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R. Triolo, L. Blum, and M. A. Floriano
The Journal of Physical Chemistry, Vol. 82, No. 72, 7978
which may have serious consequences for the tritium inventory in the CTR.
References and Notes (1) This work was supported by the U S . Energy Research and Development Administration. (2) For general reviews see, e.g., (a) F. L. Vook, H. K. Birnbaum, T. H. Biewitt, W. L. Brown, J. W. Corbett, J. H. Crawford, A. N. Goland, G. L. Kulcinski, M. T. Robinson, D. N. Seidman, and F. W. Young, Jr., Rev. Mod. Phys., Suppl. 3 (1975); (b) F. L. Vook, Phys. M a y , 28, 34 (1975); (c) G. L. Kulcinski and G. A. Emmert, J . Nucl. Mater., 53, 31 (1974); (d) S. 0. Dean et ai., Status and Objectives of Tokamak Systems for Fusion Research, WASH-1295 (1974). (3) G. L. Kulcinski, R. W. Conn, 0. Lang, L. Wittenberg, J. Kesner and D. C. Kummer, UWFDM-108, University of Wisconsin (Aug 1974). (4) G. L. Kulcinski, R. W. Conn, and G. Lang, J . Nucl. Fusion, 15, 327 (1975). (5) G. P. Lang and V. L. Holmes, J . Nucl. Fusion, 16, 162 (1976). (6) G. A. Beitel, J . Vac. Sci. Techno/., 6, 224 (1969). (7) K. Fiaskamp, G. Stockiin, E. Vietzke, and K. Vogeibruch, "Sticking Coefficient of Atomic Hydrogen on Graphite", paper presented at the V I Intematlomal Symposium on Molecular Beams, Nordwijkerhout, April 18-22, 1977. (8) K. J. Dietz, E. Geissler, F. Waelbroeck, J. Kirschner, E. A. Niekisch, K. 0. Tscherisch, 0. Stocklin, E. Vietzke, and K. Vogeibruch, J . Nucl. Mater., 63, 167 (1976).
(9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24)
(25)
M. Balooch and D. R. Oiander, J . Chem. Phys., 63, 4772 (1975). B. J. Wood and H. Wise, J . Phys. Chem., 73, 1348 (1969). R. K. GouM, J . Chem. Phys., 63, 1825 (1975). P. S. Gill, R. E. Tomey, and H. C. Moser, Carbon, 5, 43 (1967). P. L. Walker, F. Rusinko, andL. 0. Austin, A&. &tal., 11, 133(1959). S. K. Erents, C. M. Braganza, and G. M. McCracken, J. Nucl. Mater., 63, 399 (1976). R. Behrisch, J. Bohdansky, G. H. Oetjen, J. Roth, 0. Schilling, and H. Verbeek, J. Nucl. Mater., 60, 321 (1976). N. P. Busharov, E. A. Gorbatov, V. M. Gusev, M. I. Guseva, and Yu. V. Martynenko, J. Nucl. Mater., 63, 230 (1976). J. Roth, J. Bohdansky, W. Poschenrieder, and M. K. Sinha, J. Nucl. Mater., 63,222 (1976). M. Kaminsky in "Radiation Test Facilities for the CTR Surface and Materials Program", C. J. Persiani, Chairman, 1975, p 16 ff. R. W. Conn and Y. Kesner, J . Nucl. Mater., 63, 1 (1976). 8. Diehn, A. P. Wolf, and F. S. Rowbnd, Z. Anal. Chem., 204, 112 (1966). See, e.g., In 0. Friedhnder, J. W. Kennedy, and J. M. Miller, "Nuclear and Radiochemistry", 2nd ed, Wiiey, New York, N.Y., 1966, p 95 ff. See, e.g., R. Wolfgang, Prog. React. Kinet., 3, 97 (1965). J. B. Lewis in "Modern Aspects of GraphRe Technology", L. C. F. Blackman, Ed., Academic Press, New Ycfk, N.Y., 1970, pp 129-199. L. L. Mantell in "Carbon and Gaphke Handbook", Interscience, New York, N.Y., 1968, pp 323-377. J. Chenion and F.-M. Lang, Bull. SOC.Chim. Fr., Part 7 , 62 (1975).
Simple Electrolytes in the Mean Spherical Approximation. 2. Study of a Refined Model R. Triolo,'t L. Blum,t and M. A. Florlanot5 Isfitufo dl Chirnica-Fisica, Universifa' di Palerrno, 90 123 Palerrno, Italy end Physics Department, College of Natural Sciences, Universlfy of Puerto Rico, Rlo Piedras, 0093 7, Puerto Rico (Received April 22, 7977; Revised Manuscript Recelved March 3, 1978)
The agreement between mean spherical approximation (MSA) osmotic coefficients of the primitive model of simple electrolytes and corresponding experimental osmotic coefficients is surprisingly good in the range 0.2 5 M 5 2.0. It is shown here that when a density dependent hard core diameter is introduced into the mean spherical approximation, the differences between the MSA osmotic coefficients and the experimental osmotic coefficients are within the uncertainties in the experimental values. The density dependence of the hard core diameters is attributed to the softness of the pair potential. Osmotic and activity coefficients of aqueous solutions of 19 ionophores uni-univalent are compared at 25 "C.
In recent works, the mean spherical approximation' was shown to possess a general analytical solution,' that, as was later shown by H i r ~ i k eand , ~ Blum and H d ~ eis, just ~ the consistent, statistically correct version of the DebyeHuckel theorya5 In the consistent, statistically correct version of DH theory, the excluded volumes of the ions in the ionic atmosphere are taken into account in the calculation of the screening length. This very simple theory gave6 surprisingly accurate osmotic coefficients for 23 monovalent salts in H 2 0 a t 25 "C. We must stress that, during the fitting procedure, only one parameter was varied, namely, the diameter of one of the ions. The agreement with experimental data was generally good, but not as good as the experimental accuracy (0.1% or better). The motivation behind that work was the goal of providing a convenient (Le., easy and accurate) theory for the discussion of experimental data. With this idea in mind, in the present note we discuss some plausible modifications of the primitive model which incorporate two + Universita' di Palermo. *University of Puerto Rico. Part of these data submitted by M. A. Floriano in partial fulfillment of the requirements for the degree of Dr. in Chemistry, University of Palermo.
0022-3654/78/2082-1368$01 .OO/O
effects which we feel play a major role in the deviations of the real electrolytes from the primitive model. These effects are the effect of soft repulsions and attractions, and the effect of solvation. In a more elaborate study made by Ramanathan and Friedman: these effects were lumped into one short-ranged contribution to the ion-solvent interaction called the "Gurney term". The parameters of this extra contribution were chosen to fit the experimental data over a range of concentrations. In the present work we want to use the same physical idea, but rather than adjusting the parameters of the potential directly, we introduce a density dependent hard core. This procedure was suggested by the "Blip Function Approximation" of Andersen, Chandler, and Weeksee The blip function approximation (BFA) is usually applied to repulsive pair potential, and it has been shown that, for a given repulsive potential U ( r )the hard core diameters, u, are a function of density and temperature. Our procedure does not neglect the intricacies of the complicated ion-solvent interactions represented by the Gurney POtential, but rather is a simple parametrization or effective range representation of those interactions. The hard core diameters used here are functions only of density. They are defined by o ( p ) = a o ( l + z:a,'pn) (1) G 1978 American Chemical Society
The Journal of Physical Chemisfry, Vol. 82, No. 12, 1978 1369
Simple Electrolytes in the Mean Spherical Approximation
or
TABLE I: Best Hard Core Diameters Found by Least-Squares Refinement against Experimental McMillan-Mayer (MM) Osmotic Coefficients' Salt
where a,, a,', and b, are coefficients to be found from the data. Of course form (2) is a more flexible expression than form (1) as it contains additional features and does converge for all p, but we found that the coefficients b, are very small for all the salts.gb The hard sphere diameters used here are not necessarily the hard sphere diameters of the BFA. The BFA concept is used here only to show that density dependent hard sphere diameters are a reasonable way to account for soft core interactions. A second way to include some of the effects left out by the primitive model is to use a density dependent dielectric constant t. This is justified from the physical point of view, since the dielectric screening must be less in more concentrated solutions, where the ions are closer, on the average, and therefore less shielded by the solvent. A less intuitive but more convincing argument can be made by considering the MSA for the mixture of charged hard sphere and hard dipolesg in which the same Waisman-Lebowitz' results appear, albeit with a concentration-dependent dielectric constant. The consideration of nonprimitive (non-McMillan-Mayer) models, such as the hard ion-hard dipole m i ~ t u r estrongly ,~ suggests a density dependent dielectric constant. To simplify things, one could approximate the functional dependence of the dielectric constant to a simple first-order polynomial e=
Eo(1
+
POP)
(4) where
LiCl LiBr LiI NaF NaCl NaBr NaI KF KC1 KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CSI
4.44 t 4.36 i 4.89 t 3.29 i 3.33 i 3.42 i 3.62 i 3.91 i 2.84 i 2.82 i 2.96 i 4.73 * 2.57 i 2.34 * 1.99 i 4.82 i 2.02 i 1.73 t 1.51 i
0.02 0.02 0.07 0.04 0.05
0.05 0.04 0.04 0.04 0.05 0.04 0.02 0.02 0.02 0.02 0.05 0.02 0.03 0.01
Concn SD% range, M
cy
-77 -38 -126 -477 -132 -101 -101 -71 -176 -184 -216 -148 -121 -180 -194 -78 -32 -71 -276
t
4
t 6 f
i
t i i i
*
i t i
* +_
t
t i i f
15 21 15 14 10 10 14 15 11 38 98 10 12 10 8 20 4
0.18 0.21 0.73 0.11 0.30 0.30 0.25 0.23 0.21 0.24 0.20 0.14 0.12
0.11 0.12 0.40 0.07 0.15 0.04
0.2-2.0 0.2-2.0 0.2-2.0 0.1-1.0 0.2-2.0 0.2-2 .o 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0 0.2-2.0
' u o + and cy are defined in e q 1 0 of the paper, SD% is the standard deviation percent. the relatively low concentrations we are dealing with, it will suffice to take the Percus-Yevick results'O for the reference osmotic coefficient (via PY compressibility):
(7) -I
L
(3)
T o compute the pertinent thermodynamic excess functions, we use the simplified f ~ r m u l a ~ , ~
OO+
where
t, = 2ipiuin A, =
1 - :13
from the same equation, we derive r is the shielding parameter already d i s c ~ s s e d , ~ ~Also, ~~~
to= x i p i where pi is the number density of ion i, ao2= 4ae2/thT, where e is the electron charge and t is the dielectric constant of the solvent. The parameters P, and & are complex functions of I', pi, and ui,but generally very small. For the average excess activity coefficients we obtain
(5) where the excess energy is given by
where, again, R is a complex function4 of I', ui,and pi. In eq 4-6, the terms containing P, are very small and can be usually neglected. Therefore we have the DebyeHuckel-like formulas A @= - I T 3 / 3 n t o
(4b)
&OZ
A In y+ = - -
4nto
Evidently the reference fluid is the hard sphere fluid. For
In y o = @ "
-
1- In A ,
71 + ___
2AOtO
(9)
As we will see in the numerical section of this paper, the activity coefficients have been obtained through analytical integration of Gibbs-Duhem equation, rather than by means of eq 5 and 9. The motivation in doing so was that we obtained both experimental and MSA activity coefficients with the same technique, and could avoid any differences due to different treatment of data. Our interpretation is that for every concentration u and t are constants fixed, as for example, by the blip function optimization procedure or other similar ones. That is, at each concentration the short range interactions are represented by a constant u. Truly density and dielectric constant dependent potentials would produce very complex expressions that defeat the main purpose of this approach: simplicity. Since these variations are small, we will ignore the effects on the thermodynamic formulas. Numerical Results The model we used in this study was a very simple one, in which the cation hard core u' was adjusted according to the formula u+ =
U,+(l
+ Lyp)
(10)
while the anion hard core diameter was held constant and
1370
The Journal o f Physical Chemistry, Vol. 82, 1.4 v 7 -
1.3
No. 12, 1978
I
I -
f.2 - ! . I --
---
0.4
1.2
0.8
MOLALITY
-
1.6
Figure 1. Computed (-) and experimental (circles) osmotic coefficients for three typical cases. The data for LiI (a),KBr (e),and CsCl (0) are given in the MM system.
R. Triolo, L. Blum, and M. A. Floriano
difference between the calculated and the experimental osmotic coefficients” (corrected for Lewis-Randall to McMillan-Mayer conversion12)in the concentration range 0.2-2.0 M. For most salts the agreement is within the experimental accuracy, and several typical cases are plotted in Figure 1. Figure 2 shows the comparison between experimental MM activity coefficients and MSA activity coefficients, for the same electrolytes shown in Figure 1. In both cases, activity coefficients have been obtained by means of analytical integration of the Gibbs-Duhem equation, using a slightly modified version of the method suggested by Lietzke.13 A quick analysis of the data of Table I shows several interesting results: (i) the hard core diameter decreases with increasing the concentration; (ii) the model with linearly dependent hard core diameters works better the bigger the cation; (iii) the ionic hard sphere diameter is smaller the more polarizable the ion. All these features are in accordance with either the results of the blip functions approximation or physical intuition. Clearly the higher the density and/or the ionic polarization, the closer the ions come together and thus the ionic hard sphere diameter must be smaller. More discussion will be presented in a forthcoming paper in which results obtained by varying the dielectric constant will be reported.
Acknowledgment. The authors acknowledge Mrs. I. Ruffo for the help in the initial stage of the work, Dr. E. Johnson for critically reading the manuscript and for useful suggestions, and Professors C. V. Krishnan and H. L. Friedman for useful advice and discussions during the preparation of this manuscript. One of us (R.T.) gratefully acknowledges the Consiglio Nazionale Delle Ricerche for a NATO Fellowship and CRRNSM for partial financial support through Grant 76 PASMFAD. L. B. acknowledges partial support through Grant NSF CHE-77-04597. References and Notes
0.50
I
0.4
0.0 MOLALITY
-
1.2
1.6
Figure 2. Computed (-) and experimental (circles) MM activity coefficients for the same electrolytes as in Figure 1. The same identification code is used for the circles.
equal to the Pauling crystallographic value. The results for 19 alkali halides in water at 25 “C are shown in Table I. The zero density hard core diameters uof and (Y were adjusted to minimize the square of the
(a) J. K. Percus and G. Yevich, Phys. Rev. 6, 136, 290 (1964); (b) J. L. Lebowitz and J. K. Percus, Phys. Rev., 144, 251 (1966); (c) E. Waisman and J. L. Lebowitz, J . Chem. Phys., 66, 3086 (1972); (d) E. Waisman and J. L. Lebowitz, bid., 66, 3093 (1972). L. Blum, Mol. Phys., 30, 1529 (1975). K. Hiroike, Mol. Phys., in press. L. Blum and J. S.H&e, J. Phys. Chem., 61, 1311 (1978). P. Debye and E. Huckel, Z. Phys., 24, 183, 305 (1923). R. Trido, J. R. Wigera, and L. Blum, J. phys. Chem., 80, 1858 (1976). P. S. Ramanathan and H. L. Friedman, J . Chem. Phys., 54, 1086 (1971); E. Bich, W. Ebeling, and H. Krienke, Z.Phys. Chem. (Lebzb), 257, 549 (1976). H. C. Andersen, D. Chandler, and J. D. Weeks, Adv. Chem. Phys., 34, 105 (1976). (a) L. Blum, J. Chem. Phys., 61, 2129 (1974), and unpublished material; (b) R. Triolo, M. A. Floriano, I. Ruffo, and L. Blum, Ann. Chim., in press. R. J. Baxter, J . Chem. Phys., 52, 4559 (1970). W. J. Hamer and Y. C. Wu, J . Phys. Chem. Ref. Data, I , 1047 (1972). H. L. Friedman, J . Solution Chem., 1, 387 (1972). M. H. Lietzke and R. W. Stoughton, J . Phys. Chem., 65, 508 (1961).
