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Rotating-frame relaxation rates of solvent molecules in solutions of paramagnetic ions undergoing solvent exchange
JOURNAL
OF MAGNETIC
RESONANCE
59, 36 l-372
(1984)
Rotating-Frame Relaxation Rates of Solvent Molecules in Solutions of ParamagneticIons Undergoing Solvent Exchange S. CHOPRA, R. E. D. MCCLUNG, AND R. B. JORDAN Department
of Chemistry,
University
of Alberta, Edmonton, Alberta T6G .?G2, Canada
Received December 2 I, 1983 The theory is developed for the dependence of the rotating-frame relaxation rate (R,,) of bulk solvent nuclei on the magnitude of the spin-locking field H, (=--w,/y) for a dilute solution of a paramagnetic ion dissolved in a solvent which undergoes exchange between bulk and coordinated sites. When certain conditions are satisfied R,, will vary with w, according to RI, = R;“, + $(R, - R&o;‘, where R;“, = Rzs + PmrmR2m(R2m + r,,,)-‘, .!$ = (R,, + r&R,,,, + rm)-‘{(R2,,, + r,,# + A&}, RI,, R2,,, are the relaxation rates in the coordinated site, Aw, is the chemical shit? between the bulk and coordinated sites, r, is the solvent exchange rate from a coordinated site, Rsr is the sum of the pure solvent and outer-sphere relaxation rates, and P, = (no. coordinated solvent molecules) X (metal-ion molality/solvent molality). Data for cobalt(R) in CHsOD and nickel(I1) in CHsCN are used to show that the R,, measurements are especially useful in the determination of AU,. The latter can be combined with bulk solvent shifts to determine the number of coordinated solvent molecules. INTRODUCTION
The exchange of solvent between bulk (S) and metal-ion coordinated sites (S’)
111
[MS’,] + S = [MS’,-,S] + S
can be determined from transverse relaxation rate (&) measurements of the bulk solvent nuclei. For such systems containing paramagnetic metal ions, the modified Bloch equations (I) have been solved by Swift and Connick (2). Since then, the theory has been refined (3) and expanded (4) and has seen wide application. However, rotating frame relaxation rates (RI,) have been used relatively little since the early developments by Meiboom (5) and Deverell et al. (6), as noted by Reeves (7). In this paper, a theoretical description of RI, is developed from the modified Bloch equations for systems such as in Eq. [ 11. The conditions under which RI, exhibits a dependence on the magnitude HI of the spin-locking field are elucidated, and the influence of the bound solvent chemical shift (AU,) and solvent exchange rate (rm) on R,, are shown. Experimental measurements of R,, for two systems are reported to show how parameters of interest can be extracted from the variation of RI, with oI (= -yH,). THEORY
The system under consideration constitutes a two-site exchange problem in which the population of the bulk solvent site (S) is very much greater than the population 361
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362
CHOPRA,
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JORDAN
of the bound solvent site (S’). The modified Bloch equations for the magnetization components M = (Ma My, Mz, Mr, Mb, K), 121 where Mk and Mi, are the k components of the bulk solvent and coordinated solvent magnetizations in the frame rotating at the Larmor frequency of bulk solvent molecules, and can be written dM = -MW dt
+ (0, 0, R,sM,
0, 0, R,&f:O)
131
with f’,r,
+ & 0 0 -Pmrm 0 0
w=
0 Pmrm + hs -WI 0 -p,r,
0 WI p,r, + RI, 0 0
0
0 -rm 0
-r, 0 0 R2m
+
rm
Awn,
-Aw,
R2m
0
-p,r,
0 0 -rm 0
+
rm
R,,,,wi r,,,
-w1
[41 Here R1,2sand R1,2m are the longitudinal and transverse relaxation rates in the bulk and coordinated sites, respectively, r,,, is the reciprocal of the lifetime (7,) of a coordinated solvent molecule, P,,, is the ratio of the populations in bound and bulk sites, Aw, is the difference in resonance frequencies of bound and bulk sites, and w , = -YH, characterizes the magnitude of the spin-locking field in the r,, experiment. The time dependence of the magnetization in the R,, experiment is characterized by the eigenvalues of the matrix W. Although W has six eigenvalues, the one of interest here lies near Rl,2, because Pm 4 1 for all cases of interest. The eigenvalue A satisfies the secular equation IW - All = 0 PI where 1 is the 6 X 6 unit matrix. Equation [5] can be written explicitly as (Pd,
+
R2s
-
A)
P,.,,r,,,+ Rzs - A WI P,,,r,,, + RI, - A -WI
0 -Pdm 0
X
0 0 -pmrm
0
-rm
0
0
0 Awn,
-rm 0
R2,,, + r,,, - A
-Aw, 0
R2,,, f r,,, - A R,, +“:,
-01
- A
-Pd, 0 P,,,r,,, + Rzs - A X
-WI -P,r, 0
0 WI
-h 0 P,,,r,.,,+ R,, - A 0 0 -Aw, 0 -P,r,
0 -rm 0
0 0 -rm
= 0.
R2,,, + r,,, - A -w1
R,, +“:,
- A
bl
ROTATING-FRAME
RELAXATION
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363
If one assumes, for small P,,, , that RI, - A and RZs - A are proportional to P,,,, so that the elements P,.,,r, + R,,z, - A are also proportional to P,,, , and that terms in Pf, and higher order terms in Eq. [6] can be neglected, the complexity of the secular equation is reduced considerably. (All elements in the determinants in Eq. [6] which contain P,,, can be neglected.) Furthermore, because of the dominance of electron nuclear relaxation effects in paramagnetic systems, and the kinetic lability of the metal ions, one expects that R2m
RI,,, + r,,, B A
+ rim,
[71
so that Eq. [6] can be reduced to a linear equation in A with solution A = R,, = RZs + P,,,r,,,
&,,,[UL, + G,,)(&, + r,,J+ &I + ~d,Uh,, + r,J (Rz,,, + r,,,)[(& + r,,,)(R,, + 4 + &I + &,4&, + rm) ’
[*I
In an RI,, study, one will measure values of R,, for a number of values of wI . It will be useful to analyze Eq. [8] further to determine the conditions under which R,, exhibits a significant w1 dependence, and to identify those magnetic and/or kinetic parameters which can be determined from the RI, data. It is convenient to rearrange Eq. [8] to the form
RI, = ks + p,r,
from which it is clear that R,, will not change significantly with wI if 9 rm
or
Rz,,, % Awf,
Wal
rf, % w:
or
AWL % wf
[10’-4
&,
The conditions in [ lOa] will be imposed by the kinetics and relaxation mechanisms of the system. The limitations in [lob] depend on the practical upper limit on the magnitude of the H, field. For a typical spectrometer, the upper limit on wI probably is -3 X lo6 s-’ (which corresponds to a 1 ps * pulse, and H, N 100 G for protons). Thus r,.,, and Aw, must be 63 X lo6 s-’ to if one is to observe an wI dependence of R,,. The lower limit on w1 generally will be set by the inhomogeneity of the main field (A&), since it is necessary that AH0 d H, = -w&y. From Eq. [9] it can be seen that, in the limit of large w,(w~ - CD),
[Ill while in the low wI limit (w, - 0),
% = & + Mm
R2,,,(&,, + c-d + Ad-, (R2m+ rmj2 + Awf, = R2-
iI21
364
CHOPRA,
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JORDAN
This expression for R,, is identical to that for R2 obtained by Swift and Connick (2). It is noteworthy also that the expression for R;“, bears strong resemblance to that for RI in such systems (9), i.e., [I31 In all these equations, Ris may be considered to be the sum of the relaxation rate in the pure solvent and the outer-sphere contribution (Rio) from the paramagnetic complex. Rearrangement of Eq. [9] and substitution from Eqs. [ 1 l] and [ 121 yields
RI,=
iI41
where s2 = (RI, + rm)
p
(R2m + r ) Ill
{@2m
+ rnJ2 + Add
.
A nonlinear least-squares fit of the variation of R,, with o1 to Eq. [ 141 can be used to obtain best fit values of R2, R ;“, and S,. Since R2 can be measured independently, it may be more convenient to rearrange [ 141 to obtain R,, = R;“, + S:(R2 - R,,,)w;~.
[161
A plot of RI, versus (R2 - Rl,)o;2 should be linear with slope S; and intercept R;*,. It is noteworthy that RZs does not appear in [ 141 and [ 161, and that S,, is independent of P,,,. The latter is a very important and unique feature of the RI, - wI results. To illustrate the practical aspects of an RI,, - w, study, the temperature dependence of R2 (=Ry,,), R;“, and RI is shown in Fig. 1 for a typical solvent exchange system.’ The various limiting regions of the RI and R2 curves have been discussed previously (2). The RI, - wI measurements will be useful only when R2 is significantly different from Rc, which occurs in regions B and C of Fig. 1. In these regions r,,, % Rzm 3 RI, so that to a good approximation S; = (rf, + Auf)
1171
The temperature dependence of S, and AU, is also shown on Fig. 1. In region B, where r,,, % AU,,,, then S, = r,,, and has the exponential temperature dependence of r,,,. In region C, where AU, > r,,,, S, = Aw, and has the slower T-’ dependence expected for PO,. Hence RI, measurements in region C provide a convenient method for the determination of AU,,,. It is clear from Fig. 1 that S, falls below AU, as one enters region D. This occurs because the inequality r,,, % RI,,,, Rz,,, is no longer valid, and S, now approaches ’ The data plotted assume [m]/[S] = 1.5 X 10e3, n = 6, AH* = 14 kcal mol-‘, AS* = 8 cal mol-’ RsO = ([m]/[S])(4 X lO’/Z’) degI; R,, = (1 X 104/T) exp(lOOO/RT); R2,,, = (2 X 104/T) exp(lOOO/RT); exp(lOOO/RT), Aw, = 5 X 106/T. Ri obs = Ri + Rm.
ROTATING-FRAME 105
1
RELAXATION
365
RATES
I
I
_---
SP
r __----
A
B
‘04 c -
-
t
102'
I
2.6
I
r,>Ao,r\A.o,>r,
I
I
103/T,
D
-I
I
I
3.4
3.0
C
-m
I
3.8
I
I
I
4.2
‘K-1
FIG. 1. Typical temperature dependence of RI, R$, R2, S,, and AU, for solvent nuclei undergoing exchange with a bound site on a paramagnetic metal ion. Parameters used in the calculations are given in (IO).
the limiting form (Rl,,.,/&,,$‘*A~, to be expected when Ri,, R2,,, % r,,,, and AU, % Rz,, r,,,. This limiting value is not observable because the difference between R2 and R;“, is too small in region D. There is some difference between AU, and S, even in region C, but it is less than 10% for the typical system modeled in ,Fig. 2, for which (Rzm/Rlm) = 2. However, the difference in region C might be substantial if Rzm were larger so that Rz,,, could not be neglected relative to r,,,. In such circumstances, the difference between AU, and S, can be estimated since, in region B where r,,., %-RI,, R2,,,, (Rlm/R2,,,) = (RJRE) so that this ratio can be determined,
366
CHOPRA,
MC
I”“““““‘1
CLUNG,
AND
JORDAN
I
100 -
80 -
6
60-
6 ri-
-
0”““” 0
FIG. 2. Variation 46.0" (0).
of R,,
40 with
80
I”’
120
(R2 - RI&;*
for
160 cobalt(I1)
’
200
240
in CH30H
at 9.7”
‘1 (Cl),
280 18.3”
(A),
and
and, since it is not expected to be strongly temperature dependent it can be used to obtain Aw, from S, = (R1m/R~m)‘f2Awm. It is apparent from the above discussion that an RI, - wI study can be used to determine r,,, (region .B) and/or AU, (region C) depending on the temperature region accessible to the measurements. The main advantage is that these values can be determined without a knowledge of the number or concentration of exchanging sites. From a kineticist’s viewpoint, this advantage is somewhat offset by the more limited temperature region over which RI, depends on r,,, (region B) compared to R2 (regions B and C). A major advantage of the RI, - w, results is that they can be used to determine the number of coordinated solvent molecules, if the chemical shift of the major site is measured in regions A and/or B. In these regions, the shift is given by (2) Apw* = P,,,Aw, = Mm1 AU
PI
m
1181
_ [mlKw KS’where n is the number of exchanging sites on the minor species of molal concentration [m], [S] is the molal concentration of the major site, and a simple Curie law temperature dependence has been assumed for AU,. The measured shifts can be used to obtain K,, and n can be calculated from
ROTATING-FRAME
RELAXATION
RATES
367 [I91
where T is the temperature at which the R,, - wI measurements were performed to determine AU,. The general features of RI, - o, studies will be demonstrated by results from the cobalt(II)-CHsOD system. A more limited study will be used to demonstrate the determination of the solvation number of nickel(R) in acetonitrile. EXPERIMENTAL
The relaxation rates were measured on a Bruker SXP4-100 spectrometer at 89.4 MHz. A 24 kG Varian electromagnet and flux stabilizer were used. The experiments were controlled by a Nicolet Instrument Corp. Model 1180 computer and 293A programmable pulser. Temperature was controlled by a Bruker 8-ST 100/700 unit, and measured in a glycerol sample with a copper-constantan thermocouple. The RI values were determined by a 180-7-90 pulse sequence, and the CarrPurcell-Meiboom-Gill pulse sequence was used to measure Rz. The data were collected and analyzed on the Nicolet 1180 computer using locally developed pulse programs and iterative least-squares software. Day-today reproducibility was about +3% for R, and +5% for RZ. The RI, experiments were performed by using the gated attenuated output on the power amplifier on the spectrometer to generate the rotating H, field. The magnitude of H, was determined before each experiment by measuring the length of a 180” pulse (t& from the attenuated output, and w1 calculated from wI = (7r/1~lt& = -Y&Z,. The tlsa could be changed from about 8 to 400 ps by a simple adjustment of the attenuator. The software for the RI, determinations allows repetitive generation of the 90” ,-H,(r),-acquire sequence, so that the phases can be optimized by maximiiing the FID after such a sequence. The 90” pulse length was taken to be half of the 180” pulse length determined by observing a null signal after a 180“~-H)(T), sequence. The RI, measurements were performed for durations 7oftheH1pulse10msto -2T,,,, in 10 to 15 equal increments. Samples for relaxation rate studies were prepared by weight on a vacuum line using standard techniques, and were sealed off before use. The CO(CH~OH)~(CIO& was prepared by dehydration of hydrated cobalt(R) perchlorate in methanol with trimethylorthoformate, followed by evaporation of the solvent. Cobalt(II)-methanol solutions were tested for water by observing the low temperature NMR of the coordinated solvent. The best way to ensure that no detectable bound water was introduced during handling was to add l-2% trimethylorthoformate to the solvent. The Ni(CH,CN),(ClO& was prepared by dehydration of Ni(OH&(CiO,), in acetonitrile with 3A-molecular sieves followed by titration to remove the sieves, and evaporation of the solvent. The process was repeated three times with fresh solvent and sieves. The absence of an- O-H stretch in the infrared spectrum was used to confirm that the material was anhydrous. The solid was handled in a dry nitrogen atmosphere in a glove bag.
CHOPRA,
368
MC CLUNG,
AND JORDAN
Solvents were distilled from 3-A molecular sieves, and stored under vacuum over sieves. The acetonitrile was initially dried by distillation from calcium hydride. RESULTS
The solvent proton relaxation rates and shifts in the cobalt(II)-methanol system have been studied by Luz and Meiboom (8, ZO), and the shifts of the bound methyl protons in Co(HOCH&+ by Plotkin, Copes, and Vriesenga (11). A solvation number of 6 has been determined from the relative intensities of the bound and free methanol resonances (IO), and from the tcp dependence of R2 (12). This substantial previous work makes the cobalt(II)-methanol system ideal for testing the predicted dependence of R,, on wI, and to evaluate the accuracy of AU, determined from R,, studies. TABLE I . 1
R IpValues for Co(ClO&’
in CH,OD
Temp. (“C)
hb (A4
26.8
11.1 14.5 17.9 27.0 29.9 38.9 40.9 58.7 84.1 106.5
12.1 12.8 16.0 29.9 33.3 44.6 47.0 74.7 104 132
11.3 13.8 16.8 27.3 31.2 44.4 47.5 75.5 110 134
31.0
12.0 15.0 20. I 25.0 34.1 50.0 66.5 89.9 119
13.5 17.2 24.8 32.9 48.6 78.2 95.9 130 164
14.2 17.2 23.5 30.8 46.4 76.5 106 139 167
37.2
11.9 14.4 20.0 25.2 34. l 47.2 75.3
19.8 23.1 34.9 48.5 66.0 91.5 131
19.8 23.8 34.3 45.5 65.9 94.4 138
R ,p (W
R 1pWC)
’ [Co(II)]/[CH~OD] = 2.17 X lo-‘. b The length of a 180” proton pulse on the line producing the rotating field; w, = n/tlso.
ROTATING-FRAME
RELAXATION
369
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The variation of RI, with wI has been measured at six temperatures between 9.7 and 46°C. The earlier work (8, 20) indicates that the maximum RZ (see Fig. 1) should occur at ~32°C at our operating frequency of 89.4 MHz. Thus, the measurements span the range r,,, > AU, at high temperature to AU, > r,,, at low temperature. Typical results at 9.7, 18.3, and 46.O”C are plotted according to Eq. [16] in Fig. 2. The slopes (Sz) of the lines are approximately equal at the lower temperatures as expected, because AU, > r,,, so that S,’ = Aw$. At 46”C, the slope is much greater because r,,, > Aw, and Sz w rf . The RI, data for 26.8, 31.0, and 37.2”C are given in Table 1, where they are compared to the values calculated from a least-squares fit to Eq. [ 141. The agreement between the observed and predicted values shows that the data is well represented by Eq. [14]. The experimental values of RI and R2, and the results of the least-squares fits of the R,, data are summarized in Table 2. The RZ values are in good agreement with values calculated from the results of Luz and Meiboom (8).
TABLE 2 Summary of RI, Rz, and R,, Results for Co(ClO&
in CHsOD
Temp. (“C)
& (s-l) RI (s-l)
9.1
18.3
26.3
31.0
31.2
46.0
59.8
123
209
229
203
143
3.15
3.73
3.66
3.59
3.51
3.51
6.14 f 0.28 6.75
7.43 f 0.51 7.40
7.72 k 0.61 7.81
8.72 + 0.85 7.99
10.9 + 0.96 8.21
8.29 f 2.69 8.48
0.48 I
1.05
2.07
2.89
4.65
8.85
524 582
604 602
611 619
687 628
854 640
642 651
13.2 +I 0.7
11.7 k 0.7
14.6 f 0.7
17.6 + I.0
34.1 + 1.7
3.70 3.51 3.21
3.29 3.48 3.12
3.20 3.40 3.03
3.01 3.36 2.99
3.48 3.31 2.93
R E (s-l) ObSd
calcd” lo+ X r, (s-l)* R2m (s-7
obsd’ calcdd
10-n x s; (s-2)
101 + 6
1O-4 X AU, (tad/s) ObSd
calcd’ calcd b
4.72 3.24 2.85
’ Calculated from r,,, and the Rim (calcd) values. b Calculated from parameters in (8) after correction to 89.4 MHz, for 6 X [Co”]/[CH,OD] = 1.30 x 10-Z. ’ Obtained from Rlpco and r,,, values at each temperature. dCalculated from Rzm = (1.39 X 106/T) exp(-12OO/RT) to give a reasonable representation of the observed values. ‘Calculated from parameters in (12) based on slow-exchange bound methanol shifts.
CHOPRA,
370
MC CLUNG,
AND JORDAN
The “observed” AU,.,, values in Table 2 were calculated by solving Eq. [ 151 using the tabulated values of S:, r,,, , and &,. In addition, RI,,, was calculated from R r,.,,= RI X [CH@D]/6 X [Co*+]. Any uncertainties in R,, are of little consequence because (RI, + T,,,)/(R~~ + r,.,,) > 0.95. The values of Aw, determined from the R,p study agree (within 10%) with the results from the two studies (8, II) of chemical shifts in this system. One exception is the value of AU,,, at 46”C, which deviates substantially from the other values from the R,, measurements and from the chemical-shift studies. This discrepancy arises because the system is approaching the limit of S,, = r,,,, so that (Sz - rfn) is not well defined. In fact, an 8% increase in r, would give AU,,., in the expected range. It is clear that the RI, - wI study gives reliable values of Ao, in region C of Fig. 1. At the two lowest temperatures, the value of AU, obtained directly from S, is within 5% of that calculated after the corrections for RI,,,, R2,,,, and r,,,. Thus good values of AU,,, can be obtained from RI, measurements in region C without a detailed knowledge of the other parameters. The nickel(II)-acetonitrile system also provides a good test for the R,, - ml measurements. There has been an extensive recent study by Newman et al. (13) in
60
0
0
40
80
120
160
IoQx(R~-R~~)~T*~S FIG. 3. Variation of RI, with (R2 - R,,)w;* for nickel(l1) in CHpCN at 26.4” (A), 32.0" (O), and (0).
38.0”
ROTATING-FRAME
RELAXATION
371
RATES
which much of the previous work is summarized. The solvation number of 6 for nickel(I1) has been determined from a comparison of the chemical shifts of bound and bulk solvent resonances. Although an earlier RZ - tcp study (24) gave a solvation number of 4, recent RI - tw data (IS) confirm that the correct value is 6. The parameters given by Newman et al. (13) indicate that the maximum R2 should occur at ~45’C at 89.4 MHz for this system. The R,, - wI studies have been done in the slow-exchange region at temperatures of 26.4, 32, and 38°C. Figure 3 shows appropriate plots of the data according to Eq. [ 161. The results of least-squares fits to Eq. [ 141 are summarized in Table 3. The values of S, differ by ~5% from the AU, calculated by solving Eq. [ 151 at the two lower temperatures. This shows again that reliable values of AU, can be calculated directly from S, in the slow-exchange region without a precise knowledge of r,,,, RI,,,, and R2,,, . Of course, this results because A& & (Rzm + r,.,,)’ and r,,, > RI,, RZm. The AU,,, values obtained here can be combined with the bulk solvent shifts reported by Newman et al. to calculate the solvation number for nickel(I1) in acetonitrile. The observed shifts (13) between 335 and 346 K are characterized by TABLE 3 Summary of R,, R?, and R,, Results for Ni(ClO&
in CHrCN”
Temp. (“C)
& W’) RI W’)
26.4
32.0
38.0
53.2
74.4
99.2
7.33
6.86
6.45
10.2 + 0.34 10.1
10.16 rfr 0.12 9.85
9.53 f 0.25 9.55
3.24
5.32
8.86
362
351
340
2.81 + 0.19
2.85 k 0.07
3.30 k 0.12
RI,, W’) ObSd
calcd b lo-+,
(~6)~
R2m (s-‘Jb
1o-8 x s: (s-2)
IO+ X AU, (tad/s) ObSd
calcd’ n
1.69 1.79
1.62 1.75
1.59 1.70
5.91
6.05
6.04
’ The ratio [Ni”]/[CHsCN] = 2.705 X lo-‘. b Values are calculated from parameters in (14) except the preexponential factors for R2,,,, Rm, and R,, have been changed to be consistent with the measurements: RI,,, = 69.8 exp(980fR7); R2,, = 65.7 exp(1960/R7’); R,, = 29 exp(980/R7). cCalculated by extrapolation of slow-exchange coordinated solvent shifts. The persistent difference between observed and calculated values may be due to ion pairing effects in the 10 times more concentrated solutions used to observe coordinated solvent.
372
CHOPRA,
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AND JORDAN
K, = 2.99 X lo7 radians s-’ K (see Eq. [18]) at 89.4 MHz. This value of K,, and the AU, values from the R,p measurements were used in Eq. [ 191 to calculate the solvation numbers given in Table 3. The results clearly indicate that n = 6 for nickel(I1) in acetonitrile. CONCLUSIONS
The theoretical developments and experimental results show that an RI, - wI study can be used to determine the solvent exchange rate (r,,,) and bound solvent chemical shift (AU,.,,) in systems described by Eq. [l]. This information can be obtained without knowing the number of exchanging sites. The Ao, values can be combined with bulk solvent shift measurements to determine the number of exchanging sites, i.e., the solvation number. The data analysis clearly indicates the conditions under which an RI, - oi study can be used to determine the parameters of interest. One important limitation of the RI, - wI measurements is the requirement that r,, Aw, 9 Rz,,,. Therefore, the measurements will not be useful for systems with a very efficient inner sphere relaxation mechanism, i.e., large R2,,,. This generally occurs when the electron spin relaxation rate on the metal ion is relatively small ( lo*-lo9 s-i) and a scalar relaxation mechanism is dominant. Thus systems with manganese(II), copper( iron(III), and vanadium(IV), to name some common examples, are unlikely to show a dependence of RI, on o, , unless the metal ion is complexed by some ligands which increase the electron spin relaxation rate to 810” SC’. REFERENCES 1. H. M. MCCONNELL, J. Chem. Phys. 28,430 (1958). 2. T. J. SWIFT AND R. E. CONNICK, J. Chem. Phys. 37, 308 (1962). 3. (a) J. S. LEIGH, JR., J. Magn. Reson. 4, 308 (197 1); (b) J. JEN, Adv. Mol. Relax. Processes 6, 17 1 (1974); (c) J. P. CARVER AND R. E. RICHARDS, J. Magn. Resort 6, 89 (1972). 4. (a) N. S. ANGERMAN AND R. B. JORDAN, Inorg. Chem. 8, 1824 (1969); (b) J. S. LED AND D. M. GRANT, J. Am. Chem. Sot. 99, 5845 (1977); (c) J. JEN, J. Mu@. Reson. 30, 111 (1978). 5. S. ME~B~~M, J. Chem. Phys. 34, 375 (1961). 6. C. DEVERELL, R. E. MORGAN, AND J. H. STRANGE, Mol. Phys. 18, 553 (1970). 7. L. W. REEVES, in ‘Dynamic Nuclear Magnetic Resonance Spectroscopy” (L. M. Jackman and 8. 9. IO. II.
Z. N. Z. K. 12. I. 13. K. 14. I. 15. M.
F. A. Cotton, Eds.), pp. 125-127, Academic Press, New York, 1975. Luz, AND S. MEIB~~M, J. Chem. Phys. 40,2686 (1969). BLOEMBERGEN AND L. 0. MORGAN, J. Chem. Phys. 34,842 (1961). Luz AND S. MEIB~~M, J. Cheti. Phys. 40, 1058 (1964). PLOTKIN, J. COPES, AND J. R. VRIESENGA, Inorg. Chem. 12, 1494 (1973). D. CAMPBELL, P. E. NIXON, AND R. E. RICHARDS, Mol. Phys. 20, 923 (1971). E. NEWMAN, F. K. MEYER, AND A. E. MERBACH, J. Am. Chem. Sot. 101, 1470 (1979). D. CAMPBELL, R. A. DWEK, R. E. RICHARDS, AND M. N. WISEMAN, Mol. Phys. 20,933 (1971). TIPTANATORANIN, Ph.D. Thesis, University of Alberta, 1983.
OF MAGNETIC
RESONANCE
59, 36 l-372
(1984)
Rotating-Frame Relaxation Rates of Solvent Molecules in Solutions of ParamagneticIons Undergoing Solvent Exchange S. CHOPRA, R. E. D. MCCLUNG, AND R. B. JORDAN Department
of Chemistry,
University
of Alberta, Edmonton, Alberta T6G .?G2, Canada
Received December 2 I, 1983 The theory is developed for the dependence of the rotating-frame relaxation rate (R,,) of bulk solvent nuclei on the magnitude of the spin-locking field H, (=--w,/y) for a dilute solution of a paramagnetic ion dissolved in a solvent which undergoes exchange between bulk and coordinated sites. When certain conditions are satisfied R,, will vary with w, according to RI, = R;“, + $(R, - R&o;‘, where R;“, = Rzs + PmrmR2m(R2m + r,,,)-‘, .!$ = (R,, + r&R,,,, + rm)-‘{(R2,,, + r,,# + A&}, RI,, R2,,, are the relaxation rates in the coordinated site, Aw, is the chemical shit? between the bulk and coordinated sites, r, is the solvent exchange rate from a coordinated site, Rsr is the sum of the pure solvent and outer-sphere relaxation rates, and P, = (no. coordinated solvent molecules) X (metal-ion molality/solvent molality). Data for cobalt(R) in CHsOD and nickel(I1) in CHsCN are used to show that the R,, measurements are especially useful in the determination of AU,. The latter can be combined with bulk solvent shifts to determine the number of coordinated solvent molecules. INTRODUCTION
The exchange of solvent between bulk (S) and metal-ion coordinated sites (S’)
111
[MS’,] + S = [MS’,-,S] + S
can be determined from transverse relaxation rate (&) measurements of the bulk solvent nuclei. For such systems containing paramagnetic metal ions, the modified Bloch equations (I) have been solved by Swift and Connick (2). Since then, the theory has been refined (3) and expanded (4) and has seen wide application. However, rotating frame relaxation rates (RI,) have been used relatively little since the early developments by Meiboom (5) and Deverell et al. (6), as noted by Reeves (7). In this paper, a theoretical description of RI, is developed from the modified Bloch equations for systems such as in Eq. [ 11. The conditions under which RI, exhibits a dependence on the magnitude HI of the spin-locking field are elucidated, and the influence of the bound solvent chemical shift (AU,) and solvent exchange rate (rm) on R,, are shown. Experimental measurements of R,, for two systems are reported to show how parameters of interest can be extracted from the variation of RI, with oI (= -yH,). THEORY
The system under consideration constitutes a two-site exchange problem in which the population of the bulk solvent site (S) is very much greater than the population 361
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362
CHOPRA,
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of the bound solvent site (S’). The modified Bloch equations for the magnetization components M = (Ma My, Mz, Mr, Mb, K), 121 where Mk and Mi, are the k components of the bulk solvent and coordinated solvent magnetizations in the frame rotating at the Larmor frequency of bulk solvent molecules, and can be written dM = -MW dt
+ (0, 0, R,sM,
0, 0, R,&f:O)
131
with f’,r,
+ & 0 0 -Pmrm 0 0
w=
0 Pmrm + hs -WI 0 -p,r,
0 WI p,r, + RI, 0 0
0
0 -rm 0
-r, 0 0 R2m
+
rm
Awn,
-Aw,
R2m
0
-p,r,
0 0 -rm 0
+
rm
R,,,,wi r,,,
-w1
[41 Here R1,2sand R1,2m are the longitudinal and transverse relaxation rates in the bulk and coordinated sites, respectively, r,,, is the reciprocal of the lifetime (7,) of a coordinated solvent molecule, P,,, is the ratio of the populations in bound and bulk sites, Aw, is the difference in resonance frequencies of bound and bulk sites, and w , = -YH, characterizes the magnitude of the spin-locking field in the r,, experiment. The time dependence of the magnetization in the R,, experiment is characterized by the eigenvalues of the matrix W. Although W has six eigenvalues, the one of interest here lies near Rl,2, because Pm 4 1 for all cases of interest. The eigenvalue A satisfies the secular equation IW - All = 0 PI where 1 is the 6 X 6 unit matrix. Equation [5] can be written explicitly as (Pd,
+
R2s
-
A)
P,.,,r,,,+ Rzs - A WI P,,,r,,, + RI, - A -WI
0 -Pdm 0
X
0 0 -pmrm
0
-rm
0
0
0 Awn,
-rm 0
R2,,, + r,,, - A
-Aw, 0
R2,,, f r,,, - A R,, +“:,
-01
- A
-Pd, 0 P,,,r,,, + Rzs - A X
-WI -P,r, 0
0 WI
-h 0 P,,,r,.,,+ R,, - A 0 0 -Aw, 0 -P,r,
0 -rm 0
0 0 -rm
= 0.
R2,,, + r,,, - A -w1
R,, +“:,
- A
bl
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363
If one assumes, for small P,,, , that RI, - A and RZs - A are proportional to P,,,, so that the elements P,.,,r, + R,,z, - A are also proportional to P,,, , and that terms in Pf, and higher order terms in Eq. [6] can be neglected, the complexity of the secular equation is reduced considerably. (All elements in the determinants in Eq. [6] which contain P,,, can be neglected.) Furthermore, because of the dominance of electron nuclear relaxation effects in paramagnetic systems, and the kinetic lability of the metal ions, one expects that R2m
RI,,, + r,,, B A
+ rim,
[71
so that Eq. [6] can be reduced to a linear equation in A with solution A = R,, = RZs + P,,,r,,,
&,,,[UL, + G,,)(&, + r,,J+ &I + ~d,Uh,, + r,J (Rz,,, + r,,,)[(& + r,,,)(R,, + 4 + &I + &,4&, + rm) ’
[*I
In an RI,, study, one will measure values of R,, for a number of values of wI . It will be useful to analyze Eq. [8] further to determine the conditions under which R,, exhibits a significant w1 dependence, and to identify those magnetic and/or kinetic parameters which can be determined from the RI, data. It is convenient to rearrange Eq. [8] to the form
RI, = ks + p,r,
from which it is clear that R,, will not change significantly with wI if 9 rm
or
Rz,,, % Awf,
Wal
rf, % w:
or
AWL % wf
[10’-4
&,
The conditions in [ lOa] will be imposed by the kinetics and relaxation mechanisms of the system. The limitations in [lob] depend on the practical upper limit on the magnitude of the H, field. For a typical spectrometer, the upper limit on wI probably is -3 X lo6 s-’ (which corresponds to a 1 ps * pulse, and H, N 100 G for protons). Thus r,.,, and Aw, must be 63 X lo6 s-’ to if one is to observe an wI dependence of R,,. The lower limit on w1 generally will be set by the inhomogeneity of the main field (A&), since it is necessary that AH0 d H, = -w&y. From Eq. [9] it can be seen that, in the limit of large w,(w~ - CD),
[Ill while in the low wI limit (w, - 0),
% = & + Mm
R2,,,(&,, + c-d + Ad-, (R2m+ rmj2 + Awf, = R2-
iI21
364
CHOPRA,
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This expression for R,, is identical to that for R2 obtained by Swift and Connick (2). It is noteworthy also that the expression for R;“, bears strong resemblance to that for RI in such systems (9), i.e., [I31 In all these equations, Ris may be considered to be the sum of the relaxation rate in the pure solvent and the outer-sphere contribution (Rio) from the paramagnetic complex. Rearrangement of Eq. [9] and substitution from Eqs. [ 1 l] and [ 121 yields
RI,=
iI41
where s2 = (RI, + rm)
p
(R2m + r ) Ill
{@2m
+ rnJ2 + Add
.
A nonlinear least-squares fit of the variation of R,, with o1 to Eq. [ 141 can be used to obtain best fit values of R2, R ;“, and S,. Since R2 can be measured independently, it may be more convenient to rearrange [ 141 to obtain R,, = R;“, + S:(R2 - R,,,)w;~.
[161
A plot of RI, versus (R2 - Rl,)o;2 should be linear with slope S; and intercept R;*,. It is noteworthy that RZs does not appear in [ 141 and [ 161, and that S,, is independent of P,,,. The latter is a very important and unique feature of the RI, - wI results. To illustrate the practical aspects of an RI,, - w, study, the temperature dependence of R2 (=Ry,,), R;“, and RI is shown in Fig. 1 for a typical solvent exchange system.’ The various limiting regions of the RI and R2 curves have been discussed previously (2). The RI, - wI measurements will be useful only when R2 is significantly different from Rc, which occurs in regions B and C of Fig. 1. In these regions r,,, % Rzm 3 RI, so that to a good approximation S; = (rf, + Auf)
1171
The temperature dependence of S, and AU, is also shown on Fig. 1. In region B, where r,,, % AU,,,, then S, = r,,, and has the exponential temperature dependence of r,,,. In region C, where AU, > r,,,, S, = Aw, and has the slower T-’ dependence expected for PO,. Hence RI, measurements in region C provide a convenient method for the determination of AU,,,. It is clear from Fig. 1 that S, falls below AU, as one enters region D. This occurs because the inequality r,,, % RI,,,, Rz,,, is no longer valid, and S, now approaches ’ The data plotted assume [m]/[S] = 1.5 X 10e3, n = 6, AH* = 14 kcal mol-‘, AS* = 8 cal mol-’ RsO = ([m]/[S])(4 X lO’/Z’) degI; R,, = (1 X 104/T) exp(lOOO/RT); R2,,, = (2 X 104/T) exp(lOOO/RT); exp(lOOO/RT), Aw, = 5 X 106/T. Ri obs = Ri + Rm.
ROTATING-FRAME 105
1
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365
RATES
I
I
_---
SP
r __----
A
B
‘04 c -
-
t
102'
I
2.6
I
r,>Ao,r\A.o,>r,
I
I
103/T,
D
-I
I
I
3.4
3.0
C
-m
I
3.8
I
I
I
4.2
‘K-1
FIG. 1. Typical temperature dependence of RI, R$, R2, S,, and AU, for solvent nuclei undergoing exchange with a bound site on a paramagnetic metal ion. Parameters used in the calculations are given in (IO).
the limiting form (Rl,,.,/&,,$‘*A~, to be expected when Ri,, R2,,, % r,,,, and AU, % Rz,, r,,,. This limiting value is not observable because the difference between R2 and R;“, is too small in region D. There is some difference between AU, and S, even in region C, but it is less than 10% for the typical system modeled in ,Fig. 2, for which (Rzm/Rlm) = 2. However, the difference in region C might be substantial if Rzm were larger so that Rz,,, could not be neglected relative to r,,,. In such circumstances, the difference between AU, and S, can be estimated since, in region B where r,,., %-RI,, R2,,,, (Rlm/R2,,,) = (RJRE) so that this ratio can be determined,
366
CHOPRA,
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I”“““““‘1
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I
100 -
80 -
6
60-
6 ri-
-
0”““” 0
FIG. 2. Variation 46.0" (0).
of R,,
40 with
80
I”’
120
(R2 - RI&;*
for
160 cobalt(I1)
’
200
240
in CH30H
at 9.7”
‘1 (Cl),
280 18.3”
(A),
and
and, since it is not expected to be strongly temperature dependent it can be used to obtain Aw, from S, = (R1m/R~m)‘f2Awm. It is apparent from the above discussion that an RI, - wI study can be used to determine r,,, (region .B) and/or AU, (region C) depending on the temperature region accessible to the measurements. The main advantage is that these values can be determined without a knowledge of the number or concentration of exchanging sites. From a kineticist’s viewpoint, this advantage is somewhat offset by the more limited temperature region over which RI, depends on r,,, (region B) compared to R2 (regions B and C). A major advantage of the RI, - w, results is that they can be used to determine the number of coordinated solvent molecules, if the chemical shift of the major site is measured in regions A and/or B. In these regions, the shift is given by (2) Apw* = P,,,Aw, = Mm1 AU
PI
m
1181
_ [mlKw KS’where n is the number of exchanging sites on the minor species of molal concentration [m], [S] is the molal concentration of the major site, and a simple Curie law temperature dependence has been assumed for AU,. The measured shifts can be used to obtain K,, and n can be calculated from
ROTATING-FRAME
RELAXATION
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367 [I91
where T is the temperature at which the R,, - wI measurements were performed to determine AU,. The general features of RI, - o, studies will be demonstrated by results from the cobalt(II)-CHsOD system. A more limited study will be used to demonstrate the determination of the solvation number of nickel(R) in acetonitrile. EXPERIMENTAL
The relaxation rates were measured on a Bruker SXP4-100 spectrometer at 89.4 MHz. A 24 kG Varian electromagnet and flux stabilizer were used. The experiments were controlled by a Nicolet Instrument Corp. Model 1180 computer and 293A programmable pulser. Temperature was controlled by a Bruker 8-ST 100/700 unit, and measured in a glycerol sample with a copper-constantan thermocouple. The RI values were determined by a 180-7-90 pulse sequence, and the CarrPurcell-Meiboom-Gill pulse sequence was used to measure Rz. The data were collected and analyzed on the Nicolet 1180 computer using locally developed pulse programs and iterative least-squares software. Day-today reproducibility was about +3% for R, and +5% for RZ. The RI, experiments were performed by using the gated attenuated output on the power amplifier on the spectrometer to generate the rotating H, field. The magnitude of H, was determined before each experiment by measuring the length of a 180” pulse (t& from the attenuated output, and w1 calculated from wI = (7r/1~lt& = -Y&Z,. The tlsa could be changed from about 8 to 400 ps by a simple adjustment of the attenuator. The software for the RI, determinations allows repetitive generation of the 90” ,-H,(r),-acquire sequence, so that the phases can be optimized by maximiiing the FID after such a sequence. The 90” pulse length was taken to be half of the 180” pulse length determined by observing a null signal after a 180“~-H)(T), sequence. The RI, measurements were performed for durations 7oftheH1pulse10msto -2T,,,, in 10 to 15 equal increments. Samples for relaxation rate studies were prepared by weight on a vacuum line using standard techniques, and were sealed off before use. The CO(CH~OH)~(CIO& was prepared by dehydration of hydrated cobalt(R) perchlorate in methanol with trimethylorthoformate, followed by evaporation of the solvent. Cobalt(II)-methanol solutions were tested for water by observing the low temperature NMR of the coordinated solvent. The best way to ensure that no detectable bound water was introduced during handling was to add l-2% trimethylorthoformate to the solvent. The Ni(CH,CN),(ClO& was prepared by dehydration of Ni(OH&(CiO,), in acetonitrile with 3A-molecular sieves followed by titration to remove the sieves, and evaporation of the solvent. The process was repeated three times with fresh solvent and sieves. The absence of an- O-H stretch in the infrared spectrum was used to confirm that the material was anhydrous. The solid was handled in a dry nitrogen atmosphere in a glove bag.
CHOPRA,
368
MC CLUNG,
AND JORDAN
Solvents were distilled from 3-A molecular sieves, and stored under vacuum over sieves. The acetonitrile was initially dried by distillation from calcium hydride. RESULTS
The solvent proton relaxation rates and shifts in the cobalt(II)-methanol system have been studied by Luz and Meiboom (8, ZO), and the shifts of the bound methyl protons in Co(HOCH&+ by Plotkin, Copes, and Vriesenga (11). A solvation number of 6 has been determined from the relative intensities of the bound and free methanol resonances (IO), and from the tcp dependence of R2 (12). This substantial previous work makes the cobalt(II)-methanol system ideal for testing the predicted dependence of R,, on wI, and to evaluate the accuracy of AU, determined from R,, studies. TABLE I . 1
R IpValues for Co(ClO&’
in CH,OD
Temp. (“C)
hb (A4
26.8
11.1 14.5 17.9 27.0 29.9 38.9 40.9 58.7 84.1 106.5
12.1 12.8 16.0 29.9 33.3 44.6 47.0 74.7 104 132
11.3 13.8 16.8 27.3 31.2 44.4 47.5 75.5 110 134
31.0
12.0 15.0 20. I 25.0 34.1 50.0 66.5 89.9 119
13.5 17.2 24.8 32.9 48.6 78.2 95.9 130 164
14.2 17.2 23.5 30.8 46.4 76.5 106 139 167
37.2
11.9 14.4 20.0 25.2 34. l 47.2 75.3
19.8 23.1 34.9 48.5 66.0 91.5 131
19.8 23.8 34.3 45.5 65.9 94.4 138
R ,p (W
R 1pWC)
’ [Co(II)]/[CH~OD] = 2.17 X lo-‘. b The length of a 180” proton pulse on the line producing the rotating field; w, = n/tlso.
ROTATING-FRAME
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The variation of RI, with wI has been measured at six temperatures between 9.7 and 46°C. The earlier work (8, 20) indicates that the maximum RZ (see Fig. 1) should occur at ~32°C at our operating frequency of 89.4 MHz. Thus, the measurements span the range r,,, > AU, at high temperature to AU, > r,,, at low temperature. Typical results at 9.7, 18.3, and 46.O”C are plotted according to Eq. [16] in Fig. 2. The slopes (Sz) of the lines are approximately equal at the lower temperatures as expected, because AU, > r,,, so that S,’ = Aw$. At 46”C, the slope is much greater because r,,, > Aw, and Sz w rf . The RI, data for 26.8, 31.0, and 37.2”C are given in Table 1, where they are compared to the values calculated from a least-squares fit to Eq. [ 141. The agreement between the observed and predicted values shows that the data is well represented by Eq. [14]. The experimental values of RI and R2, and the results of the least-squares fits of the R,, data are summarized in Table 2. The RZ values are in good agreement with values calculated from the results of Luz and Meiboom (8).
TABLE 2 Summary of RI, Rz, and R,, Results for Co(ClO&
in CHsOD
Temp. (“C)
& (s-l) RI (s-l)
9.1
18.3
26.3
31.0
31.2
46.0
59.8
123
209
229
203
143
3.15
3.73
3.66
3.59
3.51
3.51
6.14 f 0.28 6.75
7.43 f 0.51 7.40
7.72 k 0.61 7.81
8.72 + 0.85 7.99
10.9 + 0.96 8.21
8.29 f 2.69 8.48
0.48 I
1.05
2.07
2.89
4.65
8.85
524 582
604 602
611 619
687 628
854 640
642 651
13.2 +I 0.7
11.7 k 0.7
14.6 f 0.7
17.6 + I.0
34.1 + 1.7
3.70 3.51 3.21
3.29 3.48 3.12
3.20 3.40 3.03
3.01 3.36 2.99
3.48 3.31 2.93
R E (s-l) ObSd
calcd” lo+ X r, (s-l)* R2m (s-7
obsd’ calcdd
10-n x s; (s-2)
101 + 6
1O-4 X AU, (tad/s) ObSd
calcd’ calcd b
4.72 3.24 2.85
’ Calculated from r,,, and the Rim (calcd) values. b Calculated from parameters in (8) after correction to 89.4 MHz, for 6 X [Co”]/[CH,OD] = 1.30 x 10-Z. ’ Obtained from Rlpco and r,,, values at each temperature. dCalculated from Rzm = (1.39 X 106/T) exp(-12OO/RT) to give a reasonable representation of the observed values. ‘Calculated from parameters in (12) based on slow-exchange bound methanol shifts.
CHOPRA,
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MC CLUNG,
AND JORDAN
The “observed” AU,.,, values in Table 2 were calculated by solving Eq. [ 151 using the tabulated values of S:, r,,, , and &,. In addition, RI,,, was calculated from R r,.,,= RI X [CH@D]/6 X [Co*+]. Any uncertainties in R,, are of little consequence because (RI, + T,,,)/(R~~ + r,.,,) > 0.95. The values of Aw, determined from the R,p study agree (within 10%) with the results from the two studies (8, II) of chemical shifts in this system. One exception is the value of AU,,, at 46”C, which deviates substantially from the other values from the R,, measurements and from the chemical-shift studies. This discrepancy arises because the system is approaching the limit of S,, = r,,,, so that (Sz - rfn) is not well defined. In fact, an 8% increase in r, would give AU,,., in the expected range. It is clear that the RI, - wI study gives reliable values of Ao, in region C of Fig. 1. At the two lowest temperatures, the value of AU, obtained directly from S, is within 5% of that calculated after the corrections for RI,,,, R2,,,, and r,,,. Thus good values of AU,,, can be obtained from RI, measurements in region C without a detailed knowledge of the other parameters. The nickel(II)-acetonitrile system also provides a good test for the R,, - ml measurements. There has been an extensive recent study by Newman et al. (13) in
60
0
0
40
80
120
160
IoQx(R~-R~~)~T*~S FIG. 3. Variation of RI, with (R2 - R,,)w;* for nickel(l1) in CHpCN at 26.4” (A), 32.0" (O), and (0).
38.0”
ROTATING-FRAME
RELAXATION
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which much of the previous work is summarized. The solvation number of 6 for nickel(I1) has been determined from a comparison of the chemical shifts of bound and bulk solvent resonances. Although an earlier RZ - tcp study (24) gave a solvation number of 4, recent RI - tw data (IS) confirm that the correct value is 6. The parameters given by Newman et al. (13) indicate that the maximum R2 should occur at ~45’C at 89.4 MHz for this system. The R,, - wI studies have been done in the slow-exchange region at temperatures of 26.4, 32, and 38°C. Figure 3 shows appropriate plots of the data according to Eq. [ 161. The results of least-squares fits to Eq. [ 141 are summarized in Table 3. The values of S, differ by ~5% from the AU, calculated by solving Eq. [ 151 at the two lower temperatures. This shows again that reliable values of AU, can be calculated directly from S, in the slow-exchange region without a precise knowledge of r,,,, RI,,,, and R2,,, . Of course, this results because A& & (Rzm + r,.,,)’ and r,,, > RI,, RZm. The AU,,, values obtained here can be combined with the bulk solvent shifts reported by Newman et al. to calculate the solvation number for nickel(I1) in acetonitrile. The observed shifts (13) between 335 and 346 K are characterized by TABLE 3 Summary of R,, R?, and R,, Results for Ni(ClO&
in CHrCN”
Temp. (“C)
& W’) RI W’)
26.4
32.0
38.0
53.2
74.4
99.2
7.33
6.86
6.45
10.2 + 0.34 10.1
10.16 rfr 0.12 9.85
9.53 f 0.25 9.55
3.24
5.32
8.86
362
351
340
2.81 + 0.19
2.85 k 0.07
3.30 k 0.12
RI,, W’) ObSd
calcd b lo-+,
(~6)~
R2m (s-‘Jb
1o-8 x s: (s-2)
IO+ X AU, (tad/s) ObSd
calcd’ n
1.69 1.79
1.62 1.75
1.59 1.70
5.91
6.05
6.04
’ The ratio [Ni”]/[CHsCN] = 2.705 X lo-‘. b Values are calculated from parameters in (14) except the preexponential factors for R2,,,, Rm, and R,, have been changed to be consistent with the measurements: RI,,, = 69.8 exp(980fR7); R2,, = 65.7 exp(1960/R7’); R,, = 29 exp(980/R7). cCalculated by extrapolation of slow-exchange coordinated solvent shifts. The persistent difference between observed and calculated values may be due to ion pairing effects in the 10 times more concentrated solutions used to observe coordinated solvent.
372
CHOPRA,
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AND JORDAN
K, = 2.99 X lo7 radians s-’ K (see Eq. [18]) at 89.4 MHz. This value of K,, and the AU, values from the R,p measurements were used in Eq. [ 191 to calculate the solvation numbers given in Table 3. The results clearly indicate that n = 6 for nickel(I1) in acetonitrile. CONCLUSIONS
The theoretical developments and experimental results show that an RI, - wI study can be used to determine the solvent exchange rate (r,,,) and bound solvent chemical shift (AU,.,,) in systems described by Eq. [l]. This information can be obtained without knowing the number of exchanging sites. The Ao, values can be combined with bulk solvent shift measurements to determine the number of exchanging sites, i.e., the solvation number. The data analysis clearly indicates the conditions under which an RI, - oi study can be used to determine the parameters of interest. One important limitation of the RI, - wI measurements is the requirement that r,, Aw, 9 Rz,,,. Therefore, the measurements will not be useful for systems with a very efficient inner sphere relaxation mechanism, i.e., large R2,,,. This generally occurs when the electron spin relaxation rate on the metal ion is relatively small ( lo*-lo9 s-i) and a scalar relaxation mechanism is dominant. Thus systems with manganese(II), copper( iron(III), and vanadium(IV), to name some common examples, are unlikely to show a dependence of RI, on o, , unless the metal ion is complexed by some ligands which increase the electron spin relaxation rate to 810” SC’. REFERENCES 1. H. M. MCCONNELL, J. Chem. Phys. 28,430 (1958). 2. T. J. SWIFT AND R. E. CONNICK, J. Chem. Phys. 37, 308 (1962). 3. (a) J. S. LEIGH, JR., J. Magn. Reson. 4, 308 (197 1); (b) J. JEN, Adv. Mol. Relax. Processes 6, 17 1 (1974); (c) J. P. CARVER AND R. E. RICHARDS, J. Magn. Resort 6, 89 (1972). 4. (a) N. S. ANGERMAN AND R. B. JORDAN, Inorg. Chem. 8, 1824 (1969); (b) J. S. LED AND D. M. GRANT, J. Am. Chem. Sot. 99, 5845 (1977); (c) J. JEN, J. Mu@. Reson. 30, 111 (1978). 5. S. ME~B~~M, J. Chem. Phys. 34, 375 (1961). 6. C. DEVERELL, R. E. MORGAN, AND J. H. STRANGE, Mol. Phys. 18, 553 (1970). 7. L. W. REEVES, in ‘Dynamic Nuclear Magnetic Resonance Spectroscopy” (L. M. Jackman and 8. 9. IO. II.
Z. N. Z. K. 12. I. 13. K. 14. I. 15. M.
F. A. Cotton, Eds.), pp. 125-127, Academic Press, New York, 1975. Luz, AND S. MEIB~~M, J. Chem. Phys. 40,2686 (1969). BLOEMBERGEN AND L. 0. MORGAN, J. Chem. Phys. 34,842 (1961). Luz AND S. MEIB~~M, J. Cheti. Phys. 40, 1058 (1964). PLOTKIN, J. COPES, AND J. R. VRIESENGA, Inorg. Chem. 12, 1494 (1973). D. CAMPBELL, P. E. NIXON, AND R. E. RICHARDS, Mol. Phys. 20, 923 (1971). E. NEWMAN, F. K. MEYER, AND A. E. MERBACH, J. Am. Chem. Sot. 101, 1470 (1979). D. CAMPBELL, R. A. DWEK, R. E. RICHARDS, AND M. N. WISEMAN, Mol. Phys. 20,933 (1971). TIPTANATORANIN, Ph.D. Thesis, University of Alberta, 1983.