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+
C6H6 (CzH&NH, is unusual in that the ratio of observed values H E / G E = 4.5, whereas for random mixtures it may reach a value of about 2.1 and for nonrandom mixtures it should decrease below this va1ue.l For a proper combination of the values of to, qlk, and vlk, the value of HEo/GEo may become even less than unity. The agreement obtained for such a system, CCl, CH&N, shows that specific interactions in pure CH&N consist of multipole interactions that are conformal with to(r) of an inert solvent. However, if the association in the pure liquid is due to strong hydrogen bridges, as in CH3NH2 or the alcohols, the above theory fails for mixtures of such liquids with inert solvents as well as with electron donors or acceptors. The systems given in Table I11 would in absence of DA interactions be at least approximately conformal. We believe that the present values of GEo and H E o are more realistic than any previously estimated values because the ordering effects (particularly on HEo)were in earlier works ignored. One may tentatively classify as weak the DA interactions that produce GDAranging from zero to about -500 J mol-l ( x = 0.5) near room temperature (0 > G D A / ( R T )2 -0.20). Among such systems, the most interesting are the C6F6 C6H6 system and the last seven systems of components that usually behave like electron donors. The function GEo for the C6F6 C6H6 system appears to be smaller than for C6F6 C-CgH12 and accordingly we obtain a less negative value of GDAthan did Gaw and Swinton by a direct comparison of the two systems. Their results are confirmed in that GEo and HEo are asymmetrical in a direction opposite to that of the observed G E and HE so that GDAand HDAare
+
+
+
+
nearly symmetrical functions of x i (Figure 1).The excess functions of the last system in Table 111, l-C6HI2 CsH6, were evaluated assuming qii = qijo = 0 because qii of l-hexene is certainly very small. The excess volumes VE are in most cases wrong. When component i is an inert solvent and qjj is large, the calculated values of VE are too large (part I). When there are DA interactions and qij increases, VE decreases below the observed value (see, e.g., Table 11). This excessive effect of 7 on VE, and also on VLoof the pure components, is almost certainly due to neglecting the temperature dependence of collision diameters u in the hard-sphere term $(() of the equation of state. For certain assumed values of the function u(T)we have found that I$(() moderates the effect of q on Vio.However, we did not yet try to apply these concepts to mixtures of fluids.
+
Acknowledgments. This study was partially supported by the American Petroleum Institute Research Project 44 and the Thermodynamics Research Center Data Project of the Thermodynamics Research Center. References and Notes (1) A. Kreglewski and R. C. Wilhoit, J. Phys. Chem., 78, 1961 (1974). (2) H. H. Vera and J. M. Prausnitz, J. Chem. Eng. Data, 16, 149 (1971). (3) D. L. Andersen, R. A. Smith, D. B. Myers, S. K. Alley, A. G. Williamson, and R. L. Scott, J. Phys. Chem., 86, 621 (1962). (4) W. J. Gaw and F. L. Swinton, Trans. Faraday SOC.,64, 2023 (1968). (5) H. G. Harris and J. M. Prausnitz, lnd. Eng. Chem., Fundam., 8, 180 (1969). (6) R. Philippe and P. Clechet, "Third International Conference on Chemical Thermodynamics," Vol. Ill, Baden, Austria, 1973, p 118. (7) A. Kreglewski, Bull. Acad. Polon. Sci. Ser. Sci. Chim., 13, 723 (1965). (8) A. Kreglewski, J. Phys. Chem., 78, 1241 (1974).
The Debye-Bjerrum Treatment of Dilute Ionic Solutions J.-C. Justice Laboratoire d'Electrochimie, Universite Paris Vl, 75230 Paris Cedex 05, France (Received November 20, 1973; Revised Manuscript Received February 75, 1974)
It is shown that the generalization of conductance equations based on the Debye-Huckel correlation functions to the Bjerrum concept, as originally proposed by Bjerrum for the Debye-Huckel activity coefficient law, does not constitute an alteration of the original hard-sphere model as assumed by Fuoss. The consequences of what we call the Debye-Bjerrum treatment are discussed in detail and the analysis of experimental data using this approach is shown to be more satisfactory than with other methods used previously.
I. Introduction In the preceding contribution1 to this journal a critical analysis of a method proposed by the author2 for processing experimental conductance data has been presented. The criticisms of Professor Fuoss concentrate on two points. First, the replacement of the parameter a by R, the Bjerrum critical distance, is theoretically interpreted as an improper change in the initial model. I t is claimed that the new treatment would describe the properties of a model where ions are represented by hard spheres of radius R/2. The Journal of Physical Chemistry, Vol. 79, No. 5, 7975
Secondly, it is concluded that the new equation is inadequate for analysis of experimental data because it does not lead in general to a unique evaluation of the adjusted parameters. An answer to these two criticisms is presented. It is shown that the substitution of a by a critical distance R is a direct consequence of the introduction of an association concept to correct for the failure of the Debye-Huckel treatment of electrolyte solutions in solvents of low dielectric constant or for electrolytes of high valency. Also the criticisms concerning the supposed adulteration of the orig-
Debye-Bjerrum Treatment of Dilute Ionic Solutions inal model are analyzed in detail and rejected. Then the advantages of analyzing experimental data by this new equation are discussed together with interesting possibilities of deriving information concerning short-range specific ionic interactions.
11. Theoretical Discussion The study of the necessity for and the consequences of the combination of the Bjerrum association concept3 and the original treatment of Debye and Huckel necessitates a brief review of the fundamental equations used by Debye and Hucke14 together with a careful examination of the meaning of the distance parameter a. Otherwise it is easy to criticize the Bjerrum model by a grotesque argument, the origin of which is the apparent ease with which the Bjerrum-Debye functions are obtainable from the original Debye-Huckel (DH) functions. This has often resulted in the concealment of the important basic ideas which the Bjerrum association concept contains or has been used to discredit these ideas. 1. Excess Thermodynamic Functions. The original idea of the DH treatment was to divide the space around each ion into two regions. In the first region, defined as r < a, the correlation functions are fixed at zero
for Y < a (1) It was assumed that there could be no other ion in the sphere of radius a because of the hard-core nature of the ions but (1) does not explicitly take short-range repulsive fotces into consideration and consequently must be considered more generally as a mathematical postulate which may, as we shall see below, be given a different physical meaning. In the outer region ( r 2 a ) , the DH derivation of correlation functions is carried out through a series of approximations: (a) use of the Poisson-Boltzmann (PB) equation; (b) expansion and linearization of the P B equation (LPB); and (c) truncation at the third term (for symmetrical electrolytes) of the Taylor series expansion of the barometric equation. One of the two undeterminicies of the solution of the LPB equation is lifted by introducing condition 1 in the electroneutrality equation. The three approximations underestimate the effects of the short-range multi-particle configurations. The third one particularly underestimates the effects of the shortrange binary configurations so that the DH results are in error after the first term of the limiting laws. Bjerrum, recognizing the inadequacy of the DH correlation functions, proposed the addition of a chemical model which would better represent the effects of the short-range anion-cation pairs which are the most probable of the binary configurations. Bjerrum’s important propositions can be stated as follows. (I) In dilute solutions a twofold partition is made in the ensemble of all anion-cation pairs for all configurations: class I will concern all pairs such that gij = 0
r+-< €2
class I1 which is represented by a fraction y of the total ionic population is obviously defined by y+-
2 R
(11) Every pair of class I is considered as no longer having any electrical interaction with any other ion of class I or 11. (We shall also assume later that they have no interaction
455
with an external field either.) (111) The interactions of all the ions which belong to class I1 can be adequately treated in terms of an average distribution such as that of Debye and Htickel. (IV) The probabilities that pairs of ions of the same sign or that aggregates containing more than two ions can exist within a sphere of radius rii < R are considered negligible. To derive the new function for the potential it is necessary, following Debye and Huckel, to proceed from the LPB equation and the electroneutrality condition where c must be replaced by c y (and x by ~ y l ’ ~with ) a new postulate that will replace eq 1. This is simple because, according to the model, for an ion of class I1 there cannot be any charge closer than R. (cf. propositions 1 and 2). This leads to gijlI = 0
for Y
R the original DH treatment is more advantageous. Onsager’s answer to this question15 . . . “the distinction beHowever, conductance theory is not at present complete tween free ions and associated pairs depends on an arbienough to allow any value to be chosen for R and certainly trary convention. Bjerrum’s choice is good, but we could vary it with reason. In a complete theory this would not not R = a as suggested by Fuoss. matter; what we remove from one page of the ledger would (5) The confusion mentioned above concerning the physbe entered elsewhere with the same effect.” The most suitical meaning of (vu - vj; ) resulted in the FUOSS’ suggestion able critical distance R , corresponds to the zero point of that only ions in contact are associated. According to this model, it follows that a in the J and J 3 / 2 coefficients must the dotted curve in Figure 1. Bjerrum estimated the absciskeep its contact meaning even the association concept (of sa llf(R c ) of this point as equal to 2. ( R,= --e le2 /2DkT.) Fuoss) is added. For the “contact” pair association conThis critical distance could be varied at will within the stant, Fuoss uses the relation interval f6R where the total error A keeps within the experimental error band -go < A < bo. Clearly the magnitude of 6R depends on the quality of the two theories which are combined and will be the larger the better these theories. Figure 1 shows that if the contact distance of the R P model which in fact not only does not represent “contact pairs”
The Journal of Physical Chemistry, Voi. 79, No. 5, 1975
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but is relative to a completely unrealistic physical situation deals with the mean velocity vector v,j and not with tion. the mean velocity modulus so that an ion can still move Fuoss clearly states that the model chosen is of conducteven in a direction where a component of this vector is zero. The expression (vLl r)r=R= 0 does not imply that the ing spheres for the cations and charged point masses for sphere r = R 12 around each ion is a hard core or that the the anions (or the reverse), "which can penetrate the catregion outside that sphere is solid! ionic spheres ." It is strange to see that the hard-core nature of the distance parameter a is not respected in his derConsequently all these boundary conditions may be used ivation of the association constant of pairs of ions supposed with the Bjerrum model as well. to be in contact. It has also been shown that the best way to process exI t is not surprising to note that experimentally deterperimental data is to use the expanded forms of the conmined values of K A do not confirm eq 23. A plot of log K A ~ ductance equation and to adjust Am, K A , and 5 3 1 2 fixing us. 1/D is a straight line whose slope and ordinate interthe distance parameter in the J coefficient and in the accept would constitute two independent determinations of tivity coefficient law. In the present state of the theory, and a. The literature abounds in results which show that a congiven the observed relative freedom in the choice of R, the cave down curvature in this plot is always 0b~erved.l~ This choice of the Bjerrum critical distance seems quite reasoncurvature has very often been attributed to "solvation" efable. The information derived from K A should not be affects, but a significant part of this curvature is predicted by fected provided the same distance is used in the upper the Bjerrum relation (8) so that solvation effects deterlimit of the Bjerrum integral. mined this way would systematically be overestimated. Therefore FUOSS'criticisms cannot be accepted. The present method of improving the Debye-Huckel treatment (6) Until a more complete conductance theory is availof electrolyte solutions by inserting the Bjerrum correction able it is better arbitrarily to fix Rf = R J = q and adjust for short-range electrostatic interactions is the most simple A m , K A , and 5312. and realistic way to handle this difficult problem. The Since R can be varied somewhat without significantly changing the value of u'i, this parameter must not be adsame combination should also be made to the theories of justed at least in the most fundamental coefficient J of the transport numbers21 and diffusion coefficients,22,23which conductance theory. Otherwise the least random experiare also based on the Debye-Huckel correlation functions. mental error will result in a random determination of R within the error band and K A would consequently also be Acknowledgment. The author is grateful to Dr. A. D. numerically changed accordingly. The information which is Pethybridge for his help in suggesting corrections to the of interest to chemists is found in K A . Once the theoretical language used in the manuscript. J ( R ) function and the value of RJ are clearly indicated a References and Notes fruitful comparative study can be made for electrolytesolvent systems. The ultimate aim is then to analyze the R. M. Fuoss, J. Phys. Chem., 76, 1383 (1974). J. C. Justice, J. Chim. Phys., 65,353 (1968). clearly defined K A values in terms of specific short-range N. Bjerrum, Kg/. Dan. Vidensk. Selsk., 7,no. 9 (1926). ionic interactions. Indeed the most interesting aspect of the P. Debye and E. Huckel, Phys. L,24, 185 (1923)(cf. also "The ColBjerrum association concept is that it may easily be generlected Papers of Peter J. W. Debye," Interscience, New York, N.Y., 1900,p 217). alized to any short-range pair potential curve by introducE. Pitts, Proc. Roy. Soc., 217,43 (1953). ing in eq 7 more sophisticated U+-* ( r ) functions which R. M. Fuoss and L. Onsager, J. Phys. Chem., 61,668 (1957). R. M. Fuoss and K. L, Hsia, Proc. Nat. Acad. Sci. U. S., 57, 1550 are characteristic of the variation of free energy involved by (1967). the overlap of Gurney cospheres.18-20 R. M. Fuoss and F. Accascina, "Electrolytic Conductance," Interscience, New York, N.Y.. 1959,D 121,eq 1 1 . ( 7 ) Last but not least, the set of eq 21, 22, and 4 shows Cf. ref 8,p 127,eq 35. that the present conductance theory is consistent with that J. E. Lind, Jr., and R. M. Fuoss, J. Phys. Chem., 65,999 (1961). for activity coefficients (Rf = RJ and the same K A ) so that D. F. Evans, C. Zawoyski, and R. L. Kay, J. Phys. Chem., 69, 3878 11965) ---, it should be possible to calculate stoichiometric activity J. C. Justice, R. Bury, and C. Treiner, J. Chim. Phys., 65, 1708 (1968). coefficients from the values of K A derived from conducJ. C. Justice, J. Chim. Phys., 66,1193 (1969). J. C. Justice, Nectrochim. Acta, 16,701 (1971). tance measurements. The difficulty of obtaining excess L. Onsager, Proceedings of the 19th Meeting of the Societe de Chimie free-energy data directly from thermodynamic experiments Physique at Montpellier, Sept 1968,p 86. in solvents of lower dielectric constants makes a close reCf. ref 12, p 210;also R. M. Fuoss, J. Amer. Chem. Soc., 60, 5059 (1958). consideration of the Bjerrum model worthwhile.
-
I
\
T. L. Fabry and R. M. Fuoss, J. Phys. Chem., 66,971 (1964);68,974
(1964).
IV. Conclusion It has been shown that none of the boundary conditions used in the DH model treatment for the determination of the potential of average field, activity coefficients, and conductance are restricted to the hard-core nature of the sphere of radius a. In particular the hydrodynamic condi-
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R. W. Gurney, "Ionic Processes in Solution," McGraw-Hill. New York, N.Y., 1953. J. C. Rasaiah and H. L. Friedman, J. Chem. Phys., 48,2742 (1968). J. C. Rasaiah, J. Solution Chem., 2, 301 (1973). R. L. Kay and J. L. Dye, Proc. Nat. Acad. Sci. U.S.,49,5 (1963);G. A. Vidulich, G. P. Cunningham, and R. L. Kay, J. Solution Chem., 2, 23
(1973). P. Turq, Chem. Phys. Lett., 15,579 (1972). H. Latrous, P. Turq, and M. Chemla, J. Chem. Phys., 11-12, 1650
(1972).
