Reverse pulse voltammetry. Application to second-order following

285

J. Phys. Chem. 1981, 85, 285-289

seen in Figure 7, the presence of another factor is suggested to affect the reaction rates of the present oxidation. It is speculated that the most favorable steric conditions are met in the case of n = 1 for the attack of Ce(V). For example, the methyl group in AO+-CH3 may be directed toward the inside of surfactant layer. As a result, the carbon atom Cg,which is considered to be the site attacked by Ce(1V) ion, is exposed toward the water phase. On the other hand, the interaction of AOH+ with surfactant molecules is mostly due to the dispersive force between

n-electron groups. Thus the benzene ring in AOH+ is faced with the surfactant layer to result in the blocking of the Cg atom by the bulky surfactant molecules. These steric factors may cause the higher reactivity of AO+--CH3as well as the lower reactivity of AOH+ than expected from their hydrophobicity scale. Acknowledgment. We express our thanks to Dr. T. Suzuki of Hokkaido University for his kind advice in synthesizing the dyes.

Reverse Pulse Voltammetry. Application to Second-Order Following Reactions Janet Osteryoung, Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214

D. Talmor, J. Hermolln, and Emllla Klrowa-Elsner Institute of Chemistry, University of Tel-Aviv, Tel-Aviv, Israel (Recelved: August 28, 1980)

Reverse pulse voltammetry is applied to the quantitative treatment of second-order chemical reactions following reversible electron transfer. Simple algebraic and graphical procedures based on the results of simulations are employed to estimate the backward rate constant of the chemical reaction from an approximationto the equilibrium constant (K'J which is obtained experimentally. One iteration produces accurate values of the forward and backward rate constants and the true equilibrium constant as shown by detailed fitting of experimental results to simulated results. For the following dimerization of N-methyl-2-carbomethoxypyridinylradical the technique yields values of K = 104.e8M-', kf = 104.03M-' s-l. The treatment is applicable for -1.3 5 log (K'C) I4.

Introduction Previous work has described the technique of reverse pulse polarography (RPP) (or reverse pulse vo1tammetry)l and its application to the study of homogeneous reactions coupled with electron-transfer processes.24 The purpose of this paper is to describe the quantitative treatment of RP data for second-order following reactions. To our knowledge the case of reversible following dimerization reactions has not been treated b e f ~ r e . ~ Consider the reaction sequence

potential until the waves for oxidation of the radical and dimer are obtained. If (7 - t,) is sufficiently long in comparison with t, then the concentration profiles for both radical and dimer should be very flat on the time scale t,, and therefore the RP limiting currents, i w ,and ~ iwp, for the radical and dimer, respectively, can be said to_arise from average concentrations in the diffusion layer, CR and CD,respectively, which are nearly equal to the surface concentrations of these species. Thus we may write eq 2 and 3, where i R p 8 is the RP diffusion-limited current for ~RP,R

An approximate equation for the equilibrium constant for dimer formation can be derived by using the RP currents to estimate the concentrations of radical (R) and dimer (D) in the diffusion layer. The experiment is the following. The potential is held for a time (7- tP) at a value for which the material initially present, 0, is reduced at the diffusion-limited rate. During that period radical and dimer are formed near the electrode. Then a pulse of duration t, is applied in the positive direction. The experiment is repeated with successively more positive values of the pulse (1) Osteryoung, Janet; Kirowa-Eisner, Emilia, Anal. Chem. 1980,52, 62-6. ( 2 ) Kashti-Kaplan, S.; Hermolin, J.; Kirowa-Eisner, E., submitted to J.Electrochem. SOC. (3) Osteryoung, Janet; Hermolin, Joshua; Kashti-Kaplan, S.; KirowaEisner, Emilia, submitted to J. Electroanal. Chem. (4).Hermolin, Joshua; Kashti-Kaplan, S.; Kirowa-Eisner, Emilia, submitted to J. Electroanal. Chem. ( 5 ) Elving, Philip J. Can. J. Chem. 1977, 55, 3392-412. 0022-3654/81/2085-0285$01.OO/O

FACR[DR/(~~,)I'/~

(2)

i ~ p =, ~~ F A C D [ D D / ( ? ~ ~ , ) ] ~ / ~ (3)

the radical, R (R. of eq 1);iw,D is the RP diffusion-limited current for the dimer, D (eq 1);F is the Faraday constant; A is the electrode area; and D Rand DD are the diffusion coefficients of R and D, respectively. The factor of 2 in the equation for iRp,D arises because we assume that the oxidation of dimer requires two electrons per mole. The equilibrium constant for dimer formation (eq 1) is then given by

K' =

CD/(~R)'= ~'R~,DFADR/[~~R~,R~(~DD~,) (4)

The normal pulse (NP) current for the reduction of 0 is just iNp

= FACb[Do/(~tp)]'/2

(5)

where Cbis the bulk concentration of 0, and Do the diffusion coefficient of 0. If Do = DR = DD, then we can combine eq 4 and 5 to obtain 0 1981 American Chemical Society

286

The Journal of Physical Chemistry, Vol. 85, No. 3, 1987

K’ = iRP,DiNP/ (2iRP,R2Cb) (6) There are cases for which the reactions of eq 1apply but for which the dimer does not exhibit a diffusion-controlled oxidation wave. In such cases it is useful to estimate the R P current for the dimer by using the relation between the DC limiting current for reduction of 0 and the R P current for oxidation of the products (eq 7),l where -iRP/iDC = ( T / t , ) ’ / , - 1 (7) spherical effects have been ignored. Note that this equation is not that which applies to polarography, but rather that for voltammetry at a stationary (planar) electrode. Because within this approximation ( T / t , ) l / , is just the ratio iNp/iDc,eq 7 becomes simply eq 8, where iNp (8) -iW = iNp- iDc is the limiting cathodic current for reduction of 0. Thus in eq 6 we use eq 9. (9) ~ R P , D= ~ N + P ~ D C- ~ R P , R The notation K’in eq 6 is used to indicate that the value calculated is only an approximation to the true value of K = kf/kb. The assumptions of eq 6 are that (1)the pulse duration is sufficiently short that the homogeneous reaction does not procede significantly on that time scale, (2) the concentration profiles are sufficiently flat near the electrode surface that at the time of pulse application concentrations in the diffusion layer corresponding to the pulse duration are independent of distance from the electrode, (3) the time at the initial potential is sufficiently long that homogeneous equilibrium has been reached by the time of pulse application, and (4)the electrochemical reaction (eq 1)is reversible, and diffusion coefficients of radical, dimer, and starting material are the same. When these assumptions hold, the pulse currents, iRp,Rand iw,D, are simply used as a measures of the uniform concentrations of radical and dimer, respectively. Because of the complexity of the system, it is not obvious where the range of accuracy of eq 1lies. Furthermore, if equilibrium is not attained by the time of pulse application, it should be possible to extract kinetic information. In attempts to do so, K’might prove to be a useful kinetic parameter. We have employed digital simulation using the methods of FeldbergeJto calculate the currents in eq 1and thus to calculate K’for various values of kf, kb, and cb.The time dependence of the resulting values is then used to assess the difference between K’ and the true value, K, expressed as log (K’IK). Values of K and kb can be obtained by using the results of the simulations by two procedures. The first and most general procedure employs detailed curve fitting of the experimental points to simulated curves. The second employs simple algebraic and graphical procedures based on the results of the simulations to estimate k b from K’and then using kb to estimate log (K’IK). Both procedures are applied to data obtained by NP and RP for N-methyl-2-carbomethoxypyridinium ion., Experimental Section The experimental system employed is N-methyl-2carbomethoxypyridinium in acetonitrile (nominally dry) with 0.25 M LiC104as supporting electrolyte. A modified PARC Model 174 polarographic analyzers was used in (6) Feldberg, S.W.In “Electroanalytical Chemistry”; Bard, A. J., Ed.; Marcel Dekker: New York, 1971; Vol. 3. (7) Hanafev. M. K.:Scott, R. L.: Ridmay, - - T. H.; Reilley, C. N. Anal. Chem. 1978,k0,116-37. (8) Abel, R. H.; Christie, J. H.; Jackson, L. L.; Osteryoung, Janet; Osteryoung, R. A. Chem. Instrum. 1976,7, 123-138.

Osteryoung et al.

r a i .C

-5c

-ioc -15C

.-171

E , V vs Ag/Ag* Figure 1. DC and RP waves for 2 mM N-methyl-2-carbomethoxypyridinium ion in acetonitrile with 0.25 M LiCIO,. T = 0.333 s; t, given in figure in ms; SMDE electrode area: 0.013 cm2. Arrows indicate the initial potential.

conjunction with a PARC Model 303 static mercury drop electrode (SMDE) with drop area 0.013 cm2. Other experimental details were routine and are discussed more completely e l s e ~ h e r e . ~ , ~ Digital simulations were carried out according to the procedure of Feldberg.6J All programs are coded in Fortran IV, and program listings as well as tabulations of values of K’ are available on request.

Results and Discussion Under the experimental conditions chosen the model reaction has the following mechanism:, py+

* -4

Py.

2Py.

=Py, kr kb

where Py+ indicates the N-methyl-2-carbomethoxypyridinium ion. Figure 1illustrates the DC wave for reduction of Py+ and the RP waves for the reoxidation of the radical, Py, and the dimer, Py,. Notice that the oxidation wave of Py, is distorted by adsorption. Therefore in this case the limiting current imp was obtained through eq 9. The RP experiment has two time parameters, T and t,. If the RP currents are to be used to estimate concentrations in the diffusion layer, then it is necessary that the equilibrium be frozen on the time scale t,. That is, the concentrations established during the period T - t should not change because of readjustment of the ckemical equilibrium. During the RP experiment, the reaction scheme of eq 1 is turned around; the reactant is R., and a second-order preceeding reaction is taking place (eq 10). Assume that kb

DZ2R. kr

-e-

R.-0

e-

(10)

equilibrium has been attained in the diffusion layer during the period T - t,. Then we can associate with the equilibrium concentration of the dimer, D, a kinetic current, ik, that will flow during the period t,. The requirement (9) Galus, Z. “Fundamentals of Electrochemical Analysis”; Wiley: New York, 1976; p 332.

Reverse Pulse Voltammetry

that the equilibrium be frozen during the period t, can then be formulated approximately by putting some limits on the magnitude of ik Because we wish only to estimate errors and the situation is very complicated, a crude calculation is justified, and we use simply the standard polarographic treatment for a second-order preceding reaction.8 We choose as the kinetic parameter a = (kbtp/ (KCD)'Iz)'I2 where C D is the equilibrium concentration of dimer at r - t,. The error due to ik depends on both a and K. For large values of a,ik/idJ), the ratio of kinetic current to the diffusion-limited current which would be obtained for the dimer, is large; but if K is small, CD is small, and hence the contribution of ik to the total current due to the radical is small. On the other hand, if K is large, CR is small, and even when a is fairly small the kinetic current can be a significant contribution to the current due to the ~ , be inradical. In order for the limiting current, i R P , to creased by less than 10% by the kinetic effect, we find that, for log (KCb)= 0, 1, 2, and 3, ik/id,D must be less than 0.2, 0.05,0.014, and 0.0045, respectively. For log (KCb) I -1, the kinetic effect is unimportant. When one uses the approximate relation for the kinetic current (eq 11),8the maximum values of kb for less than ik/id,D = 0.81a(l - ik/id,D)3/4 (11) 10% error in iRP,Rcan be calculated. For log (KCb) = 0, 1, 2, and 3 and t, = 1 ms, these values are k b < 42, 8, 2, and 0.7 s-', respectively. How is this definition of frozen equilibrium to be translated into practical experimental criteria without requiring knowledge of K and kb? If the effect of kinetic current is sufficiently large, it will be reflected in the time dependence of the limiting current (Le., d log i/d log t # 0.5). However in many cases the range of t , over which frozen conditions can hold is too limited to test this relationship satisfactoril , Digital simulation for a variety of sets of values of KC!-r and kb show that the criterion described above corresponds to a requirement that neither Ellz nor dE/d In [(i/(im - i)] changes with changing t,. Realizing that treatment in terms of kinetic currents is approximate, we identify the criterion of less than 10% change in the current for the radical with the criteria that one operate in a range of t, for which E l l 2 changes by no more than 1.5 mV and dE/d log [(L iRp,R)/(iDc - i)] differs by no more than 1 mV from its reversible value. Comparison of results of simulations for frozen and nonfrozen conditions in the following discussion will illustrate the reasonableness of this set of criteria. For even moderately fast reactions the criteria given above can require values of t, small enough that variation of t,. will not be practical. In the following we do not consider t, as a parameter, but rather fix its value at 1 ms, which is a reasonable lower limit for routine work. Digital simulation has been used to calculate values of K'Cb for various values of K and kb by determining concentration profiles, calculating the resulting currents, and using eq 6 to calculate K'. A set of results is presented in Figure 2. As would be expected, the general trend is that K' is a better a proximation to K for larger values of k7 where k = kpCf + k b , and for a given value of kr, the approximation is better for smaller values of K. The detail is more complex. For example, the error in the approximation is not a function of kr alone or even of kr and KCb alone. From the experimental point of view it is useful to examine the dependence of K'on 7,for these quantities are determined experimentally. If the data of Figure 2 are used to make a plot of log (K'Cb) vs. log 7 for values of 7 in the range 0.36-5.5 s, it is seen that the curves are nearly

The Journal of Physical Chemlstty, Vol. 85, No. I

I

I

+2

3, 1981 287

P 3'

-i

-2i -1

w

2

i

0

-2

1

3

log kz Figure 2. Approximate equilibrium constant vs. kinetic parameter calculated from simulated currents, t, = 1.1 ms. The number on each family of curves is the true value, log (KCb). Each Individual curve is based on five points for the same k , and k b but values of T of 0.38, 0.57, 1.12, 2.24, 5.47 s.

TABLE I: Slope (a), Intercept ( b ) ,and Correlation Coefficient ( r ) for the Straight Lines log ( K ' / K )= a log T + b for the Calculations of Figure ZU hCb, S-

1 10 100 1 10 100 1 10 100 1 10

kp, S-

0.1 0.1 0.1

1 1 1 10 10 10 100 100 100b

a

0.91 0.72 0.61 0.54

0.44

b -1.36 -1.35 -1.44 -0.56 -0.55 -0.61 -0.15 -0.15 -0.18 -0.063 -0.10 -0.18b

r

0.998 0.998 0.999 0.981 0.989 0.995 0.976 0.980 0.981 0.972 0.970 0.961b

0.41 0.14 0.13 0.12 0.026 0.025 100b 0.016b a r has units of seconds. For this case the equilibrium is not frozen during the time t,.

linear and that they fall into groups of lines of similar slope according to the value of kb. Therefore we determined the parameters describing the family of straight lines log (K'/K) = a log 7 + b (12) in which the units of 7 are seconds. The results of linear least-squares fit to the calculated points are given in Table I. Correlation coefficients are included to suggest the quality of the linear relationship. Because the slopes are very nearly the same for a given value of kb, we use a second regression equation to estimate k b from the slope. Using average values of a from Table I for each value of kb, we obtain eq 13, where for the first estimation the log kb = 2 log ( A @ 4-

c)

(13)

coefficients are A = 2.14, B = -0.47, and C = -2.17. The estimate of kb also gives an estimate of kf, k&'. Using the values of kfobtained, one can obtain a better estimate of kb by using the coefficients A = 2.03 + 0.01(kfCb)1.06, = -0.48 + 0.001(kfcb)1~07, and C = -1.70 - 0.119(k&b)o.6.Thus

The Journal of Physical Chemistty, Vol. 85, No. 3, 1981

288

Osteryoung et ai. I

I

log K‘C’

I

I

I

I

I

I

I

I

1

I 4.5

I

I 2

I

I

s”

-7

-2 0

I

7 I

-15

-10

-

2

I

I

I

I

I

I

I

I

-05

0

05

i0

(5

20

25

30

log k,?‘

Flgure 3. Plot of log (K’Cb)vs. log ( k g )for various values of log (KCb) and k f . This graph is used to estimate the value of k f . Numbers on each family of curves are values of log (KCb).

1

without other knowledge we can estimate k b from the experimental values of iNp and iRp,R as a function of 7. Of course, eq 13 is obtained from the simulations for t , = 1 ms. Having obtained the value of kb, we can find the value of K and hence the true value of kf in the following way. Figure 3 shows a plot of log (K’Cb)vs. log ( k b T ) using the calculations of Figure 2. The method of obtaining kf is illustrated by using the set of experimental points displayed along with the calculated curves. Values of 0.42, 0.57, 0.78,0.99, and 1.16 were obtained for log (K’Cb)at T = 0.36,0.57, 1.12, 2.24, and 5.47 s, respectively, for the pyridinium system. These values were fitted to the equation of Table I to give a = 0.64, from which, using eq 13, log k b = -0.65, where k b is in units of s-l. Using this value, we made the plot of log (K’Cb)w. log ( k b T ) in Figure 3. The simplest analysis considers the point at T = 5.47 s. This oint lies between the curves for log (KCb)= 1 and log (KC ) = 2, each having log k b = 0 ( k b in s-l). The difference between the estimate K’of eq 6 and the true value, expressed as log (K’IK),is -0.48 for log (KCb)= 1 and -0.54 for log (KCb)= 2 at the value of log ( k b d , +0.09. The average difference, weighted by the distance from the experimental point to each curve, is -0.52. Thus log (KCb) = 1.16 + 0.52 = 1.68, log K = 4.68 ( K in M-l), and log kf = 4.03 (kf in M-’ s-l). The same data can be analyzed by utilizing extensive simulations to fit the experimental curves. Examples of the quality of the fit and the change obtained with change in rate parameters are shown in Figure 4, which contains the same experimental points as Figure 3. The values of log kf and log k b derived from the direct comparison of experimental and simulated curves are 4.11 and -0.70, respectively, with log (KCb) = 1.81, log K = 4.81. Thus the two methods of analysis yield answers agreeing within normal experimental error for such chemical systems. Let us return to the problem of specifying a frozen homogeneous equilibrium during the period t,. Examining Figure 3, we see there is a regular pattern in the trend toward smaller values of - log (K’/K)with an increase in the value of kbT for a given value of KCb. The value of log (K’/K)does not depend on kbr alone, for the members of each family of curves do not overlap exactly. Consider,

!

g 1 2 1 ,’I2

Flgwe 4. Values of log (K’Cb)vs. T”*. Solid lines are from simulations. t, = 1.1 ms; C b = 1 mM. Points are experimental for the pyrMinlum system for the same conditions as the simulation. (top) k f = 1.3 X lo4 s-‘ M-I; numbers on the curves are values of kb in s-’. (bottom) kb = 0.2 s-’; numbers on the curves are values of k, in s-‘ M-I.

for example, the case log (K’Cb) = 1. There are three separate curves having log (K’Cb) = 1, corresponding to the values log (kfCb)= 0, 1, 2 and log kb = -1, 0, 1, respectively. At log ( k b T ) = -0.3 the curves with log (kfCb) = 0 and 1 give different values of log (K’Cb),+0.26 and +0.32. The larger value corresponds to the larger value of kf. This pattern is repeated for each set of curves. An exception to this pattern is seen for log kb = 2 (1.56 < log kb7 < 2.74) with log (KCb)= 0. This is the only simulated data set in Figure 2 for which the condition of frozen equilibrium during the time period t, is not maintained according to the criteria established above. Recall that the maximum value of k b permitted for 10% error in imp with log (KCb) = 0 is 42 s-l, while the value here is 100 s-l. For k b = 100 s-l, &/idP = 0.28 while the limiting value for 10% error is 0.2, and the calculated shift in is 2.2 mV from the reversible value. The magnitude of the resulting discrepancy is at least 0.12 in log (K’Cb),which corresponds to an error of at least 30% in K’or at least 15% in iRp,R. Simulations for more extreme cases (e.g., log (KCb)= 0, log.kb = 3) produce larger errors as anticipated (ca. 250% in w,R). It would be useful to have an equally simple semiquantitative description of the range of values of KCb,kf, kb, and T over which the values of log ( K ’ / K ) were acceptably close to zero. Unfortunately this does not appear to be possible, probably because of the complexity of interaction of diffusion with approach toward homogeneous equilibrium during the period T - t when the radical is being generated electrochemically. gome insight is gained by examining concentration profiles for various values of T as shown in Figure 5. For T = 0.36 s, the diffusion layer thickness is -30 pm, and the concentration profiles for the radical, R.,and the dimer, R2, exhibit considerable curvature. The system is clearly not in homogeneous equilibrium. For T = 5.47 s, the diffusion layer thickness

J. Phys. Chem. 1981, 85,289-294

I

”

”

/

1

t,,

I

/’

1

/ “

? ..: 0

I

/

O.8t

I

eo

.:--, , -- ---,__ _ 60

40

I

80

,

I

100

x, pm

Concentration profiles for the system of eq 1 with kf = lo5 s-’ M-’,kb = 10 s-’, Cb = 1 mM. Solid lines, 7 = 5.47 s; dashed lines, 7 = 0.36 s. FI ure 5.

-

is 130 pm, and the concentration profiles for Re and R2 are much flatter. On the time scale of the pulse, for t, = 1 ms the diffusion layer thickness is -2 pm. As shown in the inset of Figure 5, the concentration profiles are quite flat on this distance scale for T = 5.47 s. Estimation of the equilibrium constant using the surface concentrations at 7 = 5.47 s gives K’Cb = 8.1, whereas the true value is 10. The simulation yields for this case K’Cb = 7.9.

289

Limits of Application. It has already been shown in the discussion leading to the choice of tp = 1 ms that this treatment is restricted to reactions which have sufficiently large values of KCb or sufficiently small values of [kbt / (KCb)1/z]1/2 that the homogeneous reaction is frozen on t i e time scale of the pulse. On the other hand, during the period T - t, there must be sufficient approach to equilibrium to see a measurable effect if K is to be determined. For a first-order reaction we could describe the approach to homogeneous equilibrium by In [C(O)- C ( ~ ) ] / [ C (-T ) C(m)] = k~ where C(0) and C(m) are the initial and equilibrium concentrations, respectively. If we require that [C(O)- C ( ~ ) ] / ( C (-TC) ( - ) ] > 1.1, then for measurement of K we must have log ( k ~ >) -1. There are also experimental limits on the value of log (KCb). From eq 6 and 9, it is necessary to obtain accurate measurements of iRp,D and iRP,R or of i ~ p~ ,D C ,and i w , to ~ calculate K ! For small values of log (K’Cb),iRP,,-,becomes small and is difficult to measure accurately even if it can be measured directly rather than by resorting to eq 9. We choose as a conservative experimental limit the requirement iRp,D 2 O.liNp. Then, since under those conditions iRp,R iNP,log (K’Cb) L -1.3. On the other hand, if K is large, LRP,D N tNp, and measurement of iRp,R becomes difficult. Again we choose a conservative limit, iRp,R 2 iDc/2, which gives log (K’Cb)5 4. Thus this treatment is experimentally accessible over the range -1.3 5 log (K’Cb) I 4. Acknowledgment. This work was supported in part by the National Science Foundation under Grant No. CHE 7917543.

Mixed-Alkali Effect in Aqueous Silicate Solutlons Malcolm D. Ingram, * Klrwan Klng, Davld Kranbuehl, Department of Chemlstty, College of Wllllam and Maws Wllllamsburg, Virginia 23 185

and Mohamad Adel-Hadadl Vitreous State Laboratoty, Catholic University of America, Washington,D.C. 20064 (Received: May 16, 1980)

The fluidity (4) and the conductivity ( 0 ) of strongly supercooled solutions of sodium and potassium metasilicate and their mixtures, of general formula xK2SiO3-(l-x)NazSiO3-7Hz0, have been measured to -70 “C. The Arrhenius plots were curved, and both 4 and u data were fitted to the equation 4 (or a) = A exp[-B/(T - To)], where Tois a “zero mobility temperature” obtained by an iterative least-squares method. The viscosity data were measured over nine orders of magnitude and were well described for all solutions by this three-parameter equation. When the conductivity was plotted as a function of composition at constant fluidity, a conductance minimum was found at low temperatures which was identified as the appearance of the “mixed-alkali”effect. The results offer quantitative support for the idea that the mixed-alkali effect is only found when the structural relaxation time, ( 7g), is much longer ( 100 times) than the conductivity relaxation time, ( 7 c ) . N

Introduction The “mixed-alkali effect’’1,2covers a number of remarkable properties of glass, but the most spectacular effect involves the ionic conductivity (a). Thus in glasses of constant alkali content, e.g., xK20-(1-x)Na20-3Si02 where all of the current is carried by Na+ and/or K+ ions, the conductivity of mixed glasses can be reduced by a factor of lo4 or lo5 compared with single-alkali glasses measured at the same temperature. Such large departures (1) J. 0. Isard, J.Non-Cryst. Solids, 1, 235 (1969). (2) D. E. Day, J. Non-Cryst. Solids, 21, 343 (1976).

0022-3654/81/2085-0289$01 .OO/O

from additivity clearly constitute a major breakdown of the principle of independent migration of ions, which works well for dilute aqueous electrolytes. In an attempt to bridge these two extremes of behavior and also to get an improved understanding of the mixed-alkali effect, we have examined the behavior of highly supercooled alkali silicate solutions. As a starting point, we have taken the recent paper by Moynihan3 which discusses conductivity in hydrate melts of composition xNaN03-(0.2-x)MN03-0.8Ca(N03)z~

~~

(3) C. T. Moynihan, J. Electrochem. SOC.,126,2144 (1979).

0 1981 American Chemical Society