The diffusion constants of thallous (Tl+) and lead (Pb++) ions in potassium chloride

J. Phys. Chem. Solids

THE AND

Pergamon Press 1961. Vol. 18, No. 4,

DIFFUSION LEAD (Pb”)

Printed in Great Britain.

CONSTANTS OF THALLOUS (Tl+) IONS IN POTASSIUM CHLORIDE A. GLASNER

Department

pp. 345-355.

and R. REISPELD*

of Inorganic and Analytical Chemistry, (Received

4 August

The Hebrew University,

Jerusalem,

Israel.

1960)

Abstract-Diffusion constants of monovalent thallium and divalent lead into potassium chloride (in the low temperature range of the latter) were evaluated by a new spectral method. Results obtained are summarized in Table 5. Q, the activation energy of thallium ions, was found to be somewhat greater than that of the lead ions, the latter being about equal to the activation energy of the self-diffusion of the potassium ions. It is suggested that the comparatively small pre-exponential factor De for lead is due to its complexation with cation vacancies. The results obtained on heating the pressed disks differed widely from those with the mixed powders. The difference is accounted for by assuming a prevalence of “grain boundary diffusion” in the pressed disks, whilst “volume diffusion” is deactivated by the high pressure previously applied. The energy of activation with pressed disks is found to be about 10 per cent larger than the latent heat of fusion of the salts employed, namely thallium chloride and thallium sulphate. 1. INTRODUCTION

A KNOWLEDGE of the diffusion constants of ions in the ionic lattice is greatly desired, but the accumulation of data is slow because of the experimental difficulties(l) encountered with the currently accepted methods. In a previous paper(s) a new method for the evaluation of the diffusion constants in alkali halides was proposed, making use of the characteristic optical absorption of certain cations in the ultra-violet when substituting an alkali ion in the lattice. The characteristic spectra are developed on heating the mixed powders, and the absorption is measured in pressed disks prepared from samples taken at intervals. In a preliminary study of this technique it was found that the spectra of certain cations, of comparatively volatile chlorides, are developed during the process of mixing the powders.(s) In this article we report the diffusion constants obtained by this method for monovalent thallium and for divalent lead in potassium chloride. 2. THEORErICAL It is assumed that the powder is made up of cubical grains of nearly equal size, with a mean * This article is based on a Ph.D thesis submitted by R. R. to the Senate of the Hebrew University, Jerusalem. 345

edge length 1. If each cube of the foreign salt is surrounded by cubes of the alkali halide, and there is a good contact between the grains, the boundary surface through which diffusion takes place is equal to 6 1s. On integrating Fick’s second equation : dc

dt

0%

D -+ = ( ax2

O?c

&

-+

ays

-22

)’

and introducing the limiting condition@) 4 Dt g 1, one obtains:

M% D=

(6Z2)2C02t =

(Et/Em)% 36t

(1)

where M is the total amount of foreign ions (expressed in gram mols) diffused through the boundary surface, CO is the molar concentration of the diffusing cation in its salt, t is the time of diffusion, Et is the optical density measured at the peak of absorption after time t, and Em is’the maximum density obtained for complete diffusion at time infinity. Thus, if Em and 2 are known, D may be evaluated at any temperature simply by measuring Et, or otherwise, a mean value may be obtained from the slope of a plot of Et2 versus t, from a number of measurements.

346

A.

GLASNER

and

3. EXPERIMENTAL

Matuials All salts used in Analyzed” reagents. Preparation

this

work

were

“Baker

REISFELD

The disks thus obtained were clear and transparent, transmitting 70 per cent or more light in the near ultra-violet, and somewhat less at shorter wavelengths. When kept in the desiccator the disks did not tarnish for months.

of powders

The salts were finely ground in a mechanical mullite mortar, dried at lZO”C, then accurately weighed, and mixed for 10 min in a Wig-L-Bug electric vibrator, in a plastic vial, to obtain mixtures containing one per cent by weight of the foreign salt. These mixtures were further diluted by the same procedure to give 0.02 per cent TlCl and O-01 per cent PbCls. The total weight of each batch of powders was 2-3 g, from which ten or more disks could be prepared. The time of grinding, as well as other factors which might influence the size of the grains in the powders, was always the same. Photographs taken under the microscope showed that most of the grains were of similar size, cubic form, about 20~ on the edge. Heating

R.

at constant

temperatures

In experiments up to 200°C a vapour bath (of the vacuum drying pistol type, widely used by organic chemists for desiccating under reduced pressure) with various constant boiling liquids, was employed. At higher temperatures, an oven wound with nichrome wire and regulated by a “Fielden electronic precision thermostat” was used. In both cases, the temperature was kept constant to k 1°C. The Pyrex vial containing the mixed powder was transferred from the desiccator into the preheated oven, and kept there for fixed intervals. The portions taken for the preparation of a disk were measured out by volume with a funnel-shaped glass vessel, containing approximately 200 mg potassium chloride. Pressing the disks A Hilger potassium bromide press,(s) and a Loomis 20 ton hydraulic press served for this purpose. The die was evacuated and a total pressure of 25000 lb was applied for 20 min. The die was slightly heated during the same time to about 5O”C, with the aid of a heating tape. The diameter of the circular die is 0.5 in., hence the force applied corresponds to a pressure of N 9 tons per cm2.

Measurement

of spectra and optical density

The spectra were taken with a Beckman DU spectrophotometer and photomultiplier. As a rule, points at intervals of 10 rnp were measured, against air, over the range of 200400 rnp, and in the neighbourhood of the characteristic absorption peak at intervals of l-2 mp. The disks were placed in one of the circular holes of the filter holder soon after pressing, and an additional phosphorus pentoxide tube in the cell compartment ensured a dry atmosphere at the time of measurements. After taking the spectrum, the disks were weighed. The peaks of the absorption bands were situated at 247 and at 272 rnp for the disks containing thallium and lead respectively. Disks showing a high absorbance, especially with thallium, exhibited a flat peak, supposedly because of the fluorescent nature of the materials. In order to evaluate the diffusion constants, the optical density Et was calculated as follows: The absorbance of a pure alkali halide disk, which had the same transparency in the spectral range 320400 rnp as the disk containing the foreign metal cation, was subtracted from the measured value at the wavelength of maximum absorption. The result was then corrected for the weight of the disk to a standard of 200 mg, assuming direct proportionality between weight and absorbance. The actual maximum variance in the weight of the disks was f 10 per cent. Disks prepared from the powders, containing lead chloride, soon after mixing (but without heating) did not show any absorption bands, but those containing thallium chloride exhibited a characteristic absorption, corresponding to 4-5 per cent of the thallium in the mixture. Hence, diffusion of thallium took place to this extent during mixing and pressing. The optical density Em of a 200 mg disk after total diffusion was ascertained in two ways: (a) by heating the mixed powders for a sufficiently long period; (b) by wetting the powders, drying, and regrinding. The mean of a large number of

THE

DIFFUSION

CONSTANTS

OF THALLOUS

(Tl+) AND

LEAD

(Pb++) IONS

IN KC1

347

wave length rnf FIG. 1. A few typical spectra of doped potassium chloride disks pressed after heating the mixed powders. 1. 2. 3. 4. 5.

Pure KC1 KCl:PbCls, KCl:PbCls, KCl:TlCl, KCl:TlCl,

O-01 per cent, 35O”C, 20 min. O-01 per cent, 350°C. 60 min. O-02 per cent, 187’C, 1 hr. O-02 per cent, 187”C, 8 hr.

such measurements gave the values 1.50 and 1.20 for the 0.02 per cent KCl:TlCl and for the 0.01 per cent KC1 :PbCls mixtures respectively. 4. RESULTS Dz@imim measurements in poevders of potassium chloT& Measurements on potassium chloride powders containing 0.02 per cent thallous chloride were made in the temperature range 150-2OO”C, and

in potassium chloride powders containing 0.01 per cent lead chloride, in the temperature range 300-375°C. A few typical spectra of disks, compared to air, are shown in Fig. 1. Fig. 2 shows plots of &s vs. t in “thallous chloride” and in “lead chloride” powders at various temperatures. It may be seen that in all cases the experimental points lie on a straight line up to a high fraction of the total density, this being almost 70 per cent in the case of the thallium chloride disks and some-

A.

348

GLASNER

and

what less in the case of the lead chloride disks. It should also be noted that the plot of the latter mixtures starts at the origin, whilst the extrapolated thallium chloride lines cut the abscissa at a value equal to minus 3 hr. This is in accord with the observation that unheated “thallium chloride” disks exhibit a low absorption spectrum, i.e. some preliminary diffusion has taken place at the time of mixing. 0

40

W

60

80

R.

REISFELD

One of the eight consecutive values given in the Table differs by almost 25 per cent from the rest. Rejecting this, the mean deviation is 4.6 per cent. Considering the various possible sources of error, such as incomplete mixing, fluctuations in temperature, and instability of the readings of the spectrophotometer, this small deviation from the mean appears very satisfactory. In Fig. 4, log D was plotted vs. the inverse of time in hours

l40

120

00

Er2 1

0

1

2

5

4

3

time in hours

FIG. 2. Square of measured optical density vs. time of heating at tempera-

tures indicated (.&a vs. t). Arrows on curves indicate scales employed.

In Fig. 3, log Et vs. log t was plotted. (For the thallous chloride series the time values were corrected by the addition of 20 min to the actual times.) In this plot also the points of each series of measurements lie on a straight line with a slope equal to 0.5, as required by equation (1). Hence after substituting the appropriate values for the constants in the last equation (I = 20 x 10-d cm; &,Tl+ = 1.50 ; &,Pb++ = l-20), D, the diffusion constant in cm2 se&, at several temperatures, was calculated from the slope of the straight lines exemplified in Fig. 2. The data thus obtained are summarized in Table 1. As a matter of fact, D may be evaluated from each individual measurement, and it is of interest to compare these to the mean value obtained graphically. For this purpose a series of measurements with 0.02 per cent thallous chloride is given in Table 2 in detail.

the absolute temperature. The points for both thallous and lead chloride lie on straight lines, from the slopes of which the energies of activation Q were calculated. Substituting these in the Arrhenius equation D = Do exp (- Q/RT), one obtains the expressions : DT~+ = 7.14 exp (- 25,20O/RT) ems set-l and &b++

= 4.41 x lo-sexp

in the temperature Dr@&on

(- 22,82O/RT) ems set-1

ranges mentioned.

in powders under reduced pressure

In order to find out the effect of pressure on the calculated diffusion few series of experiments were carried evacuated vessels. The temperature periments was hand-regulated by a

atmospheric constants, a out in partly in these ex“powerstat”

THE

DIFFUSION

CONSTANTS

0 1,2-

04

I

1

08

I

,

01

OF THALLOUS

1

AND

LEAD

(Pb++) IONS

lclog t , hours. 2.4 26

1.2

I.6

2

08

1.2

1.6 log(t + l/3) hours.

I

I

-IIIlIIIIl~l 0

(Tlf)

04

FIG. 3. Log Et vs. log t plots for powders. For lead chloride mixtures use upper time scale.

lO'/T FIG. 4. Arrhenius plot of the diffusion constants D obtained with KCl:TlCl and KCl:PbCla powders (n = 13, or 12 resp.).

IN KC1

349

A.

350

GLASNER

and

Table 1. Dt@sion constants qf thallous chloride and lead chloride in pota&m chloride -__ I Thallous chloride Lead chloride I “C 150

176.5 187 200

D x lOi2 0.68 4.70 8.36 17.0

“C

’

300 325 350 375

DxlO” 0.77 2.08 444 8.79

and measured by an ordinary mercury thermometer in outer contact with the tube holding the powders. The results obtained with three batches of KCl:TlCl powders, at about 150” (I and II) and at about 155°C (III) after heating for two hours, are given in Table 3. Clearly, reduction of atmospheric pressure increases the absolute rate of diffusion, but this effect becomes smaller the higher the vacuum.

Experiments with thallous sulphate Powdered mixtures of 0.021 per cent KCl:TlsSO4 were prepared so as to contain an equivalent amount of thallium to the KCl:TlCl mixtures. The development of the spectra was slower with the thallous sulphate than with the chloride, and experiments were carried out at higher temperatures. The expected full optical density was attained only at 400°C; at lower temperatures the diffusion seemed to approach a lower limit even after prolonged heating. Fig. 5

Table 3. D@sion

R.

REISFELD

Table 2. D for 0.02 per cent KCl:TlCl, calculated from single points 2Z.Z

t*, hr

D x 1013

Et

hD x 1013

--

1.33 2.33 3.33 6.83 10.83 15.83 18.83 32.83

0.143 0.185 0.236 0.335 0.420 0442 0.533 0.707

6.63 6.33 7.21 7.08 7.02 (5.32) 6.50 6.56 mean: 6.76

-0.13 -0.43 +0.45 +0.32 +0.26 -0.26 -0.20 + 0.31

./I c(W2 1

‘VL * Corrected,

,r

J

= 0.31 x 1013

to account for preliminary

DT~~so~ = 5.2 x IO-sexp (- lO,gOO/RT) cmssec-1

of thallium in potassium chloride under reduced pressure, at 150-155”C, heating time = 2 hr. I Et x lo3

0.5 mm 0.3 p

diffusion.

shows the Et2 vs. time plots in the temperature range 190400°C. Each of the curves obtained has an initial linear portion that cuts the ordinate at a relatively high point. From the slopes of these straight lines, the apparent diffusion constants were calculated in the same way as above. Also, on plotting the logarithms of the slopes AE$/At vs. the inverse of absolute temperature (Fig. 6), the energy of activation was obtained. Thus, the results with thallous sulphate may be presented by the expression:

II

III

Pressure

O.D. prior to heating *695 mm 60 mm 9mm

at 150°C

83

/ DxlO=

-

Et x 103 87

D x 101s

-

139 214 -

0.27 0.84 -

149 225 -

0.32 0.91 -

240 297

1.09 1.75

254 -

1.23 -

* Normal barometric

pressure in these laboratories.

a x 10’ 88.5

D x 1Ols -

243 -

1.10 -

386 427 450

304 3.76 4.19

THE

DIFFUSION

CONSTANTS

OF THALLOUS

(Tl+)

AND

LEAD

(Pb’+)

IONS

IN KC1

351

as above) in the temperature range 150-45O”C. In these experiments the disks often lost their transparency after continued heating, especially at the higher temperatures, and had to be re-pressed before measuring their spectrum. This was accompanied by a small loss of weight, and an improvement in transmission. The &s vs. time curves (Fig. 7) resemble in many respects those obtained with thallous sulphate powders. The Arrhenius plot of the initial slope of these curves is shown in Fig. 6, and the expression for the apparent diffusion in the pre-pressed disks may be represented by: D&r 0

6

8

24

32

FIG. 5. Diffusion of thallium in KCI:TlzS04

Dijkibn

D&so4=

time in hours

’

’

’

1.64

I.94

2’24

103/T FIG. 6. Arrhenius plot of slope of diffusion curves: 1. KCl:TlCl,

disk; 2. KCl:TlzSOr,

-6860/M’)

ems see-1

The straight lines obtained on plotting log Et vs. log t, with a slope equal to O-5 (Fig. 3), prove that the slowest step in a probable sequence of reactions, i.e. the reaction rate measured, is actually that of diffusion. Simultaneously, Figs. 2 and 3 prove the basic validity of equation (l), but

in potassium chloride disk

’

4.91 x lo-lsexp(

ems set-1

5. DISCUSSION

powders.

Several series of measurements were made with KCl:TlCl and KCl:TlsS04 disks prepared from the unheated powders (of the same concentration

1.34

= 4.21 x lo-10 exp (-4670/W)

disk; 3. KCI:TlzS04,

powder.

A.

352

GLASNER

and

R.

REISFELD

0.8

0.6

_ Kbl :Tl2504

”

0

3

6

9 lime

12 in hours

J

15

FIG. 7. Difiusion of thallium in KCl:TlCl

not the assumptions with respect to the area of the face-boundary through which diffusion progresses. Comparing the present values for the diffusion of cations in alkali halides with those available in the literature,(s) it appears that 6Is (where I is the mean length of the edge of a powder grain) is not far wrong. Yet, for a critical evaluation of the diffusion constants, as well as other related parameters obtained in this work, a closer knowledge of the area of the face boundary is necessary. Altogether in a heap of powders one would not expect a perfect contact between the grains. There are two possible mechanisms by which contact between grains at short distances may be improved: (u) surface diffusion (to be discussed later), and (b) sublimation. The experiments at reduced pressure (Table 3) are indicative of the mechanism occurring; Et as well as D appears to increase when the pressure is reduced, and it may therefore be concluded that sublimation occurs. The latter is limited, at atmospheric pressure, by the “mean free path” in air, this being of the order of magnitude 10-s cm.(?) Hence, the very small vapour

18 and KCl:Tl&04

21

24

disks.

pressure of thallous and lead chloride, at the temperature of the experiments, suffices to bridge’ small gaps between these salts and the potassium chloride grains. The mean free path is inversely proportional to pressure, but at high vacuum the effect of change of pressure on Et becomes negligible, i.e. in spite of the very large mean free path, the foreign salt does not spread all over the surface of distant potassium chloride grains, being prevented from doing so by the wall of grains circumscribing the foreign particle. To summarize, the surface of the potassium chloride cubes facing a foreign particle, at distances less than about 1 p, are covered by the sublimate at atmospheric pressure. As pressure is reduced, the coverage increases two- or threefold, but not more, as evidently only a small fraction of the foreign salt is able to escape through the slits in the “enclosure”. Hence, when working at atmospheric pressure the assumption that the “face boundary” is equal to 61s is a good first approximation, this being a reasonable fraction of the total area of the potassium chloride grains facing a foreign particle.

THE

DIFFUSION

CONSTANTS

OF THALLOUS

With respect to the effect of increasing temperature on the extent of sublimation, neither the increase of the mean free path nor that of the vapour pressure could have an influence comparable to that of a tenfold reduction of external pressure. This may be surmised from examination of Table 4 (which lists(*) melting points (M.P.),

(Tl+) AND

LEAD

(Pb++) IONS

IN KC1

353

When a certain amount of the foreign cation has diffused, there is a change in the slope of the Et2 vs. time curves (Fig. 2) ; equation (1) is no longer valid when the limiting condition 4 Dt < 1 has been passed. In the case of thallium chloride, equation (1) seems to hold up to 70 per cent or more of the total amount of the foreign ion in the mixture.

Table 4 Substance --

M.P.

TlCl TlBr

427 460 632 498 224 277

nso4

PbCla BiCla H&la

Lf

B.P.

4260 5990 5500

807 819 -

24,420 23,800 -

5650 2600 4150

954 441 304

29,600 17,350 14,080

boiling points (B.P.) in degrees Centigrade, latent heats of fusion (Lf) and latent heats of vaporization (L,), per gram mole, of a few relevant salts). The activation energies Q obtained in this work should therefore be very nearly correct, although the absolute values of D and Do may be in error by a small factor. It must be pointed out that in a previous article(s) on the absorption spectra of mercury, bismuth and antimony halides in pressed alkali halide disks, it was observed that the spectra of these cations are fully developed without heating of the powders, owing to the volatility of their halides. This is so because the low melting points and low heats of fusion (Table 4) undoubtedly assist the surface diffusion of these salts, so that large surfaces of distant alkali halide particles are covered by them. The resulting great dispersion of these volatile salts makes them unsuitable for a diffusion study by the present spectral method. The advantages of the method with less volatile salts are evident; results may be obtained on heating for short intervals, and the techniques and the mathematical equation for calculating D, the diffusion constant, are simple. Moisture must be excluded as it increases the surface boundary by dissolving the salts. Diffusion constants may be obtained from a single measurement of the absorption peak, or a more reliable mean value results from a series of measurements (Table 2). 2

LO

With lead chloride the bend in the curves occurs at a lower percentage. This may be due to the difference in the lattice of the two salts in the mixtures. The influence of the lattice is even more obvious in the case of thallium sulphate, where as a result of the process of diffusion an intervening layer of mixed potassium-thallium sulphate is formed between the original potassium chloride and thallium sulphate layers. This explains the very low apparent DO value obtained with the thallium sulphate mixtures, as well as the inequality of the activation energies, Q, with those of the thallium chloride mixtures. It is thought that Q in this case may stand for the energy of activation of the diffusion of thallium ions in the mixed potassium-thallium sulphate lattice. Table 5 contains a summary of the constants of the Arrhenius equation obtained in this work, and a comparison with some of the scanty data available from previous publications. The constants for the diffusion of thallium and lead in powder mixtures are of the same order as those obtained by more conservative methods. The energy of activation for thallium is larger than that for lead, or for the self-diffusion of potassium. This may be due to the larger size of the thallium ion, as was to be expected with a mechanism of diffusion through vacancies. The ratio of Do values for thallium and for lead is of the order of 10s. Such, and even wider, variations had been observed with other cations. Hence,

354

A.

GLASNER

and R.

REISFELD

Table 5

Diff. ion

Temp. raw3 “C

DO cm2 set-I

Ref.

__

in KC1 Tl+ Pb++ Tl+(TlsS04) Tl+ disk Tl+ disk

150-200 300-37s 190-400 150-400 300-450

7.14 4.41 x10-s (5.2 x10-8) (4.21 x 10-10) (4.91 x10-y

25.2 22.8 10.8 4.67 6.86

(Tl2S04)

K+ in NaCl Na+ cu+ Ni++

< 460’ < 5.50”

1.6 x 10-s 0.5 0.2

any conclusion with respect to the DO values is bound to be somewhat speculative. Yet, in the case of the small observed value for lead, it could be attributed to the retarding effect of a cation vacancy complexed with the divalent lead ion. The latter may fluctuate between the two lattice points at its disposition, but its chances of advancing in any one direction are thereby reduced. For the diffusion of a divalent impurity complex one has to assume either a previous dissociation of the complex or a sequence of jumps described by LIDIAFUD), in which (a) the impurity changes place with the vacancy (with probability rus) and (b) the vacancy jumps around the impurity ion from one associated position to another (with probability WI). By assuming the latter mechanism, LIDIARD explained the much larger absolute diffusion coefficient of zinc ions in sodium chloride than that of the sodium tracer, observed by Chemla in the intrinsic range. Even if LIDIARD’S assumptions prove to be right in the intrinsic range, they may not be so at the lower temperatures of the impurity range. However, there are two conditions, sine qua non, for the above mechanism, as stated by LIDIARD (Ref. 13, p. 337): “Owing to the double charge on the impurity ion it is to be expected that the potential energy barrier opposing its movement will be increased and that the condition wa < w1 will almost always be satisfied.” But actually the activation energy for the diffusion of

19-23

9,lO

17.7 25.2 25.4

11 6 6

the zinc ion is found to be only 046 eV as against 0.77 eV for the self diffusion of the sodium ion. Hence it seems more probable that ws $ wr, and WI being “very small” “it is clear that the impurity ion merely jumps backwards and forwards without progressing very far on the average”. The large diffusion coefficient of the zinc ion should therefore simply be attributed to its low activation energy. The very low values for DO and for Q observed in the experiments with disks pressed before heating, is surprising. These results may only be due to the high pressure applied, i.e. a volume disactivation of the vacancy mechanism.(14J5) Further experiments under progress in this laboratory confirm the above conclusion. The measured activation energies Q are therefore thought to be due to a surface diffusion on the grain boundaries. The actual values 4.67 and 6.86 kcal/g mol for thallium chloride and thallium sulphate respectively are about 10 per cent higher than the molar heats of fusion of these salts (see Table 4), suggesting a straightforward relation between these two parameters. The “apparent” DO values (4.2-4.9 x lo-10 cm2 see-l), very nearly the same with both thallium salts, are a direct result of the very much smaller area of the available face boundary for surface diffusion, compared to that of volume diffusion. Hence, in the mixed powders, both

THE

DIFFUSION

CONSTANTS

OF THALLOUS

volume and surface diffusion occur, but owing to the large face boundary of the powdered salts, the part of the surface diffusion in the total diffusion of the foreign cation may be neglected, in spite of the much higher activation energy of the volume diffusion. On the other hand with pressed disks, while the volume diffusion is restricted, the spectral method affords means of measuring surface diffusion on the grain boundaries. The advance of surface diffusion in pressed pills of alkali-halides, in front of the volume diffusion, has been radiographically observed by BBNARDand LAURENT. Acknowledgement-The authors are most thankful to Professor A. HALPERIN,of the Department of Physics of the Hebrew University, for many fruitful discussions.

REFERENCES 1. MOORE W. 5.. Ann. Rw. Phvs. Chem. 10. 416-20 (1959). - ’ 2. GLANER A. and &ISFELD R., J. Chem. Phys. 25, 381 (1956). 3. GLASNERA. and REISFELDR., J. Chem. Phys. 32, 956 (1960).

(Tl+)

AND

LEAD

(Pb++)

IONS

IN KC1

355

4. JOSTW., Diffusion in Solids, Liquids, Gases, p. 22. Academic Press, New York (1952). 5. FORD M. and WILKINSON G. R., J. Sci. Instrum 31, 338 (1954). in und an festen 6. See e.g. HAUFE K., Reaktionen Stoflen, pp. 376-7, Table 17. Springer Verlag, Berlin (1955). S., Textbook of Physical Chemistv (2nd 7. GLADSTONE Ed.) pp. 274-81. Van Nostrand, New York (1950). 8. Chemical Engineers’ Handbook, pp. 210-12 (edited by PERRY J. H.). McGraw Hill, New York (1950). E. G., J. Amer. Chem. 9. PHIPPST. E. and PARTRIDGE sot. 51, 1331(1929). 10. KELTING H. and WITT H., 2. Phys. 126, 697 (1949). 11. MAPOTHERD., CROOKS H. N. and MAURER R., J. Chem. Phys. 18, 1231 (1950). 12. WAGNERC., J. Chem. Phys. l&l227 (1950). .l* LIDIARDA. B., Encyclopedia of Physics, Vol. Xx, 1.7. pp. 335-9 (edited by FLOGGES.) Electrical conductiwity II. Springer Verlag, Berlin-GGttingenHeidelberg (1957). 14. KURNICKS.-W:, J. bhem. Phys. 20, 218 (1952). D. S.. I. Phvs. Chem. Solids 5. 224 15. TANNHAUSER (1958). ’” J., J. Phys. Chem. Solids 16. LAURENTJ. F. and BERNARD 7, 218 (1958).