Free Volume Model for Transport in Fused Salts: Electrical

Chem. , 1964, 68 (7), pp 1917–1929. DOI: 10.1021/j100789a042. Publication Date: July 1964. ACS Legacy Archive. Cite this:J. Phys. Chem. 68, 7, 1917-...
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ELECTRICAL COXDCCTANCE IS

GLASS-FORJIISG NITRATE?\IELTS

1917

Free Volume Model for Transport in Fused Salts: Electrical Conductance in Glass-Forming Nitrate Melts

by C. A. Angell Department of Metallurg?J, Uniaersity of Melbourne, Victoria, Australia

(Received February 11, 1964)

A free volume model for transport in fused salts has been tested by electrical conductance measurements on Ca(N03)24- KNOa melts. Low temperature results exhibit rapidly changing activation energies and fit precisely an equation of the predicted form A = A T - ” a exp(-lc/(T - To)). The constant To agrees with the measured glass transition temperature and represents the temperature for zero free volume in the liquid. T o is shown to have a characteristic value for all molten salts, which depends on the electrostatic field strength of the constituent ions. Changes of T owith composition are the main source of the composition dependence of conductance in the present system. Experimental values of k for molten salts indicate a predisposition of holes in the liquid to be of ionic size. The model offers a n explanation of the glass-forming ability of molten ZnC12.

The results of transport measurements in fused salts have most frequently been interpreted by an application of rate process theory. The temperature coefficients for transport processes are obtained in the form of activation energies which niay be further considered in ternis of separate energies for hole formation and for site to site transition (“jumping”). However, little progress ha,s been made toward explaining the magnitudes of the energies obtained in terms of the basic properties of the constituent ions or of other measurable pro1)erties of the salt. In particular, rate theory has not been capable of giving any clear account of the variations of activation energy with temperature which have been observed by accurate measurements, particularly with low melting salts and salt mixtures. An alternative approach, based on a statistical mechanical theory of liquids due to Furth,l has recently been advanced by Bockris and Hooper.2 These authors have shown that, within a variance of some 407,, the activation energies for diffusion of cations and anions in four molten alkali halides are 3.5RTm, in agreement with the value predicted by Furth on the basis of hole formation being the rate-determining step in the transport process.2a This degree of accord with theory is as good as that of the data quoted by Furth in support of the original theory, hence in this respect may be considered satisfactory, although

refinements to Ihe theory would clearly be necessary before it could serve as a reliable means of making quantitative predictions of molten salt behavior. S o t e should be made, however, of Frenkel’s3 criticism of the plausibility of the “microbubble” model used by Furth in the developnient of his theory. It is the purpose of the present paper to show that a model based on the concepts of the free volume theory for transport in liquids proposed by Cohen and Turnb ~ l lmay ~ , ~ provide an improved basis for the interpretation of transport behavior in ionic liquids. A preliminary assessment of this work6 has already shown that the variations of transport activation energies with temperature in at least one system may be accounted for quantitatively by this model. According to Cohen and T ~ r n b u l l ,a~ constituent (1) R. Forth, Proc. Cambridge P h i l . Soc., 37, 382 (1941). (2) J. O’bI. Bockris and G. W.Hooper, Discussions Faraday Soc., 218 (1961). (2a) S-OTE A D D E D I N PROoF.-It has recently been pointed out, [L. Sanis and J. O ’ N . Bockris, J . Phvs. Chem., 67, 2865 (196311 that the same relation between activation energy for diffusion and melting point applie8, broadly, t o liquid rare gases and molten metals. (3) J. Frenkel, “Kinetic Theory of Liquid?,” Oxford University Press, 1946, p. 176.

(4) M. H. Cohen and D. Turnbull, J . Cham. P h y s . , 31, 1164 (1959). (5) D. Turnbull and M. H. Cohen, ibid., 34, 120 (1961). (6) C. A. Angell, J . Phys. Ckem., 68, 218 (1964).

Volume 68, iyumber ?

J u l y >1964

1918

C. A. ANGELL

particle of a liquid may undergo diffusive displacement when a void above a certain critical size appears adjacent to it. The critical void size is assumed to be that which just allows a neighboring molecule to jump into the position vacated by the displacement, and thus prevent the return of the first particle. The voids are considered to arise from the redistribution of "free" volume in the liquid structure, "free" volume being defined as that which can be redistributed without energy change. The probability of occurrence of such a void adjacent to a given particle mas shown to be an exponential function of the ratio of the critical void volume to the total free volume, and an expression for the diffusion coefficient was derived D

= gau

exp( - yv*/vf)

(1)

where g is a geometric factor given the value a is the jump distance, or approximately the particle diameter for molecular liquids, u is the gas kinetic velocity of the particle (3ICT/m)'/*,y is a factor to correct for overlap of free volume in the calculation of the probability of occurrence of a critical void, v* is the critical void volume, and V F is the total free volume. If it is further considered that tha free volume may be taken as approximately the total thermal expansion above the temperature To a t which "free" volume begins to appear, then v i = az',(T - TO)

(2)

where a is the mean expansion coefficient in the range T-To, and grn is the mean molecular volume derived from the molar volume. The net expression for the diff usioii coefficient is thus

D

=

ga(3kT/n~)~/' exp[-?yu*/afi,(T

-

T,)] (3)

If T o > O O K . , it follows that the probability of molecular motion decreases rapidly as T o is approached from above, and the liquid will become rigid, i . e . , will transform to a glass, in the vicinity of To if crystallization does not occur. On the other hand, if To = O°K. or T >> To,eq. 3 is formally equivalent to the expressions of rate theory. It has already been shown by Lunden' and the authors that if To is assumed to be O'K., the magnitude of the activation energies for selfdiffusion found for the constituent ions of pure fused salts are predicted within 20% by eq. 3 if the critical void volume is assumed to be 90% of the mean ionic volume. However, in view of the results reported below, this success appears to be fortuitous, hence in itself offers no support of the theory. To analyze the temperature dependence of diffusion on the basis of this model, one can write for eq. 3 T h e Journal of Physical Chemistry

D = AT""xp[-IC/(T

- To)]

(4)

where A and IC are constants, from which

where EDis the conventional (temperature dependent) Arrhenius activation en erg^.^ I n view of eq. j l a curvature in the ordinary log D us. 1/T plot is therefore to be expected, but the curvature will not be marked for temperatures well removed from TL?,e.g., for temperatures in the range 3To to 4To (650-950° for T o around room temperature) the apparent activation energy would only change 15%) whereas in the range T = 1.5To to 2.0To, the variation would be 130%. As the majority of measurements on pure fused salts have only been conducted over a range of about 200') it is not surprising to find that variations in activation energy have been small if observed a t all (see also Discussion). On the other hand, recent series of measurements of conductance-temperature relations in binary fused salt systems, with liquid stability ranges considcrably below the melting points of the components, have shown marked changes in activation energies a t the lower temperatures. Cohen and Turnbull's model could be tested followwas a linear funcing eq. 4 by showing that log tion of 1 / ( T - T o ) (rather than of 1 / T as is normally attempted). A more stringent test, however, follows from eq. 5 which predicts the temperature dependence of the slope of the semilog plot. From eq. 5 , Rdlff I/ZRT should be a linear function of [ T / ( T - To)J2, which passes through the origin. I n order to test the validity of this model for fused salts, the system KXOs was selected for study. A detailed investigation of this system by Dietzel and Poegel" has shown that compositions in the range 29-49 mole % calcium nitrate may be obtained as glasses, with minimum crystallization velocities a t 38.4% (50.4 wt. %) Ca(K03)n. As the conduct-

+

( 7 ) A. Lunden, T r a n s . Chalmers Cniv. Technol. Gothenburg, No. 241, 3 (1961).

(8) C. A. Angell, unpublished work. (9) On empirical grounds an expression equivalent to eq. 4, with the pre-exponential T ' h omitted, was proposed by Tamann and Hesse [G. Tamann and 1%'. Hesse, 2. anorg. allgem. Chem., 156, 245 (1926)] to account for the temperature dependence of viscosity in polar liquids, and has been called by Miller [A. A. Miller, J . P h y s . Chem., 67, 1031 (1963) J the Modified &rhenius (M. A.) equation. (10) B. F. Markov and A. M. Tarasenko, Zh. Fis. Khim., 32, 1333 (1958). (11) N. P. Popovskaya and P. I. Protsenko, Russ. J . Inorg. Chem., 7, 1158 (1962). (12) A, Dieteel and H. J. Poegel, Proceedings of the Third International Glass Congress, Venice, Italy, 1953, p. 219.

ELECTRICAL CONDUCTANCE I N GLASS-FORMIKG NITRATE: MELTS

00

00

T'K 00

00

00

1919

ance isotherms reported below seem to verify, this system is expected to contain only the simple ions Ca+2,E(+, and N03-.13 Hence, we may expect the transport behavior in this system to be typical of simple ionic fluids, and attribute the glass-forming capabilities of the system to low crystallization rates at the unusually low liquidus temperatures in this system (see phase diagram, Fig. 1). To test eq. 5 adequately, temperature coefficients must be measured more accurately than is possible at) the moment with diffusion measurements. Hence, the electrical conductance of the solutions was measured, and the temperature dependence of conductance was corrected to correspond to that for a diffusion process. Strictly, this procedure will be valid only if all ionic conductances have equal temperature coef-. ficients or if only one species is mobile. Although experience shows the former is the case for cations and anions in pure fused salts,2~14~16 it is not certain that it also holds for mixtures; the results reported below are compatible with either, but exclude intermediate cases. The activation energy for specific conductance, E,, was converted to that for equivalent conductance, EA, by the relationlo

EA I

I

=

E,

+ nRT2

(6)

I

I

where CY is the coefficient of expansion of value 3.9 X 10-4 for pure KS0317 and 3.85 x lop4 for the 38.1% Ca(XO3)*mixture.12 Then from the Nernst-Einstein. equation, one derives, for equal or very different ionic conductance activatjon energies

t

o'2

I

I

Thus the corrected activation energy, called here-, after E,,,, which should be a linear function of [T/(T T0)I2if the free volume model is valid (eq. 5) is given by

I

Eo,,

-"

0

IO

20

30

40

50

60

MOLE 'lo C 1 ( N 0 3 ) 2

+

Figure 1. ( i ) Phaije diagram for KNO, Ca(NOa)z, showing T oi n relation t o liquidus; (ii) specific Eonductance isotherms; (iii) specific conductances a t equal intervals above T O ;(iv) specific conductances at constant T/To (value for pure KNOJ by extrapolation below melting point).

=

E,

+ nRT2 + l/zRT

(8)

(13) I t is noted though that Raman frequencies not present in the spectra of the components were found in compositions in the glassforming range by Djetzel and Poegel, who suggested that some covalent character had been imparted to the Ca+Z-NOa- bond. I t seems more likely, however, that the additional frequencies come from a slight distortion of the nitrate ions in the unbalanced cation force field in the mixture. (14) A. D. Dworkin, R. B. Escue, and E. R. Van Artsdalen, J . P h y s . Chem., 64, 872 (1960). (15) C. A. Angel1 and J. W. Tomlinson, to be published. (16) R. L. Martin, ,J. Chem. SOC.,3246 (1954). (17) H. Bloom, I. W. Knaggs, J. J. Molloy, and D. Welch, Trans. Faraday SOC.,49, 1458 (1953).

Volume 68, Number 7

Julg, 1064

C. A. AXGELL

1920

Experimental A p p a r a t u s . The capillary-type cell used for the majority of the measurements (Fig. 2) was constructed of Pyrex glass, with electrodes of platinized platinum. The cell constant, determined with 0.1 N KC1 solution, was 36.40, the relatively low value being chosen to permit nieasurenients of the lorn values of K encountered in the glass-forming melts a t low teinperatures. The lower electrode was a 0.3-cni.2 platinum sheet rigidly supported by the platinum lead wire sealed into the Pyrex wall of the cell, while the upper electrode was of coiled platinum foil supported by the platinum lead wire which was fixed in position at the top of the cell by an araldite seal. A thermocouple sheath of identical construction with the cell was fixed alongside the cell so that measurements could be made with continuously changing tenipcrature without introducing a temperature differential between the thermocouple in the sheath and the salt in the cell. For measurements in KSOa-rich melts a t temperatures above 350°, a silica capillary cell of cell constant 630.8 n-as used, as in these cases the conductance through the

walls of the Pyrex cell could not be neglected (0.1% of the total resistance). The temperature was nieasured t o .t0.lo by means of a fine gage chromel-alumel thermocouple which had been calibrated a t the melting point of lead and the boiling point of water. The output of the thermocouple was determined with a Cambridge portable potentiometer. A Philips Model PR9500 conductance bridge was used to balance the resistance of the salt in the cell against an equal resistance set by means of external precision decade boxes. The balance point could be determined in this manner to *0.05% although the quoted accuracy of measurement for the bridge under these conditions was only 0.2%. Resistance measurements were made at 1000 c.p.s. and as no difference in the specific conductance could be detected when using cells of constants differing by a factor of 20, it could be assumed that polarization resistances were a negligible fraction of the measured resistance. For higher resistances encountered in glass-forming melts at low temperatures frequencies of 50 c.p.s. were used to reduce capacitance blurring of the balance point. The melt, approximately 20 cc. in volume, was held in a Pyrex glass vessel seated in an aluminum block, the latter serving to keep the temperature of the melt uniform. This assembly was supported in a nichroniemound furnace, the temperature of which could be raised or lowered by means of a Tariac power supply. The furnace mas provided with a window and light

T

or

I 1.4

Figure 2. Conductance cell: A, Pt electrodes; B, Pyrex glass cell: C, thermocouple sheath; D, thermocouple; E, Pyrex glass vessel.

The Journal of Phusical Chemzstry

Figure 3. Log

1.6

K

1.8

!IS. 1 / T

2.0

1

x T

for Ca(NOx)?

2.2

2.4

103

+ KXOI melts.

2.6

ELECTRICAL CONDUCTANCE I N GLASS-FORMING 'NITRATEMELTS

source above the aluminum block so that the cell could be raised and inspected for absence of bubbles, etc., without removal from the furnace. Materials and Procedure. Melts were prepared by mixing weighed amounts of Analar potassium nitrate and LR anhydrous calcium nitrate (maximum impurities, Mg = 0.002%, Na < 0.02%, K < 0.005%), fusing, and filtering through a no. 3 porosity sintered glacts filter. Tbe melts were held a t 300' for 2 4 4 8 hr. during which the conductances fell 0.5-1% to steady values as traces of water escaped from the melt. Thereafter, measured conductances were independent within 0.2% of time and of the cell used, and within 0.1% of the circumstances of measurement, e.g., continuous heating or cooling. Final measurements were madle during continuous cooling a t rates of 1-2'/min., thle temperature and resistance being measured simultaneously a t approximately 10' intervals, more frequent measurements being made for the glass-forming melts a t lower temperatures when the resistance was rising very rapidly.

Results Accuracy. Melt compositions were shown by analysis to be within 0.1% of the composition intended. Errors due to retained water may amount to 0.3% and, with allowances for errors in the bridge and cell constant determination, a n over-all accuracy of f0.55% is expected. More important for this project, however, was the consistency of results during any single conductance us. temperature run, and here it was found that points showed a scatter about the best fit curve of less than O.ly0. Representative results, omitting those of the 33.5, 35.0, and 36.5 mole 70Ca(IY03)zfor clarity, are plotted in the common form of log K us. 1/T in Fig. 3. The departures from linearity increase as the liquidus temperature falls, being most notable for the 38.1 and 45.0% Ca(N03)2 melts, which failed to crystallize. Complete results for the 33.5% Ca(NO& composition are given in Table I. The present results for pure KN03 are identical with those of Bloom, et al.," and IGoger and Weisgerberi8 a t 400°, although the activation energies reported by these groups are 0.3 and 0.2 kcal., respectively, lower than the mean values found in the present work and in that of Cowen and Axonig (see Fig. 1in ref. 6). Also, the self-capsistency of the results of Bloom, et d., and Kroger and Weisgerber did not permit detection of any temperature dependence of the activation energies. The specific conductance isotherms for this systeim are plotted in Fig. 1 (ii). The conductance decreases

1921

Table I T,OC. 366.2 347.9 341.0 329.2 319.9 306,4 292.5 279.3 269,9 253,6 247.9 242.7

K,

ohm -1 om. -1

T,'C.

0,3935 0.3733 0.3551 0.3265 0,3046 0,2726 0.2403 0.2110 0.1901 0.1559 0.01442 0,01338

233.7 222.3 210.5 198,7 188.4 178.8 166.4 155.5 142.8 135.9 131.5 125.8

M,

ohm-1 om. - 1

0.01169 0.09642 0,07687 0,05938 0.04590 0.03477 0,02276 0,01457 0,00769 0,00516 0,00382 0,00252

steadily with increase in % Ca(D;O&. The decreahe, however, is not strictly linear, and the concavity relative to the abscissa may be interpreted in terms of the weak ordering tendency shown to exist in the vicinity of the composition 4KNO3Ca(XO3) z,by the heat of mixing measurements of Kleppa and Hersh.20 Dietzel and Poegel12 suggested that the glass-forming ability of these melts in the 30-50% Ca(SO,J2 composition range was due to a change from sixfold to fourfold coordination of X03- about Ca+2. I n this case, one would expect maximum effects in the conductance isotherms a t 33.3% Ca(iLTO&, but the closely spaced measurements around this composition reveal no irregularities. The isotherms, then, are most notable for their lack of special features, suggesting that the glass-forming ability of the melts is not to be accounted for in terms of structural abnormalities, such as Ca+2-(N03--)2 chains or networks, in the liquid, as envisaged previously.12 The conventional activation energies for specific conductance, E,, calculated from successive pairs of points (average 10' interval), plotted against the mean temperature of the interval in Fig. 1 of ref. 6, show a rapid and accelerating increase a t the lower temperatures, which is particularly striking. These E , values, corrected to E,,, values as described in the introduction, are then tested for linearity with the function IT/ ( T - T O ) l 2for the most appropriate choice of TO. The sensitivity of the plot to choice of To is demonstrated in Fig. 4 for the composition 45% Ca(N03)2. It is noted that for T o = 330'K. the plot is linear within

-

(18) C. Kroger and P. Weisgerber, Z . P h y s i k . Chem. (Frankfurt), 5 , 192 (1955). (19) H. C. Cowen and H. J. Axon, Trans. Faraday SOC.,52, 242 (1956), did not report specific conductances for KNOs. (20) 0. J. Kleppa and L. S. Hersh, Discussions Faraday SOC.,No. ,32, 99 (1961).

V o l u m e 68, Number 7

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C. A. ANGELL

1922

Figure 4.

2'0of E,,,

Test of 45% Ca(iVO8)z results for sensitivity to [ T / ( T - To)l2plot.

VS.

1

20 1

+

Plot, of log %TIiZ us. l / ( T - TO) for Ca(N03)2 KNOBmelts. Inset: enlargement of high temperature section of plot. Figure 5 .

experimental error for all values of [ T / ( T - T o ) ] 2> 6 ( T < 1.7To), a positive deviation in the E,,, values being found a t higher temperatures. Also, the sensitivity to T oonly becomes large for the low temperature points with E,,, > 15 kcal. With the present experimental uncertainty, this means that To can only be established precisely for melts which do not crystallize. The best choices of To, obtained from plots such as Fig. 4, for the three glass-forming compositions studied are shown in Fig. 1 together with the phase diagram for this system. 'I he relationship between T oand composition is linear within the uncertainty of determination. When the Fig. 4 type plots for these choices of To are superimposed it is found that they are indistinguishable. Due to the curtailment of measurements T h e Journal of Physical Chemistry

by crystallization it mas not possible to estimate accurate values of T O for the remaining compositions studied. However, if TO for these compositions is taken for the linear To us. composition plot shown in Fig. 1, then the E c o r vs. [ T / ( T - ?',)I2 plots for these compositions are also coincident with the others. From the slope of the combined plot (Fig. 2 of ref. 6) one obtains k (eq. 5 ) = 690 (see also Fig. 5). The significance of these observations is discussed subsequently . Using these values of To, a plot, Fig. 5, which is the free volume model equivalent of Fig. 3, may be constructed. In plotting log KT'" against 1/(T - T o ) , the difference in temperature dependence between equivalent conductance and diffusion has been taken into account, but not that between specific conductance and equivalent conductance. The latter difference, however, is a very weak function of temperature and does not affect the linearity of the plot detectably. The slope of the plot, k', however, will differ slightly from the value of k obtained from the slope, Rk, of Fig. 2 of ref. 6. Figure 2 of ref. 6 shows that increases in slope in the plot (Fig. 5) are to be expected above 1.7To for each composition. Since 1/(T - T o ) a t T = 1.7T0 has a different value for each composition, the plot (Fig. 5) is expected to "fan out" from different points. This is only observed when the plot is enlarged (inset in Fig. 5 ) , which emphasizes the advantage of the E,,, us. [ T / ( T - To)l2 plot as an indicator of the salt behavior. From the divergence of E,,, from the theoretical value a t T > 1.7T0 (Fig. 2 of ref 6), the variation of k with T / T o may be estimated. At T I T , values of 2, 3, 4, and co , the per cent increases of IC over the constant value found below 1.7To are 12, 39, 53, and 100, respectively. At the same time the values of A , the pre-exponential constant, must be increasing. However, not too much emphasis can be placed on these changes in k and A until further systems have been studied. The pre-exponential constants for the linear region of Fig. 5 are shown in Table 11.

Table I1 Mole

410 C a ( N 0 s h

2 0 . 0 (TITO 30.0 33.5 38.1 45.0

50.0

1.76)

A

81.1 80.4 80.2 80.3 76.5 72.1

ELECTRICAL C o s ~ CTANCE v

IX

1923

GLASS-FORMIN G NITRATEMELTS

I n the composition range 30.0-38.lyG Ca(N03)2,A iE, invariant, hencc. the variation in isothermal conductivity of the solutions in this region is due entirely to the composition dependence of To. Although slightly outside the linear region, the value of A for 20% C a ( r \ T 0 ~a)t~the lowest temperaLure measured is included in the table as it suggests that had the nieasurenients in this solution not been interrupted by crystallization the same value of A would have been realized a t lower temperatures.

Discussion The correlation of results achieved by this analysis, and shown in Fig. 5 and Fig. 2 of ref. 6, leaves little doubt that the exponential dependence of transport in this system on k/(T - To) is a correct formulation I n general terms, an exponential dependence of this type, expressible as exp(-E/R(T - To)), may be regarded as reflecting the probability of occurrence of an event at temperature T, when the possibility of the event vanishea below some temperature To greater than O O K . This type of temperature dependence may prove to be generally important in transport processes and it may have direct applications to other processes such as nucleation and growth in amorphous oxide films, where temperature-dependent activation energies have been observed. 2 1 It seems from the present results that the term “activation energy” used to denote the quantity derived from the slope of the ordinary Arrhenius plot may be misleading. For example, Fig. 2 of ref. 6 indicateri that although E,,, for KKOs only varies by 4.2%) over 80°, its value may still be some 30Yc greater than the temperature-independent value which, in theory, is only approached a8 T + 0 0 . Thus, if the term “activation energy” is to be used, it could be applied more appropriately to the factor 12 which is at least temperature independent in the low temperature regime (Fig. 5). However, even k becomes temperature dependent at higher temperatures. The direction of this departure from theoretical behavior is such as to decrease the apparent temperature dependence of the ordinary Arrhenius activation energy, and, to judge from the data for halides of Li and Ka,22the factors responsible niay even lead to a reversal of the temperature dependence. Hence, the significance of the slope of the Arrhenius plot beconies obscure, and we are led to adopt hereafter the noncommittal term “Arrhenius coefficient” to replace “activation energy.” Before further discussion of the present results, it is pertinent to note that behavior of the type described here is found generally in fused salt systeins, although only rarely does 4 system remain liquid to a low enough temperature to enable the glass transformation to bo

observed. It may be expected that the clearest cases will be found in eutectic systems where the melting points of the pure salts are brought down toward To by the addition of the other components, as shown in Fig. 1 (i). This expectation is well supported by the specific conductance measurements on binary molten salt systenis exhibiting deep eutectics, shallow eutectics, and no eutectics, reported by hfarkov and Tarasenko’O and by Popovskaya and Protsenko. lipz3 Some of the log K us. l / T plots of these workers and the “Arrhenius coefficients” a t different temperatures derived from, them are shown in Fig. 6 together with similar plots from the present work, other work in progress in this l a b o r a t ~ r yand , ~ ~from results reported by Bockris, et aLZ5 A consistent pattern of behavior iiapparent in both sections of Fig. 6, but unfortunately, apart from the interesting case of zinc chloride, which is complicated by incomplete dissociation (vide infra), only in the system Cd(N03)2 NaX03 are changes in E , sufficiently great and the data available accurate enough to permit a determination of To with any confidence by the method of Fig. 4. This yielded To = 340 i lO”K., k = 530 k 80 for this system, this value of k being some 25Tclower than the constant value found for the Ca(K03)2 KKO, system. The data of Fig. 6 are treated quantitatively in the following sections in which the three constants of eq. 4 are examined. Physical Origin of To. To interpret To, it is first noted that the “glass transition” temperature, T,, i . e . , the temperature near which changes in the physical properties of d a s s y solids are first observed, and arbitrarily defined by-log (viscosity) = 13, was found by Dietael and Poegel12 to be 329°K. for the 38.1% C F L ( N O ~composition. )~ This is to be compared with the value of 316°K. for To calculated from the present work. A difference is expected, since the direct nieasurement involves the properties of the material near T o when the relaxation tiines for reduction in free volume become exceedingly long, i.e., behavior appropriate to zero free volume will be realized before true To is reached. In this light, the agreement be-

+

+

(21) W. T. Denholm, private communication. (22) I. S. Jaffe and E. R. Van Artsdalen, J . Phys. Chem., 6 0 , 1125 (1956). (23) These workers, observing that the linear plots for pure salts, and systems with no minimum in the liquidus, were replaced by nonlinear plots in many eutectic systems, attempted t o explain the deviations from linearity in terms of structural alterations in the liquid anticipating the eutectic remtion. This led to some contradictions, however, since systems with shallow eutectics did not show any appreciable deviations from linearity, for reasons which the present model makes clear. (24) C. A. Angell, J. A. Corbett, and R. C. Gifkins, to be published. (25) J. O’M. Bockris, E . H. Crook, H. Bloom, and N. E. Richards, Proc. Roy. Soe. (London), A255, 558 (1960).

Volume 68, Number 7

J u l y , 1964

C. A. ANGELL

1924

25 -

20

-

En kcal 15

-

10

-

5-

100

200

300

400

500

600

700

800

900

T OC Figure 6. ( i ) Log K us. l / T plots for various molten salts and mixtures; (ii) dependence on temperature of for various molten salts a n d mixtures.

tween the two values appears very good and constitutes strong evidence for the general correctness of the model. To account for the linear increase of T o with Ca(1\’03)2,we observe that, due to the double charge and sinal1 radius of the C a f 2 ion, the mean “cationic potential” or “cationic strength” of the melt, B N , . x d ~ ,( N , = lnole fraction, z , = ionic charge, r , = ionic radius, of cation species i) , increases (linearly) The Journal of Physical Chemdatry

EK

with % ’ Ca(N03)2. To is plotted against 2”,x,lr, in Fig. 7. For the hypothetical case of zero cation strength, or absence of couloinbic attraction in the structure, a value for To of approximately 100°K. is indicated, which may be compared with the T , values of 80-1OO0K. found for simple molecular liquids.26 (26) D Turnbull and h/I. H Cohen, “Modern Aspectb of the Vitreous State,” J. D. MacKenzie, Ed., Butterworth’s Scientific Publications, -Ltd., London, 1960.

ELECTRICAL CONDUCTANCE

IS

GL-4SS-FoRhtIR.GNITRATE

/ /

/

500.

/ / / /

/

1

/

/

/

/

400. To “K

300.

/I m

s Y

//

200-

/ I’

/ /

/ /

Figure 7 . Relationship between To a n d mean cationic strength

for various molten salts a n d salt mixtures: legend as for Fig. 6.

This plot should, in principle, give To for any pure or mixed nitrate system, and the value obtained for the system Cd(NOa)z N a y 0 2 is in reasonable accord with this prediction (Fig. 7 ) . For halide and other anion hysteins, plots with similar To a t zero cation strength but with diff ereiit slope depending on anionic strength might be expected. Included in Fig. 7 are approximate values of To for the other systems of Fig. 6 calculated on the assumptions that (i) eq. 5 applies to all systems, i.e., at equal TITO, E,,,/Rk is the same for all systems, and (ii) all systems exhibit the same departures from theoretical behavior a,t high T/To found in the Ca(NO& KNOa system (Fig. 2 of ref. 6). To obtain To for the other systems on this basis, a graph of E,,, us. T / T owas constructed from Fig. 2 of‘ ref. 6. Z’/T,, arid hence T o values were then found froin the E , values of Fig. 6 at different teniperatures after correction to E,,, values. Generally, Tothus calculated was dependent on the temperature a t which E’, was taken, which is expected if IC

+

+

1926

MELTS

(eq. 4) differs from the value for the Ca(P\’O& -f I(N03 system (as was found to be the case for the Cd(NO& NaKOs system). Adjustments were therefore made to the relative value of IC until To was independent of the temperature for which it was calculated. The values of k obtained were within 30y0 of those for the Ca(SO& systeni and are tabulated and discussed below. Despite the uncertainty of these To values, it seems likely from Fig. 7 that the proposed relationship between T o and ionic strength is basically correct, 27 but confirmation must await further measurements on carefully chosen systenis. If To depends on the magnitude of the coulombic binding forces in the system, the physical origin of the constant may be siniilar to that suggested by Cohen and Turnbulls for molecular liquids. These authors, basing their discussion on a Lennard-Jones and Devonshire 6-12-type potential, V ( R ) , for a molecule in a cage forined by near neighbors, suggested free volume arises when the temperature-dependent average cage radius, RT, increases to values in the range where V ( R ) is varying approximately linearly with R. The fraction of the excess volume, Av (where Av = cage volume - molecule volume), appropriate to RT RToJ where RTo is at the edge of the range, was defined as “free” volume since it would be redistributed among the different cages without over-all energy change, the energy increase of one cage being compensated by a corresponding energy decrease of an adjacent cage. Actually the cage potential in an aggregate can be approximated by a square well potential, and T oshould be quite well defined. I n the case of an ionic liquid, where the cage particles are of opposite charge from the central particle, the cage potential should be ail inverse square fuiiction of the radius in accord with Coulomb’s law, hence the single cage potential should be a more gradual function of radius. Kirkwood, et a1.,28have shown, however, that the counterbalancing effects of neighboring cages would lead to an approximately square well potential for ionic melts also, hence the origin of To can be explained on the saiiie basis as for molecular liquids, although in the stronger force field a higher temperature will obviously be required to realize RTo. To as a Basis ,for Corresponding Temperatures. The validity of eq. 6 demonstrated for this system by Fig. 2 of ref. 6, and supported by the correlations of To with ionic strength achieved by the use of this fig-

+

+

(27) This plot is not expected t o distinguish the effects of the different anions, as these are all of about the same strength. (28) F. H. Stillinger, 3 . G. Kirkwood, and P. J. Woltowics, J . Chem. Phys., 3 2 , 1837 (1960).

Volume 68, Number Y

J u l y , 1964

C. A. AIWELL

1926

ure for calculations on other systems, indicates that ED/^ may be a constant at equal T/To for fused salts. Thus To may prove a more satisfactory basis for corresponding temperatures in fused salts than the melting point which has been used in the past, in the same way that the boiling point has proved more satisfactory than the melting point for molecular liquids. The usefulness of T o for this purpose will, of course, depend on establishing the To-ionic strength relationship (Fig. 7) more securely than can be done a t the moment. The conductances of solutions in the Ca(N03)*KKOa system have been plotted in Fig. 1 (iv) a t the corresponding temperature T,lTo = 2 . 2 5 . On this basis the conductance of the solutions increases with increasing mole yo Ca(NOa),, reflecting the extra equivalent of charge per inole of this salt. Also included in Fig. 1 for comparison is a plot of the conductance of the solutions at equal intervals above To. The Zinc Halides. The glass-forming ability of ZnClz has previously been interpreted by a "senii-network" model,2Qbut from the similar curvature of the ZnClz plot in Fig. 6 (ii) to those of other salts, it seems likely that it is simply a good example of the consequences of a very low melting point, 318", coupled with a high cation strength. From diffusion and conductance measurements on ZnBrz, it is evident3@that the zinc halides are partly associated in the liquid state, hence a chemical dissociation energy factor of the form exp(-ElRT) is to be expected in the temperature dependence of conductance for ZnClz. This should therefore have the form exp(-kl(T - T o ) )exp(-k",/ T). The E , us. T plot in Fig. 6 (ii) should therefore differ for those of other (unassociated) inolten salts by a constant factor at all temperatures. If 8.5 kcal. is attributed to the chcinical dissociation energy, and subtracted from E , a t all temperatures, the E, us. T plot has the same character as the others, and yields a value of T oof 455" in agreement with the trend of Fig. 7 and with indications from conductance ineasurenients on the undercooled Finally, the present model predicts that if a second component, which lowers the melting point of the salt relative to To, is added to molten ZnCI,, the glass-forming properties of the system should be enhanced. Bench tests with additions of Cd(N03)2 (1n.p. 350"), and reported effects of additions of alkali sulfates and halides,33 indicate that glasses are indeed formed much more readily in the binary systems. This observation is not readily explained by the semi-network model as the foreign ions mould be expected to break up the complex groups. This subject will be dealt with in more detail in a subsequent publication. The Conslunt 01 the Exponential. According to T h e Journal of Phueical Chemistry

Cohen and Turnbul14 the constant k in the exponential factor of eq. 4 is given by

where the terms have been defined in the introduction. yv*/8, determined from the present conductance nieasurements should represent the average of values for the different species. This work shows that a t temperatures below 1.7T0, k is independent of composition in the system Ca(?J03)* KXOa. Since CY varies by less than 2% in the composition range studied (vide supra) yv*/6, itself must be substantially independent of composition. This suggests that the void volume to ion volume ratio involved in ionic transport is the same for all species and hence is independent of t'he charge or size of the ions, inferring that apart from determining the value of T o , and the average disposition of catioiis about anions, the presence of electrical charges on the diffusing species has no effect on their transport behavior. This is in accord with the conclusions reached by Rice34 through a statistical mechanical analysis of transport in inolten salts. That the void volume to ion volunie ratios for cations and anions are substantially the same in pure molten salts is brought out in the General Discussion. On the other hand, the possibilit,y that the value of k is determined by a single mobile species, and consequently is independent of composition, cannot be excluded. Published data on relative ionic mobilities in binary nielts with cations of different valencies are limited to the systems PbC1, KC1 and PbCL LiC1, and, in view of the high atomic weight' of lead, these may not be representative. The evidence is, in any case, conflicting, 35 although on balance favoring Pb+, as the less mobile species. Thus in the present system E(+ would appear more likely to be the mobile species if in fact the mobilities are significantly different. As K + is the larger ion, a high mobility relative to that of Ca+Zwould conflict with bot'h Rice's view that only short range forces should decide relative t.raiisport rates in fused salt,s,3*and with the coinpleineiitary expectation that free volume alone determines the transport rates. The performance of diffusion

+

+

+

(29) J. D. McKenzie, J . Chem. Phys., 33, 366 (1960). (30) R. W. Laity, Ann. iY. Y.Acad. Sci., 79, 997 (1960). (31) H. S. Schulze, 2 . anorg. allgem. Chem., 2 0 , 338 (1899).

(32) No experimental value of the glass transition temperature for ZnCln could be found in the literature. (33) I. Schulz, Naturwiss., 44, 536 (1957). (34) S. -4.Rice, Trans. Faraday Soc.. 58, 499 (1962). (35) R. W. Laity, J . Phys. Chem., 67,723 (1963).

192’7

ELECTRICAL CONDUCTANCE IR’ GLASS-FORMING NITRATE MELTS

measurements on all ions in representative mixed valency systems is thus of paramount importance. The single mobile ion interpretation receives some support from the fact that viscosity data for the 38.17? Ca(iYOs)zcomposition in the temperature range of the present although not of great accuracy, indicate that To for viscous flow, which must involve all species, is about 10” higher than T o for conductance. The value of yv*/1Sim calculated from k for the present system haE, the value 0.25,26 whereas a value in the range 0.5-1.0 is to be expected from the theory. Depending o n the value of y, the void volume to ion volume ratio would lie in the range 0.5 to 0.25, which is about half the values found by Cohen and ‘Turnbull for molecular liquids, and seems too small to be realistic. It appears therefore that a t least a t low temperalures, T / T o < 1.7, the probability of a diffusing particle finding a suitable void is higher than would be expected from a purely random distribution of the free volume, i.e., a predisposition of voids to be of ion size is indicated. -4 nonrandom distribution itself is not surprising since, due to the requirement of local electrical neutrality, a random distribution of the cation free volume fraction (see General Discussion) mould be incompatible with an independently random distribution of the anion free volume fraction. The local neutrality factor may render intermediate hole sizes energetically unfavorable, and unbalance the negative exponential distribution of free volume proposed by Cohen and Turnbull in favor of ion-size holes. The distributed volume in this case would not be “free” volume in the strict sense of the latter authors. As described in Results (Fig. 5, inset), the value o€ k increases as T / T o increases above the value of 1.7, which rn ould be expected if the free volume distribution becomes increasingly random as the temperature increases. From these results a fully random distribution would only be approached at T * . However, if the differences normally found between EDand E , -1RT27could have been taken into account in obtaining E,,,, IC would probably have been found to have reached the value appropriate to random distribution at about 4To. An alternative possible explanation of the changes in k is that the effective value of T o decreases with increasing temperature due to changes in the nature of the cage potential with increasing specific volume, Le., Fig. 5 is incorrecfJlycalculated for the high temperature points. However, this postulate is not well supported by calculations of the necessary changes in To -f

to maintain A and k constant with increasing temptrature. Approximate values of k obtained in the other molten salt systems of Fig. 6 are given as ratios, Rk, to k for the Ca(NOa)2 KSOa system in Table 111, together with estimates of the mean expansion coefficients, a , and derived yv*/b, values.

+

Table I11 RI,

Salt

Ca(?;Oa)z

+ KT\;Oa

+ NaNOz + KaKOs eut. LiCl + KC1 eut.

KNOs

1 0

0 85

+ LlNOs Cd(KO& + NaNO2 Cd(?;Oa)* + KXOi Cd(5Oa)z + C S ~ O a CdClz + PbClz eut.

1 0 1 4 078 078 078 078 082

AgiYOa

070

ZnClz

070

SrIz Cd( SOa)z

OL

3 3 3 1 2 2 2 2

6 5 3 9 7 8 9 9

yv*/.ir,

0.25 .21 .23 ,18 ,145

2 8 2 5 2 3

.l5 ,155 ,155 .16 .12

.ll

A

80 106 230

50 102

70 64

36 165 90 163

The low values of yw*/flm in Table I11 show that the failure to achieve a random distribution of free volume is apparently general in molten salts. A full explanation of this feature must await further investigations. Particularly notable are the low values of yv*/’l?m for the salts containing cations of periodic table group B elements. These ions have nonrare gas electronic configurations which result in unusually high van der Waals energies in structures containing them. Consequently, abnormally close approach between these ions, and therefore movement into abnormally small holes in the liquid structure, is possible, so the lower void volume t o ion volume ratios shown in Table 111 may be expected. This ability in the liquid state parallels the well-known tendency of nonoctet cations such as Ag+ and Cu+ to form Frenkel defects in the solid state. In both solids and liquids, therefore, the ease of finding diffusion sites is the source of the abnormally high mobilities of these species. T h e Pye-exponential Constant. The pre-exponential constant suggested by Cohen and Turnbull and described in the introduction implies a direct dependence on particle diameter and an inverse dependence on the square root of the particle mass. In fused salts the jump distance would have to include the diameters ( 3 6 ) The value of 0.26 quoted in the preliminary report of this work was based on the value of the expansion coefficient at high temperatures rather than the mean value from TOt o T as it should have been. ( 3 7 ) C . A. Angell, in press.

Volume 68, Number 7

J u l y , 1964

C. A. AIVGELL

1928

of both cations and anions since a n ion must cross the first coordination shell of oppositely charged ions to reach its new equilibrium position. Hence the preexponential constants for cations and anions in a pure fused salt should be insensitive to the particle radii, and depend only on the relative ionic masses. Since the “Arrhenius coefficients’’ of diffusion for the different ions in pure fused salts have consistently been found to be equal, it would therefore be expected that diffusion coefficients of cations and anions in the same salt should be in inverse proportion to the root ionic masses. However, d i f f ~ s i o n ~and ! ’ ~ mobility38 measurements indicate a n inverse dependence on ionic radius a t least for uni-univalent salts. Values of A calculated from the Cohen and Turiibull formulation by introducing the Kernst-Einstein equation are one order of magnitude higher than the measured values, this difference being greater than can be accounted for by approximations made in the development of the expression. They also vary incorrectly with composition; hence, a formulation more appropriate to the case of fused salts seems to be required. We note that this must reconcile a n inverse dependence on radius with the required dimensions of cme2sec.-l. Estimates of A for the other systems considered in this discussion are given in Table 111. They are similar in magnitude to those of the Ca(X03), KT03 system. The constancy of A for the intermediate compositions is not of great significance since A was computed froni specific conductance rather than from equivalent conductance values. For equivalent conductance, A would decrease with increasing % Ca(K03)2. The inverse radius dependence noted for the pre-exponential term in the diffusion equation for uni-univalent salts does not apply to the only case of a di-univalent salt studied (PbClg15,3g)so the decrease of A with increasing % Ca(T\’O& is not surprising. On the other hand, the decrease could be interpreted on the basis that a mobile potassium ion determines the parameters of the conductance equation, since A would then decrease with increasing 7 ’ Ca(XO3)*due to “dilution.”

+

General Discussion The principal achievement of this work is the establishment of T oas a factor capable of accounting quantitatively for the special temperature dependence of transport observed in the present system, and its identification with the zero-point temperature of the free volunie concept of glass transformation. Through the correlations achieved with other systems, this zero point now emerges as a basic featurc of transport in fused salts, but the failure of the expressions for k and A The Journal of Physical Chemistry

in the Cohen and Turnbull treatment to account properly for the measured values of these constants means a detailed model of transport in fused salts based on the free volume concept has yet to be established. While the niodel in its present state seems to predict reliably the characteristics of transport in fused salts a t low temperatures, T / T o 7 2 , due to the deviation from theoretical behavior a t high T / To, we can only expect to predict trends in behavior iii the temperature range of most general interest. Thus me can predict, correctly, that the “Arrhenius coefficients” for equivalent conductance and diff usioii of alkaline earth halides ( T , = 2-2.5T0) in the temperature range 0-300” above the melting point will be 50-100~‘ greater than for the corresponding alkaline metal halides ( T , = 3-3.5To),but we cannot account for the relative values of “Arrhenius coefficients” within the individual series. Future analysis may make use of the observation that the temperature dependence of the “Arrhenius coefficient’’ for conductance in alkali metal halides changes from increase with falling temperature of the form predicted by the 1/(T - T o ) relation for CsCl halides to actual increase for S a and Li halides. The most important question raised by the free volume concept for fused salts concerns the distribution of the total free volume among the different ionic species. This can only be satisfactorily answered by diffusion measurements. I n pure fused salts the “hrrhenius coefficients” for diffusion of cation and anion species are the same. If the temperature dependence is described by a term of general form exp(-v*/vf), then the total free volume in a pure fused salt must be divided between the cation and anion species in proportion to their ionic volumes, since v* must in each case be characteristic of the volume of the ion.40 In binary and multicomponent systems, no diffusion measurements suitable for indicating the free volume distribution have been performed. Attention has already been drawn to the diagnostic importance of such measurements in mixed valency systems. Another question of great interest concerns the possibility of differences in T o between different species in the same melt. Again, diffusion measurements are required but, for this purpose, these would have to extend well into the transformation range where conventional techniques would be limited by the long ~~

(38) F. R. Duke and A. L. Bowman, J . Electrochem. Soc., 106, 626 (1959). (39) G. Perkins, R. B. Escue, J. F. Lamb, and J. W. Wimberley, J . Phys. Chem., 64, 1792 (1960). (40) This suggests the bizarre concept of a pure fused salt as two

separate ionic liquids, essentially independent with respect to their transport properties, within the same volume element.

1929

THERMODYNAMICS OF AQUEOUS SOLUTIONS OF HYDRIODIC ACID

diffusion periods required. Chronopotentiometric measurements in Ca(XO& K N 0 3 solvents seem a more promising approach in this case. Finally, we note the interesting prospect of using the glass-forming associated halides such as ZnCh and the Be halides as a bridge for exploring the rela-

+

tionship between simple molten salts and glass-forming oxide melts.

Acknowledgment. The author is indebted to Mr. G. M. Willis for many valuable discussions, and to Professor H . W. Worner for his interest in the work.

Thermodynamilcs of Aqueous Solutions of Hydriodic Acid from Electromotive Force Measurements of IIydrogen-Silver Iodide Cells

by Hannah B. Hetzer, R. A. Robinson, and Roger G. Bates National Bureau of Standards, Washington, D. C.

(Received February T& 1964)

Electromotive force measurements of the cell Pt; Hz(g), HI(m), AgI; Ag have been made a t eleven temperatures from 0 to 50'. The standard e.m.f. (EO) is given within 0.05 mv. by the equation E o = -0.15242 - 3.19 X 10-4(t - 25) - 2.84 X 1Op6(t - 25)2, where 1 is the temperature in degrees Celsius. These values are in excellent agreement with those obtained by Owen from studies of borax-buffered K I solutions at 5, 10, 30, 35, and 40' but differ by 0.14 to 0.17 niv. at 15, 20, and 25'. The activity coefficient (yk) of H I a t molalities (m)from 0.005 to 0.9 has been derived. The relative partial molal enthalpy (E,) of H I at 25' was calculated and compared with that for HC1 and HBr. At 25' and m = 0.1, y* is 0.811, and E, is 130 cal. mole-'.

Introduction From measurements of the e.m.f. of the cell Hz; HI, AgI; Ag containing 0.0980 and 0.0300 m hydriodic, acid, Noyes and Freed' deduced a value of - 0.14778 v. for the standard potential of the silver-silver iodide electrode at 25". The value given in the International Critical Tables2is -0.151 v. which, according to Owen,3 is based on the data of Pearce and Fortsch.* Owen himself compared the e.m.f. of the cell 11,; borax buffer and KX, AgX; Ag, with X = C1 or 1; he made measurements a t intervals of 5" from 5 t o 40" and found E" a t 25" to be -0.15230 v. Hass and Jellinek5 measured the e.m.f. of the cell with liquid junction Ag; AgI, MX, satd. KC1, 0.1 N KCI; calomel, where M = Na or K and X = C1 or I. From their data we celculate E o = -0.1502 v. at 25".

Finally, Cann and Taylor,e using measurements of a cell with a flowing liquid junction, arrived a t a value of E o = -0.1510 v. a t 25", although their value of E o == -0.1507 v. obtained by another method of cxtrapolation might be preferred. The considerable variation among these several values of E" a t 25" led us to investigate the cell Hs; Hl(m), Agl; Ag, where m is molality, in some detail from 0 to 50". We have found E" = -0.15244 v. a t (1) A. A. Noyes and E. 9. Freed, J . Am. Chem. Soc., 42, 476 (1920). (2) "International Critical Tables," Vol. V I , McGraw-Hill Book Co., Ino., New York, N. Y., 1929, p . 332. (3) B. B. Owen, J . Am. Chem. SOC.,5 7 , 1526 (1935). (4) J. N. Pearce and A. R. Fortsch, ibid., 45, 2852 (1923). (5) K. Hass and K. Jellinek, 2 . Phvsik. Chem., 162, 153 (1932). (6) J. Y. Carin and A. C. Taylor, J . Am. Chem. Soc., 59, 1484 (1937).

Volume 68, .\'umber

7

J u l g , 1964