > 0.1 molekg-l for the contribution of the p’*)term to the osmotic coefficient is zero while that to In y+ is ,B(z)*/2k2A,Zor (3(2)/2cu22. Thus, it is not necessary to use these more complex equations for most calculations. Indeed, our initial calculations were carried out on the basis of the equations of Table I1 (constant a 2 )as described above. Supplemental calculations were made for the few data where the two methods differ. Results. The solid curves in Figures 1-3 show our calculated curves from the global fit. Below 0.1 mol-kg-’ the curves are for the equations with varying a2;there is no difference at higher m. Figure 1 shows the apparent molar heat capacity at 373 and 423 K including our measured values and those of Likke and Bromley (the latter for 423 K are averages of measurements at 413 and
The Journal of Physical Chemistry, Vol. 90, No. 5, 1986 899
Thermodynamic Properties of MgS04 TABLE IV: Calculated Activity Coefficients for Aqueous MgSOl at 20 bar mlmol-kg-' 298.15 K 323.15 K 373.15 K 423.15 K 473.15 K 0.1 0.2 0.3 0.5 1.o 2.0 3.0
0.169 0.122 0.1000 0.0775 0.0557 0.0475 0.0564
0.154 0.109 0.0891 0.0683 0.0481 0.0392 0.0439
0.111 0.0762 0.0604 0.0446 0.0293 0.021 1 0.0205
0.0664 0.0432 0.0332 0.0234 0.0142 (0.0090)
0.0308 (0.0190)
TABLE V: Calculated Osmotic Coefficients for Aqueous MgS0, at 20 bar m/mol.kg-' 298.15 K 323.15 K 373.15 K 423.15 K 473.15 K 0.1 0.2 0.3 0.5 1.o 2.0 3.0
0.596 0.562 0.544 0.527 0.527 0.663 0.925
0.581 0.545 0.528 0.509 0.503 0.616 0.844
0.529 0.490 0.469 0.444 0.419 0.482 0.634
0.459 0.418 0.395 0.362 0.320 (0.335)
0.373 (0.337)
TABLE VI: Calculated Relative Apparent Molar Enthalpies for Aqueous MgSO, at 20 bar in kJmo1-I m/mol.kg-l 298.15 K 323.15 K 373.15 K 423.15 K 473.15 K 0. I 3.802 6.825 16.20 32.89 61.77 0.2 4.240 7.812 18.22 36.07 (66.39) 0.3 4.474 8.422 19.48 37.96 0.5 4.760 9.251 21.18 40.44 1.o 5.227 10.611 23.88 44.21 2.0 6.120 12.675 27.60 (49.11) 3.0 7.261 14.729 30.95 TABLE VII: Calculated Apparent Molar Heat Capacities for Aqueous MgS0, at 20 bar in J-K-'.mol-' m/mol.kg-' 298.15 K 323.15 K 373.15 K 423.15 K 473.15 K 0.0 -287.6 -251.1 -288.4 -419.8 (-696.1) 0.1 -181.7 -112.9 -40.6 15.4 53.3 0.2 -158.6 -91.9 -19.3 40.7 (88.0) 0.3 -142.2 -78.0 -6.7 53.3 0.5 -117.9 -58.4 9.8 67.1 1.o -76.7 -27.2 34.0 82.9 2.0 -22.6 13.4 62.0 (100.1) 3.0 17.2 46.4 81.5
433 K). The error bars for the Likke-Bromley values are based on their estimate of 0.01 J.K-'.g-' for the specific heat. It is clear that their values agree with ours well within their uncertainties. In addition to the solid curves for our final equation, the dashed curve for 373 K is from a very recent ion-association treatment of Archer and Wood6 which became available during our final calculations. They give only "CP- Cpo2and the comparison is The Archer-Wood values are made by using our value of lower by about 40 J.K-'-mol- in the molality range of the experimental measurements. The two curves come together, how-
cplo?.
ever, at about 0.005 mol-kg-', with our curve very slightly lower in the extremely dilute range. The osmotic coefficients of Holmes and Mesmer3' at 383 and 413 K are shown in Figure 2. The agreement is excellent with a mean deviation of 0.006. While the Archer and Wood curve also fits very well at 383 K, it departs significantly at 413 K. Graphical comparison with heat of dilution data is complicated by the fact that there is both an initial and a finaly molality for each measurement. In Figure 3 the highest molality for a given set of dilutions is shown with a flagged symbol. The symbols at the dilute molalities show the agreement or discrepancy by their location on or off the calculated curve. The measurements at 298 K are by Snipes et shown as circles, and by Poczopko and O r z e ~ z k o shown , ~ ~ as triangles. Measurements for very dilute solutions by Lange and Streeck28cannot be shown on this graph but agree comparably well. The agreement for the heat of dilution data of Leung and Millero30at 303 K is excellent with an average deviation of only 25 Jemol-I. At 373 K the heat of dilution measurements are by Mayrath and Wood32who also report data for 423 K. Provided one uses the equations with a2proportional to A,, the agreement with the measurements is good in the dilute range at 373 K. At higher molality the agreement is also good, within 0.2 kJemol-' for the individual dilution measurements and within 0.3 kJ-mol-] for the total change in 4L from 1.99 to 0.12 molskg-I. At 423 K the agreement is good above 0.03 mol-kg-I with maximum differences for individual measurements of 0.2 kJ-mol-] above 0.2 mol-kg-I and 0.4 kJemol-' between 0.03 and 0.2 kJ.mol-l. Indeed, the agreement above 1 mol-kg-' is much better for our equation than for Archer and Wood's. Below 0.03 mobkg-' there are somewhat larger differences which may exceed the experimental uncertainty at 423 K. In this very dilute range there is no confirmation from the osmotic coefficients or from our heat capacities. Hence, there is considerable uncertainty in our calculated enthalpies at very low molalities and above 400 K. Since Rard and Miller considered the freezing-point-depression data (converted to 298 K) in their evaluation of the parameters for 298 K, we did not include these freezing point data in our calculations. But our equation yields osmotic coefficients in good agreement with these freezing-point data. The equations are simple enough that users desiring precise values will want to calculate the thermodynamic functions at the particular temperatures and molalities of interest. Thus extensive tables of various functions seem unnecessary. For convenience of users, however, we do include in Tables IV-VI1 broadly spaced arrays of values over the range to 473 K and to 3 molakg-' or saturation molality. Table VI11 gives values of these functions for very dilute solutions obtained by using the equations for a2 proportional to A,.
Discussion It is clear that the electrostatic attraction of Mgz+and S042or similar ions in water is so strong that the linearization approximation of Debye-Huckel theory is inadequate. In this respect
TABLE VIII: Thermodynamic Properties for Very Dilute Solutions of Aqueous MgSO, at 20 bar
m/ mol. kg-' function
T/K 298.15 373.15 423.15
0.0001 0.967 0.957 0.941
0.001 0.896 0.855 0.789
0.005 0.796 0.728 0.637
Y+
298.15 373.15 423.15
0.907 0.884 0.850
0.732 0.656 0.551
0.520 0.414 0.297
$t/ kJ.mo1-I
298.15 373.15 423.15
0.19 0.77 1.9
0.74 3.49 8.9
-W,/ J.K-l.mo1-l
298.15 373.15 423.15
4
284 275 387
272 221 256
1.67 8.02 19.3 247 139 99
0.01
0.02
0.05
0.746 0.677 0.599
0.697 0.635 0.572
0.638 0.578 0.5 18
0.422 0.320 0.219
0.331 0.243 0.160
0.230 0.161 0.100
2.19 10.23 23.5 233 106 57
2.7 1 12.18 26.6 220 82 32
3.35 14.44 30.1 200 58 7
900
The Journal of Physical Chemistry, Vol. 90, No. 5, 1986
the radial distribution functions calculated by Valleau et al.33for the primitive model by the Monte Carlo method are of interest. The population of unlike ions at distances just above the hard-core diameter is much larger than that for a Debye-Hiickel distribution. But the distribution for the 2:2 case is continuous; there is no natural or unambiguous division into ion pairs and dissociated ions. A model including realistic water structure would show a distinction between contact ion pairs and a distribution with at least one water molecule between ions. Sound absorption meas u r e m e n t ~support ~~ this picture and give an indication of the contact ion population and that of solvent-separated ion pairs. Thermodynamic treatments, however, have not used the sonic absorption basis, and it is questionable whether the Debye-Huckel distribution would be adequate even if the contact ion population were treated separately. From the preceding discussion it is apparent that there is no unambiguous ion-pair population for MgS04 in the absence of an auxiliary and often arbitrary definition. In thermodynamic treatments this usually takes the form of an assumed expression for the activity coefficient for the dissociated ions, for example, an assumed hard-core diameter for a particular extended Debye-Huckel expression. Hamer35and others36have noted that, for a case such as MgS04, the association constant determined from a given set of data depends on the assumed activity-coefficient expression. These considerations show that a virial-series or ion-interaction treatment has as much theoretical justification as an ion-association treatment for a case where the association is not too strong. The empirical success of the ion-interaction method had been demonstrated7sZ6for M g S 0 4 at 298 K. The present calculations extend this range of empirical success to 373 K with relatively high accuracy. The evidence is the agreement with heat of dilution data below 0.1 molakg-I. At 423 K the picture is less clear; there is good agreement above 0.01 mol-kg-' but there are substantial differences below that molality. The ion-association treatment of Archer and Wood6 also shows deviations in this very dilute range at 423 K, although theirs are somewhat smaller (0.5 kJ-mol-I) than for our equation where there are differences as large as 2
kJ-mol-' . The virial-series method is expected to fail eventually as the association constant becomes large and to fail sooner for a derivative quantity such as the enthalpy than for the Gibbs energy or activity coefficient. Thus we have the encouraging situation that the present method should be quite accurate for the activity add osmotic coefficients through 423 K and very probably to somewhat higher temperature. The relationship between an ion-association treatment and a virial-coefficient treatment is complex except in the limit of low that molality where it can be (37) Here K is the association constant. Differentiation with respect to temperature to obtain the change in enthalpy and heat capacity with association yields
These equations apply for both the cases with a2 constant and ~~
~~~
Phutela and Pitzer with a2proportional to A, (with (3C2)*, etc.). Thus, to the extent that the identification of K with -2/3(2) is accurate, AH and AC, can be obtained from the temperature dependency of /3C2) or /3C2)*. In practical calculations both j3(2)and K are determined by fitting over a range of concentration (rather than at the limit m = 0); hence, one has no reason to expect agreement between the K , AH, and AC, implied from /3(2) by eq 37-39 and the values obtained from a particular ion-pairing treatment. Nevertheless, there should be semiquantitative agreement and a comparison is useful. There are two recent ion-association treatments for aqueous MgS04. The first, by Woolley and H e ~ l e r is, ~limited to 298 K while Archer and Wood6 present a more general treatment extending to 423 K. Woolley and Hepler select the values K = 200 and A H = 5.0 kJ.mol-I from earlier publications and then find AC, = 125 J.K-l.mo1-I. From their comprehensive treatment, Archer and Wood determine K = 126.4, A H = 5.97, and ACp = 203 at 298 K. These may be compared to our K = -2/3(2) = 65.5, AH = 4.9 kJ-mol-', and AC, = 134 J.K-'.mol-' calculated on the basis of constant a2. On the basis that CY, is proportional to A,, the value of K remains unchanged at 298 K while AH increases to 7.4 kJ.mo1-l and AC, = 170 J.K-'.mol-'. The earlier paper of Leung and Millero30reports AH values from 4.8 to 5.7 kJSmo1-l from varying interpretations of calorimetric measurements. Some of these values agree fairly closely while others agree only within factors of two or thereabouts. On the basis that a, is proportional to A,, our value of K will rise more rapidly and our AH will rise less rapidly than Archer's; thus, our respective parameters will come closer together at higher temperature. We turn now to a discussion of the overall representation of thermodynamic properties by our equation in comparison with that of others. In addition to Archer and Wood's equation,6 we consider that of Holmes and M e ~ m e r . ~They ' used the same form of equations as we have (with constant a,) but did not attempt to represent the properties in the very dilute range above 298 K. We agree with both of these other investigations in concluding that the vapor pressure measurements of Liu and Lindsay3*and of Fabuss and K ~ r o s are i ~ not ~ accurate enough to be useful in this connection. We have our own new heat capacity measurements which were, of course, unavailable to the others. Also the other investigators chose to exclude the heat capacities of Likke and Bromley,25 whereas we find them to be correct well within their stated uncertainty of 0.01 J.g-'.K-' for the total fluid. All three of the treatments fit the osmotic coefficients very well at 383 K; however, the agreement for Holmes' and our equations remains excellent at 413 K whereas Archer's does not agree as well at that temperature. The agreement of all three treatments is also good for the activity coefficient. At 383 K and 3 mobkg-l, for example, we obtain 0.0171 while Archer has 0.0163 and Holmes has 0.0181. Holmes and Mesmer stated a large uncertainty for their value, but it is now apparent that their equation is quite accurate at 383 K. For the enthalpy we included the data of Poczopko and Or~ e s z k which o ~ ~ extend to 3.16 mobkg-' that the others did not cite. Other enthalpy data extending to 2 molskg-' were fitted by all three investigations. Holmes and Mesmer do not claim high accuracy of agreement on enthalpies. The agreement in our case is good except for very dilute solutions ( m < 0.01 mol-kg-I) a t 423 K as noted above. Apparently, Archer did not attempt to fit the enthalpy above 2 mobkg-' and his equation diverges substantially in that range. The possible complication of hydrolysis of the Mgz+ ion should be discussed. For the reaction Mgz+ + H,O = MgOH'
+ H'
~~
(33) Valleau, J. P.; Cohen, L. K.; Card, D. N. J . Chem. Phys. 1980, 72, 5942. (34) Atkinson, G.; Petrucci, S. J . Phys. Chem. 1966, 70, 3122. (35) Hamer, W. J. "The Structure of Electrolyte Solutions"; Wiley: New York, 1959; Chapter 15. (36) Pitzer, K. S.;Brewer, L. revision of 'Thermodynamics", Lewis, G. N., Randall, M.; McGraw-Hill: New York, 1961; p 587. (37) Pitzer, K. S.;Silvester, L. F. J . Phys. Chem. 1978, 82, 1239.
Baes and Mesmer40 report log K = -1 1.44 at 298 K. From their (38) Liu, C.; Lindsay, W. J. Report 722 to the Office of Saline Water, US. Department of the Interior, 1971. (39) Fabuss, B. M.; Korosi, A. Desalination 1966, I , 139. (40) Baes, C. F., Jr.; Mesmer, R. E. Am. J . Sci. 1981, 281, 935.
The Journal of Physical Chemistry, Vol. 90, No. 5, 1986 901
Thermodynamic Properties of MgS04
---293
323
373
423
473
T/K Figure 4. The heat capacity of MgC1, at zero molality. The squares are for 175 bar, the triangles are for 100 bar, the circles are for 6 bar or for saturation pressure, and the curve is interpolated for 20 bar. See the Appendix for details.
value for AH one may calculate log K = -8.2 at 423 K and -7.4 at 473 K. Alternatively, one may consider the reaction Mg2+
+ SO4,- + H 2 0 = MgOH' + HSOi
and, with the dissociation constants of Marshall and Jones4I for HS04-, one obtains log K values of -9.4 at 298 K, -4.6 at 423 K, and -3.2 at 473 K. One notes also the appropriate activity coefficients which are very much smaller for Mg2+ and SO?- than for singly charged ions. When these data are combined, one concludes that neither hydrolysis reaction will be significant in our experimental range above 0.09 mobkg-I. Hydrolysis may become significant, however, at lower molality. It should be considered if accurate comparisons are to be made for heats of dilution below 0.01 mol-kg-' at 423 K or higher temperature. In summary, our equation has the advantage of guidance from heat capacity data extending to 473 K and of enthalpy data at 298 K above 2 mol-kg-' which were not available to or considered by the other studies. While our equations have somewhat more parameters than Archer's, our (and Holmes') equations have the advantage of direct solution without the need for iteration to determine the molalities of assumed ion pairs, triplets, and sextuplets. For calculations of full accuracy below 0.1 molskg-', additional terms must be added to our equations for the enthalpy and the heat capacity but these terms are fully determined and do not constitute additional disposable parameters. Harvie et al.839have shown that the ion-interaction (Pitzer) equations are remarkably effective in predicting the properties of complex mixed electrolytes at high concentration at room temperature. Since the parameters in these ion-interaction (41) Marshall, W. L.; Jones, E. V. J . Phys. Chem. 1966, 70, 4028.
equations are now known to high temperatures for NaC1,I0 other alkali halides,42 Na2S04," MgCl, (ref 12, 43, and 44 and the Appendix), and M g S 0 4 from this research, it is now possible to make good predictions for the various thermodynamic properties of mixed brines containing these various ions. Values at more moderate temperatures are available for many other i0ns.4~ There are also small terms arising from the mixing of ions of the same sign which are known to high temperature in a few cases46but ~.~ calculations& are known only for 298 K in other c a ~ e s . Solubility for the systems NaC1-Na2S04 and NaCl-MgC1, yielded good agreement with experiment at high temperature when these small mixing parameters were assigned their room-temperature values. While experiments to determine more generally these mixing parameters at high temperature are desirable, in the interim considerable confidence can be placed in calculations using room-temperature values of these small terms at least to 473 K. Thus, it is now possible to extend general predictions of the properties of complex natural (or artificial) brines to relatively high temperatures.
Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Division of Engineering, Mathematics, and Geosciences of the U.S. Department of Energy under Contract No. DE-AC0376SF00098. Appendix There are heat capacity measurements for aqueous MgC12 at saturation pressure by Likke and Bromley,' to 180 "C and at 6 bar by Saluja13 to 100 OC in addition to those at 1 bar and 25 "C by Perron et al.47,48 Although the precision claimed by Likke and Bromley is only 0.01 J-K-I in the specific heat, their results generally agree with recent and more precise values for other systems within a substantially smaller uncertainty. Thus we believe that considerable confidence can be placed in their values for MgC12at temperatures above the range of Saluja's measurements. Equation 14 was fitted to all of these low-pressure data with appropriate temperature dependence for the virial coefficients and the theoretical Debye-Huckel parameter. These results will be described in detail elsewhere', along with similar results for CaC1, are shown and SrC1,. The limiting values at zero molality of Cpo2 in Figure 4 along with the values of WoodI4 for 100 and 175 bar. Values for 20 bar for use in this paper were interpolated; the 20-bar curve is shown in Figure 4. Registry No. MgS04, 7487-88-9. (42) Holmes, H. F.; Mesmer, R. E. J . Phys. Chem. 1983, 87, 1242. (43) Holmes, H. F.;Baes, C. F.,Jr.; Mesmer, R. E. J . Chem. Thermodyn. 1978, 10, 983. (44) de Lima, M. C. P.; Pitzer, K. S. J . Solution Chem. 1983, 12, 187. (45) Silvester, L. F.; Pitzer, K . S. J . Solution Chem. 1978, 7, 327. (46) Holmes, H. F.; Baes, C. F., Jr.; Mesmer, R. E. J . Chem. Thermodyn. 1979, 11, 1035. (47) Perron, G.; Roux, A.; Desnoyers, J. E. Can. J . Chem. 1981, 59, 3049. (48) Perron, G.; Desnoyers, J. E.; Millero, F.J. Can. J . Chem. 1974, 52, 3738.
