3669
HYDRATED CEROMIC IONS IN H,O AND HzO-DaO MIXTURES
Proton Spin Relaxation and Exchange Properties of
Hydrated Chromic Ions in M,O and H,O-D,O Mixture@ by B. F. Meltonlb Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74074
and V. L. PollaklC Department of Physics, Case Western Reserve University, Cleveland, Ohio, 4.4106 (Keceived December $6, 1068)
E;xperimental results are presented on proton spin relaxation in Cr(II1) solutions of HzO and mixtures of HnO and DzO,in fieldsfrom 1.1 G to 14 kG, and at temperatures from 0 to 9 9 O . Water solutions containing 0.10 M HClOI give results which reflect the relaxation and exchange properties of Cr (HzO)eaealone, uncomplicated by the effects of hydrolysis products or polymers which cause thermal hysteresis or by acid-catalyzed proton exchange which is present in solutions of high acidity. Measurements on weakly acidified solutions confirm that Cr(HzO)sa+is a more strongly relaxing species than Cr(H~O)~0Kze, but the latter undergoes more rapid proton exchange with the solvent water. A homogeneous (local)proton exchange mechanism involving the coordinated OH- ion is proposed to account for this result. Measurements on 0.10 M HClOd solutions containing varying mole fractions of H20 and DzO indicate that the principal effect of deuteration at low temperatures is an increase in the lifetime of protons in the primary hydration sphere of Cra+,resulting from the greater basicity of HzO as compared to DzO. At high temperatures TZremains constant, whereas T1 decreases with deuteration due to slower rotational motion of the hydrated complex. The latter is correlated with the marked increase in viscosity which occurs in going from HnO to DzO.
+
+ p (T1, +
Introduction
(TlobS)-1 =
Although the basic mechanism for proton spin relaxation in aqueous chromium(II1) solutions is well understood,2 relaxation times reported by different observers for apparently identical systems do not Thermal h y s t e r e ~ i s , solution ~~' aging,3 specific anion eff ectsJ8 and pH d e p e n d e n ~ e ~have r ~ ~ all been noted. Some of these effects appear to be a result of the unusual features of chromium(II1) chemistry. In aqueous solution Cr3+ hydrolyzes readily, forming CrOH2+ and higher hydrolysis products, which have been showng to have a pronounced effect on proton relaxation times in the solution. Chromium(II1) also forms stable coordination compounds with a large number of other ligands, and at high temperatures long-lasting hydrolytic polymers are easily formed.ll Once formed, these polymers break down very slowly, even in highly acid media, giving rise to the thermal hysteresis effects which have been reported.' Chromium is one among several paramagnetic ions in which the lifetime for protons in the primary hydration sphere a t low temperatures is relatively long, so that the observed relaxation rate of the solution is limited by the rate of proton exchange between the hydration sphere of the ion and the bulk water. At high temperatures exchange is fast and the solution relaxation rate is limited by the rate of relaxation in the ion hydration sphere. The observed proton relaxation time in a solution containing only Cr(H20)63+is given by the expression12
where the subscripts x and w refer to water of hydration in C ~ ( H , O ) Band ~ + bulk water, respectively. The quantity TxwBis the proton relaxation time in the bulk water due to dipolar interaction with the Cra+ spins, and Tlw0is the relaxation time due to other mechanisms in the solvent, The latter quantity can be equated to the relaxation time in the solvent not containing paramagnetic ions, but otherwise under conditions identical with those of the studied solution. The time rxw is
( T l W O ) -l
( T I W S ) -I
r x w ) -1
(1)
(1) (a) This work was supported in part by the U.6. Army Research Office (Durham). (b) NASA Trainee 1966-1969. To whom all correspondence should be addressed. (0) On leave from Oklahoma State University, 1967-1968. Now at Department of Physics, University of North Carolina, Charlotte, N. C. 28206. (2) N. Bloembergen and L. 0. iMorgan, J . Chem. Phys., 34, 842 (1961). (3) G.Laukien and J. SchlBter, Z.Phya., 146, 113 (1956). (4) R. Hausser and G. Laukien, Arch. Sci. (Geneva), 11, 252 (1958). (6) R. Hausser and G. Laukien, 2.Phys., 153,394 (1959). (6) L. 0.Morgan and A. W. Nolle, J . Chem. Phys,, 31,365(1959). (7) T.H.Brown, R. A. Bernheim, and H. S. Gutowsky, ibid., 33, 1593 (1960). (8) R. Hausser and F.Noack, 2.Phys., 182,93 (1964). (9) C. E.Manley and V. L. Pollak, J . Chem. Phys., 46,2106 (1967). (10) T.J. Swift ana T. A. Stephenson, Inorg. Chem., 5 , 1100 (1966); T. J. Swift, T. A. Stephenson, and G. R. Stein, J . Amer. Chern. SOC., 89,1611 (1967). (11) J. A. Laswick and R. A. Plane, ibid., 81,3564 (1959). (12) A more general expression for Tlobs,valid for solutions containing Cr(HzO)aOH2+ in addition to Cr(HzO)s3+, is given in the Appendix, Also reproduced there are theoretical expressions for 2'1, and @ ', which will be used later in interpreting the experimental data. Volume 79, Number 11 November 1960
36’70
B. I?. MELTON AND V. L. POLLAK
the average lifetime of a proton in the hydration sphere of Cr(HzO),?+, and p is the probability of finding a
particular proton in Cr(H,O)&+. We assume a relation similar to eq 1 holds for T20”’,although in general there will also be transverse relaxation by the so-called “Au effect.”l3 However, on the basis of the parameters derived herein, one can show that this relaxation mechanism is negligible in the case of Cr(Hz0)6s+solutions up to 14 kG,which was the highest field studied. Protons in Cr(H20)2+ are known to exchange with the bulk water by transfer aCTOSS hydrogen bonds rather than by exchange of whole water molecules. The latter have hydration lifetimes of the order of hours. l4 In intermediate acid concentrations the dominant exchange mechanism is believed to be the first acid hydrolysis reactionQJ5 ki
Cr(HzO)ea+
+ H2O J_ Cr(Hz0)60H2++ HsO+ k- 1
(2)
I n highly acidified solutions there is an additional acidcatalyzed proton exchange mechanism, lo but its effect is negligible in the range of acid concentrations studied herein. I n weakly acidified or nonacidified solutions complications arise due to the formation of Cr(HzO)aOH2+, which is believed to have a pronounced effect on the observed relaxation rate due to its rapid rate of proton exchange with the bulk waterag The effects of isotopic substitution of deuterons on the proton relaxation times in aqueous solutions of Cr(NO& have been studied by Mazitov.16 The observed increase in the proton relaxation times with deuteration was interpreted on the basis of an increase in the lifetime of protons in the primary hydration sphere of CrS+. It has been recently suggested that changes in proton relaxation times as the solvent is changed from HzO to D,O can also be expected on the basis of a change in the probability of finding a proton in the ion hydration sphere.17 This arises due to the difference in vibrational zero-point energies of protons and deuterons, which can give rise to a proton fraction in the hydration sphere which differs from that in the bulk. I n this case the observed relaxation time would be given by the equation ( T l O b S ) -1
+( + ~(TIx + T x w ) - ~ [ -( ~ KIP + K1-’
= (TlWO) -1
T1wS) -l
(3)
where P is the fraction of protons in the solution. The constant K is the equilibrium constant for the exchange reaction
H,
+ D w Z ? D x + H,
(4)
where K is assumed to be independent of p. l8 Its temperature dependence is given by the relation K are aHo’RT)7 where ”” and exp(Aso’R) between the the differenceS in entropy and products and reactants in their standard states. This The Journal of Physical Chemist?@
theory was in partial agreement with published experimental results, but definitive conclusions were not possible on the basis of available data. I n the present study, measurements of the field and temperature dependence of proton spin relaxation times were undertaken on a 0.10 M HC10, solution of Cr(NO& for the purpose of determining the rate oonstants of reaction 2 and the relaxation parameters of the hexaaquo species Cr(H20)e3+. Measurements were made as a function of mole fraction of DnO to determine the quantity K in eq 3 and to study the effect of deuteration on the other parameters influencing proton spin relaxation. Measurements were also made on weakly acidified solutions of Hi0 which contained appreciable quantities of the first hydrolysis product Cr(H20)SOH2+. Its relaxation and exchange properties were examined and compared with those for Cr(HzO)6a+.
Experimental Section Measurements of proton TIin the range from 1.0 to 520 G were made using the earth’s field free precession technique.19*20 In fields between 1 and approximately 250 G the method is as follows: The sample, approximately 400 ml in volume, is placed inside a coil whose axis is oriented perpendicular to the earth’s magnetic field, Be. At time t = 0 the current is turned on through the coil, establishing a polarizing field B, of the order of 500 G along the axis of the coil. The magnetization within the sample grows exponentially toward its equilibrium value with a time constant Ti,, the spin-lattice relaxation time in B,. After a time t = &, long compared to TI,,the polarizing field is reduced to some intermediate value Bin At time t = t, ti the intermediate field is reduced suddenly to zero, leaving the magnetization at right angles to the earth’s magnetic field. The initial amplitude of the ensuing free precession signal is a measure of the amount of relaxation that took place in the interval ti. Variation of t i in successive measurements allows calculation of the relaxation time Tli in the intermediate field. To measure TIin the field B,, the magnetic field is reduced to zero a t time t = t,-ie., ti = 0. By varying t,
+
(13) T.J. Swift and R. E. Connick, J . Chem. Phys., 37, 307 (1962). (14) J. P. Hunt and R. A, Plane, J. Amer. Chem. SOC.,76, 5960 (1954). (15) R. G.Pearson, J. Palmer, M.M. Anderson, and A. L. Allred, 2.Elektrochem.964, 110 (1960). (16) R.K.Mazitov, Dokl. Akad. Nauk SSSR, 156,135(1964). (17) V. L. Pollak and R. R. Slater, 2. Naturforsch., A , 2 2 , 2110 (1967). , . (18) Although not stated explicitly in the derivation in ref 17,eq 3 holds only if the concentrations of the species AH,Dn-,, where m = 0,l.. .n,are related to one another by statistical factors alone [see A. E. Brodsky, Trans. Faraday SOC.,33, 1180 (1937)I. This is a very good approximation for solvent water and should be expected t o hoId for coordinated water as well. (19) M. Paokard and R. Varian, Phys. Rev., 93,941 (1954). (20) A. Abragam, “The Principles of Nuclear Magnetism,” Oxford University Press, London, 1961,p 64.
HYDRATED CHROMIC IONB IN H,O
AND
36'11
H20-Dp0 MIXTURES
the relaxation time T I , can be determined. Measurements a t 14 kG (60 MHz) were made using standard spin-echo nmr techniques.*l The sample temperature was controlled, in the earth's field technique, by pumping heated or cooled water around the sealed glass sample bottle. The tempering liquid was doped with Mn2+ ions so as to suppress its contribution to the nmr signal. At 14 kG the sample temperature was controlled by a regulated gas flow cryostat. Temperature stability was approximately zkO.5" with both systems. The chromium solutions were prepared from reagent grade Cr(NOJ3 and were acidified with HC104. All were prepared by dilution of the same master solution, which was 0.10 l&' in HC104. No measurable change could be detected between a freshly prepared master solution and one stored for over a year at room temperature. Values of the viscosity of H20 used in calculating rotational correlation times were taken from standard tables;2* those for DzO were calculated from q(DzO)/ 77(H20) ratios taken from the data of Hardy and CottingtonZ3 and ]Lewis and McDonald.24 Viscosities for mixtures of HzO and DzO were estimated by assuming a linear variation of viscosity with proton fraction.24 For the calculation of Tlws,self-diffusion coefficients for H2O were taken from the data of Simpson and Carr.26 Self-diffusion coefficients for Cr3+do not appear to have been measured and were approximated by the values for La3+, which were calculated from conductivity data using the Nernst equation. 26 Errors introduced by this approximation are not serious since diffusion coefficients for the ions are less than 25% of those for water, and only the sum appears in the calculations. Self-diffusion coefficients for protons in mixtures of H20 and DzO have been measured by Douglass and McCa1lZ7at 25". Values at other temperatures were calculated using the Stokes-Einstein equation
DCe,T) = kT/67i.r(P)q(PjT) Values of q(P,T) were known experimentally, and the p dependence of the Stokes-Einstein radius r could be calculated from the known variation of D and q with proton fraction at 25". Self-diffusion coefficients for Cr3+ in deuterated solutions were calculated from the Stokes-Einstein equation by assuming that the ratios r(P)/r(P = 1)were the same for Cr3+as for HzO.
Results and Discussion Solutions of Moderate Acidity. The experimental field and temperature results for chromic ions in 0.10 M perchloric acid solution of water are shown in Figures 1-3. The observed proton relaxation times in the solution have been corrected for relaxation in water not containing paramagnetic ions using the relation (TPO')-l E (Tlobs)-l- (Tiw')-'
(5)
i
i.0'C
I
1.0
,
l I l I I 1
I
I
10 FIELD (Gauss)
I
l 1 1 1 1 1
I
I
I
/
I
100
Figure 1. Field dependence of proton relaxation times a t different temperatures in a 3.0 X M Cr(NOa)a solution of 0.10 M HC104. Observed values have been corrected for relaxation in water using eq 5 and normalized by multiplying by the probability of finding a proton in the hydration sphere of a paramagnetic ion. The p T p r values were independent of the concentration of paramagnetic ions. The solid lines are least-squares fits t o the experimental data using the equations given in the text.
with a similar definition for TCor. According to eq 1, pTIOor should be independent of the concentration of paramagnetic ions, which was indeed found to be the case. The experimental curves can be accounted for satisfactorily by using eq 1 along with the standard theoretical expressions for T I , and TIWswhich are given in the Appendix. I n all, the equations contain five unknown quantities which are to be determined by fitting the theoretical expressions to the experimental data. They are the proton lifetime T ~ the ~ rotational , correlation time rl.of the Cr(HzO)&+complex, the correlation time for electron relaxation r y , the scalar coupling constant A , and the constant C appearing in the BloembergenMorgan equation for r l S ,which is related to the asymmetry of the crystalline field around a Cr3+ ion. For the ion-proton distance in C ~ ( H Z O ) Bwe ~ + used the value r = 2.74 obtained by Hausser and Noack.8 We used the same value for the closest distance of approach of a proton in the bulk to the ion spin since larger values gave slightly poorer fits to the data. According to the Debye model,28 the temperature (21) H.Y. Carr and E. M. Purcell, Phys. Rev., 94, 630 (1954); S. Meiboom and D. Gill, Rev. Sci. Instr., 29, 688 (1958). (22) "International Critical Tables," Vol. V, E. W. Washburn, Ed., McGraw-Hill Book Co., Ino., New York, N. Y.,1926,p 10. (23) R. C. Hardy and R. L. Cottington, J . Chem. Phys., 17, 509 (1949). (24) G. N.Lewis and R. T. McDonald, J . Amer. Chem. Soc., 55,4730 (1933). (25) J. H. Simpson and H. Y. Carr, Phys. Rev., 111, 1201 (1958). (26) B. Ottar, "Self-Diffusion and Fluidity in Liquids," Oslo University Press, Oslo, 1968,p 118. (27) D. C. Douglass and D. W. McCall, J . Chem. Phys., 3 1 , 569 (1959). (28) N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev., 73, 679 (1948). Volume 73,Number 11 November 1969
3672
B. E'. MELTON AND V. L. POLLAK
'
O
O
i-
;.'...
I-
I
T
-I
/
l
2.8
/
l
3.0
L
I
1
3.4
3.2 1 0 3 1 ~P
I
3.6
I
/
/2xwcp=l.o1
I
3.8 1 0 % ~(OK"')
K-~
Figure 2. Temperature dependence of p T p r for Cr(N08)8solutions at various fields and acid concentrations. For solutions containing 0.10 M HClOd, p T p r values are denoted by 0 at 520 G and by 0 a t 1.1 6. The solid lines result from the least-squares fit to the experimental data and also indicate t,he temperature dependences of the proton relaxation time Xlws in the bulk water due to dipolar interaction with the ion spins, the proton lifetime T ~ ~ , the electron relaxation times TI,P~, and the rotational correlation time cr for Cr(H20)Ea+. The constants P and G are given by F = 4/30 X(X 1) y 1 ~ y s ~ f i ~and r-6 G = 2/3 S(S l)(A/fi)z (see eq A2 of the Appendix). For a 0.001 M HClO, solution of 3.0 X M Cr(NOs)8,the p T p values are denoted by V at 410 G and by A at 1.1 G. This solution exhibited thermal hysteresis, as shown by the values V and A, which were measured at room temperature after the sample had been heated to 75". The p T p values for this solution were dependent on the concentration of paramagnetic ions due to the shift in equilibrium of reaction 2 to the right with decreasing metal ion concentration.
Figure 3. Temperature dependence of p T p r and p T p at 14 kG for 0.10 M HCIO4 solutions of Cr(NO&. Values for a solution containing 100% Ha0 (P = 1.0) are denoted by 0 and 0 . Those for a 80% 4 0 - 2 0 % HsO solution (p = 0.2) are denoted by A and A. The solid lines result from the leastsquares fits to the experimental data. Those for the = 0.2 solution were calculated assuming only changes in rotational correlation times and in the activation energy for proton exchange in going from HZOto a mixture of €120and DzO. The notation is the same as that in Figure 2.
+
+
dependence of the rotational correlation time can be computed from the expression rT = 4nqa3/3kT
(6)
where 7 is the bulk viscosity of the solution at temperature T, and a is the radius of the coordinated paramagnetic ion. For rxw and rv we assume exponential temperature dependences of the forms rxw = rxwoexp(V,,/RT) and r y = rvOexp(V,/RT). The solid curves through the data points in Figures 1-3 represent leastsquares fits of the theoretical equations to the experimental data, using as adjustable parameters the unknown quantities T ~ V,,,~ rye, ~ VTn , a, C , and A . All are determined unambiguously from the combination of field and temperature results and are presented in Table I. The remaining curves in Figures 2 and 3 The Journal of Physical Chemistry
Table I : Parameters Obtained from the Least-Squares Fit of the Equtions in the Text to Proton Relaxation Data on Cra+Solutions of Ha0 4 . 1 X 10-14exp(ll.2/RT) x lO-l*exp(2.l/RT) a = 4.4A C = 1 . 3 X lozosec-z A/h = 2 . 1 MHa rxw = T~
= 9.3
result from the least-squares curve fits and may be used to synthesize all the experimental data. At low fields where ws2rv2 T , , at O D and 380 G since at low temperatures T~~ competes with T~ similar Droton transfers have been Dostu_ ~ four-center . ~ for the correlation time of the dipolar interaction. At high temperalated p r e v i o u s ~ y ~But ~ such mechanism w o u ~ ~ a l ~ o w tures T~ ?< ~1~ for both Cra+ and CrOHZt, which makes Tiy = Fix in high fields, since T~ is expected to be approximately the same for only one of the eleven protons in Cr(H20)50H2+to be both species. exchangeable with the bulk, and the increase in proton (39) N.,Bjerrum, Z . Phya. Chem. (Leipzig), 73,724 (1910). rate with the formation Of CroH2+ (40) H. Kwart,, L. P. Kuhn, and E. L. Bannister, J . Amer. Chem. only be accounted for if CrOH2+were a more strongly floc., 76, 5998 (1954); J. Hine, “Physical Organic Chemistry,” McGraw-Hill Book Co., Inc., New York, N. Y., 1962,p 112. relaxing sDecies than Cr3+, which has been shown not (41) C. G. Swain and J . F. Brown, J . Amer. Chem. Soc., 7 4 , 2634 to be the case. Four-center proton transfer involving (1952): E. S. Gould, “Mechanism and Structure in Organic Chemistry,” Holt-Dryden, New York, N. y., 1959, 542. coordinated water molecules is not important for Cr-
YH’
~
~
Y
L
The Journal of Phyaical Chmistru
3675
HYDRATED CHROMIC IONS IN H 2 0 AND H20-DzO MIXTURES
'1 ; 0.2
4
H
0.4
4
0.6
1 0.8
.(('cI
1.0
PROTON FRACTION
Figure 4. Proton relaxation a t 14 kG in 0.10 M HClOd solutions of Cr(N03)$of varying proton fraction 0. Values of p T l , p at 0" are denoted by 0 and 0 ; by A and A a t 28'; and by 0 and at 99'. The values denoted by X represent the p dependence of the proton lifetime in the hydration sphere of Cr3+ at 28'. The solid curve was calculated using the ~ ( p ) / ~=( p1) ratios from the theory of Bunton and Shiner and the value 701 = 1) = 6.3 psec.
result was noted by Baker,4z who observed that the viscosity of heavy water equals that of light water a t a temperature 8.5" less, from 16" to the highest temperature studied, 35". At low temperatures where rxw >> Tlx,TZX, both pTY' and p T Y increase with decreasing proton fraction p. From eq 3 this could be due either to an increase in the term in brackets with decreasing p (ie., K > l ) , an increase in rXw, or both. The relaxation time TIw' contributes significantly to TlCOr a t O", but it gives an isotope effect in the opposite direction due to the decrease in self-diffusion coefficients of the solvent protons with partial deuteration. Near 100" where rXw ( T ~ ~ / T ~ ~T) (~T ~~~(All ) T ~modulation ~ of the quadratic crystalline field splitting by collisions with water molecules outside the complex. where the subscripts 2, y, and w refer to the species The resulting equation was C T ( H ~ O ) ~Cr(Ht0)50H2+, ~+, and the bulk water, respectively. The quantities x, y, and w are the molar concentrations of the various species in the solution; r j Wis the average lifetime of a proton in the species j before transfer to the bulk water; and 21' , is the proton where T~ is the characteristic correlation time for the relaxation time in environment j. The quantity relaxation, and C is a constant for the hydrated com(TlWs)-l is the relaxation rate in the bulk water due to T expect ~ ~ rzS < plex. I n high fields where w ~ 2~1 we dipolar interaction with the spins S and is proportional 7 I a , but in such high fields the terms involving ~2~ in to the number of ion spins per unit volume; ( Z ' I ~ ~ ) - ~eq A2 and A3 are negligible. is the relaxation rate due to other mechanisms in the Theoretical equations for Tl,zwshave been derived bulk water and can be equated to the relaxation rate in by Pfeifer.b1 His model includes modulation of the the solvent not containing paramagnetic ions, but dipolar interaction by translational diffusion as well as otherwise under conditions identical with those of the by relaxation of the ion spin. With the usual approxistudied solution. The proton lifetime T~~ is related to the forward rate constant of reaction 2 by the equation T~~ = 12/lc1.50 The quantity T~~ is included to allow (50) Note that the rate constant kl of this paper corresponds t o kiw for transfer of protons from the species y = Cr(H2O)sof ref 9. (61) H. Pfeifer, Ann, Phys., 7, 1 (1961). OH2+ to the water system. Such transfer is assumed
+
The Journal of Physical Chemistrtd
3679
DIFFUSION OF HALOGENATED HYDROCARBONS IN HELIUM mations 01 to
