599
NOTE8
where
T~
is the longitudinal ultrasonic relaxation time
a is the degree of dissociation, and yk is the mean activity coefficient of the electrolyte. If the formard process is diffusion clontrolled its rate constant may be calculated by the Smoluchon ski-Debye2 formula modiSed3 by the intioduction of the Stokes-Einstein relation
where N is the Avogadro number, k the Boltzmann constant, yl the solvent viscosity, b the Bjerrum parameter
ZA and ZU the ionic charges, e the electronic charge, D the dielectric constant of the solvent, and a the distance of approach of (he two ions in the ion pair. Then
where K ( A ) is the association constant for ion pair formarion, determined for instance by conductance measurements (K(A:i = ki/icn). On the other hand, the dielectric wlaxation time associated 1% ith the dipolar orientation of I he ion pair is given by the Debye relation4
Introducing (15’) rind (VI) into (V), and neglecting e-b in comparison to 1 gives
estimation of the distance of approach of ions can be made through the Fuoss formula,* k r A = K ( A ) = (4rNa3/3000)eb. This gives a = 3.d1 A. [It is believedg that the a value obtained from the J parameter of the conductance equation (a, = 5 b) is less reliable, since the J2c3/2 term was not considered in the analysis and its neglect may affect the determination of J . ] With the above parameters, using D 20.7, 9 = 0.00302 P, the determined T~ = 0.77 X sec,6 and calculating 0 = 3.9 X A I , for c = 0.05 34, one can obtain from eq VI1 T L =- 1.6 X sec. The ultrasonic experimental value, Irl(exp), is 2.0 X 10-9 sec. On the other hand, by retaining UJ = a = 5.0 A, 6 becomes 5.7 X M for G = 0.05 M , and the calculated value of T L becomes 0.5 X see. The agreement between these two results is rather. good considering the approximations involved in the Debye relation,1° in the Smoluchowski-Debye relat i ~ n and , ~ in the evaluation of 0 (as pointed out elsewhere5). The above results, while reinforcing the interpretation based on ionic association of our previous ultrasonic work,G more generally indicate an interesting correlation between the relaxation times of two molecular motions, valid when the common energy barrier is the one for viscous flow. 1=
( 2 ) M. von Smoluchowski, Z . Phys. Chem. ( F m n k f ~ i r am t M a i n ) , 92, 179 (1917); P. Debye, Trans. Eleelrochem. Soc., 8 2 , 265 (1942). (3) S. I’etrucci, 1.P h y s . Chem., 71, 1174 (1967). (4) P. Debye, “Polar Molecules,” Chemical Catalog, Kew York, N.Y., 1929, Chapter V.
( 5 ) G. 6. Darbari and S.I’etrucci, J . Phys. Chem., 74, 268 (1970). (6) E. A. S. Cavell, T m n s . I“aiado~/Soc., 6 1 , 1578 (1965). (7) 1’1. B. Reynolds and C. A. Kraus, J . Amer. C‘hem. Soc., 70, 1709 (1948); M. J. McDowdl and C. A. Kraus, ibid,, 73, 3293 (1951). (8) R . M.Fuoss, ihid., 80, 5059 (1958). (9) J. C. Justice, J . Chim. P h ~ s . 6, 5 , 353 (1968): R . XI. Fuoss and K. L. I-Isia, Proc, >\+-at.A c n d . Sei., 57, 1550 (1966); 5 8 , 1818 (1967). (10) C. 1’. Smyth. “Dielectric Constant and Structure,” McGrawHill, New York, N. Y . , 1955, Chapter 11.
Axial Coordination in the Vanadyl Ion
I n the above, it i s assumed that the ions’ center to center distance a (according to the Debye-Huckel definition of contact, distance of the two spheres of radius a/2) is equal to the radius of the dipole taken as a sphere. We may tesl eq VI1 by applying it to a case where both 71,and 7 D have been reported, namely, the system BuqKBr in acetone at 2 5 ” . Ultrasonic measurements5 were interpreted in terms of diff usion-controlled ion pair formation idectric relaxation data6 were interpreted in terms of dipolar orientation of the ion pairs. Since both the ionic translation for ion pair formation and dipolar orientation are presumed to be opposed in this case by the activation energy barrier for viscous flow, eq VI1 should bse applicable. Using K ( d ) -- 264 M-l from conductance data7 an
by Amos J. Leffler Department of Chemistry, Villanora C n i i m s i t v , Villnnoca, Pennsylcania 19085 (Receiccd Srptcmlier 17, 1870) Publication costs assisted hi/ Villanoaa University
The nature of the aqueous solvation sheath around the vanadyl ion has been the subject of considerable. recent investigation. 1-4 It has been shown conclusively (1) R. Ilausser and G . Laukien, Z . Phiis., 153, 391 (1959). ( 2 ) R. K . M a z i t o v and A . I. Iiivliind, Dok’. Akad. N a u k SSSR, 166, 654 (1966). (3) J. Reuben and D. Fiat, I n o r g . Chcm., 6, 579 (1967).
The Journal of Physical Chemistry, Vol. 7’5, N o . 4, 2.9’71
60
NOTES
that below ambient temperature the four equatorial water molecules are comparatively tightly bound. However, it is uncertain whether 'H and I7Q shift and relaxation effect>sin the bulk solution are due to weak coordination in the axial position opposite the vanadyl oxygen or on the four faces of the pyramid formed by V02+and the four more tightly bound equatorial water molecules. The purpose of this note is to describe the application of a recently reported technique5 for the study of the second coordination sphere to this problem. I n the earlier work it was found that methylene chloride dissolved in deuterium oxide did not enter the first coordination sphere of hexaaquo metal ions. Proton peak bmadenings were due to dipolar coupling between the unpaired electrons and the protons, and only bulk susceptibility shifts were observed. From the data an estinnated distance of approach of the methylene chloride to the metal ion of 0.354 nm mas found. In the present work the peak broadenirigs of methylene chlorid~and H @ were measured in a 0.115 Pi eolutiori of VQSO, 2HzQ dissolved in deuterium oxide between 4 and 32'. Using the second coordination sphere distance measured earlier and assuming the axial proton distance of 0.314 nm estimated in ref 4, values of the correlation times were calculated assuming dipolar c~oupliiig The values of T2Y were calculated from the rclal ion
Table I : Calculated Correlation Times of iblethylene Chloride As a Function of Temperature in Seconds" Second cooid
a
All values are
x
10-10
Axial
sphere
3.82 3.89 3.51 2.21
1.38 1.43 1.27 0.80
see.
-
(2)
All of the correlation time values are within a factor of 3 of each other and therefore they do not serve to indicate the site of relaxation in themselves. A comparison with the correlation times of hexaayuo metal ions measured earlier5 shows that the present second coordination sphere values are approximately an order of magnitude greater. Since there appears to be no reason to expect methylene chloride molecules in the second coordination sphere to behave differently in the presence of vanadyl ion compared with other species, the results tend to rule out second coordination sphere sites as important in the peak broadening mechanism. Rased on the present measurements the axial position is open to both bulk water and methylene chloride molecules. The calculated axial correlation time values are a function of the chosen distance of approach of the proton to the vanadium ion, and the latter value is undoubtedly too great. A similar conclusion was reached in the earlier work where second coordination sphere distance of 0.383 nm was estimated for the V-H distance for water. We can estimate an axial V-H distance for methylene chloride by assuming a correlation time of 2 X IO-" sec found in the earlier work. The value is 0.208 nm, which is unreasonably short on geometrical grounds. Therefore the true correlation time must be between the two values, and it can be inferred that some type of interaction occurs between the vanadyl ion and methylene chloride. A possible mechanism i s a dipolar interaction between the vanadium and chloride atoms. This would have the proper temperature dependence since the thermal motion of the molecules would increase with increasing temperature resulting in shortened correlation times.
where the terms have their usual meanings. These are shown in Table I.
(4) K. Wuthrich arid R. E. Connick, Inorg. Chem., 6, 583 (1967); 7, 1377 (1968). (5) A. J. Leffler, J. Phus. Chem., 74, 2810 (1970).
vihere the N values are the numbers of moles of each substance present and n is the number of positions of each type. The values of NcR~cI~ were those determined earlier5 while N D was ~ ~corrected from pure deuterium oxrdc flor the amounts of vanadyl sulfate and methylene chloride present. Values of n were taken to be 1 for the axial position and 4 for second coordination sites 5s in ref 4. The values of T 2 were those measured by peak broadenings corrected for the broadening observed in pure deuterium oxide. Using ~ correlation times were the caIcuia,ted values of T 2 the calculated from the equation ._ -
_-4- S(S I
2'2M
3
+
l)g2P2gN2PN2To
n2r6
The Journal of Physical Chemislrv, Vol. 76,No.
4, 1971
