450
NOTES
of the most rapid rate of coagulation to the actual, slower rate. I n order to predict the magnitude of this quantity theoretically, it has been usual to compare the rapid rate derived by Smoluchowski‘ with a slower rate from Fuchs’ equation,2 thereby obtaining the expression3 25
where V is the energy of interaction of two spherical particles of radius a and s = R/a; R is the distance between particle centers. I n using Smoluchowski’s value for the rapid rate of coagulation, however, one is assuming that no attractive forces are in operation until the particles are in contact. It is this assumption which accounts for the fact that theoretical curves of W as a function of surface potential always fall below unity at low potentials and thus cannot, easily be compared with ex~eriment.~)5 Fuchs provides a general equation for the flux of particles in a force field around a central particle, this force field not necessarily involving any element of repulsion. This leads to the following expression for the number of collisions J per second with one particle
J =
8rDNa e x P ( y ds
where D is the diffusion coefficient for a primary particle and N is the number of particles per cubic centimeter. Thus, if we define a new rapid rate such that the energetic interaction between the particles is entirely attractive and compare the corresponding flux with that when there is a superimposed repulsive force, we obtain a new expression for W
-
4-
8 e
3 2.0C
.Y
r 0
1.5-
8 1.01 100
I
I
200
500
I
1,000 radius(&
I
ZOO0
I
5000
I
lO,O(
Figure 1. Correction factor as a function of particle radius: (a) 1 X lo-” erg; (b) 5 X 10+2 erg; (c) 1 X erg; and (d) 5 X 10-l3 erg.
and in Figure 1 is shown plotted as a function of particle radius for four values of Hamaker constant. The attractive energy has been calculated using the limiting form of the Hamaker equation with the retardation corrections of Schenkel and Kitchener.6 (1) M. Von Smoluchowski, Z . Physik. Chem., 92, 129 (1917). ( 2 ) N. Fuchs, 2. Physik, 8 9 , 736 (1934). ( 3 ) E. J. W Verwey and J. Th. G. Overbeek, “Theory of Stability of Lyophilic Colloids,” Elsevier Publishing Co., Amsterdam, 1948, p 166.
(4) See ref 3, p 176. ( 5 ) K. E. Lewis and G . D. Parfitt, Trans. Faraday Soc., 6 2 , 1652 (1966). (6) J. H. Schenkel and J. A. Kitchener, ibid., 56, 161 (1960).
The Proton Magnetic Resonance Spectra of Ammonia Nickel Cyanide Clathrates
where VA is the attractive potential energy. When so defined, W falls to unity only when repulsion is entirely absent and would thus be expected to be more in accord with experimental reality. I n terms of the magnitude of W , the new expression gives rise to values which are higher than previously predicted by a factor which is constant for a given particle radius and Hamaker constant.
The ratio f between corrected and uncorrected values of W is such that
by Kimiko Umemoto and Steven S. Danylukl Department of Chemistry, University of Toronto, Toronto, Canada (Receiued July 85, 1966)
The wide-line proton magnetic resonance spectra for ammonia nickel cyanide clathrates of benzene, aniline, and pyridine have been reported recently by Nakajima, Bhatnagar, and Cole,2 and by Nakajima.3 I n each (1) Address correspondence to this author at Argonne National Laboratory, Argonne, Ill.
The Journal of Physical C h a i s t r y
--
NOTES
case the derivative curves were found to be asymmetric and although the line width, A H , and secondmoment, Sz, variation with temperature were attributed qualitatively t o the motional behavior of the enclathrated molecule, this interpretation is open to some question because of the difficulties inherent in evaluating these parameters for superimposed sets of signals, Le., those for ammonia protons and aromatic ring protons. The situation is further complicated by the possibility of electron-nucleus interaction between paramagnetic (octahedrally coordinated) Ni(I1) atoms and the ammonia ligands. Such an interaction would act t o shift the resonance frequency, Ho,for the ammonia protons from the value expected for a purely diamagnetic compound, I n order to assess the relative contribution of the host and enclathrated molecules t o the observed signal, we have investigated the proton spectra for complexes in which the ammonia and aromatic molecules have been selectively replaced by their deuterio analogs. Since the observed signals are now solely due to the enclathrated molecules and host molecules, respectively, the determination of Sz values for the individual components is simplified considerably.
+
451
b
0
u
IO Gauss
-O 4
0
Ha
Experimental Section Details for the preparation of deuterated Ni(CN)2n'D3.11 and :!Ji(CN)2NH3.C6D6clathrates are reported e l s e ~ h e r e . ~Fully deuterated chemicals, except for the enclathrated molecules, were used in the preparative steps throughout and the preparations were carried o u t in a drybox. The nmr spectra were measured with a Varian DP 60 spectrometer and the field was calibmted with a Harvey-Wells G 502 gaussmeter. Care nas taken to avoid modulation broadening of the signal and modulation amplitudes were usually maintained below 1.7 gauss.
Results and Discussion The spectra observed for the Ni(CN)2.NH3.C6DB (I), Ni(C1;)2 KD3.C6H6(II), and Ni(cN)z "3. CaH6 (111) Clathrates at 298 and 100°K are shown in Figure 1 (a-e) and a summary of A H , SZ, and (H* - H o ) ~ values derived from these spectra is given in Table I. The spectrum for Ni(CN)2NH4.C6D6at 298°K (Figure la) is typical of a symmetric three-spin system16 apart from tho narrow signal located a t the center. The latter is probably due to small amounts of H20 which could not be removed from the starting NiS04 despite repeated attempts to prepare the anhydrous salt.' From the line-width and second moment values, 12.6 f 0.4 gauss2 for I at 298"K, it can be concluded that the NH3 groups are reorienting about the
Figure 1. Proton magnetic resonance derivative curves of: ( a ) Ni(CN)2NH3-C6D~ a t 298°K; ( b ) a t 100'K; (c) Ni(CN)2NDa.CsH6a t 298°K; ( d ) a t 100°K; a t 298°K; ( f ) a t 100'K. (e) Ni(CN)2NH3.C~H~ The zero field has been taken as the field value of a purely diamagnetic reference sample.
C3v axis at this temperature.s At 100°K both AH and Sz for I are much larger than the values at 298°K. Although the Sz value is less accurate at 100°K because of the marked asymmetry of the signal, the magnitude indicates that the NH3 groups are most likely rigid at this temperature. The line-shape changes with
(2) H. Nakajima, V. M. Bhatnagar, and A. R. H. Cole, J. Phys. SOC. Japan, 17, 1194 (1962). (3) H. Nakajima, ibid., 20, 555 (1965). (4) K. Takahashi (Umemoto), Ph.D. Thesis, University of Toronto, 1965. (5) Ho represents the field value for a purely diamagnetic sample while H* is the field value for the proton signal studied, Le., NHa or C8H0. (6) E. R. Andrew, "Nuclear Magnetic Resonance," Cambridge University Press, Cambridge, England, 1958, Chapter 6. (7) A small residual signal of approximately the same line width and intensity was noted for the fully deuterated complex, Ni(CN)2. ND3. C6Ds. Since the starting materials were all better than 95% deuterated the residual signal must be due to tightly bound hydrate water which does not exchange during the synthesis steps. (8) The theoretical rigid lattice second-moment value for the NH, groups is 30.2 gauss*.
Volume 71, Number 2 January 1967
NOTES
452
temperature for I are similar to changes noted previously for nickel aminess and can be attributed to an electron-nucleus interaction between paramagnetic nickel atoms and the ammonia protons in the clathrate. Magnetic susceptibility measurements have shown that more than 50% of the Ni atoms are paramagnetic in the Ni(CN)&H3iCI clathrates.losll An electron-nuclew interaction of the Fermi contact type is further confirmed by the large shift of -10 gauss of the resonance frequency upon lowering of the temperature.
Table I: Second Moments, Line Widths, and Field Shifts for Proton Signals in Ammonia Nickel Cyanide Benzene Clat hr:ttesa Ni(CN)zND:* CsHs
Ni(CN)zNH:. C,Ds
(theoretical), gaussa 4.3 Rigid lattice Hexad axis, 1 . 1 Czv axis, Rot'ation S2 (observed), gaussZ 9.6 100°K 1.6 298°K A H , gauss 8.5 100°K 2.6 298°K H* - H . gauss 1.2 100°K 0.0 298°K S2
Q
30.2
7.6 39.5 12.6 17.3 2.1
9.8 2.0
Observed Sz values are accurate to 3~0.4gauss;2 line widths
are given &s the distance between the maximum and minimum of the derivation curves and are accurate to f 0 . 2 gauss; H* denotes the field value a t the zero value of the derivative spectrum.
The spectrum for the benzene protons in I1 at 298"K, Figure le, shows a single slightly asymmetric derivative curve with a second-moment value of 1.6 f 0.2 gauss2. This value is appreciably less than the rigid lattice value, 4.3 =t 0.2 gauss2, and the benzene molecules are therefore reorienting about their hexad axis12,13 a t 298°K. At lOO"K, the signal for I1 shows a pronounced asymmetry and the line width increases to 8.5 gauss. The second moment, S2 = 9.6 f 0.2 gauss2, evaluated by the method of Andrew and Tunstalll* for asymmetric lines, is more than twice the value calculated for a rigid lattice, Table I. This difference can be attributed to an "indirect" electron dipole-nuclear dipole interaction between the Ni(I1) atoms and the benzene protons. The mechanism for this interaction differs from that for the Ni(II)-NH3 The Journal of Physical Chemistry
interaction since the latter is due to the presence of unpaired electron spin density at the ammonia protons, a situation which is unlikely for the nickel-benzene interaction. The present second-moment value for benzene in clathrate I1 at 100°K is appreciably larger than the value reported by Nakajima3 for benzene in the fully protonated clathrate, I11 at 77°K. The discrepancy cannot be attributed to the temperature difference or to an additional intermolecular contribution in I11 since both effects would act in the opposite direction to the observed discrepancy. Since the values given by Nakajima3were apparently derived from line shapes obtained by fitting the observed curve to two assumed component curves, it is likely that at least part of the discrepancy arises from this procedure. The spectra for the fully protonated clathrate, I11 (Figure le, f), are markedly asymmetric at both of the temperatures reported with the asymmetry most pronounced at 100°K. The present spectra also show considerably more detail at both temperatures than the earlier spectrum reported at 90°K by Nakajima.2 Comparison of the spectra in Figure la, c, and e shows that the asymmetry for I11 is due t o a superposition of two signals, i e . , NH3 and C6H6, with different H* ~ a l u e s . 1 ~The difference of approximately 2 gauss is presumably due to the electron-nucleus contact interaction between Ni(I1) and XH3 which acts to shift the NH3 protons upfield. Because of the asymmetry, a satisfactory estimate of the second moments is not feasible. However, it is not likely that the magnitude will differ very much from the sum of the values for the two deuterio clathrates. The motional properties of the enclathrated benzene in I11 are not expected to differ from those in the deuterated cases. Acknowledgments. The financial support of the National Research Council of Canada and the Ontario Research Foundation is gratefully acknowledged. Part of this work was completed at Argonne National Laboratory with the support of the U. S. Atomic Energy Commission.
(9) P. H.Kim, J. Phys. SOC.Japan, 15, 445 (1960). (10) M. Kondo and M. Kubo, J. Phys. Chem., 61, 1648 (1957). (11) R. S. Drago, J. R. Kwon, and R. D. Archer, J. Am. Chem. SOC., 80,2667 (1958). (12) Other modes of reorientation, e.g., about the twofold axis are restricted by the geometry of the cavity.'a (13) H. M. Powell, G. Huse, and P. W. Cooke, J. Chem. Soc., 153 (1943). (14) E. R. Andrew and D. P. Tunstall, Proc. Phya. Soc., 81, 986 (1963). (15) H* is taken as the field a t the zero of the derivative curve for the given signal.
