4783
The Thermodynamics of Ion Association in Solution. I. An Extension of the Denison-Ramsey Equations L. D. Pettit and Stanley Bruckenstein Contribution f r o m the Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455. Received M a y 4,1966
Abstract: The Dension-Ramsey treatment of ion-pair formation has been extended, taking into account in the po-
tential energy term the various possible interactions between ions and their induced dipoles. In addition, orientation entropy effects have been included. Internuclear distances calculated in this way from published ion-pair constants are smaller than those calculated by the Denison-Ramsey or the Bjerrum equation and were only slightly larger (-15 %) than the crystallographic radii. Further extension to include higher ionic aggregates was made and the result applied to ion triplet and quadrupole formation. In these ion aggregates the calculated internuclear distances between ions are nearly the same as in ion pairs, in contradistinction to results obtained using earlier treatments. In solvents of dielectric constant less than 7 (benzene, dioxane, anisole, chlorobenzene, and tetrahydrofuran) quaternary ammonium salts do not appear to form solvent-separated ionic aggregates. of low dielectric constant ( D ) such as benzene ( D = he association of ions in solvents of low dielectric 2.274) solvation energies are very low and contact ion constant t o form ion pairs, triplet, or higher aggregates is an accepted and well-documented fact. Such pairs are to be expected. A consideration of the Bjerrum equation immediately indicates that it should not ion aggregates are assumed in the quantitative treatbe applied to such solutions at experimentally realizable ment of many properties and reactions of electrolytes in concentrations. The Bjerrum association constant is these solvents, e.g., colligative properties, conductance expressed in terms of (1 - a), the degree of association, minima, and reaction and polymerization mechanisms. and is derived assuming a to approximate to unity, Ion-pair formation in particular has been considered at length in a review by Kraus.lB2 An essential requirei.e., in very dilute solution. This situation is unobtainment of all such treatments is a knowledge of the extent able experimentally. Even in solutions as dilute as 1W6 M in benzene, extensive ion-pair formation, of association, i.e., the associated species formed and the together with higher association, is found. l o The D.R. association constants. Such constants can often be treatment does not suffer from the same failing but, determined by direct study of the particular systems as Fuoss has pointed out,6bJ1it does not include any involved using one or more of the recognized techniques for measuring association constants. Once obtained, orientation entropy terms. Fuoss and Accascina have these constants are of only limited value in estimating made an approximation to allow for this in their the degree of association in other systems. Attempts revised treatment of ion triplets. These are certainly have been made to relate association constants to important. It therefore seems probable that both properties of the solvent and solute, but all are limited in treatments are limited in accuracy, and the apparent their application and can only be used under restricted agreement between them is due to a coincidental balance conditions if the results are to have any meaning. Even of errors. Theoretically the Bjerrum treatment is less it is probable that the results are misleading since they satisfactory, involving as it does an impractical definithen indicate a physical picture which has much greater tion of an ion pair. Fuoss has considered the Bjerrum interionic separations than expected. function critically and pointed out its shortcomings.6b The major equations have been derived by B j e r r ~ m , ~ Further, both treatments consider only ion-pair or extended by Fuoss and Kraus4,5and Denison and Ramtriplet formation although the Bjerrum treatment has sey,ea and improved by FuossGband Gilkerson.’ They been extended to cover quadrupole^.^ We have now have been discussed critically by Flaherty and Sterns extended the D.R. treatment to allow for entropy and by F ~ o s s . Internuclear ~~ separations for ion changes on association and the formation of higher pairs calculated by the Bjerrum equation are shown to aggregates up to sexapoles. be surprisingly close to those obtained from the Denison-Ramsey (D.R.) treatment for a range of solvents, Discussion but, in every case, the separation is more than one Denison and Ramsey considered the free-energy would expect for two ions in contact, strongly suggesting changes in the cycle that the ions are somehow solvent separated. Such species have been detected elsewhere,g but in solvents IV ion + ion at infinity in medium -+ion pair at r2 in medium
T
(1) C. A. Kraus, J. Phys. Chem., 60, 129 (1956). (2) See also R. M. Fuoss and F. Accascina, “Electrolytic Conductance,” Interscience Publishers, Inc., New York, N. Y.,1959. (3) N. Bjerrum, Kgi. Danske Videnskab. Selskabs, 7 , No. 9 (1926). (4) R. M. Fuoss and C. A, Kraus, J. Am. Chem. SOC.,55, 2387 (1933). (5) R. M. Fuoss and C. A. Kraus, ibid., 57, 1 (1935). (6) (a) J. T. Denison and J. B. Ramsay, ibid., 7 7 , 2615 (1955); (b) R. M. Fuoss, ibid., 80, 5059 (1958). (7) W. R. Gilkerson, J. Chem. Phys., 25, 1199 (1956). (8) P. H. Flaherty and K. H. Stern, J. Am. Chem. SOC.,80, 1034 (1958).
ion
+ ion in vacuo
I1
+
ion pair in vacuo
(9) E. F. Caldin, “Fast Reactions in Solution,” Blackwell, Oxford, 1964. I) 94 ff.
(10)-E. D. Hughes, C. K. Ingold, S. Patai, and Y. Packer, J. Chem. SOC..1206 (1957). (li) R. M. Fuoss, J. Am. Chem. Soc., 79, 3301 (1957).
Pettit, Bruckenstein 1 Thermodynamics of Ion Association in Solution
4784
and calculated the free-energy change, AFIV, from the relationship
+
+
AFIV = A F I AFII AFIII (1) The quantities AF, and AFIII will include the freeenergy changes on solvation of the free ions and the ion pairs, respectively; the solvation energy changes were considered to be approximately the same and therefore were omitted from the calculation. This assumption is reasonable in solvents of low dielectric constant since the absolute magnitude of the solvation energy would normally be small, and the solvation energy of the ion pair, while probably being lower than that of the free ions, would probably approach the latter. Accepting this assumption, the relationship derived by Denison and Ramsey is
where N is the Avogadro number and AVz the change in potential energy on formation of the ion pair. Inspection of the treatment shows that this relationship is equally applicable t o more extensive ion associations and is a general expression irrespective of the degrees of association of steps I and I11 of the cycle, provided the assumptions concerning changes in solvation energies can still be accepted. Equation 2 will apply rigorously only at absolute zero. T o calculate the free-energy change at temperature T, we use the enthalpy AH"
=
AFIv
_ + RT2['_en,"],
and entropy
to obtain AGO = AFI, - RTln Q
where Q is a function of partition functions of the products and reactants. The term R In Q ( = A S ) includes the changes in orientational entropy (i.e., changes in vibrational, rotational, and translational entropy associated with the reaction). Thus, eq 2 becomes AGO
=
AGIv
NA V, D
= __
- TAS
(3)
and sexapoles.16 For an equation to be of use in calculating potential energy values, all the quantities involved must be available. We therefore set up a potential energy expression for an ion pair (AB), which allowed for all possible interactions betwsen ions and their induced dipoles, but which included the various other forces in a general term B/rn,where B was obtained by minimizing the expression for the energy as in the Born treatment of ionic lattices, i.e.
B rn
where r is the internuclear separation and p A and pB are the induced dipole moments. Values for these dipole moments were calculated by the expressions used by Rittner. This expression for Vz is somewhat simpler than Rittner's and does not require all the experimental quantities necessary in his treatment. Pauling calculated the potential energies of alkali halide ion pairs using a similar term, BITn, but n was a variable depending on the ions concerned and polarizabilities were omitted." The value of n in eq 5 was adjusted by us empirically to obtain the best fit with measured potential energies of the alkali halide ion pairsIY using reported polari~abilities'~and internuclear separation.?O It was found that the value of n = 6.9 gave the best agreement, particularly for the larger ions such as rubidium or cesium. For smaller ions smaller n values are required but, since most ions studied in solution are large ions, n = 6.9 was selected as the most generally reliable value. Table I shows a comparison of values for the potential energies at 0°K calculated by eq 5 compared with the values calculated from thermodynamic data, l 6 the values shown in parentheses. Considering approximations inherent in the Denison-Ramsey treatment, this agreement is completely adequate for intermolecular separations greater than 2.0 A. Substantial differences are found only in the lithium and sodium fluorides which are much smaller than species encountered in solution studies. This value of iz = 6.9 was assumed to apply to ion triplets, quadrupoles, sexapoles, as well as to ion pairs, and was used in all calculations described below. Table I. Potential Energies of Gaseous Alkali Halide Ion Pairs at 0°K (kcal)
i. e. -2.303 log K, =
F
NAV, A S -DRT R
(4)
__
where both AV, and A S will generally be negative. Association constants can therefore be calculated provided AV, and A S are known. Potential energy values for ion aggregates in vacuo have been rigorously calculated for the alkali halides. Rittner in 1951 considered ion-pair formation and various other workers have considered quadrupole^^^-'^ (12) (13) (14) (15)
E. S. Rittner, J . Chem. Phys., 19, 1030 (1951). C. T. O'Konski and W. Higuchi, ibid., 23, 1174 (1955). T. A. Milne and D. Cubicciotti, ibid., 29, 846 (1958). J. Berkowitz, ibid., 29, 1386 (1958).
Journal of the American Chemical Society
88:21
(5)
C1
Br I
Li
Na
K
Rb
cs
195.4 (178.2) 152.1 (156.8) 143.9 (148.0) 132.3 (138.8)
159.8 (144.4) 127.7 (128.2) 121.5 (123.1) 113.4 (116.0)
139.1 (136.5) 112.1 (114.9) 106.5 (110.9) 98.9 (103.7)
133.7 (131.3) 107.4 (109.9) 101.9 (104.6) 94.9 (98.4)
129.9 (127.3) 103.5 (102.3) 98.0 (98.7) 91 . O (89.6)
(16) T. A. Milne and D. Cubicciotti, ibid., 30, 1625 (1959). (17) L. Pauling, Proc. Natl. Acad. Sci. (India), A25, Part I (1956). (18) D. Morris, Acta Cryst., 9, 197 (1956); B. Barrow and A. Caunt, Prdc. Roy. SOC.(London) A219, 130 (1953). (19) J. E. Mayer and M. G. Mayer, Phys. Rec., 43, 605 (1953). (20) L. Pauling, "The Nature of the Chemical Bond," 2nd ed, Cornel1 University Press, Ithaca, N. Y . , 1Y40.
1 November 5, 1966
4785 Table 11. Ion-Pair Formation at 25' ~~
Log Solute
Bu4N.picrate
Solvent
D
Kz
Ref
Benzene Benzene Benzene HAC HAC HAC THF THF THF THF Dioxanewater
2.27 2.27 2.27 6.13 6.13 6.13 7.38 7.38 7.38 7.38 2.38 2.56 2.90 3.48 4.42 5.80 4.34 5.04 5.63
17.00 16.66 16.85 5.48 4.87 6.88 4.37 4.07 4.50 5.72 15.70 14.00 12.00 9.60 7.50 5.80 8.90 8.60 7.70
7 7
Anisole m-CsH4CI? CsHsCI
~
BJerrum5
C
d d d e e e e 11 11 11 11 11 11 7 7 7
5.8 5.8 5.9 6.4 7.6 4.9 7.0 8.0 6.6 4.9 6.0 6.2 6.4 6.6 6.7 6.5 4.9 4.9 4.8
~~~
A ___ D.R. Eq 4b 12,
6.2 6.3 6.3 7.2 8.1 5.8 7.5 8.1 7.5 5.8 6.5 6.8 7.0 7.3 7.4 7.3 6.3 5.6 5.6
F
4.6 4.6 4.6 3.7 3.8 3.2 3.4 3.7 3.5 3.1 4.5 4.6 4.6 4.5 4.2 3.8 3.7 3.3 3.2
AS,eu -13.6 -15.4 -15.0 -17.4 -18.7 - 16.8 -17.5 -15.4 -16.0 -16.1 -16.5 -16.5 -16.5 -16.5 -16.5 -16.5 -18.1 -18.1 -18.1
a As pointed out by FuossGbthe Bjerrum function from which these values are calculated is not entirely satisfactory. Values are included here for comparison since this function has been used by many other workers in this field. b In general, internuclear distances in ion pairs are about 15 more than separations in crystals (cf. alkali halides). Estimated contact internuclear distances (A) are: KC1 = 2.7, NaC104 = 3.8, BuaN.CI = 4.1, Bu4N .c104 = 4.5, and Bu4N.picrate = 3.7 A (ref 7). S. Bruckeni C. A. Kraus,J. Phys. Chem., 60, 129 (1956). stein and I. M. Kolthoff, J . Am. Chem. SOC.,78, 2974 (1956). 6 D. N. Bhattacharyya, C. L. Lee, J. Smid, and M. Szwarc, J . Phys. Chem., 69, 608 (1965).
Ion-Pair Formation in Solution. Using eq 5 in the way outlined above, potential energy values for ion pairs may be calculated. Where polarizability values are not available, the volume of the ion, in cubic angstroms, will be satisfactory, since the induced dipole interactions contribute only a minor part of the total energy which is therefore not very sensitive to small errors in polarizability values. As the interionic separation increases, the influence of polarization terms decreases so that, in some cases, they can be neglected altogether. The expression for the potential energy of ion pairs then reduces to AV2 = -[1 - (l/n)]. ( e 2 / r ) . Accurate values for the entropy change accompanying stage I1 of the D.R. cycle for ion-pair formation can be calculated, provided the vibrational frequency of the ion pair is known, using the standard equations of statistical thermodynamics. Assuming the interaction between the ions to be purely Coulombic the force constant, k, and hence the frequency, w, may be expressed in terms of the second differential of the potential energy expression, i.e. k = d2V/dr2
whenr
=
(6)
re w = - 1- k
2nc
p
where p is the reduced mass. Havingobtained a value for the frequency, w, the translational, rotational, and vibrational entropies of an ion pair (&, SR,and Sv, respectively) may be evaluated by eq 7, 8, and 9 where M is the molecular weight and I the 5
+ In (2nMkT)"/'V/h3N SR = R[l + In 8n21kT/h2]
ST = R[i
moment of inertia of the ion pair. Equation 4 may now be used to obtain a value for log Kz. Alternatively, from experimental association constants the value of r2, the interionic separation in an ion pair, may be derived for comparison with the values calculated by the Bjerrum and original D.R. equations. Such a calculation is possible since, although both A S and A V are functions of r, the influence of small changes of r on the total entropy change is small, allowing an approximate value to be used in a the calculation of A S which can then be used to calculate a more accurate value for r from eq 4. This assumption is acceptable at room temperatures since eq 7 is independent of r, eq 8 is dependent on r only in that it will influence the moment of inertia, and eq 9, while being strongly dependent on w and hence r, contributes only 1 or 2 % toward the total entropy at room temperature. Equation 4 has been used to calculate r2 from reported association constants for a number of different ion pairs in solution. The results for a representative selection of the data available are shown in Table I1 where they are compared with results from other treatments. Internuclear distances likely to be found in contact ion pairs are given in the footnote of Table 11. Entropy values are estimated to be within f 2 eu, which corresponds to an error of f0.4 log unit in the value of Kz. In general, ions which are approximately spherical have been chosen. It will be noticed that, in all cases, the distances are shorter than previously calculated values and correspond closely to the separations expected from two ions in contact. This result indicated that there is no solvent separation as had been previously thought, and many problems in interpreting these r values have been removed since more consistent values are obtained. Perchloric acid appears to have a larger ionic separation than sodium perchlorate indicating that the solvated proton (in acetic acid) is slightly larger than the sodium ion. This result is in full agreement with the crystallographic study of
Pettit, Bruckenstein
Thermodynamics of Ion Association in Solution
4786 hydroxonium perchlorate carried out by Truter2* who reported a unit cell for HC1O4.Hz0 of volume 93 A3, compared with volumes of 81 A 3 for NaC104 and 100 A 3 for NH4C104,22suggesting that, in acetic acid, the solvated protons should certainly be larger than the sodium ion. In calculating the entropy change on ion-pair formation the species is assumed to be (H HAC)+ CIOr-. Results in tetrahydrofuran (D = 7.38) indicate definite solvation (e.g., Na+ > Cs+). Specific solvation invalidates assumptions made in the original Dension-Ramsey cycle and, in general, this treatment should be restricted to solvents of lower In any case, dielectric dielectric constant (zk.
