ELECTRICAL BIREFRISGENCE AT HIGHFIELDS
Dec., 1963
neutron irradiation of crystalline nitrogen-containing has provided two interesting organic molecules, insights into the chemical processes which possibly occur during the recombination period of the recoil radiocarbon atom into stable chemical species. (i) The carbon-14 recoil atom does not appear to be stabilized within the ljolid matrix in the form of “sample” compounds or in the form of a chemical species which would undergo reaction, such as decomposition or rearrangement, in the presence of water to produce “simple” compoundrr. Subsequent to a replacement 3-4,8t11
2691
reaction, a fragmentation of the molecule occurs in which only one or two chemical bonds are broken. Then the large molecular fragments containing the recoil carbon-14 atom may undergo additional reactions with the parent molecule or other molecular fragments present within the immediate vicinity to produce the final chemical species observed in solution. (ii) Molecules of similar structure appear to undergo replacement reactions which lead to the formation of similar matrix-stabilized species, containing the recoil carbon-14 atom.
:ECLECTRICAL BIREFRINGENCE A T HIGH FIELDS’ BY D. N. HOLCOMB~ AND IGNACIO TINOCO, JR. Department of Chemistry, University of California, Berkeley, California Received June 1’7, 1963 Calculations of O’Konski, et al., for steady-state electric birefringence a t high-field strength have been extended to the most general molecular model in which both the electrical and optical polarizabilities need have no symmetry and the permanent dipole moment may have any orientation. The steady-state electric birefringence has been calculated as a function of electric field strength. Values of the electrical orientation energy up to 30 timeai kT have been considered. Because the orientation energy due to the permanent dipole is linearly dependent on field strength, while that due to the induced dipole depends upon the square of the field strength, it is found that the normalized birefringence ( A n ( E ) / A n ( E +.m )) may pass through B minimum or a maximum when plotted against the field strength. These calculations allow a new interpretation of the data previously obtained by Haschemeyer and Tinoco from electric birefringence studies of fibrinogen. The application of the results to magnetic birefringence, electric dichroism, and optical rotation of electrically oriented samples is discussed.
htroduction For low electric-field intensities, the magnitude of the birefringence observed for many substances may be expressed by the Kerr3 law An = K E 2
(1)
where An is the birefringence, E is the external field intensity, and K is the Kerr coefficient characteristic of the given sample. For high-field strength, the proportionality will not, in general, be maintained. 4 ~ 6 O’Konski, et aZ.,6 have done numerical calculations of the high-field birefringence of particles having cylindrical symmetry of their electrical and optical properties. Their model, in which the directions of the dipole moment and the largest principal polarizability coincide, appears to represent a number of rigid macromolecules such as, for example, tobacco mosaic virus. Shah’J has extended the calculations to a cylindrically symmetric model in which the dipole moment and the largest principal electric polarizabilities are perpendic(1) Presented in part a t the 144th National Meeting of the American Chemical Society in Loa Angeles, California, April, 1963. Supported in part by U. S. Public Health Service Grant G M 10840, b y an unrestricted grant from Research Corporation, and by the U. S. Atomic Energy Commission. (2) National Science Foundation Postdoctoral Fellow. (3) J. Kerr, Phil. Mae., 60, 337, 446 (1875); 8 , 85, 229 (1879): 18, 53, 248 (1882); 87, 380 (1894): 38, 144 (1894). (4) H. A. Stuart, “Hand- und Jahrbuch der Chem. Physik,” Vol. 10, part 3B, Akademisohe Verlagsgesellschaft, Leipaig, 1939; A. Peterlin and H. A. Stuart, ibid., Vol. 8, part 113, 1943. (5) C. G. LeFevre and R. J. W. LeFevre, Rev. Pure A p p l . Chem., 6 , 261 (1955). (6) C. T. O’Konski, K. Yoshioka, and W. H. Orttung, J. Phys. Chem., 68, 1558 (1959). (7) M. J. Shah, “Proceedings of the 6th International Congress for Optics,” Munich, 1962, p. 209. (8) M. J. Shah, I B M Technical Report, TR-02-250 (1963); J . P h y s . C h e m . , 67, 2215 (1963).
ular. This model has been used to explain the observed electric birefringence of bentonite su~pensions.~-~’ For other materials, the cylindrically symmetric model does not appear to be an adequate representation. Recent measurements of Haschemeyer and Tinocol2013 on the protein, fibrinogen, have been interpreted by assuming a permanent dipole moment which is perpendicular to the symmetry axis of the molecule’s electrical polarizability. The present work presents a survey of the steady-state or equilibrium birefringence of a more general model which is schematically represented in Fig. 1.
Theory It is possible to represent the electrical properties of any molecule by three principal electric polarizabilities and three components of an electric dipole moment related to three perpendicular axes.I4 These polarizabilities and dipole moment components are designated as a 2 and ~ f ,respectively, where i (= 1, 2, or 3) specifies a particular axis of the molecule. (See Fig. 1.) If the molecule is placed in an electric field, there (9) M. J. Shah and C. M. Hart, I B M J. Res. Develop., 7 , 44 (1963). (10) M. J. Shah, D. C. Thompson, and C. M. Hart, J . Phy9. Chem., 67, 1170 (1963). See also H. Meuller, Phys. Rev., 55, 508, 792 (1939); F. J. Norton, ibid., 66, 668 (1939); H. Meuller and B. W. Sakmann, ibid.. 5 6 , 615 (1939); B. W. Sakmann, J. O p t . Soc. A m . , 86, 66 (1945). (11) C. Wippler, J. Chzm. Phys., 69, 328 (1956), has explained the orientation of bentonite in a n electric field (as measured by light mattering) by assuming two types of particles, one possessing only a permanent electric dipole moment and the other only a n electrical polarizability. While Shah’s explanation, involving only one type of particle, seems more realistic, i t does not appear t h a t a distinction between the two may be readily made. (12) A. Haschemeyw, Ph.D. Theais, University of California, Berkeley, 1961. (13) A. Haschemeyer and I. Tinoco, Jr., Bioehem., 1, 996 (1962). (14) See, for example, P. Debye, “Polar Molecules,” Dover Publications, Inc., New York, N. Y., 1928.
D.
2692
s.HOLCOMB ASD
IGIV~C TINOCO, I O JR.
1-01. 67
and
CdkT
=
y21sin2 0 cos2
x-
yZ2cos2 e
- Qi223;
(5)
respectively, where
/2
Fig. 1.-Schematic representation of the model. The principal axes of the optical and electronic polarizabilities are taken to be coincident. The molecule may have any hydrodynamic shape.
exists an orientat’ionforce upon t’he inolecule due to the interactions of its permanent and induced moments with the field; the potential energies of these interactions are given by eq. 2 and 3, respectively.
L,Tp = -w.E
(2)
uI= - I / ~ E ~ . E
(3) Here p is the permanent electric dipole moment vector, w E is the electric polarizability tensor, and E is the electric field vector.I6 (In solutions, the quantities one actually deals with are the effective internal field strength, EefE, and the increment Aa: (= Qisolllte - Qisolvent) of the solute’s polarizability over that of the solvent. The equations and results are directly applicable to solutions on making the substitution of these quantities for E and a, which rigorously apply only to molecules in a vacuum.) The molecule will tend to orient so that the potential energy of orientation (I;= Up VI) is minimized. Model.-The model (Fig. 1) used in this work is one in which the principal axes of t’he optical and electrical polarizabilities coincide.16 The orientation of t’he molecule is expressed in t’ermsof three Eulerian angles, 0, x, and 4, which are measured with respect to an external Cart’esian coordinate system, one axis of which is taken as the direction of the electric field. The angle 6, the colatitude, is between the electric field vector and any one arbitrary axis of the molecule, here chosen as axis 3. The angle x measures rotation about this axis, and 4 is the longitude of this axis.” Steady-State Orientation.-Equations 2 and 3 may be written explicitly in terms of Euler’s angles18 as
+
I;,/kT = PI sin 0 cos
x
-
p2 sin 0 sin x - P3 cos R
(4) (15) See, for example, C. J. F. Bottcher, “Theory of Electric Polarisation,” Elserier Publishing Go., IVew- York, K. Y., 1952. (16) Coincidence of the optical and electrical axes is a hypothesis which is not necessarily ralid. (See ref. 15 and 20.) The electric polarizability refers to the polarioahility a t the frequency of the applied electric field, while the optical polarizability refers t o the frequency of the incident light heam. The magnitudes and the principal directions of these polarizabilities need not coincide. Furthermore, each polarizability will still be frequency dependent. The electrical polarizability may depend on the range of frequencies characterizing the electric pulse and the optical polarizability will vary with the wave length of the light. I n general, however, particularly with macromolecules, one cannot determine a l l of the parameters necessary TO specify a completely general model. (17) I. Tinoco, Jr., J . A m . Chem. So?., 77, 4486 (1955). I n this reference + was incorrectly called the colatitude. (18) A convenient table of dot products between molecule and space fixed axes is giren by S. Bhagavantam, “Scattering of Light and the Raman Effect,” h n d h r a University, Waltair, India, 1940, p. 31.
IC is the Boltzmann constant, and T is the temperature. The steady-stat,e distribution function describing the orientation of molecules in an electric field is given as ,.-U/kT
Since U , the orientation energy, does not depend upon the longitude, 4, one may directly integrate over $I in eq. 7 to obtain the distribution function as a function of only 0 and x, The exponential term involving energy may be written as cos 0 e - U / k T = (e -61 sin 0 cos x) (ep2 sin e sin x) ) X (e - 7 2 1 sin2 8 cos2 x )eY8*cosl e (8) where the ea2zE term which would result from eq. 5 has been dropped, as this is a constant factor which would appear in both numerator and denominator in eq. 7 . A final expression for exp(- U / k T ) is obtained by expanding each of the exponents on the right hand side of eq. 8 in an infinite Taylor’s series and then taking the term-by-term product of these five series m
e-U/kT
=
2
K (cos e ) z ( n + q ) (sin e ) 2 ( l + n + p j
x
l,m,n,p,q=0
(cos X)2(z+P)(sin XI’”
(9)
where p
K =
22
2m
2n
(-1) P I Pz P3 Y2IPY32* (21)!(2?7%) !(2n)!p!q!
It can be shown that only terms involving even poweis of P I , pz, and P3 will enter into the final expression for the birefringence and this fact has been taken into account in writing eq. 9.19 Expression for the Birefringence.-The quantity measured in a birefringence experiment is the difference between the solution’s refractive indices parallel and perpendicular to the electric field, An = n ; - n ~ . . Since the difference in refractive index is directly proportiona14 to the difference in optical polarizability, one may consider the corresponding expression
[ k a o , k - j a 0 . j ] f ( S , x ) sin 0 dB dx d+
(10)
In this equation k a,nd j represent unit vectors parallel and perpendicular to the electric field, respectively. Writing the optical polarizability tensor, no in terms of three principal comporlellt,sand unit vectors along three (1s) Due
t o the limits of 8 and x , terms involving odd powers of the
p ’ s are found to be zero in the subsequent integration (eq. 11). If odd powers of p did occur, i t would imply a dependence of An upon the sign of the field.
ELECTRICAL BIBEFRIXGENCE
Dec., 1963 mutually perpendiculan axes (e" 01~~Okk) eq. 10 becomesz0
+
=
alloii
+
- l)f(O,x) sin 0 dB dx
A~O/ALY.O
==
pz
=
=
P(I
2 2 ( c + m +( ~ I ) m p)! X m n p .ta) ll!
2683
(am - l )-! = 1 form = 0 (m - l ) ! m !
pn-1
+ +
+ + + + ( 2 + m + p + d![2(n + Q) + 11! (n
(11)
The first and second integrals on the right hand side of eq. 11 are designated @I and GPz,respectively. The general form of is analogous to that of the @(P,r) defined by O'Konski, et u Z . , ~ for the case of cylindrically symmetric molecules. @1 measures the orientation of axis 3 (taken a,s the symmetry axis for cylindrically symmetric particles) and aZmeasures any preferential orientation around this axis, For cylindrically symmetric particles there will be no preferential orientation of the transverse axes (1 and 2) and, with (x) = 45* and (2 cos2 (x) - 1) = 0, Q2 will always be zero. Even for asymmetric particles @? will approach zero as a high-field limiting value, if the saturation orientation is with axis 3 parallel t o the field. On the other hand, if the orientation is such that axis 1 or axis 2 is parallel to the saturating field, then e = 0 and sP1 will approach -0.5 as a high-field limiting value. From eq. 7, 9, and 11 one may write an expression for the ratio of Aao at E to AaSo (corresponding to a saturating voltage E +- ). as @ =
HIGHFIELDS
a220jj
P3
(1 - cos2 6)(2 cos2 x
AT
-t n)!
S1P(n + a) + 11 + m + n + p + 4 ) + 31 &(I + m + p + 1 ) M n + y) + 11 D S1[2(1+ m + + p + + 3 l W l + p ) - 11 D Sl[2(l + p ) + I l M n + a) + 11 D D ( I + m + p + 1)[2(1+ m + n + p + d +31
sz -8 3
[2(Z
+
=
n
Q)
=
The value of Aa,O will depend upon the particular molecular model and will be given as some function of the principal optical polarizabilities of the molecule. Appropriate forms of Aa,O are given in the following discussions of particular cases. Low Orientation Energy,,-For sufficiently low orientation energy, U / k T 0), the orientation a t saturation must be with axis 3 parallel to the field and consequently @$ must eventually approach zero. Thus, a plot of a2vs.
3 - ( 0 2 2 - 011) 4
) f(0,x) sin
3 ~ 0 ~2 e1
(
JO2"J"
J* ;JoT
0 d6 dx
(1 - cos2 0)(2 cos2 x
(26)
- 1)
f(0,x) sin 0 de dx
One sees by comparison of this equation with eq. 11 that the preceding calculations are again directly applicable. Magnetic Fields.-For magnetic birefringence,* one need only replace the electrical parameters by the corresponding magnetic ones. The permanent dipole moment is set equal to zero, and the electric polarizability tensor is replaced by the magnetic susceptibility tensor, x . As the increment in susceptibility of the solute over the solvent is generally small, magnetic birefringence has not been widely studied. However, as high magnetic fields are becoming commercially available, this should provide a useful technique for studying oriented molecules in electrically conducting solutions. Polydispersity.-.The precedin.; equations and results all refer to monodisperse systems. The effect of polydispersity on the rise, reverse, and decay times in (23) L. I. Schiff, "Quantum Mechanics," XIcGraw-Hi11 Book Co., Inc., New York, N. Y.,1949, p. 245. (24) I. Tinoco, Jr., Aduan. Chem. Ph?~x., 4, 113 (1962). (25) I. Tinoco, Jr., J. Am. Chem. Soc., 81, 1540 (1959).
Dee., 1963
ELECTRICAL BIREFRINGENCE AT HIGHFIELDS
2697
electric birefringence at low-fiefd strengths has been considered by Tinoco and Yamaoka.26 The steadystate or equilibrium effect, Aaoe, may be written as the sum of the equilibriuni effects for each component
where Vi is the volume fraction of component i. Acroe,, will be given for each species by eq. 16 in the lowfield limit and by eq. 11 in general. Each a and ,u will then refer to that value for the particular species. The analysis of experimental curves will be difficult and, in general, one can not distinguish a monodisperse system orienting by a complex mechanism from two or more species orienting by different mechanisms. Time Dependence.-These calculations have not dealt with the transient parts (rise and decay) of the birefringence response to an applied electrical pulse, but have been limited to the equilibrium birefringence. Benoit” considered the transient response (to a rectangular pulse of a small electric field) of ellipsoids of revolution having a permanent dipole moment along the axis of revolution. Tinocol’ subsequently extended the theory a t low fields to a more general model, in which the direction of the permanent dipole moment is arbitrary, and which includes the Kirkwood-SchumakerZSconcept of polarizability due to mobile protons on the surface of the macromolecule. The equations for the rise and decay of the birefringence are explicitly solvable only for cylindrical symmetry of the hydrodynamic form. Xumerical computations concerning the transient birefringence of a more general model a t high fields would be interesting, but it would require so many parameters to describe the results that computation should probably await particular experimental results. Application of the Results.-The calculations presented here can provide an alternative interpretation of the fibrinogen data obtained by Haschemeyer and Tinoc0.12,~3 With materials for which one can approach saturation and experimentally obtain An/Ans, the choice of a model is simplified considerably as one may use the depth of the minima [(a (minimum)] as a criterion for fitting the data. With many smaller molecules, such as fibrinogen and other globular proteins, one cannot experimentally attain the high voltages necessary to bring about complete orientation. One then needs to characterize the curves by other parameters. Useful parameters which are readily available from the fibrinogen data, include, for example, the voltage a t which the minimum occurs [V(X) 16,000 v./cm.] and the voltage corresponding to the intercept of the 6 = 0 axis [V(I) 25,000 v./cm.]. The curves in Fig. 6 have been constructed for several of the models in section D. From these curves amd other similar ones, one may obtain a preliminary fit to the experimental data. Having thus limited the possibilities to only a f m models, one may complete a more careful point-by-point comparison. The magnitude of the transverse permanent dipole moment is immediately given from the intercept voltage [V(I) ] since p12(I) = ,ulz[V(I)J2/k2T2. (26) I. Tinoco, Jr., and K. Yamaoka, J . Phys. Chem., 69, 423 (1959). (27) H. Benoit, Ann. Phys., 6, 561 (1951). (28) J. G. Kirkwood and J. 13. Shumaker, Froc. Nutl. Acad. Sci., 38, 855 (1952).
Pd4
Fig. 6.-p1~(I)/p1~(M)us. &/PI for models having a transverse dipole moment. In choosing a model, these curves may be directly compared with the experimental data. (I corresponds to the 6 = 0 intercept and M to the minimum of the saturation curve.)
-0.038 20
40
60 E~ x
Fig. ‘I.-Comparison
80
I03
IO-^ C V O L T W C M ) ~
of a theoretical curve with Haschemeyer’s12,1* fibrinogen data (see ref. 29).
The fibrinogen data and the computed curve which seemed to give the best agreement with the data (pl2/y32 = 10 and ,83/p1 = 0.2) are shown in Fig. 7. Cylindrical symmet’ry of the optical and electronic polarizabilities of the fibrinogen molecule has been assumed. The resultsz9are ,ul = 2,500 D., ,uz = 500 D. and - a$) = 3 X ~ m . ~These . values may be compared with the original, more qualitative interpretation which was based on the time dependence a t low fields and the assumption that more than one species was present in solution. This led to one particle whose main orienting force (producing a negative An) was (29) The values given here for the electrical parameters may be open t o question, due to some uncertainties in the experimental data. Non-ohmic behavior of the carbon resistor volt,age divider used in determining the values of the applied electric field was discovered after the data were taken and, consequently. the original values of the voltage may be in error. (We are indebted to Mr. C. Paulson for bringing this to our attention and for providing us with a calibration curve t o correct the original voltages.) Further, because only very dilute solutions of fibrinogen were of sufficiently low conductivity t o permit high fields to be established across the cell, the resultant birefringence signals were not large compared to those of the solvent alone.30 Consequently, there is some uncertainty in the original values of the retardation. (30) The Kerr effect in Fater hari recently been investigated by W. H. Orttung and J. A. Meyers, J . Phys. Chem., 6’7, 1905 (1963).
EPHRAIM BEN-ZVI
2698
characterized by pl = 1200 D. and another particle whose main orienting force (producing a positive An) Was A ~ a c t e r Z e dby (maE - ~ 2 =~ 1.2 ) x lo-’’ ~112.~.
Conclusiono--Riearl~ and many low molecular weight globular proteins orient so rapidly that one cannot measure the rise and decay of these times of their birefringence. The calculations show that, so long as one can measure the equilibrium birefringence throughout a sufficiently wide voltage range, electric birefringence studies
VOl. 67
should still yield useful information about their electric and optical properties. As consideration of only cylindrically symmetric models may be inadequate in many cases, whenever possible, the optical and electrical polarizabilities should be completely specified. Acknowledgments*-The authors are indebted to Dr. M. J. Shah for providing manuscripts of some of his work Prior to Publicatsion, to Mr. E. T’inoco for his assistance with some of the Fortran programming, and to Dr. IC. Yamaoka for his helpful criticism.
OXIDATION OF PHOSPHOROUS ACID BY PEROXYDISULFATE. OF THE REACTION I N NEUTRAL SOLUTION’
I. KINETICS
BY EPHRAIM BEN-ZVI~ Department of Chemistry, Immaculate Heart College, Los Angeles 27, California Received June l Y , 1963 Kinetics of the oxidation of phosphorous acid by peroxydisulfate Fas investigated in neutral solution. The experimental rate law, d(S208-2)/dt = [kl kz (ZH~P0~)]’/z(S~O~-2)8/~, was adequately explained by a chain mechanism, initiated by two parallel reactions. The rate constants were determined a t three teniperatures between 48 and 57’, and an attempt was made to estimate activation energies of the elementary processes.
-
+
Introduction Peroxydisulfuric acid, H2S20s,and its salts are powerful oxidizing agents. The standard potential of the half-reaction
SzOs-2
+ 2e-
= 2S04-2
(1) is 2.01 v . ~ ;yet, many of its reactions are slow. Kinetically, peroxydisulfate presents an interesting case as its reduction can occur by means of several distinct mechanisms. Kumerous reactions of peroxydisulfate have been investigated. The reader is referred to two reviews which appeared re~ently.4.~ Phosphorous acid, H3P03, is a good reducing agent particularly in basic solutions. Standard potentials for the reduction of &PO4 to H3P03are -1.12 and -0.276 v., in alkaline and acidic solutions, respectively.6 Xot many kinetic investigations have been reported on the oxidation of phosphorous acid. It appears that here, too, several distinct mechanistic paths are available for the process. Mitchell studied the oxidation of phosphorous acid by iodine in acid solution.7 The work was extended by Griffith and McKeown.8 Similarly, with the rethe oxidation of actions of hypophosphorous HaP03in an acidic solution involves a general acidcatalyzed equilibrium between the “normal” and the “active” forms of the acid. (1) Presented, in part, a t the 144th National Meeting of the American Chemical Society, Los Angeles, Calif., April, 1963. (2) (a) A grant from the Petroleum Resesrch Fund is gratefully acknowledged: (b) Patricia Perez assisted in part of the experimental work. (3) W. M. Latimer, “The Oxidation States of the Elements and their Potentials in Aqueous Solutions,” 2nd. Ed., Prentice-Hall, N e d York, N.Y., 1952, p. 78. (4) D. A. House, Chem. Rev., 62, 183 (1962). (5) W. K. Wilmarth and A. Haim, in “Peroxide Reaction Mechanisms,” edited by J. 0. Edwards, Interscience Publishers, New York, N. Y . ,1962, PP. 175-226. (6) Reference 3, p. 107. (7) A. D. Mitchell, J . Chem. S a c , 1’23,2241 (1923). (8) R. 0. Griffith a n d A. MoKeown, Trans. Faraday Sac., 36, 766 (1940). (9) W. A. Jenkins and D. M. Post, J . Inorg. Nuel. Chem., 11, 297 (1959), and references quoted in it.
acid
HJ’Os(normal)
+iodine
HsPOa(active) -+ products
(2) pH dependence of the reaction in neutral solution indicated that only the monohydrogen phosphite ion, HP03-2, is being oxidized; no measurable reaction between 1 2 and H2P08-was detected.* Certain other reactions of phosphorous acid involve, probably, free radicals.10-11 We have observed that peroxydisulfate oxidizes H3P03 over a wide range of hydrogen ion concentration. The present paper reports the results of the investigation in neutral solution, where phosphorous acid is present as the HzPO3- and HP03-2ions.12
Experimental Materials.-Preliminary experiments indicated that the reaction was slower in redistilled water than in the commercial distilled water. Therefore, redistilled water was used in all kinetic measurements. It was prepared as described elsewhere,13 except that the KHSQa step was omitted. Stock solutions were prepared from reagent grade chemicals. Potassium peroxydisulfate was recrystallized from the redistilled water. All other chemicals were of the best available grade and were used without purification. Phosphorous acid was neutralized by the addition of solid NaQH before diluting to mark in the volumetric flask. Xitrogen was the Matheson prepurified gas.13 Procedure.--Predetermined volumes of all the reagents, except peroxydisulfate, were pipeted into the reaction flasks, which were Pyrex gas washing bottles with fritted-glass dispersion tubes. KzS~OE solution was placed in a separate flhsk. Nitrogen was bubbled through all the solutions during the duration of an experiment. rlfter deaeration a t room temperature and equilibration with a constant temperature bath, the reqction was started by pipeting an appropriate volume of KzSzOs solution into the reaction vessel. (10) N. Kornblum, A. E. Kelley, and G. D. Cooper, J . Am. Chem. Sac., 74, 3074 (1952). (11) K. Nertz and C . Wagner, Ber., 70B,446 (1937). (12) HsPOs is a dibasic acid. cf. textbooks of inorganic chemistry, e.g., W. 3%.Latimer and J. H. Hildebrand, “Reference Book of Inorganic Chemistry,” 3rd Ed., Macmillan and Co., Ltd., London, 1951, p. 229. (13) E. Ben-Zvi a n d T. L. Allen, J . Am. Chem. Soc., 88,4352 (1961).
