4710 probably because the Co-X bond in that case is not too different from the other Co-N bonds, as far as the hydrogen atoms are concerned.
Experimental Section All chemicals were analytical grade. Dimethyl sulfoxide was dried over calcium hydride for 24 hr or more and was vacuum distilled over Molecular Sieve 3A. Ammonia, methylamine, and dimethylamine were passed through a 9-ft column packed with iVolecular Sieve 3A into DMSO to make up solutions. The concentration was determined by diluting the DMSO solution with water and titrating with standard HCl solution. Pyrrolidine and ethylenediamine were directly weighed for making up solutions in DMSO. The concentrations of the amines were 0.1-1.0 N ; in this range the NH proton chemical shift of each amine did not change with the concentration within experimental error. The perchlorate salts were prepared by passing the corresponding gases (ammonia, methylamine, and dimethylamine) or adding the liquid amines (pyrrolidine and ethylenediamine) to solutions of perchloric acid in ethanol, filtering, washing with ether, and drying under vacuum. The cobalt(II1) complexes were prepared according to standard procedurelo and were recrystallized from water. The concentrations of the salt solutions were 0.2-0.5 N . The proton nmr spectra of these solutions did not change with concentration. Proton nmr spectra were taken with a Varian A-60A spectrometer a t 35". Acknowledgment. This work is supported by the National Institutes of Health. (10) W. C. Fernelius, Inorg. &n., 2, 221 (1946).
Conductance of Thallous Nitrate in Dioxane-Water Mixtures at 25" by Alessandro D'Aprano' and Raymond M. Fuoss Sterling Chemistry Laboratory, Yale University, New Haven, Connecticut 06680 (Received August 6 , 1068)
Ion association in aqueous solutions of the alkali halides is undetectable by conductance a t concentrations less than about 0.04 N , but if the range of concentration is extended to 0.1 N (the upper limit of applicability of the present theory), association constants of the order of unity are found.2 Nitrates have been assumed to be more highly associated than halides. I n order to test this assumption, we have measured the conductance of a series of nitrates in dioxane-water The Journal of Physical Chemistry
NOTES mixtures a t 25". Here we present the results for thallous nitrate. I n water, the association constant was found to be 3.2 f 0.1, with a limiting conductance of 146.20. From the latter, the single-ion conductance of the thallous ion is 74.75, in excellent agreement with an earlier value3 of 74.71.
Experimental Section Fischer's Purified grade of thallous nitrate was dried for 24 hr at 100"; portions were then weighed in platinum boats on the microbalance to make the initial solutions for the conductance determinations. Dilutions were made by weight and were calculated to volume concentrations c (equiv/l.) = po(l yw), where po is the solvent density and w is the weight concentration of the salt (equiv/kg of solution). For thallous nitrate in water, y = 0.232, from a density of 1.035 g/ml a t w = 0.1638. For solvent mixtures 2,3, and 4, y = 0.212, 0.190, and 0.141, respectively. For mixtures 5-8, where the concentrations were less than 0.007, we used y = 0.14. Water was laboratory supply distilled water, boiled vigorously and then cooled under nitrogen; its conductance was (1-2) X 10-6 mho. Dioxane was refluxed under nitrogen over potassium hydroxide at least 24 hr and was then distilled under nitrogen from silver nitrate just before use. (It had been found that dioxane distilled from potassium hydroxide still contained a trace impurity, undetectable by vaporphase chromatography, which reduces silver and thallous nitrate solutions to give colloidal or mirror metal.) Viscosities of the solvents were determined in an Ubbelohde viscometer whose water flow time was 473.1 sec. Dielectric constants were determined at 1 Four conductance cells were used, with constants 5.1313 h 0.0001, 0.85696 f 0.00002, 0.51361 f 0.00004, and 0.14464 f 0.00001. The first three were calibrated using potassium chloride solutions.6 The fourth was calibrated by comparison with the second and third, using 0.0024 N tributylammonium pictrate in 2propanol. The conductance bridge and general technique were as described by Lind and F u o ~ s . Solvent ~ properties are summarized in Table I, where the p is the density, D is the dielectric constant, 1007 is the viscosity in centipoises, and uo is the solvent conductance. The conductance data are summarized in Table 11, where c (equiv/l.) is concentration and A (cm-2 ohm-l equiv-') is equivalent conductance.
+
(1) On leave of absence from the University of Palermo, Palermo, Italy. (2) R. M. Fuoss and K. L. Hsia, Proc. Nat. Acad. Sei. U.S., 57, 1550; 58, 1818 (1967). (3) R. A. Robinson and C. W. Davies, J . Chem. Soc., 139,574(1937). (4) J. E.Lind, Jr., and R. M. Fuoss, J . Phys. Chem., 65, 999 (1961). (5) J. E.Lind, Jr., J. J. Zwolenik, and R. M. Fuoss, J . Amer. Chew. Soc., 81, 1657 (1959).
NOTES
4711
Table I : Solvent Properties Wt %
sys-
temno. dioxane
0.0 7.8 16.0 36.2 52.5 59.2 70.0 76.4
1 2 3 4 5 6 7 8
p,
g/ml
0.99707 1.0039 1.0109 1.0264 1,0343 1.0361 1.0369 1.0362
D
1001, OP
78.54 71.89 64.49 47.60 33.60 27.96 18.95 14.34
0.8903 1.046 1.210 1 674 1.960 2.014 1.953 1.810
lo%,
mho
1.68 0.52 0.55 0.48 0.13 0.25 0.122 0,107
I
Table 11: Conductance of Thallous Nitrate in Dioxane-Water Mixtures at 25’ 10%
A
D = 78.54 56.370 138.438 97.110 135.866 155.385 133.019 224.598 130.291 322.34 127.167 438.60 124.142 644.36 119.756
D = 71.89 39.204 66.071 103.978 151.812 211.177 287.46 420.91
122.001 119.940 117.764 115.619 113.465 111.136 107.858
D = 64.49 62.667 104.068 94.016 102.334 128.629 100.748 175.854 98.986 232.352 97.219 350.94 94.265
D = 47.60 16.773 27.241 40.153 59.360 83.616 110.507 160.351
75.858 74.717 73.599 72.252 70.888 69.647 67.780
104~
A
D = 33.60 7.211 60.528 11.792 59.392 17.982 58.190 24.999 57.088 34.855 55.816 46.319 54.608 69.513 52 718 I
D = 27.96 2.7519 4.3712 6.650 9.301 12.962 16.616 25.351
56.123 55.324 54.383 53.464 52.411 51.509 49.714
D = 18.95 1.3330 47.726 2.1763 46.302 3.2159 44.876 4.6545 43.298 6.737 41.492 9.312 39.742 14.321 37.233
D
=
1.0511 2.0150 2.6780 3.5200 4.6963 6.6832
14.34 39.817 36.647 34.880 33.237 31.361 29.009
Discussion Equivalent conductance depends on concentration through an equation with three parameters: Ao, the limiting conductance; a, the center-to-center contact distance between the spheres which represent the ions; and K A , the association constant. Symbolically A
= ?(A0
- A N [1 + ( A . X I X > I / [ 1+ (3P/2)1
(1)
where y is the ratio of the free-ion concentration to the stoichiometric, AA is the electrophoretic term, A X / X is the relaxation field (limiting value at low concentrations, - Q C ~ ’ ~ ) , and the term in the denominator allows for the detour effect. An explicit expansion of eq 1 is available.2s6 Defining 6A by 6A = A(ca1cd) - A(obsd)
(2)
the standard deviation is given by u2 = 26Aj2/(n
- 3)
(3) where n is the number of data points. I n order to determine the three parameters from the data, eq 1 is written in the form A
A(C;
no f
13 f Y, KA -k
(4) where A,, 8, and K A are preliminary estimates of the parameters and x = A&, y = Au, and z = AKA are increments in their values which are chosen to satisfy the equations which minimize the standard deviation du2/dx
=
=
0;
2,
da2/dy = 0;
2)
dd/dz = 0 (5)
The calculation is an iterative process of successive approximations, which can be programmed for an electronic computer. Our preferred method of analyzing the data is to give the computer a sequence of d values and ask for the pairs of values of A. and K A which will minimize u2 for each value of d. Then u is plotted against d . Ideally, a curve with a sharp minimum will result, which locates the desired contact distance. Then from plots of K A and of A0 against the corresponding d values, the association constant and limiting conductance are interpolated, to correspond to that value of it which minimizes a. An example is shown by curve 1in Figure 1 for thallous nitrate in water (system 1 of Table I). The u-it curve has a sharp minimum at d = 7.8 where u = 0.020 A unit. At d = 7.0, u = 0.078, and at d = 8.5, a = 0.092; both of these values of a are well beyond the experimental error. The analysis of the data for the other systems is not so unambiguous, as shown by the other seven curves of Figure 1, which are numbered to correspond to the system numbers of Table I. For D = 71.89 and 64.49, minima appear at 7.0 and 8.0, respectively, but they are not as sharp as the one in the water curve. At D = 47.60, the a-d curve actually shows a slight maximum, but the entire variation in u as d goes from 5.0 to 9.0 is only 0.010 A unit; within the experimental error, the a-d curve is flat. In other words, any value of d between 5.0 and 9.0 will reproduce the observed data within k0.04 A unit, if the corresponding K A and A, values are used. The change of K A with d for system 4 is shown as the lower curve in Figure 2 (ordinate scale to left) ; A, does not change very much (6) Y . C. Chiu and
R. M. Fuoss, J . Phys. Chem., in press. Volume 72,Number 13 December 1968
4712
L
OO
L-
5
9
4
8
Figure 1. The method for the determination of the minimizing contact distance.
5.0'
1
I
4
5
6
I
8
I
I
8
9
Figure 2. The values of the minimizing association constants for a range of d values.
(79.792 at = 4.0 to 79.866 at & = 9.0). It is easy to see what the difficulty is: an increase in & increases conductance through the higher terms in electrophoresis and relaxation, while an increase in KA decreases conductance by decreasing y a t a given concentration. Around D = 50, for any it value in a reasonable range, the pair of values of & and KA which minimize u have opposing effects on conductance which almost exactly compensate each other, and so we get a band of solutions instead of a unique solution as we did for the water case. For D > 50, the long-range terms involving & control and the system of equat,ions (5) can be solved for the three parameters. For D < 50, the balance between K& effects and K A effects is again upset; KA now dominates the change of conductance with concentration, and the minimum reappears. However, the nature of the curves has also changed. The minima now are broad and shallow. The Journal of Physical Chemistry
25
IOO/D
75
Figure 3. The dependence of association constank on the dielectric constant.
I n other words, 8. can again be chosen over a fairly wide range without changing c by morc than a few hundredths of A unit, if matching values of A0 and KA are chosen. As an example, the minimizing KA values for system 6 (D = 27.96) are shown as the upper curve (ordinates to the right) of Figure 2. This insensitivity of u to it at lower diclectric constants might be expected; most of the conductance change with concentration is due to y, and when KA is large, ~1 small change in it will therefore cornpensate a rclatively largc change due to a change in the contact distance. A self-consistent set of parameters for all the data can be obtained by setting it = 8.0, the average of thc values found for t>hethree systcms of highest dielectric constant, and using the values of KA and .io for the other systems which minimize u for d = 8.0. These constants are summarized in Table 111. A plot of log KA against the reciprocal dielectric constant is shown in Figure 3. As in the case of the alkali halides, there is distinct curvature :Lt the waterrich end; presumably, energy and entropy terms in K A due to differences in interaction between solvent and ions and ion pairs as the structure of the solvent is changed from hydrogen-bonded water to an essentially unstructured water-dioxane mixture are responsible.
Table 111: Derived Constants for d System no.
= 8.0
Ao
Kh
U
146.195 128.02 111.64 79.90 64.17
3.2 4.0 4.9 10.9 35.0 6
