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Mn(D2O)62+ in D2O solution
Volume 128, number 5,6
CHEMICALPHYSICS LETTERS
8 August 1986
A ‘Li NMR I~~GA~~N OF THE ION PAIRS Li “/ ‘ON(SO&ANT) Li ~/~n~~O)~+ IN I&O SALOON P.H. FRIES, J. RENDELL, E.E. BURNELL and C.N. PATEY ~~~rtMe~~ af CkemMy, W~iu~rs~~of British Columtta, Vamwuver, British Colirmbia, Canada V6T I Y6 Received 12 May 1986
‘This paper describes an investigation of the relative dynamics of the ion pairs Li4/QN(SOs)$ - and Li’/Mn(Dzo>$’ in dilute D20 solutions at 25°C. We report the ~te~~ti~ie spin-lattice relaxation rates of the ‘Li nuclei due to their magnetic dipolar ~teraction with the ~~~ne~c ions *OH(SO& - and Mn(D20):‘. The rn~s~erne~s were carried out at diierent ionic strengths and the experhnentai results are compared with theoretical calculations for model solutio~~s.
In several recent articles [ l-4] we have described nuclear magnetic resonance (NMR) studies of the relative motion of various ion pairs in dilute aqueous solution. The NMR experiments consist of measuring the relaxation rate of nuclear spins located on diamagnetic ions due to their magnetic dipolar interaction with the electronic spins S of a charged paramag netic species. The measured relaxation rates can then be compared with detailed ~eoreti~ copulations [5] irr order to evaluate different models for electrolyte solutions. The present paper is a further contribution to this work in which we examine the motion of Li+ ions relative to the nitroso~s~fonate anion ‘ON(SO&and the hexaquom~g~ese(II~ cation Mn(D20@ in D20 solution. We note that Mn(D20)p is the stable form [6, p. 146; 7] of Mn*’ in acidic or neutral systems. ~per~ent~ly, we measure the relaxation rate at 155.46 MHz of the ‘Li+ nuclei due to their interaction with the paramagnetic ions ‘ON(SO& (S = l/2) and Mn(DzO)$+ (S = .5/2). The measured values are compared with theoretical results obtained for both hydrated and u~y~ated Li+ models. For ‘ON(SO&- the hydrated model gives rel~ation rates in good agreement with experiment, whereas calculations for the unhydrated picture are roughly forty times too iarge. This provides evidence that pe532
netration of the Lii hydration sphere by the anion ‘ON(SO&- is not suf~ciently frequent as to be important on the time scales probed in the NMR experiments. For Mn(D~O)~ the hydrated and unhydrated models give similar results and both are in good accord with the exper~ent~.me~uremeuts.
2. The experimentalmfssurement The observed total spin lattice rel~ation rates, f/T1 ,of the ‘Li* nuclei in the presence of ‘ON(SO& and Mn(D20$ are given in table 1. ‘I%@=measurements were made at 155.46 MHz and 25 z 1’C. The required ~te~~ticle rel~ation rates l/Tin* are given by l/Tpt= = l/T1 - t/e,
(0
where l/e is the relaxation rate of 7Li’ in the absence of any p~amagnetic species. For the relatively dilute solutions considered in the present work l[Tsf does not depend upon the concentration of diamagnetic ions. All solutions were prepared using freshly distilled D,O (Merck, Sharp and Dohme, 99.8 at% D). The salts used were LiCl (BDH, a~ydrous, 99%), &CO3 (Fisher, certified ACS), and MNC12*4H,0 (BDH, “analar”). The K,ON(SO,), was prepared as in ref. [9], Li2C03 rather than LiCl was used in the nitroso0 a~-2614~86~$03.50 0 Elsevier Science Publishers B.V, (North-Holland Physics Publishing Divison)
Volume 128, number 5,6
CHEMICAL PHYSfCS LETTEZRS
Table 1 The experimentally measured relaxation rates l/Z’1 of ‘Li+ at 155.46 MHz a) (A) The Li+rON(SO&
- results
P2CO31 Q-l)
fK,oNfS03)21 (molQ_I)
i
ID-1
(mol
(molQ2-1)
(s-‘)
0.005 0.005 0.005 0.005
0.005 0.01 0.02 0.04
0.030 0.045 0.075 0.135
0.313 0.588 1.06 1.82
(mol a-')
i (mol Q-l)
1lTt (s-l)
0.005 0.01 0.02 0.04
0.030 0.045 0.075 0.135
0.402 0.806 1.69 3.72
(B) The Li’~n{~~O)~+ [Liq (mol
results
[MnC12.4HzO] Q-l)
0.015 0.015 0.015 0.015
a) The observed relaxation rate of ‘Li+ in the absence of any paramagnetic species is l/e = 0.027 8-l [8].
disulfonate experiments in order to provide the basic medium necessary for the stability of the radical anion ‘ON(S03)$-. No attempt was made to remove the coordinated water in MnCi2*4H20 or to exchange it with D20. The protons added to the D20 solutions from this source amount to only aO.4 at% H at the salt concentrations used, and will have no measurable effect upon the relaxation rates. Dissolved oxygen was removed from all samples by seven freezepump-thaw cycles [lo] and the tubes were sealed under 1 atm of nitrogen. All mea~rements were made on a Bruker WH4OO spectrometer and the 7Li relaxation times were obtained using the inversion recovery method [lO,ll]. The T1 values were determined by a three-parameter least-squares analysis [ 1 l] of the signal intensity versus recovery time measurements.
3. ~ornp~n
with theory
The spin relaxation formalism is described in ref. [5]. We assume that the measured interparticle relaxation rate is completely dominated by the long-range
8 August 1986
magnetic dipolar coupling between the electronic and nuclear spins. In principle there is also a short-range scalar hyperfme contact interaction (lo], but in the present experiments the Larmor angular frequency of the electronic spins os is large and we would not expect a s~~~c~t scalar contact cont~bution to 1-i Tlmte* [5 131. This will be particularly true for the repulsive iois Li+ and Mn(D20$ which are rarely in close contact. Furthermore in the case of the attractive ions Li+ and ‘ON(SO3)21- the interparticle dipolar coupling has been found [14] to be the dominant relaxation mechanism of the 7Li nuclei even at low values of ws. The dipolar relaxation rate is given by [5,12] l/Tptm = f&$vs2h2S(S+ 1) X (Nsr/7rb3)[72(WLir) + g 72(Us7)1 9
(2)
whereT2(W~) is a dimensionless spectral density,l\rs is the number density of the electronic spins, and 7 and o represent, respectively, the gyrom~netic ratios and the Larmor frequencies associated with the different spins. Also in eq.(2), b is the distance of closest approach between the ion centers and r = b2/D, where D is the relative diffusion constant for the ion pair. In the actual calculations D is taken to be the sum of the ionic self-diffusion constants Dt. For present purposes it is convenient to express all variables in cgs units such that eq. (2) can be written in the form
X (NAl103nDb)[j.2(WLiT) + $ 72(97)3,
(3)
where NA is Avogadro’s number and cs is the, spin con~ntration in mole/& The quantity l&I’~“‘r can be compared directly with experimental results. The theoretical calculations ofj2(ar) and hence l/Tr@r were carried out as discussed in ref. [ 11. That is,j2(or) is determined by solving the Smoluchowski equation Including a term depending upon the average force between the ions. The potentials of mean force used in the present calculations were obtained by solving Integral equation theories for model electrolytes. Our models differ from the usual continuum approximations in that the solvent is included as a discrete molecular species [5 ,151. We refer to these as molecular solvent models. Also in the present calcula533
CHEMICALPHYSICS LETTERS
Volume 128, number 5,6
8 August 1986
Table 2 Physical parameters used in the theoretical calculations. The quantities d, IpI, Dt and D* are, respectively, the hard-sphere diameter the length of the eccentricity vector, the translational diffusion constant, and the rotational diffusion constant of the various ionic species. If the diffusion constants have been measured in HaO, the Da0 values are estimated by dividing the Hz0 results by the viscosity ratio nDa~/uHs~ = 1.23, as in refs. [l-4] Ion _. -
Mn(D#)e+ ‘ON(SOs# LS*(hydrated) Li+(unhydrated)
10s d
108 IPI
(cl4
bq
6.16 7.17 7.39 1.90
1.6 (electronic spin)
10-m Dr
lo5 Dt (cm2 s-l)
(s-l)
1.2 0.58 [ 171 a) 0.84 [ 181 a) 0.84 [18]3
2.3 [19]
a) Using the viscosity correction.
tions it is necessary to take spin eccentricity effects Into account [ 161. These effects arise from the fact that for ‘ON(S03)s- the unpaired electron is not located at the ion center. The inclusion of spin eccentricity requires the rotational diffusion constant D* and the distance I p I giving the approximate location of the spin with respect to the ion center. The physical parameters used in the present calculations are given in table 2. All ions are taken to be spherical in shape and for ‘ON(S03);- the diameter d and the distance IpI were estimated with the aid of molecular models [ 11. It should be noted that for ‘ON(S03$ the parameters given in table 2 are consistent with those used in our earlier work [ 1,2]. As mentioned above, in neutral solution Mn2+ exists as the hexaquomanganese(I1) ion Mn(D20)F. We take the diameter of this species to be 7.17 A which is approximately 2 X (0.80 + 2.80) = 7.20 A where 0.80 A is the crystal radius [6, p. 1461 of Mn2+ and 2.8OW is the diameter, d,, of the water molecule. The value 7.17 A rather than 7.20 A is used in the calculation simply because it is compatible with the finite grid width (i.e. O.O2d,) used in the solution of the integral equation theory [ 151. This hard-sphere diameter gives fair agreement between theory and ex eriment [4] for the ion pairs CH,COO-/Mn(D,O), 8 and in D20 solution. We have (CI-I&&/Mn(D&@ carried out calculations assuming both hydrated and unhydrated Li+ ions. For Li+(hydrated) we take the diameter to be =2 X (0.92 t 2.80) = 7.44 A, and for Li+(unhydrated) d = 2 X 0.92 = 1.84 W where 0.92 A is the crystal radius of Li+ as determined from X-ray electron density maps [20]. Again it can be seen from table 2 that the numbers used in the actual calculation 534
are adjusted slightly in order to be compatible with the finite grid width. Experimental values for 10d2/cs T,rnter as a function of j1j2 where 1 is the total ionic strength are compared with theoretical calculations in figs. 1 and 2. The potentials of mean force used in the calculation
0.0
._._._._.-.-____._._.-.-.-._.__._._._._._._._._._._.-.-._._._._I-I 0.0
I 0.1
1
I
0.2
.
I
.
0.3
o.t4 h
I
l/2
Fig. 1. The quantity 10-2/c~7’~ter (Q mol-r s-l) defined by 9q. (3) as a function of the square root of the ionic strength Zln for the ion pair Li+/‘ON(SO&- in D20 at 25°C. The crosses are the experimental values, and the solid and dashdotted curves represent theoretical results for the molecular solvent electrolyte model and for an uncharged hard-sphere system, respectively. The curves labelled h are for Li+(hydrated) and the remainder for Li+(unhydrated). For the molecular solvent model the theoretical values obtained for Li+(unhydrated) are about 40 times greater than the experimental results and are not included in the fiiure.
Volume 128, number 5,6
CHEMICAL PHYSICS LE’MERS
8 August 1986
EpO 0
3.0
lo-2
44
w
2.5 I---- GJ’?
0
44
._*_._._*_._._l_._ y?
o.ou, 0.0
0.1
0.2
-3.
j
1. 0.3
0.4 a
l/2
I
, 2.
*
, 3.
I
,
I
4.
, 5.
R/d,
I
Fig. 2. The quantity 10-2/c~T~ter @mole-’ s-l) as a function of the square root of the ionic strengthiiR for the ion pair Li’/Mn(~O)~~ in DzO at 25% The cnrves are as in fig. 1.
at infinite dilution are shown in figs. 3 and 4. Theoretical curves are included for two model solutions. The solid curves are for the molecular solvent picture mentioned above. In this model the ions are taken to be charged hard spheres and the solvent is a fluid
Fig. 3. The potentials of mean force at infinite dilution for the ion pair Li+/‘ON(SO&-. The curves are as in fig. 1.
Fig. 4. The potentials of mean force at infinite dilution for the ion pair Li+/Mn(D~O)~+. The ewes are as in fig_ 1.
of water-like particles [ 151. The dash-dotted lines are obtained if all electrostatic interactions are ignored and the ions and solvent molecules are treated simply as hard spheres. These curves are included in the figures to illustrate the influence of the electrostatic forces. The results for Lit~*ON(S03)~-are shown in f@. 1. It can be seen that for this system the theoretical calculations for Li+(hydrated)/‘ON(S03)~- are in good agreement with the experimental measurements. For Li’$mhydrated)/‘ON(SO&the theoretical values obtained for l/c~@~ are ~40 times lar er than those found for ~t(hyd~ted)/*ON(S03)2-. 9 These results, which are ob~ously in d~r~ment with the experiments, are not shown in the fgure. The ex erimental and theoretical values for Li+/ Mn(D20)6& are compared in fq_ 2. For this repulsive ion pair the spectral density is largely determined by the long-range part of the potential of mean force, and hence the theoretical calculations give similar results for both Lit~hy~ated) and Lit(~ydrated). The Li+(u~ydrated) ~c~atio~ do lie a little closer to the experimental points but the present theory is far too crude to allow us to distinguish between the hydrated and unhydrated pictures for pairs of repulsive ions. 535
Volume 128, number 5,6
CHEMICALPHYSICS LETTERS
There is a further important point concerning the theoretical calculations which should be addressed. In a sense,the explicit consideration of the Li+(hydrated) model is inconsistent with our molecular solvent picture. The water molecules are not covalent& bonded to Li+ and interact with this cation essentially through a paiwise electrostatic potential 1211. Hence in a true molecular solvent model hydration effects should be implicitly accounted for. However, it should be kept in mind that in the present calculations the dynamical theory [S] is carried out at the Smoluchowski level. This means that the relative tmn~ation~ diffusion of the ion pair, and consequently the calculated spectral density is determined by the equilibrium or long-time average ion-ion potential of mean force. For attractive ion pairs such as Li+/‘ON(SO&- the NMR measurements probe the relatively short-time (i.e. <, 70 ps) behaviour of the relative ion motion [S] . If on this time scale the anion ‘ON(SO& does not frequently penetrate the Li+ hy~ation sphere, then we would expect a theory employing the equilibrium potential of mean force for Li’(unhydrated)/*ON(SO&to give relaxation rates which are much larger than the experimental values. As discussed above, this turns out to be the case with the Li+ (unhydrated) calculations being some forty times greater than the experimental measurements. On the other hand, the Ll~~ydrated) calculations, which assume that the ‘ON(SO&- ion never penetrates the Li+ hydration sphere, are in good agreement with the experiments. Of course, it is also possible that the discrepancy between our calculations and the experimental results for Li+(unhydrated)/‘ON(SO&is not due to the Smoluchowski theory. For example, the various pair ~tera~tions may be overs~p~ed in the present model [5,15] , or the equilibrium theory used to obtain the ion-ion potential of mean force may not be sufficiently accurate for Li+(unhydrated)/‘ON(SO& Eventually, it should become possible to evaluate both the equilibrium and dynamical parts of the theory with careful computer simulations. The NMR relaxation results could then be used to test and improve various models.
8 August 1986
4. Conclusions This paper describes an investigation of the relative motion of the ion pairs Li’/‘ON(SO&and Li+/ Mn(D20)~ in dilute D20 solution. The relative dynamics were studied by measu~g the spin-lattice relaxation rate of the 7Li nuclei due to their magnetic dipolar interaction with the aramagnetic species ‘ON(SO&- and Mn(D20)68 . The experiments were carried out at 25°C and the measured relaxation rates were compared with theoretical results. Theoretical calculations were carried out assuming both hydrated and ~ydmted Li* ions. For the attractive ion pair Li+/‘ON(SO&- the ~+~ydrated) results are in good agreement with the experiments whereas the relaxation rates obtained for Li+(unhydrated) are much larger than the experimental values. This strongly suggests that ‘ON(SO& does not penetrate the Li+ hydration sphere on the time scales probed by the NMR me~urements. Of course, this is not a sunrise result since there is co~derable evidence [6, pp. 109,140] * (e.g., conductivity, diffusion constant measurements, X-ray diffraction data) that Li+ behaves as a hydrated species in aqueous solution. It is worth noting that in our earlier studies [l] involving larger ions (e.g., (CH&P[ON(SO&-) good agreement between theory and experiment was. obtained w~r~o~~explicitly taking hydration spheres itllto account. Thus the model and theory described in ref. [5] is less satisfactory for Li+ since the hydration effects implicitly included in the theory are not sufficient to explain the experimental results. As discussed in section 3 this failure is possibly due to the fact that the relative ion dynamics is treated only at the Smoluchowski level. For the ion pair Li~~M~(D~O)~ both the Li+~y~ated) and the ~+(~y~ted) calculations agree rather well with experiment. This is in accord with our earlier observations [2] for pair of repulsive ions.
Acknowiedgemmt We are grateful to the Natural Sciences and Rngi* The valuesof the self-diffusionconstants are given by the Nernstexpressionfrom conductivitymeasurements in ~$5. [ 17,181.
536
Volume 128, number 5,6 neering
CHEMICAL PHYSICS LETTERS
Research Council of Canada for fmancial sup-
port.
References [l] PH. Fries, N.R. Jagannathan, F.G. Herring and GN. Pa&y, J. Chem. Phys. 80 (1984) 6267. [ 21 P.H. Fries, 3. RendelI, E.E. BurnelI and G.N. Patey, J. Chem. Phys. 83 (1985) 307. [3] P.H. Fries, N.R. Jagannathan, F.G. Her& and G.N. Patey, 3. Phys. Chem. 89 (1985) 1413. [4] PH. Fries, N.R. Jagannathan, F.G. Herrhtg and G.N. Patey, to be published. [5] P.H. Fries and G.N. Patey, J. Chem. Phys. 80 (1984) 6253. [6] J. Burgess, Metal ions in sohrtion (Wiley, New York, 1978). [ 7 J FAA, Cotton and G. W~n~n, Advanced morgauic chemistry (Interscience, New York, 1%2) p. 694. [ 81 B.C. Hertz, R. Tutsch and H. Versmold, Ber. Bunsenges. Physik. Chem. 75 (1971) 1177. [ 9] E.G. Rozantsev, Free nitroxyl radicals (Plenum Press, New York, 1970). [ 10) ML. Martin, J.J. Delpuech and GJ. Martin, Practical NMR spectroscopy (Heyden, London, 1980).
8 August 1986
[ 111 G.C.Levy and I&Peat+ J.Magn. Reson. 18 (1975) 500. [ 121 A. Abragam, The prmciples of nuclear magnetism (Cktrendon Press, Oxford, 1961). [ 131 L. Helm and H.G. Hertz, Ber. Bunsenges. Physik. Chem. 85 (1981) 158. f 141 A. Landesman, Compt. Rend. Acad. Set. (P&s) 246 (1958) 1.538. [IS] G.N. Patey and S.L. Caruie, J. Chem. Phys 78 (1983) 5183; P.G. Kusahk and G.N. Patey, J. Chem. Phys. 79 (1983) 4468. [16] J.P. Albrand, MC. Taieb, PH. Fries and E. Belorizky, J. Chem. Phys. 78 (1983) 5809; P.H. Fries and E. Belorizky, J. Phys. (Paris) 39 (1978) 1263; Y. Ayant, E. Belorizky, P.H. Fries and J. Rosset, J. Phys. (Paris) 38 (1977) 325. [ 171 G. Jander, Chr. Blohm aud B. Grttttner, 2. Anorg. Chem. 258 (1949) 205. [ 181 R.A. Robinson and R.H. Stokes, Electrolyte solutions ~B~tterworths, London, 1965). [19] R.G. Kooser, W.V. Volland and J.H. Freed, J. Chem. Phys. 50 (1969) 5243. [20] D.F.C. Morris, Struct. Bonding 4 (1968) 63; D.M. Adams, Inorganic soiids (Wtiey, New York, f974). [ 211 E. Clementi and H. Popkie, J. Chem. Phys. 57 (1972) 1077.
537
CHEMICALPHYSICS LETTERS
8 August 1986
A ‘Li NMR I~~GA~~N OF THE ION PAIRS Li “/ ‘ON(SO&ANT) Li ~/~n~~O)~+ IN I&O SALOON P.H. FRIES, J. RENDELL, E.E. BURNELL and C.N. PATEY ~~~rtMe~~ af CkemMy, W~iu~rs~~of British Columtta, Vamwuver, British Colirmbia, Canada V6T I Y6 Received 12 May 1986
‘This paper describes an investigation of the relative dynamics of the ion pairs Li4/QN(SOs)$ - and Li’/Mn(Dzo>$’ in dilute D20 solutions at 25°C. We report the ~te~~ti~ie spin-lattice relaxation rates of the ‘Li nuclei due to their magnetic dipolar ~teraction with the ~~~ne~c ions *OH(SO& - and Mn(D20):‘. The rn~s~erne~s were carried out at diierent ionic strengths and the experhnentai results are compared with theoretical calculations for model solutio~~s.
In several recent articles [ l-4] we have described nuclear magnetic resonance (NMR) studies of the relative motion of various ion pairs in dilute aqueous solution. The NMR experiments consist of measuring the relaxation rate of nuclear spins located on diamagnetic ions due to their magnetic dipolar interaction with the electronic spins S of a charged paramag netic species. The measured relaxation rates can then be compared with detailed ~eoreti~ copulations [5] irr order to evaluate different models for electrolyte solutions. The present paper is a further contribution to this work in which we examine the motion of Li+ ions relative to the nitroso~s~fonate anion ‘ON(SO&and the hexaquom~g~ese(II~ cation Mn(D20@ in D20 solution. We note that Mn(D20)p is the stable form [6, p. 146; 7] of Mn*’ in acidic or neutral systems. ~per~ent~ly, we measure the relaxation rate at 155.46 MHz of the ‘Li+ nuclei due to their interaction with the paramagnetic ions ‘ON(SO& (S = l/2) and Mn(DzO)$+ (S = .5/2). The measured values are compared with theoretical results obtained for both hydrated and u~y~ated Li+ models. For ‘ON(SO&- the hydrated model gives rel~ation rates in good agreement with experiment, whereas calculations for the unhydrated picture are roughly forty times too iarge. This provides evidence that pe532
netration of the Lii hydration sphere by the anion ‘ON(SO&- is not suf~ciently frequent as to be important on the time scales probed in the NMR experiments. For Mn(D~O)~ the hydrated and unhydrated models give similar results and both are in good accord with the exper~ent~.me~uremeuts.
2. The experimentalmfssurement The observed total spin lattice rel~ation rates, f/T1 ,of the ‘Li* nuclei in the presence of ‘ON(SO& and Mn(D20$ are given in table 1. ‘I%@=measurements were made at 155.46 MHz and 25 z 1’C. The required ~te~~ticle rel~ation rates l/Tin* are given by l/Tpt= = l/T1 - t/e,
(0
where l/e is the relaxation rate of 7Li’ in the absence of any p~amagnetic species. For the relatively dilute solutions considered in the present work l[Tsf does not depend upon the concentration of diamagnetic ions. All solutions were prepared using freshly distilled D,O (Merck, Sharp and Dohme, 99.8 at% D). The salts used were LiCl (BDH, a~ydrous, 99%), &CO3 (Fisher, certified ACS), and MNC12*4H,0 (BDH, “analar”). The K,ON(SO,), was prepared as in ref. [9], Li2C03 rather than LiCl was used in the nitroso0 a~-2614~86~$03.50 0 Elsevier Science Publishers B.V, (North-Holland Physics Publishing Divison)
Volume 128, number 5,6
CHEMICAL PHYSfCS LETTEZRS
Table 1 The experimentally measured relaxation rates l/Z’1 of ‘Li+ at 155.46 MHz a) (A) The Li+rON(SO&
- results
P2CO31 Q-l)
fK,oNfS03)21 (molQ_I)
i
ID-1
(mol
(molQ2-1)
(s-‘)
0.005 0.005 0.005 0.005
0.005 0.01 0.02 0.04
0.030 0.045 0.075 0.135
0.313 0.588 1.06 1.82
(mol a-')
i (mol Q-l)
1lTt (s-l)
0.005 0.01 0.02 0.04
0.030 0.045 0.075 0.135
0.402 0.806 1.69 3.72
(B) The Li’~n{~~O)~+ [Liq (mol
results
[MnC12.4HzO] Q-l)
0.015 0.015 0.015 0.015
a) The observed relaxation rate of ‘Li+ in the absence of any paramagnetic species is l/e = 0.027 8-l [8].
disulfonate experiments in order to provide the basic medium necessary for the stability of the radical anion ‘ON(S03)$-. No attempt was made to remove the coordinated water in MnCi2*4H20 or to exchange it with D20. The protons added to the D20 solutions from this source amount to only aO.4 at% H at the salt concentrations used, and will have no measurable effect upon the relaxation rates. Dissolved oxygen was removed from all samples by seven freezepump-thaw cycles [lo] and the tubes were sealed under 1 atm of nitrogen. All mea~rements were made on a Bruker WH4OO spectrometer and the 7Li relaxation times were obtained using the inversion recovery method [lO,ll]. The T1 values were determined by a three-parameter least-squares analysis [ 1 l] of the signal intensity versus recovery time measurements.
3. ~ornp~n
with theory
The spin relaxation formalism is described in ref. [5]. We assume that the measured interparticle relaxation rate is completely dominated by the long-range
8 August 1986
magnetic dipolar coupling between the electronic and nuclear spins. In principle there is also a short-range scalar hyperfme contact interaction (lo], but in the present experiments the Larmor angular frequency of the electronic spins os is large and we would not expect a s~~~c~t scalar contact cont~bution to 1-i Tlmte* [5 131. This will be particularly true for the repulsive iois Li+ and Mn(D20$ which are rarely in close contact. Furthermore in the case of the attractive ions Li+ and ‘ON(SO3)21- the interparticle dipolar coupling has been found [14] to be the dominant relaxation mechanism of the 7Li nuclei even at low values of ws. The dipolar relaxation rate is given by [5,12] l/Tptm = f&$vs2h2S(S+ 1) X (Nsr/7rb3)[72(WLir) + g 72(Us7)1 9
(2)
whereT2(W~) is a dimensionless spectral density,l\rs is the number density of the electronic spins, and 7 and o represent, respectively, the gyrom~netic ratios and the Larmor frequencies associated with the different spins. Also in eq.(2), b is the distance of closest approach between the ion centers and r = b2/D, where D is the relative diffusion constant for the ion pair. In the actual calculations D is taken to be the sum of the ionic self-diffusion constants Dt. For present purposes it is convenient to express all variables in cgs units such that eq. (2) can be written in the form
X (NAl103nDb)[j.2(WLiT) + $ 72(97)3,
(3)
where NA is Avogadro’s number and cs is the, spin con~ntration in mole/& The quantity l&I’~“‘r can be compared directly with experimental results. The theoretical calculations ofj2(ar) and hence l/Tr@r were carried out as discussed in ref. [ 11. That is,j2(or) is determined by solving the Smoluchowski equation Including a term depending upon the average force between the ions. The potentials of mean force used in the present calculations were obtained by solving Integral equation theories for model electrolytes. Our models differ from the usual continuum approximations in that the solvent is included as a discrete molecular species [5 ,151. We refer to these as molecular solvent models. Also in the present calcula533
CHEMICALPHYSICS LETTERS
Volume 128, number 5,6
8 August 1986
Table 2 Physical parameters used in the theoretical calculations. The quantities d, IpI, Dt and D* are, respectively, the hard-sphere diameter the length of the eccentricity vector, the translational diffusion constant, and the rotational diffusion constant of the various ionic species. If the diffusion constants have been measured in HaO, the Da0 values are estimated by dividing the Hz0 results by the viscosity ratio nDa~/uHs~ = 1.23, as in refs. [l-4] Ion _. -
Mn(D#)e+ ‘ON(SOs# LS*(hydrated) Li+(unhydrated)
10s d
108 IPI
(cl4
bq
6.16 7.17 7.39 1.90
1.6 (electronic spin)
10-m Dr
lo5 Dt (cm2 s-l)
(s-l)
1.2 0.58 [ 171 a) 0.84 [ 181 a) 0.84 [18]3
2.3 [19]
a) Using the viscosity correction.
tions it is necessary to take spin eccentricity effects Into account [ 161. These effects arise from the fact that for ‘ON(S03)s- the unpaired electron is not located at the ion center. The inclusion of spin eccentricity requires the rotational diffusion constant D* and the distance I p I giving the approximate location of the spin with respect to the ion center. The physical parameters used in the present calculations are given in table 2. All ions are taken to be spherical in shape and for ‘ON(S03);- the diameter d and the distance IpI were estimated with the aid of molecular models [ 11. It should be noted that for ‘ON(S03$ the parameters given in table 2 are consistent with those used in our earlier work [ 1,2]. As mentioned above, in neutral solution Mn2+ exists as the hexaquomanganese(I1) ion Mn(D20)F. We take the diameter of this species to be 7.17 A which is approximately 2 X (0.80 + 2.80) = 7.20 A where 0.80 A is the crystal radius [6, p. 1461 of Mn2+ and 2.8OW is the diameter, d,, of the water molecule. The value 7.17 A rather than 7.20 A is used in the calculation simply because it is compatible with the finite grid width (i.e. O.O2d,) used in the solution of the integral equation theory [ 151. This hard-sphere diameter gives fair agreement between theory and ex eriment [4] for the ion pairs CH,COO-/Mn(D,O), 8 and in D20 solution. We have (CI-I&&/Mn(D&@ carried out calculations assuming both hydrated and unhydrated Li+ ions. For Li+(hydrated) we take the diameter to be =2 X (0.92 t 2.80) = 7.44 A, and for Li+(unhydrated) d = 2 X 0.92 = 1.84 W where 0.92 A is the crystal radius of Li+ as determined from X-ray electron density maps [20]. Again it can be seen from table 2 that the numbers used in the actual calculation 534
are adjusted slightly in order to be compatible with the finite grid width. Experimental values for 10d2/cs T,rnter as a function of j1j2 where 1 is the total ionic strength are compared with theoretical calculations in figs. 1 and 2. The potentials of mean force used in the calculation
0.0
._._._._.-.-____._._.-.-.-._.__._._._._._._._._._._.-.-._._._._I-I 0.0
I 0.1
1
I
0.2
.
I
.
0.3
o.t4 h
I
l/2
Fig. 1. The quantity 10-2/c~7’~ter (Q mol-r s-l) defined by 9q. (3) as a function of the square root of the ionic strength Zln for the ion pair Li+/‘ON(SO&- in D20 at 25°C. The crosses are the experimental values, and the solid and dashdotted curves represent theoretical results for the molecular solvent electrolyte model and for an uncharged hard-sphere system, respectively. The curves labelled h are for Li+(hydrated) and the remainder for Li+(unhydrated). For the molecular solvent model the theoretical values obtained for Li+(unhydrated) are about 40 times greater than the experimental results and are not included in the fiiure.
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CHEMICAL PHYSICS LE’MERS
8 August 1986
EpO 0
3.0
lo-2
44
w
2.5 I---- GJ’?
0
44
._*_._._*_._._l_._ y?
o.ou, 0.0
0.1
0.2
-3.
j
1. 0.3
0.4 a
l/2
I
, 2.
*
, 3.
I
,
I
4.
, 5.
R/d,
I
Fig. 2. The quantity 10-2/c~T~ter @mole-’ s-l) as a function of the square root of the ionic strengthiiR for the ion pair Li’/Mn(~O)~~ in DzO at 25% The cnrves are as in fig. 1.
at infinite dilution are shown in figs. 3 and 4. Theoretical curves are included for two model solutions. The solid curves are for the molecular solvent picture mentioned above. In this model the ions are taken to be charged hard spheres and the solvent is a fluid
Fig. 3. The potentials of mean force at infinite dilution for the ion pair Li+/‘ON(SO&-. The curves are as in fig. 1.
Fig. 4. The potentials of mean force at infinite dilution for the ion pair Li+/Mn(D~O)~+. The ewes are as in fig_ 1.
of water-like particles [ 151. The dash-dotted lines are obtained if all electrostatic interactions are ignored and the ions and solvent molecules are treated simply as hard spheres. These curves are included in the figures to illustrate the influence of the electrostatic forces. The results for Lit~*ON(S03)~-are shown in f@. 1. It can be seen that for this system the theoretical calculations for Li+(hydrated)/‘ON(S03)~- are in good agreement with the experimental measurements. For Li’$mhydrated)/‘ON(SO&the theoretical values obtained for l/c~@~ are ~40 times lar er than those found for ~t(hyd~ted)/*ON(S03)2-. 9 These results, which are ob~ously in d~r~ment with the experiments, are not shown in the fgure. The ex erimental and theoretical values for Li+/ Mn(D20)6& are compared in fq_ 2. For this repulsive ion pair the spectral density is largely determined by the long-range part of the potential of mean force, and hence the theoretical calculations give similar results for both Lit~hy~ated) and Lit(~ydrated). The Li+(u~ydrated) ~c~atio~ do lie a little closer to the experimental points but the present theory is far too crude to allow us to distinguish between the hydrated and unhydrated pictures for pairs of repulsive ions. 535
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There is a further important point concerning the theoretical calculations which should be addressed. In a sense,the explicit consideration of the Li+(hydrated) model is inconsistent with our molecular solvent picture. The water molecules are not covalent& bonded to Li+ and interact with this cation essentially through a paiwise electrostatic potential 1211. Hence in a true molecular solvent model hydration effects should be implicitly accounted for. However, it should be kept in mind that in the present calculations the dynamical theory [S] is carried out at the Smoluchowski level. This means that the relative tmn~ation~ diffusion of the ion pair, and consequently the calculated spectral density is determined by the equilibrium or long-time average ion-ion potential of mean force. For attractive ion pairs such as Li+/‘ON(SO&- the NMR measurements probe the relatively short-time (i.e. <, 70 ps) behaviour of the relative ion motion [S] . If on this time scale the anion ‘ON(SO& does not frequently penetrate the Li+ hy~ation sphere, then we would expect a theory employing the equilibrium potential of mean force for Li’(unhydrated)/*ON(SO&to give relaxation rates which are much larger than the experimental values. As discussed above, this turns out to be the case with the Li+ (unhydrated) calculations being some forty times greater than the experimental measurements. On the other hand, the Ll~~ydrated) calculations, which assume that the ‘ON(SO&- ion never penetrates the Li+ hydration sphere, are in good agreement with the experiments. Of course, it is also possible that the discrepancy between our calculations and the experimental results for Li+(unhydrated)/‘ON(SO&is not due to the Smoluchowski theory. For example, the various pair ~tera~tions may be overs~p~ed in the present model [5,15] , or the equilibrium theory used to obtain the ion-ion potential of mean force may not be sufficiently accurate for Li+(unhydrated)/‘ON(SO& Eventually, it should become possible to evaluate both the equilibrium and dynamical parts of the theory with careful computer simulations. The NMR relaxation results could then be used to test and improve various models.
8 August 1986
4. Conclusions This paper describes an investigation of the relative motion of the ion pairs Li’/‘ON(SO&and Li+/ Mn(D20)~ in dilute D20 solution. The relative dynamics were studied by measu~g the spin-lattice relaxation rate of the 7Li nuclei due to their magnetic dipolar interaction with the aramagnetic species ‘ON(SO&- and Mn(D20)68 . The experiments were carried out at 25°C and the measured relaxation rates were compared with theoretical results. Theoretical calculations were carried out assuming both hydrated and ~ydmted Li* ions. For the attractive ion pair Li+/‘ON(SO&- the ~+~ydrated) results are in good agreement with the experiments whereas the relaxation rates obtained for Li+(unhydrated) are much larger than the experimental values. This strongly suggests that ‘ON(SO& does not penetrate the Li+ hydration sphere on the time scales probed by the NMR me~urements. Of course, this is not a sunrise result since there is co~derable evidence [6, pp. 109,140] * (e.g., conductivity, diffusion constant measurements, X-ray diffraction data) that Li+ behaves as a hydrated species in aqueous solution. It is worth noting that in our earlier studies [l] involving larger ions (e.g., (CH&P[ON(SO&-) good agreement between theory and experiment was. obtained w~r~o~~explicitly taking hydration spheres itllto account. Thus the model and theory described in ref. [5] is less satisfactory for Li+ since the hydration effects implicitly included in the theory are not sufficient to explain the experimental results. As discussed in section 3 this failure is possibly due to the fact that the relative ion dynamics is treated only at the Smoluchowski level. For the ion pair Li~~M~(D~O)~ both the Li+~y~ated) and the ~+(~y~ted) calculations agree rather well with experiment. This is in accord with our earlier observations [2] for pair of repulsive ions.
Acknowiedgemmt We are grateful to the Natural Sciences and Rngi* The valuesof the self-diffusionconstants are given by the Nernstexpressionfrom conductivitymeasurements in ~$5. [ 17,181.
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Research Council of Canada for fmancial sup-
port.
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