1858
R. Triolo, J. R. Grigera, and L. Blum
Simple Electrolytes in the Mean Spherical Approximation R. Triolo, istituto de Chimica-Fisica, Universita di Patermo, 90 123 Patermo, ttaty
J. R. Grigera, Catedra de Biofisica, Facultad de Ciencias Exactas, Universidad Nacionai de La Piata, and Departamento de Siof~sica,iMBiCE, Casiila Correo 130, La Plata, Argentina
and L. Hum* Physics Department, College of Natural Sciences, University of Puerto Rico, Rio Piedras, Puerto Rico 0093 1 (Received March 2, 1976)
The analytical solution of the mean spherical approximation for a mixture of charged hard spheres of different size is used as an empirical equation to fit the experimental osmotic coefficients for 23 monovalent salts, mostly alkali halides. For every salt only one adjustable parameter is used, namely, the hard core diameter of one of the ions. The general agreement is good and the adjusted diameters are in general agreement with the accepted ideas about ion hydration. While the thermodynamic properties of simple electrolytes have been known with good accuracy for quite some time, it is only in recent years that the theoretical understanding of electrolytic solutions has reached the stage of quantitative agreement over wide ranges of concentrations. The classic work of Debye and Huckell was the first milestone in electrolyte theory, and provided the basis for all subsequent work. However it was good only for very dilute solutions, and its extensions to higher concentrations by researchers, such as Guggenheim, Scatchard,2and more recently P i t ~ e rmust , ~ be regarded as largely empirical, because the Debye-Huckel theory treats the hard exclusion core of the ions in a statistically inconsistent way, and a different approach, based on rigorous expansions, and pioneered by Mayer and Friedman4 has to be used. The first successful calculations for a wide range of concentrations and salts from a statistically consistent theory, the hypernetted chain equation (HNC),5 were performed by Friedman and co-workers.6 The results were in excellent general agreement with the measured thermodynamic properties, but the numerical solution of the HNC equations involved a fair amount of computation, since, as we know, it is a transcendental integral equation. Another theory that considers the core exclusion effects in a consistent way is the mean spherical approximation of Lebowitz and Percus7 (MSA). Here, the ion exclusion core is treated exactly, in the sense that the ions are not allowed to overlap, but the longrange part of the potential is used as an approximate closure of the integral equation for the structural correlation functions. For a vanishing hard core diameter this closure yields asymptotically the Debye-Huckel theory. The advantage of this approach is that it is analytically solvable,8 and gives rather good agreement with the HNC calculations, at least for the simpler salts. Furthermore it can be improved in a systematic mannerg-l1 with relatively little effort. In addition it is also solvable in models of electrolyte in which the solvent is treated as a collection of hard spheres with dipole movements,12 and also other cases of interest in colloid chemistry. In the present communication we apply the recently obtained solution of the MSA for a mixture of hard spheres of arbitrary charge and size13 as an empirical theory to fit the experimental, Lewis-Randall, osmotic coefficients of a colThe Journal of Physical Chemistry, Vol. BO, No. 17, 1976
lection of monovalent salts in aqueous solution. The motivation of this calculation was provided by a recent study,l4 which showed that the agreement between the osmotic coefficient of the MSA was very close to that of the HNC theory for NaC1. In the solution for the general case MSA, we need to find the value of a scaling parameter I’,which has the dimensions of an inverse length, and which, in fact, for the limit of zero concentration becomes identical with the Debye inverse length. For clarity, we give now a brief summary of the required equations. For a system that is a stoichiometric neutral mixture of charged hard spheres of diameter oi, and electric charge zi, the numerical density pi (in particles per A3) is (1)
Pi = Pui
where ui is the stoichiometric coefficient in the electrolyte dissociation reaction, and p = 6.0225 X
10-4c
(2)
with c the molar concentration of the salt. The scaling parameter r is obtained by solving the algebraic equation 21‘ = a
[
pi~i7-]1’2
(3)
a2 = 4re2/(c&*T)
(4)
where with e the electronic charge, to the solvent dielectric constant,
K B the Boltzmann constant, and T the absolute temperature. The parameter Mi in eq 3 is defined by
M~= pi- rpoi2~1/(2~~2)1/(1+ roil with
(5)
Simple Electrolytes in the Mean Spherical Approximation
1859
recover the simple quadratic equation first obtained and solved for the MSA by Waisman and Lebowitz.8 For most cases, and whenever the ionic sizes are not too different from each other and/or the concentration is of the order or lower than 1N, we can ignore the terms containingP1 in (5)and use the simpler equation vizi2/(i [
1
+ raip 112
2 r = (Y i
(9)
Unfortunately, both (3) and (9) are algebraic equations of an order higher than 4 and do not possess closed form solutions. However they can be solved numerically by the standard methods. In the results reported here, we solved eq 3 using the Newton-Raphson method, starting at very low concentrations, where the initial r was taken to be the Debye inverse length. However we found that the same results were obtained in almost all instances by solving (9) by iteration. The iterative solution of (9) is given by 2 ~ ( , += ~)
[
p
c vizi2/(1+ i
r(n)ai)2],I2
2ro= (Y[pcvizi2]1/2
(10) (11)
Starting the iteration with the Debye length given by eq 11 will, in our experience, make sure that we obtain, out of all the possible solutio& of the algebraic equations, those that go to the physical limit at low concentration. Our experience is that about four iterations are sufficient, and can be easily done with the aid of a pocket calculator. The excess internal energy per unit volume due to the electric charges of the system is
from which the corresponding excess free energy per unit volume can be obtained by integrating the thermodynamic relation
The result is
where {is a "turning on" parameter (essentially a2of eq 3), and r, P1,and Q are implicit functions of 3: The contribution to the osmotic coefficient can be calculated from the thermodynamic relation
with the result
Again, the case of low concentration andlor similar ionic size
is simple because then we can ignore P1 and integrate (16) to obtain the very suggestive relation
A4 = -r3/(3rp
(17)
vi)
The osmotic coefficient for the whole mixture is obtained by adding the hard core contributions. For the relatively low concentrations that we are dealing with, it will be enough to use the virial expansionlg
or, up to O(p2) 40
= 1 + PP2 + P2P3
(20)
h = (~/2)((3/3+ ( 1 ~ 2 / ~ 0 ) /33
(x2/36)[{3'
+ 6{2((1{3 +
C~')/{OI
(21)
(22)
although in the reported calculations we used the PercusYevick values, and later checked the results with the above equations. The final result for the osmotic coefficient is (23) 4 = 40 + A 4 with +o given by (18) and A 4 given by (16). The low concentration osmotic coefficient is given simply by
4 - 1 = p/32 + p 2 p 3 - r 3 / ( 3 s C vi)
(24)
As was mentioned above we have always used the more exact formulas, in the reported calculations, but found that in all but a few extreme cases such as KI a t the highest concentration (2 N) it is identical with the results of (24), which can be done on a simple hand calculator. Using a least-squares fitting we adjusted the hard core diameters of the ions of 23 monovalent salts, mostly alkali halides, to the tabulated, LewisRandall osmotic coefficients, To keep matters simple we did ignore the fact that, the way it stands, the MSA is really a Macmillan-Mayer type and the difference due to the correction, which is small, is taken care of by the adjustment of the hard core diameter. Only one of the ions was adjusted per salt. We found that for alkali halides it was more convenient to use the Pauling radii for the halogens and adjust the size of the cation, while for LiNO3, LiC104, and NaN03, we adjusted the size of the anion, using the average adjusted diameter in the lithium and sodium halide calculations. The results are shown in Figures 1 and 2. The adjusted calculated 4 was mostly within 1%of the experimental values, but for some salts with big anions (iodides) the errors were up to about 2%, for concentrations ranging from 0 to 2 N. The best values of the core diameters, together with the standard deviations, are shown in Table I, along with the Pauling diameters, shown for comparison, We notice some regularities. Small ions, such as,H+ and Li+ are strongly hydrated. For Li+, for example, the effective diameter is very closely the Pauling radii plus the diameter of water (2.76 A) although this fact has no simple geometrical interpretation. Sodium has also a larger diameter (2.6 A) than the crystallographic value (1.9 A). For these cations the hard core diameter is nearly constant in the halide series, or in other words the hard core diameters are nearly additive, which is a very pleasing feature. For the bigger cations K+, Rb+, and Cs+ the adjusted diameters are smaller than the crystallographic values given by Pauling, and also the constancy throughout the halide series is lost. The general agreement is The Journal of Physical Chemistry, Vol. 80, No. 17, 1976
1860
R. Triolo, J. R. Grigera, and L. Blum
TABLE I: Best Hard Core Diameters Found by Least-Squares Refinement against the Interpolated Data of Ref 15a
Electrolyte
Concn range, N
U+
Au+
U-
HC1 HBr HI LiCl LiBr LiI NaF NaCl NaBr NaI KF KC1 KBr KI RbCl RbBr RbI CSCl CsBr CSI LiN03 LiC104 NaN03
0-2 0-2 0-2 0-2 0-2 0-2 0-1 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-2 0-2
4.14 4.27 4.21 3.88 3.90 3.88 2.57 2.59 2.68 2.58 3.45 1.82 1.66 1.49 1.61 1.18 0.70 1.15 0.73 0.21 3.89 3.89 2.60
0.03 0.03 0.05 0.04 0.04 0.06 0.07 0.05 0.04 0.07 0.03 0.06 0.07 0.08 0.05 0.06 0.05 0.02 0.03 0.04 0.0
3.60 3.90 4.32 3.60 3.90 4.32 2.12 3.60 3.90 4.32 2.72 3.60 3.90 4.32 3.60 3.90 4.32 3.60 3.90 4.32 3.27 4.26 1.79
0.0 0.0
s,%
Au-
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
UP+
UP-
3.60 3.90 4.32 3.60 3.90 4.32 2.72 3.60 3.90 4.32 2.12 3.60 3.90 4.32 3.60 3.90 4.32 3.60 3.90 4.32
1.0 1.0
2.0 1.20 1.20 1.20 1.90 1.90 1.90 1.90 2.66 2.66 2.66 2.66 2.96 2.96 2.96 3.38 3.38 3.38 1.20 1.20 1.90
1.0
0.6 0.5 0.7 1.0 1.0
1.6 0.7 1.0 1.2
1.6 0.9 1.1
1.3 0.4 0.5
0.05 0.04
1.2 1.2 1.6
0.08
1.2
A u is the uncertainty in the hard core diamenter, while s is the standard deviation of the theoretical curve against the data. up is the Pauling diameter, shown for comparison.
I
00
04
i
1
08
12
I6
20
MOLARITY-
Flgure 2.
I
01)
04
OB MOLARITY
--.
12
16
-
-08
20
Figure 1. Osmotic coefficientsof monovalent salts: dots are experimental values interpolated from ref 15; the continuous line is the cal-
culated value.
also poorer for these salts. We believe that this is a consequence of hydration effects that result in sizable attraction between the ions. This effect is commonly known as negative hydration. The results show a general agreement with the current ideas of ion hydration,lC18 in the sense that small ions tend to hydrate more strongly than the larger ones, and that the larger ions show strong attractions a t short ranges, probably caused by a strong decrease in the dielectric constant in The Journal of Physical Chemlstry, Vol. BO, No. 17, 1976
Same caption as for Figure 1.
the vicinity of the ions, a consequence of “structure breaking”. Finally, we remark that the present results are not necessarily in disagreement with the HNC calculations on similar models made by Rasaiah,21 because we did not fit the region between 0.1 and 0.5 N very accurately and in fact, that is where the largest discrepancies occur in most cases. When we tried to fit this region to within experimental error (0.1%) we found that the agreement at high concentrations was very poor. The purpose of this work was to show that the MSA is a viable empirical theory for electrolytic solutions. We must stress the fact that the MSA is the first of a series of approximations, and that the agreement can be improved by including corrections and the use of more realistic Hamiltonian models.22However, it is interesting to see the surprisingly good
Potential Dependence of the Electrochemical Transfer Coefficient agreement that one obtains from such a simple theory with only one adjustable paramenter per salt, and in some cases, per ion. Acknowledgments. We are deeply indebted to Professor H. L. Friedman for numerous comments and his critical reading of the manuscript, and to Drs. C. V. Krishnan and A. H. Narten for their very useful suggestions. One of the authors also acknowledges the support of Oak Ridge Associated Universities (Contract No. S-1358)for part of this work.
References and Notes (1) P. Debye and E. Huckel, Phys. Z.,24, 185,334 (1923). (2) E. A. Guggenheim, Phil. Mag., (7) 19,588 (1935); G. Scatchard, Chem. Rev., 19,309 (1939); M. H. Lletzke and R. W. Stoughton, J. Phys. Chem., 66,508 (1962). (3) K. S. Pitzer, J. Phys. Chem., 77,2268 (1973); K. S. Pitzer and G. Mayorga, ibid., 77, 2300 (1973). (4) .J. E. Mayer, J. Chem. Phys., 18, 1426 (1950); H. L. Friedman, “Ionic Solution Theory Based on Cluster Expansion Methods”, Interscience, New York, N.Y., 1962. (5) A. R. Allnatt, Mol. Phys., 8, 533 (1964).
1861 (6) J. C. Rasaiahand H. L. Friedman, J. Chem. Phys., 46,2742 (1968); 50,3965 (1969); P. S. Ramanathan, C. V. Krishnan, and H. L. Friedman, J. Solution Chem., 1, 237 (1972). (7) J. L. Lebowitz and J. K. Percus, Phys. Rev,, 144, 251 (1966). (8) E. Waisman and J. L. Lebowitz, J. Chem. Phys., 56, 3086, 3093 (1972). (9) H. C. Andersen, D. Chandler, and J. D. Weeks, J. Chem. Phys., 57,2626 (1972). (10) J. S. Hoye, J. L. Lebowitz, and G. Stell, J. Chem. Phys., 61, 3253 (1974). (11) G. Stelland K. C. Wu, J. Chem. Phys., 63,491 (1975). (12) L. Blum, J. Chem. Phys., 61, 2129 (1974), and unpublished work. (13) L. Blum, Mol. Phys., 30, 1529 (1975). (14) J. R. Grigera and L. Blum, Chem. Phys. Lett., in press. (15) R. A. Robinson and R. H. Stokes, “Electrolytic Solutions”, Butterworths, London, 1959. (16) P. W. Gurney, “Ionic Processes in Solutlon”, Dover, New York, N.Y., 1953. (17) H. L. Friedman and C. V. Krishnan, “Thermodynamics of Ion Hydration” a chapter in “Water, a Comprehensive Treatise”, F. Franks, Ed., Plenum Press, New York, N.Y., 1973. (18) G. Stell, J. Chem. Phys., 59, 3926 (1973). {19) J. L. Lebowitz, Phys. Rev., 133, A895 (1964). (20) H.L. Friedman, J. Solution Chem., 1, 387, 413, 419 (1972). (21) J. C. Rasaiah, J. Solution Chem., 2, 301 (1973). (22) For an excellent discusslon see H. L. Friedman and W. D. T. Dale, “Electrolyte Solutions at Equilibrium” in “Modern Theoretical Chemistry”, Vol. IV, B. J. Berne, Ed., Plenum Press, New York, N.Y., 1976.
Potential Dependence of the Electrochemical Transfer Coefficient. Further Studies of the Reduction of Chromium(ll1) at Mercury Electrodes Michael J. Weaver and Fred C. Anson* A. A. Noyes Laboratory,’ California lnstitute of Technology, Pasadena, California 9 I125 (Received March 15, 1976) Publication costs assisted by the National Science Foundationand the U.S.Army Research Office (Triangle Park)
The electrochemical reduction rates of three complexes of Cr(II1) have been measured over an unusually large potential range in order to provide a stringent test of the theoretical prediction of Marcus that the electrochemical transfer coefficient, a , should exhibit a potential dependence. The three complexes studied, Cr(OH2)50S03+, Cr(OH2)5F2+,and Cr(OH2)63+,follow outer-sphere reaction mechanisms and bear differing charges which lead to diffuse double-layer corrections of varying magnitudes. This allowed the reliability of the diffuse-layer corrections to be established. Within experimental error, no potential dependence of a was observed with the sulfato complex under conditions where an easily detectable dependence is predicted by the Marcus theory. Some potential dependence of a was observed with the fluoro and aquo complexes but it was of the opposite sign and much smaller than that predicted by the Marcus theory and is attributed to uncertainties in diffuse-layer corrections. The present results are compared and contrasted with previous attempts to detect a potential dependence of the transfer coefficient.
Introduction Despite the considerable recent attention that has been focused on the subject,2-7 an unambiguous experimental test of the predicted8-10 potential dependence of the electrochemical transfer coefficient, a , has remained elusive. The difficulties met in previous studies include those from the following sources: (i) With multiply charged reactants, large corrections are necessary to account for the effect of the diffuse layer on measured reaction rates. As a result, large uncertainties are introduced in extracting the theoretically relevant, intrinsic transfer coefficientll from the apparent transfer coefficient obtained experimentally from slopes of rate-potential plots. (ii) Predicted changes in a are often so small that they are commensurate with experimental uncer-
tainties when the kinetic measurements are made with techniques which do not allow potentials much removed from the standard potential to be explored. (iii) The theoretical treatments were derived for simple, one-electron,outer-sphere reactions of which examples are not abundant, particularly if it is desired to restrict measurements to relatively negative potentials in order to minimize specific adsorption of the ionic components of the supporting electrolytes. It has often been suggested2bf+ that the redox couples most suited to experimental tests of the possible potential dependence of a are those with large standard rate constants because the predicted potential dependence of the transfer coefficient is more marked for larger values of the standard rate constant (i.e,, with lower reorganizational energy barriers for electron transfer). However, the predicted value of da/dE is only twice The Journal of Physical Chemistry, Vol. SO,No. 17, 1976
