736
C. T. MOYNIHAN AND C. A. ANGELL
Mass Transport in Ionic Melts at Low Temperatures.
Chronopotentiometric
Diffusion Coefficients of Silver (I), Cadmium(11) ,and Thallium(I) in Calcium Nitrate Tetrahydrate by C. T. Moynihan and C. A. Angell Department of Chemistry, Purdue University, Lafayette, Indiana 47907
(Received July %II 1969)
Chronopotentiometricdiffusion coefficients have been measured for Ag+, T1+, and Cd*+ions in molten calcium nitrate tetrahydrate over the temperature interval 15-60’. The diffusion coefficients, which are in the range 10-8 to 10-6 cm2/sec,all show non-Arrhenius temperature dependences. Despite the differences in charge and size, the Arrhenius coefficients of the diffusivities at equal temperatures are almost the same for all ions and are also very close to those previously observed for electrical conductance and shear and bulk fluidities of the pure solvent. The cooperativecharacter of the basic mass-transporting event is thus further supported. At a given temperature, D A g t and D ~ l are t about twice as large as DCdzt. Application to these data of an empirical, Stokes-Einstein type of correlation between diffusivityand fluidity for fused nitrates suggests that for D M Z+ the mobile entity is the hydrated cation Cd(Hz0)42+. In the past few years a number of investigations of the temperature dependence of transport properties have been reported for the ionic liquid calcium nitrate tetrahydrate (mp 42.7’). These include studies of cond~ctance,l-~shear v i s c o ~ i t y , ’ ~bulk ~ viscosity,S and polarographic diffusion coefficient of Cd2+ at two temperaturesU6Much of the interest in this system has been generated because it is an easily studied example of an ionic liquid in which measurements can be extended well into the metastable supercooled region. It has been found for such liquids that the first few orders of magnitude of transport properties in the supercooled region generally exhibit a non-Arrhenius temperature dependence and are well described by the three-p* rameter Fulcher equation.’
Aj, kj, and T o , jare empirical parameters and wj is a transport property, e.g., equivalent conductance (A), the reciprocal of the shear viscosity (l/qs) or bulk viscosity (l/rv), or the diffusivity/temperature quotient ( D i / T ) . 8 Equation 1 can be derived from theories which take the free volumeg or the configurational entropylo as the important quantities in setting the temperature dependence of the liquid transport properties. In these theories the Toparameter appears as an equilibrium property of the liquid, a theoretical glass transition temperature at which the free volume or configurational entropy of the liquid vanishes on cooling a t an infinitely slow rate. Inherent in the configurational entropy derivationlo of eq 1is also the notion that transport in the liquid at low temperatures occurs via a cooperative rearrangement of relatively large numbers The Journal of Physical Chemistry
of liquid particles, a conclusion which has also been reached experimentally from a comparison of transport properties and structural relaxation times for a variety of liquids.”,l2 Although the applicability of eq 1to low-temperature melts has been demonstrated many times for transport properties which reflect the combined mobilities of all ions present, it has not so far been upheld for the motion of a single species. Since the possibility that nonArrhenius transport behavior might arise from some combination of individual ionic properties has not, to date, been eliminated, and because in any case the factors determining the preexponential term in eq 1 are (1) C. T. Moynihan, J . Phys. Chem., 70,3399 (1966). (2) C. A. Angell, J . Electrochem. Sac., 112, 1224 (1965). (3) C. A. Angell, J . Phys. Chem., 70,3988 (1966). (4) C. T. Moynihan, C. R. Smalley, C. A, Angell, and E. J. Sare, ibid., 73,2287 (1969). (5) G. S. Darbari and S. Petrucci, ibid., 73, 921 (1969). (6) J. Braunstein, L. Orr, A. R. Alvarez-Funes, and 13. Braunstein, J. Electroanal. Chem., 15, 337 (1968). (7) C. A. Angell and C. T. Moynihan in “Molten Salts: Characterization and Analysis,” G. Mamantov, Ed., Marcel Dekker, Ino., New
York, N. Y . , 1969, p 315. (8) The preexponential T-I/a in aq 1 and the 5’-1 term in the wi for diffusion coefficient arise from the application of the T’/z temperature dependence of ideal gas molecular velocities t o diKusion coefficients in the liquid8 and from the Nernst-Einstein and StokesEinstein relationships among the various transport properties. These temperatures factors are frequently omitted in applying eq 1, with little effect on the derived temperature dependence parameters k j and To,j. (9, M. H. Cohen and D. Turnbull, J . Chem. Phys., 31, 1164 (1959). (10) G. Adam and J. H. Gibbs, ibid., 43, 139 (1965). (11) T. A. Litovitz and C. M, Davis in “Physical Acoustics,” Vol. 2, Part A, W. P. Mason, Ed., Academic Press, New York, N. Y . , 1965, p 281.
(12) F. J. Bartoli, J. N. Birch, N.-H.-Toan, and G. E. McDuffie, J . Chem. Phys., 49,1916 (1968).
MASSTRANSPORT IN IONIC MELTSAT Low TEMPERATURES poorly understood, the lack of self-diffusion coefficient measurements in the low-temperature region has been an obvious defect in the available data. Hence it was felt that an investigation of self-diffusion, again using calcium nitrate tetrahydrate as a trial system, should be undertaken. Conventional self-diffusion coefficient measurements for the bulk ions of the melt by tracer methods suffer from two disadvantages: (a) because of the low values of the diffusion coefficients to lo-* cm2/sec), inordinately long diffusion times, of the order of several days, are required, and (b) over this long a period there is a high likelihood that the run will be interrupted by crystallization of the metastable, supercooled melt. Consequently, it was decided to perform diffusion coefficient measurements for reducible cations (Ag+, Cd2f, and T1+)present in low concentration in calcium nitrate tetrahydrate melts by the much more rapid method of chronopotentiometry. l3 In this technique, an electrochemically active species in an unstirred, dilute solution with a large excess of inert electrolyte is electrolyzed at constant current, and the transition time, 7, required to exhaust the active species at the electrode surface is determined from the potential-time curve for the electrolysis. The diffusion coefficient of the active species i is given by the Sand equation
where I is the current, F the Faraday constant, n the number of electrons required for electrolysis of an ion of i, A the electrode area, and c, the concentration of i. Equation 2 is derived on the assumption that all of the current is consumed in the Faradaic electrolysis process, but in fact part of the current is always used in charging the electrode double-layer capacitances. A number of ~ o r k e r s l ~have - ~ ~considered this problem and shown, in effect, that eq 2 may be employed provided that the transition time, 7, is measured in a proper fashion to compensate for the effect of the double-layer charging. Laity and Mc1nty1-e’~have shown via irreversible thermodynamics that for fused salts the chronopotentiometric diffusion coefficient of an ion at low concentration can be equated to both its interdiffusion and self-diffusion coefficients.
Experimental Section Calcium nitrate tetrahydrate, cadmium nitrate tetrahydrate, and silver nitrate were reagent grade chemicals and were used without further purification. The water content of the calcium nitrate tetrahydrate was determined by dehydrating the salt a t 160’ in a vacuum oven; the two lots used gave mole H 2 0 :mole Ca(NO& ratios of 4.09 f 0.01. Thallium nitdrate (British Drug Houses, Ltd.) was recrystallized once from water and dried at 110” before use. All solutions
737
were prepared by weighing the solid components directly into the chronopotentiometric cell. The calcium nitrate tetrahydrate was then melted in a hot water bath a t about 50” and the reducible cation nitrates dissolved by agitating the melt with a magnetic stirrer. The constant current source for the chronopotentiometry apparatus was a 225-V bank of dry cells connected in series with three potentiometers (10 M, 1 M, and 100 k) which were used as adjustable resistors to regulate the current. The magnitude of the current could be adjusted initially by passing it through a dummy 500-ohm resistor, after which the chronopotentiometric cell could be switched into the circuit by means of a DPDT switch. Current was measured during the electrolysis with a Keithley 6lOB electrometer which was found by calibration to be accurate to =t0.5%. Potential-time curves were measured with an EsterlineAngus Speed Servo recorder, for which less than 0.1 sec was required for full scale recorder pen deflection. A chart speed of 0.75 in./sec was used in recording the chronopoten tiograms. The calcium nitrate tetrahydrate melts (about 90 g) were contained in tightly capped 35 or 38-mm od Pyrex tubes into which dipped two electrolysis electrodes. For experiments with Ag+ the tubes were painted black on the outside to lessen the chances of photodecomposition of silver ion. The anode and combined reference and counterelectrode was a 3-mm length of 0.5-mm diameter platinum wire emerging from a Pt-Pyrex seal in a piece of Pyrex tubing. For the melts containing Ag+ ion the cathode and indicator electrode was a platinum foil of approximate dimensions 10 X 12 mm. The foil was welded to a short length of platinum wire which emerged from a Pt-Pyrex seal in a piece of Pyrex tubing. The exact surface area of the Pt cathode (including foil edges and wire) was calculated from measurements of its dimensions and was found to be 2.54 f 0.02 cm2. Cd2+and Tlf cannot be reduced at a platinum electrode in calcium nitrate tetrahydrate melts due to the low hydrogen-on-platinum overvoltage. Hence for these two ions, a mercury pool cathode had to be employed. The mercury was contained in a J-shaped piece of 7-mm od Pyrex tubing, the short arm of which terminated in a cup formed from 14-mm od Pyrex and dipped below the surface of the melt. The long arm of the J-tube extended out through the cell cap, and contact to the mercury was made by a platinum wire in(13) ,Reviews of applications of chronopotentiometry to fused salts are given by C. H. Liu, K. E. Johnson, and H. A. Laitinen in “Molten Salt Chemistry,” M. Blander, Ed., Interscience, New York, N. Y . , 1964, p 681, and by H. A. Laitinen and R. A. Osteryoung in “Fused Salts,” B. R. Sundheim, Ed., McGraw-Hill, New York, N. Y . , 1964, p 255.
(14) R. W. Laity and J. D. E. McIntyre, J . Amer. Chem. SOC., 87, 3806 (1965). (15) M. L. Olmstead and R. 9. Nicholson, J. Phys. Chem., 72, 1650 (1968). (16) R. 8. Rodgers and L. Meites, J . Electroa~al.Chem., 16, 1 (1968).
Volume 74, Number 4 February 19, 1970
C. T. MOYNIHAN AND C. A. ANGELL
738 serted down this arm. The area of the mercury pool electrode was determined by measuring the id of the cup (1.17 cm) and the height of the mercury meniscus when the cell was filled with melt (0.15 cm) and calculating the area on the assumption that the surface had the shape of an oblate hemispheroid. This gave an area of 1.15 =k 0.02 cm2for the mercury surface. The chronopotentiometry cells were thermostated in a Lauda K-2/R constant temperature circulator. Bath temperatures were measured to k0.05' with a calibrated mercury-in-glass thermometer. The circulator was turned off briefly while recording a chronopotentiogram to avoid agitation of the melt. Kormally a third reference electrode is incorporated into a chronopotentiometric cell, and the potential of the indicator electrode is measured with respect to it. In our case, however, we were not concerned with the determination of the quarter-wave potentials of the reducible tracer cations, so we measured the potential of the indicator electrode with respect to the Pt wire anode. In calcium nitrate tetrahydrate melts oxygen is evolved at a platinum anode," so that the anode maintains a constant potential during electrolysis at constant current and, hence, is suitable as a reference electrode for measuring transition times. To check this point, however, in one of the runs with silver as the reducible cation a heavy silver wire was incorporated into the cell as a third reference electrode. No difference was found between transition times measured using the Ag wire as reference electrode and those measured using the Pt anode as reference. Blank runs were performed with both platinum and mercury cathodes on samples of melt to which no reducible cation had been added in order to detect any possible contributions to transition times from reducible impurities or dissolved oxygen. At the current densities used in these experiments, no such transition times were found in the blank runs, even after bubbling pure Hence no precautions were taken 0 2 through the melt. to exclude air from the melts, save for tightly capping the cells to avoid loss of water at the higher temperatures. Runs were begun by measuring chronopotentiograms at 50°, then lowering the cell temperature and performing measurements at several temperatures down to 15'. Finally, the cell temperature was raised again to 50°, a series of chronopotentiograms run, and a final series of measurements taken at 60'. At least three potential-time scans were run a t each temperature with agreement among transition times measured at the same temperature of 1-2%. In these viscous melts, it was necessary to wait 10-20 min, with intermittent stirring, between successive chronopotentiograms to allow the bulk tracer-ion concentration to be reestablished at the electrode surface. It was generally not possible to avoid freezing of the melt when measurements were attempted below 15'. The Journal of Physical Chemistry
t
0.50 volts
-E
I
i I
t
A r+
-/
I
I
I '
I
+I sec__
Time --c Figure 1. Chronopotentiogram of 0.00809 M Cd2f solution in Ca(N0&.4.09 HzO at 49.6". I / A = 0.436 mA/cm2, T = 3.72 sec.
Measured transition times were kept, in the range 2-5 sec, which required currents in the range 0.1-1 mA.
The quantities of melt and concentrations of reducible tracer ion were such that at these currents upward of thirty chronopotentiograms could be run on a sample without changing the initial tracer ion concentration by more than0.1%.
Results A typical chronopotentiogram is shown in Figure 1. In analyzing the chronopotentiograms we have followed the method of Olmstead and Nicholson, l6 who characterize the effect of double-layer charging on measured transition times in terms of a parameter $
where C d l is the double-layer capacitance, and 7 S s n d is the value of r which satisfies the Sand equation. They show that if is less than 0.005, experimental values of r determined by the method of Delahay and Berzinsl* deviate from 7Sand by less than 1.7%. For our experiments #, evaluated using values of c d l calculated from the slope of the initial steeply rising portion of the potential-time curves, was generally less than 0.001. Hence we have used the Delahay and Berzins method of determining 7, as is shown in Figure 1. Diffusion coefficients for Ag+, Cd2+, and T1+ were calculated from the experimental data via eq 2. Ionic concentrations were calculated from the masses of the (17) R.-P. Courgnaud and B. TrBmillon, Bull. SOC.Chim. FT., 758 (1965). (18) P. Delahay and T. Berzins, J . Amer. Chem. Soc., 75, 2486
(1953).
MASSTRANSPORT I N IONIC MELTSAT
739
LOW TEMPERATURES
components used in making up the melts and the density-temperature-composition data for Ca(NO&-HzO solutions of Ewing and Mikovsky.lg Results are tabulated in Table I and shown in the form of Arrhenius plots in Figure 2.
t i"C)
7
0
6
0
5
0
4
0
30
20
10 I
1'O.O
Table I: Diffusion Coefficients of Ag+, Cd2+, and T1+ in Ca( NO& 4.09H20 Melts I
Silver(1) c30a
Temp OC
Run 1 Run 2 = 5.42 X 10-6 C S O ~= 8.04 X 10-8 DAp+ X lo', cm*/seo
59.60 49.60 39.70 29.75 21.80 14.85
10.7 7.4 4.4 2.54 1.64 0.96
10.7 7.4 4.6 2.60 1.51 0.93
Cadmium (11) CIO=
Tomp OC
-
59.60 49.60 39.70 29 75 21.80 14.85
-
Run 1 Run 2 7.26 X 10-6 CIO' 8.09 X 10-6 Dcdz+ X 107, cm*/seo
5.0 3.1 1.9 1-09 0.57 0.32
I
Run 3 C50a
5.5 3.7 2.2 1.25 0.66 0.42
= 7.68 X l o 4
5.3 3.5 2.1
Thallium(1) ma =
Temp
O C
59.60 49.60 39.70 29.75 21.80 14.85 a
Run 1 Run 2 11.55 X 10-6 cbo5 = 10.72 X 10-6 DTI+ X lo7 oms/seo
10.4 7.0 4.2 2.30 1.26 0.71
10.9 7.3 4.4 2.47 1.42 0.75
Concn of ion a t 50' in mol/cms.
The internal precision of a set of diffusion coefficients measured during a single run, which affects the uncertainty in the calculated temperature dependences of D,, is determined by the uncertainties in the measured currents, transition times, and temperatures and is estimated to be about 3%. The relative precision expected when diffusion coefficients measured during different runs are compared depends in addition on the uncertainities in the tracer-ion concentration and melt water content and is estimated to be around 5%. Finally, the absolute accuracy of the D, measurements depends on the accuracy of the electrode area measurement as well as on the previously mentioned factors and is estimated to be 7% for D A ~and + 9% for Dcda+and
Dm+. Agreement between values of D,measured in different runs for Cd2+ and for T1+ a t lower temperatures is somewhat outside the estimated relative precision of
0.5-
0.2.
0.11
'
29
3.0
3.1
3.2 3.3 I/T x IO3
3.4
3.5
I
36
Figure 2. Arrhenius plots for diffusion coefficients, equivalent conductance, and shear and bulk viscosity for Ca(NO&.4.09HzO. Ordinates: Di, cm2/sec; AT, cm2 deg/ohm equiv; T/q. and T/qv, deg/P.
5%. The most likely source of the discrepancies is a small amount of creepage of melt between the mercury and glass in the cathode, leading to some deviation from run to run in the actual area of the electrode. The agreement between values of D A ~ for + different runs is well within the estimated relative precision, indicating that the poor relative precisions for Dcdz t and DTI+ are indeed due to the cause suggested, since the problem of poorly defined electrode area is not encountered with the Pt-foil electrode. Braunstein and coworkers6 have measured D C d 2 + by polarography in Ca(N0&-4.00€€20 at 50 and 100"; at 50" their result was (3.12 0.010) X lo-' cm2/sec. We may correct this result to the melt composition by which studied here by an empirical all of the isothermal composition dependence of transport properties in Ca(N0&-H20 solutions is lumped into a variation in the TO parameter in eq 1. This gives for the result of Bmunstein, et al., D C d l t 3.38 X
*
(19) W. W. Ewing and R. J. Mikovsky, J . AmeT. Chem. Soc., 7 2 , 1390 (1950). (20) C. A. Angell, J . Chem. Phys., 46,4673 (1967). (21) C. A. Angell, E. J. Sare, and R. D. Bressel, J . Phys. Chem., 71, 2759 (1967).
Volume 74, Number 4
FebTuaTy 10, 1070
C. T. MOYNIHAN AND C. A. ANGELL
740
Table 11: Fit of Equation 1 to Transport D a t a for Ca(NOs)t.4.09BtO with Same T o , jfor All D a t a Stand dev
Ai
5.06 X 5.84 X 4.42 X 3.97 x 10-6 4.31 x 10-8 8.92 x 8.60 x 3.063 X lo3 1.087 X l o w 4 2.47 x 10-4 a
Units: ( D , / T ) ,cm2/sec deg; AI cmZ/ohm equiv;
582.4 598.2 663.0 635.0 650.4 655.2 644.8 591.86 694.51 726.8
202.5 202.5 202.5 202.5 202.5 202.5 202.5 202.5 202.5 202.5
in In wj
0.038 0.023 0.030
0.044 0.012
0.017 0,025
0.007 0,004 0.033
q s and qvr P.
lo-' cm2/sec at 50" for Ca(NOa)zq4.09Hz0. This datum is shown plotted in Figure 2 and is in excellent agreement with our results, suggesting that there are no undetected systematic errors in our method. To facilitate comparison of temperature coefficients, our diffusion coefficient data have been computer fitted to eq 1 using the value To = 202.5, which was found previously4 to give an adequate fit for conductance and shear viscosity at this melt composition. The calculated parameters are given in Table 11. The standard deviations in In ( D , / T ) are consistent with our estimate of about 3% for the internal precision of a given run.
Discussion In Figure 2 the temperature dependences of the tracer diffusion coefficients are compared with those for the conductance and shear viscosity determined previously4 for Ca(N0&.4.09H20 and with the bulk viscosity data of Darbari and Petrucci6 for Ca(NO&.4.0 HzO. The bulk viscosity data shown considerable scatter when plotted as a function of concentration, so that no attempt has been made in Figure 2 to correct them to the melt composition used in this study. In Table I1 similar comparisons are made among the transport properties in terms of the parameters of eq 1 needed to describe the data. In this case, the bulk viscosity parameters determined for Ca(NO& e4.0 HzO have been corrected to the composition Ca(NO&. 4.09H20as b e f ~ r e . ~ , ~ , * ~ J ~ As expected from the behavior of the other transport properties, the Arrhenius plots of DIare not linear, but are curved in a direction corresponding to an apparent increase in activation energy with decreasing temperature. Similarly, as expected from the theoretical interpretation of To,,in eq 1 as an equilibrium rather than a transport property of the melt,1p4j7an adequate fit to all the transport data for the melt can be obtained with the same value of TO,,, as is shown in Table 11. The various transport properties show a remarkable similarity to one another in their temperature dependences when compared on a logarithmic scale. The The Journal of Physical Chemistry
TQ,~
kj
largest discrepancy between the IC, terms in Table I1 (582.4 for D A ~ + / Tvs. 726.8 for l/qv) is about 22%. For comparison purposes, the agreement between the temperature dependences of the transport properties could probably be improved if one could dissect the shear and bulk viscosities into contributions from the appropriate relaxation times and moduli 7s = qv =
GmTs
( K , - Ko)T,
G, is the infinite frequency modulus of shear rigidity, K , and K Othe infinite and zero frequency moduli of , the shear and compression, and T * and T ~ respectively, structural relaxation times. Litovitz and McDuffie and their coworkers11~12~22 have shown experimentally that better correlations are achieved when one compares the temperature and pressure dependences of T~ or T~ with those of other transport properties than when the similar comparison is made using q s and q v . This result is a reasonable one, since G, and ( K , - K O )measure elastic, solid-like properties, rather than fluid-like properties of the liquid. G, and ( K , - K O )both increase with decreasing temperature, so that it seems likely that part of the discrepancy between the ICj values for l / q s and l/qv and the k , values for the other transport properties would disappear if one were able to take these factors into account. The close agreement between the temperature dependences on a logarithmic scale (or activation energies) of the diffusion coefficients of various ions in the same melt seems to be a fairly general phenomenon in anhydrous fused salts, having been observed both for cation and anion self-diff usion coefficients in high-temperature pure fused ~alts7~23~2~ and mixed melts25-28and for
(22) T. A. Litovitz and G. E. McDuffie, J. Chem. Phys., 39, 729 (1963). (23) C. A. Angel1 and J. W. Tomlinson, Trans. Faraday SOC.,61,2312 (1965). (24) C. A. Sjoblom, Z . Naturforsch., 23a, 933 (1968). (25) F. Lantelme and M. Chemla, Bull. SOC. Chim. FT., 969 (1963).
MASSTRANSPORT IN IONIC MELTS AT Low TEMPERATURES polarographic diffusion coefficients in lower temperature melt^.'^^^ On the other hand, in high-temperature melts discrepancies frequently occur between the activation energies of diffusion coefficients and shear viscosityZ4and even more frequently between the activation energies of shear viscosity and equivalent conduct a n ~ e ,A~usually , ~ ~ having the lower temperature coefficient. Hence, it would appear that for ionic liquids close similarity among the temperature dependences of all the different transport properties is a phenomenon associated primarily with low temperatures. We interpret this similarity in temperature coefficients observed in our study as a manifestation of the cooperative nature of the mass transport mechanism at low temperatures, such that the “activated jump” involved in the transport of any single particle occurs in concert with a simultaneous rearrangement of a Iarge number of other particles in its vicinity. At, a given temperature, the actual magnityde of the diffusion coefficient of an ion in calcium nitrate tetrahydrate appears to depend more on its coulombic charge than on any other factor, as is seen from the relatively small difference in the diffusional mobilities of Ag+ and T1+ in comparison to that of Cd2+,which is only about half as mobile as either of the singly charged ions. A similar preponderance of charge effects in determining cationic mobilities has been observed in higher temperature anhydrous melts for tracer ion diffusion coeffic i e n t ~ , ~as~ well - ~ ~as- for ~ ~electrical ion Although the decrease in cationic mobility with increase in coulombic charge is in accord with the more or less intuitive feeling possessed by physical chemists that increased charge should lead to local electrostriction in the melt and a corresponding restriction in the “freedom of motion’’ of the ion, no one has yet put forth a satisfactory theory for the effect in anhydrous melts. The formation of “complex ions” by highly charged cations is clearly too simplistic an explanation. It does not account, among other things, for the fact that temperature coefficients of transport properties of multivalent ions are comparable to those of univalent ions and do not appear to reflect changing equilibrium constants and that electrical mobility behavior is quite similar in systems such as Ca(NO&-LiN03 and CdCl2KC1, where the “complex-forming” abilities of the ions are presumably quite different. ae A note of caution needs to be injected here, of course, in comparing our results for a hydrate melt with anhydrous systems, namely, that the presence of water may invalidate the comparison. I n this light it is interesting to interpret our diffusion results in terms of the Stokes-Einstein equation (3)
74 1
the identification of the friction coefficient, fi, of the Einstein equation
Di
=
with the hydrodynamic force impeding the steady movement of a sphere of radius ri in a continuous fluid medium of viscosity qs. k is the Boltzmann constant, and the numerical factor a is equal to 6 n if there is no slippage at the fluid-sphere interface and 4n if there is complete slippage.as Although this model has no obvious relevance to the case of ionic motion, many studies show that a relation of the form of eq 3 exists between diffusion coefficients of atomic size particles and the fluid viscosity. Forcheri and Wagner,28 for instance, have shown that a plot of univalent ion diffusion coefficients in univalent nitrates us. T/vsri gives, with considerable scatter, a straight line with a slope corresponding to a = 4 . 6 in ~ eq 3. Similar plots for molten nitrates and halides, with similar slopes, have been presented by Bockris and coworkers.39 To test this correlation for the present results, we first note that proton magnetic resonance ~ t u d i e s ~ , ~ ~ have shown that in calcium nitrate tetrahydrate large univalent ions such as K + or (CHa)dN+are unhydrated, while Cd2+ has a hydration number of 4. It seems a safe presumption that Ag+ and T1+, which are similar in size to K+, are likewise unhydrated. Hence for calculations with eq 3 we may use the “bare” ionic radii41for Ag+ and Tl+, but for Cd2+we have a choice of using the “bare” ionic radius or the “hydrated” ionic radius (= rCdz+ 2rH20). In Table I11 are shown ratios of the observed D iin calcium nitrate tetrahydrate to the Di calculated via eq 3 using a = 4 . 6 at ~ two temperatures. The results
+
(26) F. Lantelme and M. Chemla, C. R. Acad. Sci., 258, 1484 (1964). (27) P. L. Spedding and R. Mills, J . Electrochem. Soc., 113, 599 (1966). (28) S. Forcheri and V. Wagner, 2. Naturforsch., 22a, 1171 (1967). (29) M. Francini and S.Martini, ibid., 23a, 795 (1968). (30) J. P. Frame, E. Rhodes, and A. R. Ubbelohde, Trans. Faraday Soc., 5 5 , 2039 (1959). (31) H. A , Laitinen and W. S.Ferguson, Anal. Chem., 2 9 , 4 (1957). (32) H. A. Laitinen and H. C. Gaur, Anal. Chem. Acta, 18, 1 (1958). (33) C. E. Thalmayer, S. Bruckenstein, and D. M. Gruen, J . Inorg. Nuc2. Chem., 26,347 (1964). (34) F. Caligara, L. Martinot, and G. Duyckaerts, J . Electroanal. Chem., 16,335 (1968). (35) W. K. Behl and J. J. Egan, J . Phys. Chem., 71, 1764 (1967). (36) J. C. T. Kwak, Ph.D, Thesis, University of Amsterdam, 1967. (37) A. Berlin, F. Menes, S. Forcheri, and C. Monfrini, J . Phyls. Chem., 67,2505 (1963). (38) H. Eyring, D. Henderson, B. J. Stover, and E. -M.Eyring, “Statistical Mechanics and Dynamics,” John Wiley & Sons, Inc., New York, N. Y . ,1964, p 463. (39) (a) J. O’M. Bockris, S. Yoshikawa, and S. R. Richards, J . Phys. Chem., 68, 1838 (1964); (b) J. O’M. Bockris, S. R. Richards, and L. ”anis, ibid., 69, 1627 (1965). (40) C. T. Moynihan and A. Fratiello, J . Amer. Chem. Soc., 89, 5546 (1967).
The original derivation of the equation was based on
(41) C. S. G. Phillips and R. J. P . Williams, “Inorganic Chemistry,” Oxford University Press, New York, N. Y., 1965, p 152.
Volume 74,Number 4 February 19, 1970
742
C. T. MOYNIHAN AND C. A. ANGELL
Table 111: Comparison of Observed Diffusion Coefficients in Calcium Nitrate Tetrahydrate t o Those Calculated from the Stokes-Einstein Equation using a = 4 . 6 ~ Ion
AF2 +
T1+ Cd(He0)d2+
CdZ+
Tir
A
1.26 1.40 3.73 0.97
Di,obsd/Di,oalod
14.850
59.60’
2.63 2.24 3.03 0.79
1.72 1.89 2.52 0.6;
show for the first three entries that the calculated Dt are too low, the discrepancies becoming worse the lower the temperature. Since the a value of 4-87 talien from Forcheri and Wagner is strictly empirical and applies to melts with viscosities lower by two orders of magnitude than those encountered here, the low values of D Z , c a i o d simply reflect the fact already noted that the temperature coefficient of l/rs is somewhat greater than that for D,/T. The fact remains, however, that for a given value of v S there is some correlation between Di and ionic radius, and at a given temperature the agreement among ( D P , o b s d / D l , c a l c d ) for Ag+, TI+, and Cd(H20h2+ is as good as that noted by Forcheri and Wagner28for the higher temperature melts. On the other hand, ( D . l , o b s d / D i , c a ~ c dfor ) the unhydrated Cd2+ion is clearly out of line with the other three values. The implication here is that possibly one can account for the magnitude of D zof a multivalent ion in a hydrate melt in terms of ionic size, if one takes the kinetic entity to be the hydrated ion for the multivalent ions and the unhydrated ion for the monovalent ions, an*option not available in anhydrous melts. The mobile hydrated cation is not entirely unexpected, since it has previously been shown that Cd2f and Ca2+hydrate equally in their tetrahydrate melts4 and that Tovalues for the melts are consistent with the presence of cations with the charge: radius ratio of the tetrahydrate cation.3 On the other hand, it has not
The Journal of Physical Chemistry
previously been possible to provide even ambiguous evidence that this cation is also a kinetic entity. We have noted before4*’that factors determining the magnitude of the preexponential A , in eq 1 are poorly understood. The above discussion accepts that there is a difference of T / r z and a numerical factor between AD^ and but does not consider what factors may be common. To avoid circularity we must restrict comparisons to the data on Ag+ and Tlf. This, however, is sufficient to point up (Table 11) the absence of any m--Iia mass dependence of AD,, which most treatments of rate processes suggest should enter either as a particle velocity termg or as a vibration frequency,42 Again, the problem seems resolved by the cooperative rearrangement mechanism, for which the appropriate frequency is presumably a property of the cooperative region involving a complex of frequencies, rather than a property of the particular particle whose motion is being followed. In the future we hope to report chronopotentiometric diffusion coefficients for ions in an anhydrous melt capable of a large degree of supercooling, e.g., a Ca(N03)~KN03 melt,43in which the diffusion coefficients may be studied over a temperature interval sufficient to encompass at least three orders of magnitude in Di, including the “normal” high temperature values of around cm2/sec. Such a study should resolve some of the questions we have raised here in regard to the dependence of cationic mobilities on coulombic charge, as well as provide some insight into the breakdown in cooperative transport behavior which is presumed to coincide roughly with the onset of an Arrhenius temperature dependence of transport properties. Acknowledgment. This work was supported by a grant from the Department of the Interior, Office of Water. (42) C. Kittel, “Introduction to Solid State Physics,” John Wiley &
sons.New York, N. y . ,1966,I)568. (43) c. -4.Angell, J . Phys. Chem., 68,1917 (1964).
