556
Communications to the Editor
References and Notes (1) J. H. Lunsford,Catal. Rev., 8, 135 (1973). (2) K. Tanaka and G. Blyholder. Chem. Cornmun., 736 (1971); J. Phys. Chem., 76, 1807 (1972). (3) K. Tanaka and G . Blyholder, Chem. Comrnun., 1343 (1971): J. Phys. Chem., 76, 3184 (1972). (4) K . Tanakaand K. Miyahara. Chem. Commun., 877 (1973). (5) C. Naccache, Chem. Phys. Lett., 11, 1323 (1971). (6) A. J. Tench and T. Lawson, Chem. Phys. Lett., 7,459 (1970). (7) N. B. Wong and J. H. Lunsford. J. Chem. Phys., 56, 2664 (1972). (8) V. B. Kazansky. V. A. Skvets. M. Ya Kon, U. U. Nikisha, and B. N. Shelimov, Proc. 5th Int. Congr. Catal., 104 (1972)
Research Institute for Catalysis Hokkaido University Sapporo, Japan 060
Ken-ichi Tanaka
Received October 5, 7973
Solvation Numbers in Nonaqueous Solvents
Sir: In a recent paper,l Della Monica and Senatore state that in methanol monovalent cations are more solvated than divalent cations of comparable crystallographic radii. This statement was based on the Stokes radii, as calculated from conductance data2 by means of eq l.3 This conr , = 0.821ZI/(X+")q (1) clusion is contrary to that expected on the basis of Coulombic theory4 as well as the results found by this method for mono- and divalent cations in aqueous solution^,^ and is, in fact, so unexpected that it casts doubt on the estimation of solvation numbers from conductance data. Because of these discrepancies and the possibility that nonaqueous solvents, or a t least methanol, behave anomalously in their interactions with cations it seemed of value to reexamine this method of calculating solvation numbers for divalent cations in as many nonaqueous solvents as data were available. In Table I are given the X o values and the calculated Stokes radii for a number of mono- and dipositive ions in methanol, acetonitrile, propanol, and acetone. Also given
in this table are the corrected radii obtained from the Stokes radii by assuming that the crystallographic radii of the larger tetraalkylammonium ions represent their true radii in solution ( i e . , that they are unsolvated).6 From these data, it can be seen that both the Stokes and corrected radii for the divalent ions studied are greater than those for monovalent ions in all solvents. This is in agreement with expectations of a greater degree of solvation for more highly charged species and indicates that these solvents behave no differently, in this regard a t least, than water. Because the results obtained here for methanol are based on the same conductance data as those of Della Monica and Senatore, it would seem that their unusual conclusion was due to the omission of a factor of 2 (for the charge of the alkaline earth ions) in eq 1. The apparent correlation between the extent of solvation and the charge density is further substantiated by the decrease in corrected radii (and therefore in solvation) in a given solvent which is found with increasing crystallographic radii for ions of like charge. In an attempt to compare the solvation numbers found by this method with those obtained by other methods, eq 2 was applied to the data for Mg2+, using values of 0.65 A
for rcryst for Mg2+ and 50, 68, 170, and 145 A3 for the volumes of the methanol, acetonitrile, propanol, and acetone molecules, respectively. The values of h so obtained (15,14, 14, and 16, respectively) are remarkably similar although they are very dependent on the molecular volumes assumed. The value of 15 obtained for methanol is much greater than the value of 6 measured by means of nmr.8 A comparable difference occurs for the hydration and numbers of Mg2+ found from conductance ( h = nmr ( h = 6)1° data and is most likely due to the fact that the nmr results reflect the number of solvent molecules in the first solvation sphere alone,ll while conductance measurements lead to the inclusion of a t least one additional layer of the solvent sheath. In conclusion, this study gives no indication that the solvation numbers obtained from conductance data for cations in nonaqueous solvents are in any way anomalous, increasing as they do with increasing charge density of the
TABLE I: Conductivities, Stokes Radii, and Corrected R a d i i for Ions in Different Solvents
Li
3.786 3.280 2.786 2.406 2 .248 2.576 3.415 4.063 5.205
Sr2
39.6a 45.7" 53.8~ 62.38 66 . 7 b 58.2b 43.9b 36.9b 57.6' 6O.Oa 59 .Ou
5.082
5.62 5.50 5.54
Ba2+ Zn2+
59.60
5.030
5.52
+
Na K+
+
cs
+
Me4N
+
EkN+ PrlN+ BurN+ Mg2+ Ca2 +
5.000
4.73 4.425 4.122 3.89
79.gC 76.9~ 83.4. 97.6~ 94.zd 83.7d 69.6d 61.3d 94.8~
2.966 3.082 2.842 2.428 2.516 2.831 3.405 3.866 5.00
4.137 4.238 4.00 3.625
6.01
10.32' 12.45e
4.071 3.374
14.40, 2.917 15.05, 2.791 12.19' 3.446 10.17/ 4.131 9.400 8.938
4.918 4.418
8.280
72.8" 3.717 77.4gh 3.492 80.6h 3.358 96.63& 89.49' 75 .09$ 66.40' 70.20 83.60
2.801 3.024 3 ,604 4.076 7.710 8.188 6.474 7.05
8 5 . 0 ~ 6.368 94.8~ 5.00
4.625 4.42 4.294
6.950
6.01
W. Libus and H. Strzelecki, Electrochim. Acta, 17, 1749 (1969). a Reference 2.0 * E . C. Evers and A. G . Knox, J . Amer. Chem. Soc., 73, 1739 (1951). A. H. Harkness and H. M. Daggett, Can. J . Chem., 43, 1215 (1965). e T.A. Gover and P. G . Sears, J . Phys. Chem., 60, 330 (1956). f D. F. Evans and
P. Gardam, J . Phys. Chem., 7 2 , 3281 (1968). Kraua, J . Amer. Chem. Soc., 73, 3293 (1951).
0
*
P. Van Rysselberghe and R. M. Fristrom, J . Amer. Chem. SOC.,67,680 (1945). M. B. Reynolds and C. A . D. F. Evans, J. Thomas, J. Nadas, and M. A. Matesich, J . Phys. Chem., 75, 1714 (1971).
The Journal of Physical Chemistry, Vol. 78, No. 5. 1974
557
Communications to the Editor ion. They are, however, appreciably larger than the solvation numbers obtained from nmr studies and comparisons of the results obtained by these methods must be made with the knowledge that they measure different quantities, although each can provide useful information about the behavior of closely related solventsllJ2 and ions.11+13
Nonweighted linear fitb
1.156 rt 0.028
5745
119
1.234 f 0.033 1.231 f 0.045
5556 f 93 5559 89
Weighted linear
References a n d Notes
fitb
(1) M. Della Monicaand L. Senatore, J. Phys. Chem., 74, 205 (1970). (2) H. Hartley and H. R. Raikes, Trans. Faraday Soc., 23, 393 (1927). (3) R. A. Robinson and R. H. Stokes, "Electrolyte Solutions," Butterworths, London, 1968, p 44. (4) B. E. Douglas and D. H. McDaniel, "Concepts and Models of Inorganic Chemistry" Blaisdell, New York, N. Y., 1965, p 126. (5) Reference 3, p 12. (6) Reference 3, p 125. (7) J. E. Huheey, "Inorganic Chemistry, Principles of Structure and Reactivity," Harper and Row, New York, N. Y., 1972, p 73. (8) J. H. Swinehart and H. Taube, J. Chern. Phys., 37, 1579 (1962). (9) Reference 3, p 62. (IO) N. A. Matwiyoff and H. Taube, J. Amer. Chem. Soc., 90, 2796 (1988). ( 1 1) A. Fratiello in "Inorganic Reaction Mechanisms," J. 0.Edwards, Ed., Interscience, New York, N. Y., 1972, p 57 f f . (12) A . P. Zipp, J. Phys. Chem., 77, 718 (1973). (13) P. Bruno, M. Della Monica, and E. Righetti, J. Phys. Chem., 77, 1258 (1973).
Department of Chemistry State University College Cortland, New York 13045
TABLE I: Comparison of Least-Squares Fits of Spectral D a t a For Et&-& in Heptane at 150a
Arden P. Zipp
Received October 23, 7973
Some Comments on the Calculation of Equilibrium Constants and Extinction Coefficients for 1:I Complexes Publication costs assisted by the National Science Foundation
Sir: Several authors have reported methods for using nonlinear least-squares analysis to infer formation constants, K, and optical extinction coefficients, ~ D A ,for a molecular complex, DA, existing in solution in equilibrium with free donor and acceptor mo1ecules.l Standard errors in the parameters have also been computed from the shape of the error surface (the sum of squares of absorbance deviations as a function of K and eDA) in the vicinity of the optimum ~ investigators have argued values of K and c D A . ~ C .Various that numerical optimization methods are superior to graphical or linear least-squares methods based on the Benesi-Hildebrand (BH) equation2 and related linear relations. What seems not to have been properly appreciated by many workers in the charge-transfer field is that use of any of the linear forms will yield answers virtually identical with those inferred from the more complicated nonlinear analyses if data are correctly weighted. For example, a recent paper in this journal3 includes a discussion of the relative merit of various linear forms of the BH equation (in terms of differences in correlation coefficients of the least-squares fits) and a comparison of two nonlinear fitting methods with the linear forms. In our opinion, this type of discussion is unproductive, since all of the various linear and nonlinear forms based upon the same physical
Nonlinear fit
a Data taken from the Ph.D. Dissertation of J. D. Childs, The University of Oklahoma, 1971; absorbances were measured at 250 nm. Corrections have been made for the absorbance of free donor and acceptor in all three
fitting methods.
Benesi-Hildebrand equation2 was used.
and mathematical model must give consistent results if proper statistical methods are employed. To illustrate this point, we consider using the BH equation in the form
where A is the absorbance per unit path length, and where we assume that the total donor concentration ([D]) is so much greater than the total acceptor concentration ([A]) that the concentration of free donor (CD) is practically equal to [D]. We also ignore the effects of absorbance of donor and acceptor at the wavelength chosen for analysis, although these absorbances can be taken into account with little additional difficulty. The principle of weighted linear least squares4 requires that we seek a minimum in the function
where the summation extends over all sets of measured values of At, [A],, and [D]l. Standard methods are readily applied to obtain least-squares values of K and cDA and estimates of their standard deviations, uK and utDA,,provided the W[ values can be chosen properly. If the weights are chosen injudiciously (for example, if W, is by default taken to be unity for all points) unreliable answers may be obtained with the BH equation; criticism in the literature of the uncritical use of the BH equation in this way is quite a p p r ~ p r i a t e However, .~ the calculation of weights is straightforward if reasonable estimates of the uncertainties in absorbances and concentrations can be made. Decisions regarding these errors should not be made thoughtlessly since the calculated parameters can depend strongly on the weights, especially when the data do not permit a precise determination of K and eDA. Let us consider the case in which it is reasonable to assume that the absorbances are subject to equal absolute error at all points and that the donor and acceptor concentrations are known exactly. (These are the assumptions usually made in nonlinear analyses of spectral data.) The dependent variable (Yt = [A]t/A!) in the linear fit of Yt us. 1/[DILwill then have the weight
where U A is the (constant) error in absorbance.6 Using this weighting scheme, best values of K and tDA, and estimates of standard deviations in C D * - ~and (Ke~*)-l result directly from the standard linear least-squares analysis. Simple propagation of errors formulas than yield UK and DA
a
The Journal of Physical Chemistry, Vol. 78, No. 5, 7974
