J. Phys. Chem. 1988,92, 1631-1639 Using eq 11.9 relying on BI,,,on El,,,, we write
Thus (i) dividing both numerator and denominator by eo and (ii) recalling that the dielectric constant is nothing but t, = €/to, we get (21 l)(€, - 1) = (11.20) (1 l)Er 1
+ +
The expression between brackets in eq 11.1 1 becomes
- €0)
( 1 + 1)E/m - ( 1 +
(I
c0a1+2
+ l)e + le0
1Elm -
eoa/+2
(11.14)
which may be written
I+ 1
&e - €0)
1
According to eq 11.15, the fictive charge distribution 11.12 is
1631
(11.15) ulmof
eq
+
As recalled in section 1I.B the factorfll) calculated by eq 11.19 is different from the corrective factor fMsT of Miertus et Effectively for a dipole 2,for instance, the charge surface density u is exactly given by u
3€, 1 €, - 1 211 cos 8 = 2€, 1 4T a3
7-)
-( +
(11.21)
In the case of dipoles (1 = 1) the corrective factor fMST(1) of Miertus et aL60 is (11.22) Whereas ours is 3(€, - 1) 2€, 1
f(1) = --
+
Hence according to eq 11.8 Ulm
1 (21 + 1)(€- €0)=Gon(ltm) 47r (I l ) € IC0
+
+
(11.18)
In conclusion, the fictive charge surface density a-may be generated by multiplying ea+ multipolar component Gon(l,m) of the normal component of bo, (on the sphere of radius a ) by a factor AI) which only depends on I but not on m.
+ l)(€(I + l ) € + IC0
(21
f(l)=
€0)
(11.19)
3€, 2€,
-1
C,
(11.23)
€,
The reason for this apparent disagreement is that in fact Mertius et al. calculate 1 t , - 121.1COS8 1 €,- 1, no = - --= - -Go,( 1) (11.24) 47r cy r3 4 a €, '
so that the exact surface charge density = fMST(1)"O
u
is given by (11.25)
whereas we calculate u
= f(Z)BO,(l)
(11.26)
At the end, the two processes lead to the same value of
U.
Calculation of the Thermodynamic and Transport Properties of Aqueous Species at High Pressures and Temperatures: Dissoclation Constants for Supercrltical Alkali Metal Halldes at Temperatures from 400 to 800 "C and Pressures from 500 to 4000 bar Eric H. Oelkers and Harold C. Helgeson* Department of Geology and Geophysics, University of California, Berkeley, California 94720 (Received: June 15, 1987; In Final Form: October 8, 1987)
Dissociation constants from 400 to 800 O C at 500-4000 bar have been calculated for 14 aqueous alkali metal halides from experimental conductance measurements reported by Quist and Marshall' using the Shedlovsky equation and the law of mass action, together with independently calculated limiting equivalent conductances for the electrolytes generated from equations given by Oelkers and Helgeson? The requisite activity coefficients were calculated for neutral and charged species from the Setchtnow and extended Debye-Hiickel equations, respectively. Where direct comparison can be made, dissociation constants reported by Quist and Marshall3 and Dunn and Marshall4 fall within estimated uncertainties of corresponding values generated in the present study. The logarithms of the calculated dissociation constants range from - 4 . 7 at low temperatures and high pressures to --4.5 at high temperatures and low pressures.
Introduction Dissociation constants computed from experimental measurements of the conductances of supercritical aqueous electrolyte solutions have been reported by Fogo, Benson and Copland,' (1) Quist, A. S.; Marshall, W. L. J . Phys. Chem. 1969, 73, 987. (2) Oelkers, E. H.; Helgeson, H. C. Geochim. Cosmochim. Acra 1988,52, 63. (3) (a) Quist, A. S.; Marshall, W. L. J . Phys. Chem. 1968, 72, 684. (b) Quist, A. S . ; Marshall, W. L. J . Phys. Chem. 1968, 72, 2100.
0022-3654/88/2092-163 1$01.50/0
Franck? Quist et al.,' Quist and Marshall>* Ritzert and Franck: Dunn and M a r ~ h a l l Mangold ,~ and Franck,lo and Frantz and (4) Dum, L. A.; Marshall, W. L. J. Phys. Chem. 1969, 73, 723. (5) Fogo, J. K.; Benson, S . W.; Copeland, C. S . J. Chem. Phys. 1954,22, 212. (6) (a) Franck, E. U. Z . Phys. Chem. 1956,8,92. (b) Franck, E. U. Z . Phys. Chem. 1956,8, 107. (c) Franck, E. U. Z . Phys. Chem. 1956,8, 192. (7) Quist, A. S.; Franck, E. U.; Jolley, H. R.; Marshall, W. L. J . Phys. Chem. 1963, 67, 2453. (b) Quist, A. S.; Marshall, W. L.;Jolley, H. R. J. Phys. Chem. 1965, 69, 2726.
0 1988 American Chemical Society
1632 The Journal of Physical Chemistry, Vol. 92, No. 6,1988
Marshall." The dissociation constants were generated in these studies along with the limiting equivalent conductances of the electrolytes by fitting the measured conductances at a series of dilute concentrations to a combined statement of the law of mass action and the Shedlovsky equation. These same equations were used in the present study, together with the SetchEnow and extended Debye-Huckel equations. The requisite limiting equivalent conductances of the electrolytes were predicted independently by using equations and parameters given by Oelkers and Helgeson? which obviates the need for experimental measurements at more than one concentration. This approach makes it possible to retrieve dissociation constants from the conductances reported by Quist and Marshall,' which were made at a single concentration of 0.01 m in order to investigate the relative importance of ion-solvent interaction in alkali metal halide solutions. The purpose of the present communication is to summarize the results of these retrieval calculations. Retrieval Strategy Because the credibility of the calculations described in the following pages depends on the equations selected to represent the various functions involved in the calculations, these equations are summarized below. Dissociation of an aqueous complex designated as ABo into its constituent ions A+ and B- can be expressed by writing ABo = A+
+ B-
(1)
If we use the subscript n to refer to the neutral complex, we can write the law of mass action for reaction 1 as
where I stands for the molal stoichiometric ionic strength ( I = 1/2(u+2+2 vZ?)m, where m denotes the molality of the solution, and u+ and u- refer to the stoichiometric number of moles of the cation and the anion, respectively, in 1 mol of the electrolyte), a, represents the degree of dissociation of the solute, yt denotes the mean molal ionic activity coefficient of the electrolyte, and 7, designates the activity coefficient of the neutral complex. Although strictly applicable only to spherical ions in an incompressible medium with a given dielectric constant, Debye-Huckel theory has been used with considerable success to represent long-range ionic interaction in aqueous solutions at high temp e r a t ~ r e s . ~ * ~ * *The J ~ -mean ' ~ molal ionic activity coefficients used in the present study were calculated from the extended DebyeHuckel equation, which can be written for dilute solutions as
+
IZ,Z-IA.f112 ,
log y* = -
+ AB,i'/2 + b,f
I
1
I
,
(3)
where b, represents the extended-term parameter which is a function only of temperature and pressure.I2 f stands for the effective ionic strength of the solution in molality units of concentration (I = 1/2(Z+2m+Z 2 m - ) ,where m+ and m- refer to the molality of the cation and the anion in solution), Z+and 2denote the charge of the cation and anion, respectively, A refers to the ion size parameter for the electrolyte, and A , and B, represent the Debye-Huckel coefficients defined by
+
1.8248 X 106p'/2
A, =
(€T)3/2
(4)
(8) (a) Quist, A. S.; Marshall, W. L. J. Phys. Chem. 1966,70,3714. (b) Quist, A. S.; Marshall, W. L. J . Phys. Chem. 1968, 72, 1545. (c) Quist, A. S.; Marshall, W. L. J . Phys. Chem. 1968, 72, 3122. (9) Ritzert, G.; Franck, E. U. Ber. Bumen-Ges. Phys. Chem. 1968,72,798. (10) Mangold, K.; Franck, E. U. Ber. Bunsen-Ges.Phys. Chem. 1969.73, 21. (11) (a) Frantz, J. D.; Marshall, W. L.Am. J . Sci. 1982, 282, 1666. (b) Frantz. J. D.: Marshall W. L. Am. J . Sei. 1984. 284. 651. (12) Helgeson, H. C.; Kirkham, D. H.; Flowers, G'. C. Am. J . Sci. 1981, 281. 1249. (13) Pitzer, K. S. J . Phys. Chem. 1973, 77, 268.
Oelkers and Helgeson
where p represents the density of H 2 0 in g cm-3, T designates temperature in K, and c refers to the dielectric constant of H20. Values of p at high pressures and temperatures can be calculated from equations and parameters given by Haar et al.I4 Correspondingly, values of c can be generated by using equations and parameters reported by Helgeson and Kirkham,15 Bradley and Pitzer,16 Uematsu and Franck," Khodakovsky and Dorofeyeva,I8 or Pitzer.I9 If ion association is limited to ion pairs, which is apparently the case for the 1:l electrolytes considered in the present study
7 = a,z
(6)
The activity coefficients of neutral species (7")can be calculated from the SetchEnowZoequation, which is given by log 7, = b,,J
(7)
where b,,n refers to the SetchCnow coefficient for the nth neutral species. It follows from eq 1-7 that the dissociation constant for a 1:l electrolyte can be calculated if values of A,, B,, b,, A, by,n and the degree of dissociation of the electrolyte solution are known. The degree of dissociation of a dilute electrolyte solution can be calculated from21,22
(8)
an = Ac/Ae
where At stands for the experimentally determined equivalent conductance and Ac corresponds to the equivalent conductance of the hypothetical completely dissociated electrolyte in a solution ii, is commonly calculated of the same effective ionic strength from the limiting equivalent conductance of the electrolyte (A'J by using the Shedlovsky equation,23which can be combined with eq 8 to give (see Appendix A)
(n.
A, = A', - (A,/A',)(A,A',
+ B,)7'12
(9)
where A , and B, can be expressed for 1:l electrolytes as A , = 8.2053
X
105p'/2/(cT)312
B , = 82.48p112/7(cT)'12
(1 1)
where 7 refers to the viscosity of H 2 0 in poises. As indicated above, the limiting equivalent conductance of an electrolyte is typically calculated by regression of experimental conductances at several solution concentrations with a combined expression of appropriate statements of eq 2-1 1. In the absence of sufficient experimental data, the limiting equivalent conductance of an electrolyte can be estimated. Franck6 made such estimates using Walden's rule. More recently, it has been shown that A', for a given alkali metal halide represented by MX can be calculated from2
(14) Haar, L.; Gallagher, J.; Kell, G. NBSINRCSteam Tables; Hemisphere: Washington, DC, 1984. (15) Helgeson, H. C.; Kirkham, D. H. Am. J . Sei. 1974, 274, 1089. (16) Bradley, D. J.; Pitzer, K.S. J . Phys. Chem. 1979, 83, 1599. (17) Uematsu, M.; Franck, E. U. J . Phys. Chem. R e j Data 1980,9, 1291. (18) Khodakovsky, I. L.; Dorofeyeva, V. A. Geokhimiya 1981. 8. 1174. (19) Pitzer, K. S. Proc. Natl. Acad. Sci. USA 1983, 80, 4575. (20) SetchCnow, J. Ann. Chim. Phys. 1892, 25, 225. (b) Long, F. A,; McDevit, W. F. Chem. Rev. 1952, 51, 119. (21) (a) Sherrill, M. S.;Noyes, A. A. J . Am. Chem. Sot. 1926,48, 1861. (b) MacInnes, D. A. J . Am. Chem. SOC.1926,48, 2068. (c) MacInnes, D. A,; Shedlovsky, T.J . Am. Chem. SOC.1932, 54, 1429. (22) (a) Harned, H. S.; Owen, B. B. The Physical Chemistry ofEIectrolyte Solutions, 3rd ed.; Reinhold: New York, 1958. (b) Robinson, R. A,; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1959. (23) Shedlovsky, T.J . Franklin Inst. 1938, 225, 1938.
Thermodynamic Properties of Aqueous Species
The Journal of Physical Chemistry, Vol. 92, No. 6, 1988 1633
where AoNaCIstands for the limiting equivalent conductance of NaCl at the pressure ( P ) and temperature (T) of interest, bl and b2 denote correlation coefficients, and So,refers to the standard partial molal entropy of the eth electrolyte at P and T. The limiting equivalent conductance of NaCl can be calculated at temperatures from 0 to 1000 OC and pressures to 5000 bar fromZ I
where R stands for the gas constant, and AN~CI and E A , N ~ repCI resent temperature-independent functions of pressure at_ the temperatures considered in the present study. Values of ANaCl and E A N can ~ ~be calculated from equations and parameters given by Oelkers and Helgeson.2 The standard partial molal entropies of the aqueous electrolytes in eq 12 (as well as those for other aqueous species) can be generated for temperatures and pressures to 1000 OC and 5 kbar from24v25,32
TEMPERATWE, C'
f'
2
3
4
/
PRESSURE, KB
Figure 1. Calculated values of log K, (symbols) for KBr generated as a function of temperature at various pressures, and as a function of pressure at various temperatures (in "C). The curves represent graphic interpolation and extrapolation of the values represented by the symbols. -I
So, =
-2 -3 -I
-2 -3 xc
(14) where stands for the standard partial molal entropy of the eth electrolyte a t the reference pressure and temperature of 1 bar and 298.15 K (Pr,Tr),cI, c2, a3, and a4 represent equation of state coefficients for the electrolyte, 8 and denote constants characteristic of the solvent, w designates the Born coefficient of the electrolyte, and Y is given by
-1
0
s
-2 -3
-2 0 Na HALIDES -30 2000 EAR
Equation 14 was derived from equations of state for the standard and heat capacity ( C O P , , ) of the eth partial molal volume (Pe) electrolyte that are consistent with scaling laws26and solvation theory, as well as with all available experimental density, heat capacity, and dissociation constant data at high temperatures and pressure^.^^,^^ This equation was used together with eq 12 and 13 to calculate values of Aoe for the alkali metal halides, which in turn were combined with experimental conductance measurements a t 0.01 m' to generate values of log K,, (see below).
Calculations and Results The specific conductances (ue) of 0.01 m alkali metal halide solutions reported by Quist and Marshall' were converted in the present study to equivalent conductances with the aid of Values of the density and viscosity of H 2 0 were generated with equations and parameters reported by Haar et al.I4 and Watson et al.,z7 respectively. Correspondingly, values of the dielectric constant were calculated by using equations and parameters given by Helgeson and KirkhamI5 at temperatures log K,, > -4.3. Beyond these limits, the calculated values of log K,,become highly uncertain, reaching 6 log K,, > 1 at log K,, 2 -0.7 and log K,, 5 -5.7. Values of 6 log K,, computed from the properties of the electrolytes shown in Table I using eq 17-25 are given in parentheses in the table. In their calculation of dissociation constants, Quist and Marshall3 and Dunn and Marshall4assumed A", to be a linear function of solution density, which was taken to be independent of temperature at T > 350 "C. The differences between the values of A", generated in this manner and those calculated directly from the experimental data are of the order of 4%,3which corresponds to that adduced above for the predictive uncertainties in the
'
limiting equivalent conductances calculated by Oelkers and Helgeson.2 Equations 17-25 can thus be used to assess the uncertainties in the logarithms of the dissociation constants of NaClO, NaBrO, and NaIo reported by Quist and Marshall3 and Dunn and M a r ~ h a l l .Such ~ estimates of the uncertainties in these log K,, values are represented in Figure 5 by the shaded regions bracketing the solid curves, which correspond to the reported dissociation constants. It can be seen in this figure that the log K,, values computed by Quist and his co-workers (curves) differ by as much as 0.5 log units from the corresponding values generated in the present study (symbols). Nevertheless, the combined uncertainties represented by the shaded regions and the symbol brackets overlap over the entire pressure and temperature range of interest. Both approaches thus yield comparable accuracy and large uncertainties at high and low degrees of association. It follows that reliable interpretation of conductance measurements to generate log K,, values is limited to dilute electrolyte solutions that exhibit moderate degrees of association. Comparisons such as those illustrated in Figure 5 and discussed above can also be made between the values of log K,, for KCl reported by Franck6 and Ritzert and Franckg and those computed in the present study. Comparisons of this kind result in relatively poor agreement, which in the case of Franck6 can be attributed a t least in part to the fact that he assumed the Walden product (Ao,q) for KCl to be independent of temperature and pressure. Recent e ~ i d e n c eindicates ~ ~ ? ~ ~ that this assumption is probably not valid at supercritical temperatures and pressures. It should be noted that the log K,, values generated above, as well as those reported by Quist and his co-workers, are based on the assumption that only negligible quantities of weak acids or bases were present in the dilute solution used in the experiments. Whether or not this was indeed the case has yet to be demonstrated. Conclusions The dissociation constants of 14 alkali metal halides have been calculated at temperatures from 400 to 800 "C and pressures from 500 to 4000 bar by using conductance data reported by Quist and Marshall,' together with limiting equivalent conductances predicted from equations given by Oelkers and Helgeson.2 Where comparisons can be made, the values of log K,, generated above are in reasonable agreement (taking account of mutual uncertainties) with corresponding values reported by Quist and Marshall3 and Dunn and M a r ~ h a l l .The ~ uncertainties attending the calculated log K, values given in Table I are small at moderate degrees of association but increase asymptotically as a,,approaches 0 or 1. Because log K,, values were generated in the present study for such a wide variety of alkali metal halides at a myriad of supercritical temperatures and pressures, the values can be used to develop general correlations for predicting the thermodynamic properties of aqueous complexes to 800 "C and 5 kbar.32 (30) Marshall, W. M. J. Chem. Phys. 1987, 87, 3639. (31) Oelkers, E. H.; Helgeson H. C., submitted for publication in J. So-
lution Chem.
J. Phys. Chem. 1988, 92, 1639-1645
Acknowledgment. The research reported above represents part of the senior author's Ph.D. dissertation at the University of California, Berkeley. This research was supported by the National Science Foundation (NSF Grants EAR 81-15859 and EAR 8606052), the Department of Energy (DOE Contract DEAT03-83ER-13100 and DOE Grant DE-FG03-85ER-13419), and The Committee on Research at the University of California, Berkeley. We are indebted to Peter C. Lichtner, Everett L. Shock, Barbara L. Ransom, William M. Murphy, Leo Brewer, George H. Bremhall, William L. Marshall, and Dimitri Sverjensky for helpful discussions, encouragement, and assistance during the course of this study. Thanks are also due Kim Suck-Kyu for drafting figures, Joachim Hampel for photographic assistance, and Tony Wong for his computer expertise. Appendix A ShedlovskyZ3proposed the equation (32) Sverjensky, D. A.; Shock E. L.; Helgeson H. C., submitted for publication in Geochim. Cosmochim. Acta.
1639
A, an=
- +
Aoe
(
AAAoe + BA
A02
)
A,(Za,)'/2
(A-1)
to calculate the degree of dissociation of electrolyte solutions. Dividing both sides of eq A-1 by A, and combining with eq 8 yields
Substituting eq 6 into eq A-2 and multiplying both sides by Aoe results in A 0 11
I _~e -- 1 + -(Ahhoe + B,)1112
Ae
Aoe
(A-3)
which can be rearranged to give eq 9. Registry No. NaC1, 7647-14-5; NaBr, 7647-1 5-6; NaI, 7681-82-5; KC1,7447-40-7; KBr, 7758-02-3; KI, 7681-1 1-0; CsCI, 7647-17-8; CsBr, 7787-69-1; CsI, 7789-17-5; RbF, 13446-74-7; RbC1, 7791-1 1-9; RbBr, 7789-39-1; RbI, 7790-29-6; LiCI, 7447-41-8.
Laboratory-Frame Cross-Correlation Functlons in Molecular Liquids Induced by External Fields M. W. Evans7 Department of Chemistry, University College of Wales, Aberystwyth SY23 1 NE, Wales (Received: July 1 , 1987; In Final Form: September 17, 1987)
It is shown by computer simulation that new types of statistical cross correlation appear in the laboratory frame in molecular liquids subjected to external fields of force. New types of time cross-correlation functions are established (i) with an external homogeneous field; (ii) via linear flow induced by an externally applied inhomogeneous electric field; (iii) via vortex flow set up by a rotating external electric field. A simple analytical theory is developed to establish the probable effect of strong cross correlations between linear and angular momentums on the absorption and dispersion spectra of the molecular angular velocity at microwave and far-infrared frequencies.
Introduction The recent discovery has been made that new types of fundamental time cross-correlation functions (ccf s) become observable in molecular liquids when the overall isotropy of the sample is removed by an external field of force.' The ccf
was characterized for the first time by computer simulation. Here v(t) is the molecular center of mass linear velocity and o the angular velocity of the same molecule. Both vectors are defined in the laboratory frame of reference. An analytical theory has been developed' for C l ( t ) based on the Langevin equation for rototranslational molecular diffusion in the presence of a homogeneous external electric field. Both the computer simulation and the theory showed clearly that the liquid sample was also birefringent; e.g., the angular velocity and rotational velocity autocorrelation functions (acfs) in the presence of a z-axis field had a different time dependence in the z and x axes of laboratory frame. This leads to the conclusion that the techniques availablez) for the study of electric (or magnetic) field induced birefringence in molecular liquids or liquid crystals can also be used to observe fundamental new ccfs such as CI. A mathematical relation was provided between the observable elements of the matrix CI and 'Present address: Department 48B/428, IBM Data Systems Division, Neighborhood Rd., Kingston, NY 12401, and Visiting Academic, Department of Microelectronics and Electrical Engineering, Trinity College, Dublin 2, Republic of Ireland.
0022-3654/88/2092-1639$01.50/0
the angular velocity a c f s in the z and x axes of the laboratory frame. In this paper this work is extended in two directions. First, it is shown that far-infrared power absorption ~ p e c t r a ~from - ~ the computer simulation have a different frequency dependence in the z and x axes of the laboratory frame in the presence of a z axis homogeneous electric field. This property is related, through the fundamental equations of motion, to the appearance of C1 in the laboratory frame and also to new types of direct statistical correlation between the time derivative of the dipole moment c~ and its own linear velocity v. Elements of the new ccf matrix also exist therefore in the laboratory frame in the presence of birefringence. It follows that a careful measurement of the far-infrared power absorption spectrum parallel and perpendicular to the external electric field provides information on these fundamental ccfs. (1) Evans, M. W. Phys. Rev. A 1984, 3 0 , 2062; Physica 8 1985, 1 3 1 8 , 273. (2) Kielich, S . In Dielectric and Related Molecular Processes; ed.Davies, M., Ed.; Chemical Society: London, 1972; Vol. 1. (3) Beevers, M. S.; Elliott, D. A. Mol. Cryst. Liq. Cryst. 1979, 26, 41 1. (4) Evans, M. W.; Evans, G. J.; Coffey, W. T.; Grigolini, P. Molecular Dynamics; Wiley/Interscience: New York, 1982. (5) Coffey, W. T.; Evans, M. W.; Grigolini, P. Molecular Dvfusion; Wiley/Interscience: New York, 1984. (6) Memory Function Approaches to Stochastic Problems in Condensed Matter; Evans, M. W.; Grigolini, P.; Pastori-Parravicini, G., Eds.; Prigogine, I.; Rice, S . A., Series Eds.; Wiley/Interscience: New York, 1985.
0 1988 American Chemical Society
