TRANSPORT NUMBERS! IN PURE SILVERNITRATEAND LEADBROMIDE
1755
Transport :Numbers in Pure Molten Silver Nitrate and Lead Bromide by the Balance Method
by Paul Duby and Herbert H. Kellogg Department of Mineral Engineering, H e w y Krumb School of Minea, Columbia Univereitu, New York, New York (Received November lb, 1968)
An alternate method for determining transport numbers in pure molten salts is described, which is based on the measurement of the mass transport caused by electromigration of the ions. A complete mathematicaJ description is given, taking into account the mechanical return flow due to the build-up of a hydrostatic pressure difference. The experimental procedure is a semicontinuous recording of the displacement of the center of mass of a horizontal cell, which consists of two electrode compartments connected by a capillary or by a tubing with a porous disk. The results are interpreted as the net flow of the elements of the salt with respect to a fixed reference frame. They agree with data obtained by Laity and Duke’s measurement of the volume change, but a higher sensitivity of the present technique makes possible the determination of the following slight but significant temperature dependence: in AgNOI, T A = ~ 0.800 - 3.5 x lO-4(8 - 200); in PbBrz, T p b = 0.370 -- 1.66 X lOP4(8 .- 370).
Introduction The measurement and interpretat ion of transport numbers in pure fused salts is still a controversial subject as the current literature will il1ustrate.lv2 For example, a recent paper3 denies any significance to transport number measurements in pure fused salts. We believe that such measurements are meaningful and important to the understanding of the transport process, provided it is recognized lhat the electric field causes a motion of the local center of mass or center of volume relative to an exlernal reference f r a m e fixed in the laboivatory, and this same reference frame is used in the mea~urernent.~Interpreted in this light, transport numbers in pure fused salts are measurable and reproducible. Accordingly, the measurement is significant and should be accounted for by any theoretical model of {,hetransport process. To lend support to this view, we have developed an alternative method of measuring transport numbers in pure fused salts and have already reported preliminary results in an earlier communication.6 We present now a more complete account of the method, its theoretical basis and limitations.
I n contrast to the first reliable technique, which was developed by Duke and our method depends on the measurement of mass transport rather than volume transport.8 I n brief, it consists of a semicontinuous recording of the shift in the center of mass of a horizontal transport cell while the electrolysis proc e e d ~ . ~The cell contains a porous diaphragm to impede hydraulic flow, but this diaphragm need not have an unusually high hydraulic resistance. The (1) G. J. Jana, C. Salomons, and H. J. Gardner, Chem. Rev., 5 8 , 480 (1958). (2) (a) R. W. Laity, J . Chem. Educ., 39, 67 (1962); (b) T. B. Reddy, J . Electrochem. Tech., 1, 325 (1963). (3) C. Sinistry, J. Phys. Chem., 66, 1600 (1962). (4) P. Duby, to be published. (5) H. H. Kellogg and P. Duby, J . Phys. Chem., 6 6 , 191 (1962). (6) F. R. Duke and R. W. Laity, J . Am. Chem. Soc., 76, 4046 (1954); J . P h y s . Chem., 5 9 , 549 (1955). (7) R. W. Laity and F. R. Duke, J . Electrochem. Soc., 105, 97 (1958). (8) Other measurements have been reported, which are based on the observation of a volume change: (a) H. Bloom and N. J. Doull. J . Phys. Chem., 6 0 , 620 (1956); (b) H. Bloom and D. W. James, ibid., 63, 757 (1959); ( c ) F. R. Duke and J. P. Cook, ibid., 62, 1454 (1958); (d) W. Fisher and A. Klemm, 2. Naturforsch., 16a, 563 (1961); (e) R. J. Labrie and V. A. Lamb, J . Electrochem. SOC.,110, 810 (1963).
Volume 68, Number 7
J u l y , I964
1756
PAULD U B YAND HERBERT H. KELLOGG
transport number is calculatkd from the initial slope of the weight change us. time recording. Our method has the advantages of experimental simplicity, better precision than other methods, and elimination of the need for an ultrafine porous diaphragni. The high precision also makes possible the determination of the teniperature coefficient of the transport number. Because this technique is fundamentally different from other ones, we present below a theoretical treatment which explains the basis of our method and relates it to the volume-change methods.
transport number of constituent Xi, g.eq. faraday-' partial molar volume of constituent Xi in the electrolyte, cm.3 mole-l partial molar volume of constituent Xio in the electrode, ~ mmole-I . ~ equivalent volumetric transport, cm. faraday -I equivalent volume change due to electrode reaction, cm.3 faraday-' total volume change, ~111.~ sec.-l volumetric flow rate due to hydrostatic pressure, ~ m set.-' . ~ constituent of the electrolyte constituent of the electrode valence of Xi in the electrolyte mass difference between two electrode compartments, g. viscosity, poise stoichiometric coefficient density, g. ~ m . - ~ summation symbol
Theoretical Consider a liquid ionic conductor which consists of n components X , in a valence state xi. These constituents are elements or stable radicals which can be determined by chemical analysis. The actual ionic structure of the liquid js not necessarily known, and no further assumption about it need be made. The fluid is contained in a n electrolysis cell made of two cylindrical electrode compartments, connected by a horizontal capillary tubing. Before applying the electrical field, the system is in thermodynamic equilibrium. During electrolysis, temperature and pressure remain constant and no chemical reactions occur except a t the electrodes. hydraulic constant of the transport cell, sec. -I cross-sectional area of the electrode compartments, diameter of a capillary, cm. electron, base of natural logarithm Faraday constant acceleration due to gravity, cm. sec.-z hydrostatic head in the transport cell, cm. lower index representing a constituent of the liquid electrical current, amp. flux of constituent X , , g.-eq. sec.-l length of the capillary, cm. atomic or molecular mass of constituent X,, g. mole-' equivalent mass transport per faraday, g. faraday-' number of electrons iiivolved in the electrode reaction pressure difference, dyne cm.+ iiumber of constituents taking part in the electrode reaction number of constituents of the electrolyte time, see. The Journal of Physical Chemistry
Description of Electromigration. When an electrical potential difference is applied to the cell, the various constituents of the fluid migrate with different rates. The flux of any constituent Xi can be measured relative to any given reference plane normal to the capillary axis. This flux is Ji, expressed, for instance, in gram-equivalents per unit of time, and measured positively toward the cathode. The transport number Ti of constituent Xi is the number of grain-equivalents transported across the reference plane per faraday of electricity.
Ti
FJi
= -
I
The transport number has the same sign as the corresponding flux: namely, it is positive for a species migrating toward the cathode. For a purely ionic conductor, the arithmetic values of the transport numbers add up to unitylo
(9) Previous mass transport measurements have shown low precision, because the total weight change was determined after completing the electrolysis: (a) S.Karpachev and S. Pal'guev, Zh. Fia. R h i m . , 23, 942 (1949); (b) I. G. Murgulescu and L. Marta, Acad. r e p . populare Romine, Studii cercetari chim., 8 , 376 (1960). (10) If there is some electronic conductivity represented by e =
__
Ielectrone Itotal
7
then eq. 2 becomes ZITiI = 1
-
e;
the subsequent equa-
i=l
tions could be easily transformed to include this case.
TRANSPORT NUMBERS
IN PURE
SILVER KITRATE
A X D ]LEAD
Volume Change. The electromigration fluxes of the various constituents cause a volumetric flow througlh the capillary. The equivalent volumetric transport per faraday is
The two electrodes are supposedly identical, and the same electrochemical reaction occurs in opposite direction at the anode and the cathode so that the total composjtion of the system is unchanged. If the first p out of Y constituents of the system take part in the reaction, it can be written for the cathode
9
v,xi
2=1
+ ne 2 vixl0 =
a=1
(4)
This reaction causes volume changes a t the cathode and anode. If the system is closed so that no product escapes, then the volume changes are equal but with an opposite sign at the two electrodes. The equivalent volume change per faraday due to the cathodic reaction is
BROMIDE
1757
consists only of a solid phase or a molten phase below the level of the capillary and there is no compressible gaseous phase. These conditions correspond to the experiments with molten silver nitrate and lead bromide. The pressure difference is then directly proportional to the level difference between the two compartments. Let this difference be zero for t = 0, when a constant current I is suddenly applied to the system
AP(t)
=
pgAh(t)
(8) (9)
Combining eq. 6, 7 , 8, and 9 and taking the derivative d dt
2ng pD4 __ Ah(t) 128 7LA
- Ah(t) f --
=
2 A - (V,
+ V,) IF
-
(10)
The solution of the differential equation is T
R
Ah(t)
=
-?- (Vi f V,) L_ (1 - e-"t) aA F
(11)
with the hydraulic constant, all 2ng pD4 128 qLA
a=--
I n a closed system, the volume changes due to the electrochemical transport and electrode reactions produce a pressure difference between the two electrode compartments, from which a hydraulic return flow results. The volumetric flow rate is a function of time W,(t)
=
-
nD4 AI' (t) 128qL
__
It is assumed that the temperature and concentration gradients remain zero, or small enough so tha,t the corresponding fluxes are negligible with respect t o the transport due to electromigration. This assumption is true with a pure fused salt where the electrolysis does not produce any concentration change a t the electrodes. It is a justified approximation for mixtures or aqueous solutions only if the volume of the electrode compartment is large and the concentration changes remain small. Under these condlitions, the total volume change, per unit of time, in the cathodic compartment is
Equation 11 shows that the level difference is an exponential function of time. The asymptotic value is proportional to the volume change due to electrolysis and proportional to the hydraulic resistance of the capillary. Mass Transport. The equivalent mass transport per faraday is
I n a closed system where the electrodes are weighed with the electrolyte, there is no mass change due to the electrode reactions. The mass flow rate due to the hydraulic return flow is proportional to the volumetric flow rate given by (6). Using eq. 6, 8, and 13, one gets the rate of change of the mass difference between the two electrode compartments. d - AfW(t) dt
=
I ng PD4 2Mt - - 2p - __ Ah(t) F 128 qL
(14)
Using eq. 11 and 12
The experimental conditions must be defined more precisely in order to relate the pressure difference with the volume change. Let us suppose that the electrode
(11) The mathematical description also holds for a porous disk, Equation 6 then becomes Wh(t) = ( K s / $ ) A P ( t ) , with s and 2 being the apparent cross section and thickness and K a geometrical coefficient with the dimensions of a surface. The hydraulic constant becomes: a = S g ( p K a / & ) . Either K or a can be determined experimentally by measuring flow rate 8s. pressure difference.
Volume 68, Number 7
Julu, 1964
1758
d - AM(t) dt
PAUL DUBYASD HERBERT H. KELLOGG
=
I
2Mt - - 2p(Vt F
+ V,) IF (1 - e-5t)
5cm.
-
1-
-
TO BALANCE PANS
(15) The integrated form of (15), assuming no mass difference for t = 0, is
Equations 15 and 16 show that the initial slope of the recording of the mass change per unit of current is a direct measurement of the equivalent mass transport per faraday, and hence, the transport numbers.
The asymptotic slope is a function of the mass transport and the volume change. It is generally different from zero
This corresponds to a steady state when the hydraulic return flow compensates the volume change due to electrolysis and the level remains constant in the electrode compartments. The mass transport, however, is not zero because the volume of the electrode varies and so does the average density of the whole electrode compartment. It can be shown that the right-hand expression in eq. 18 is independent of the transport number in the case of a pure fused salt. This will be discussed with the results on AgNO, and PbBr2.
Experimental Apparatus. The transport cells consist of two small Pyrex cups, of 20 or 25 mm. o.d., connected by a capillary or by a tube in which a medium porosity fritted Pyrex disk (pore diameter: 10-15 k ) has been inserted. They can be suspended either by means of a small brace at the upper part of the cup or by means of a small tubing sealed through the cups. The distance between the suspension is equal to the length of the beam of an "Ainsworth" semimicro recording balance. The position of the cell below the pans of the balance is shown on Fig. 1. It hangs inside a 50-mm. diameter cylindrical steel container which fits into a split wound resistance furnace. The latter is controlled by a proportional controller. Two small steel tubes fitted through the side flanges of the container support two The Journal of Physical Chemistry
r Figure 1. Position of the transport cell inside the furnace: 1, Pyrex transport cell with fritted disk; 2 and 3, silver electrodes; 4 and 5 , chromel--alumel thermocouples; 6, steel liner; 7 , glass draft shields; 8, furnace.
chromel-alumel thermocouples, respectively, connected to the temperature control device and to a potentiometer. The tip of the measuring thermocouple is in the same vertical section as the fritted disk of the cell, at a distance of about 1.5 cm. from it. The furnace and its controller are capable of maintaining temperature to 11' over several hours. I n the temperature range of the silver nitrate experiments (215-270') the steel muffle equalizes the temperature gradient to better than 1' over the whole length of the cell (15 em.). At the higher temperature of the lead bromide experiments (370-530'), the gradient is less than 2' over the length of the connecting tube (10 cm.). It is about 5' across the width of the electrode compartment, but this does not affect the experimental results. The fine nichrome suspension wires (0.013 cm. diameter) also serve to carry the electrical current to the cell. The electrodes are niade of silver wire in the case of molten silver nitrate. For the lead bromide, they consist of molten lead and a short piece of tungsten wire to make the connection with the suspension wire outside the cell. The electrolysis circuit consists of a 400 v. regulated d.c. power supply, two variable resistances, 7000 and 500 ohms, respectively, in series, and a d.c. milliammeter. Since our technique requires the measurement of a rather small current for a short period of time, it is more accurate to measure both current and time than to use a coulometer. Because of the high resistance in series with the transport cell, the current is easy to con-
TRANSPOIRT XWMBERS I N P U R E SILVER
NITRATEAND LEADBROMIDE
trol manually; and it remains constant within 0.05 ma., or less than 0.25%, during the time of a run. Accordingly, the experimental reading error on the current may be neglected with respect to the weighing error. The maximum1 resistance of the transport cell and connecting wires is of the order of 150 ohms with a fritted disk and about 12,500 ohms with the capillary. Procedure. The apparatus shown in Fig. 1 permits direct recording of any displacement of the center of mass of the transport cell, which is caused by the transfer of mass from one side of the cell to the other. Before starting a run, one must check that the whole system is in mechanical equilibrium, and there is no friction along the suspensiion wires. If the level difference between the two electrode compartments is zero, there is no hyclraulic flow through the fritted disk and the balance beam can be brought to equilibrium by adjusting the weights on the right pan. Then, as soon as the electrical current is applied, the beam starts moving. Currents from 20 to 60 ma. are passed through the cell for periods of 10-30 min., and then reversed for an equal period of time. The weight change is recorded semicontinuously. The beam is arrested while the electrolysis proceeds and it is released a t regular time intervals during a run. These intervals are such that the addition or removal of a 10-mg. weight on the right pan of the balance approximately compensates the transport of salt due to electrodiffusion. When released, the beam is then very close to its equilibrium position and there is no hydraulic flow caused by excessive swinging of the cell, and any motion of the molten lead is also prevented. The transport numbers are computed from the weight us. time plots according to the formulas developed below. For these relatively short runs the plot yields a straight line and the average of the rates obtained in both directions is used to determine the transport number. A few longer experiments of 1-14 hr. have been carried out. They show the effect of the hydraulic return flow which results from the build-up of a hydrostatic head. This level change is due to the volume change in the clectroole compartments caused by the electrolysis. During these longer runs an a.c. current equal to about twice the amplitude of the d.c. current is superimposed in order to prevent "tree" growing on the solid d v e r electrode in silver nitrate. Several other experimental procedures were tested but they proved less reliable than the one described above. Some runs were carried out by recording the weight change continuously. The iinclination of the transport cell, however, also changed continuously since the Ainsworth balance recorded the very small displacement of the beam without applying any com-
17511
-
pensating force to bring it back to zero. This produced a gravitational flow, and a small correction had to be applied to take it into account. Some runs were carried out with the cell suspended between a fixed point and the left pan of the balance. Preliminary experiments were performed in this manner with a standard analytical balance, by observing and recording manually the weight change. These results were less accurate, but nonetheless they were in good agreement, with the one listed below. Materials. Fisher certified reagent silver nitrato was used after recrystallization from aqueous solution. The silver electrodes were made out of pure silver wire (0.3 cm. diameter) from Handy and Harman. Lead bromide was prepared from Fisher certified reagents lead nitrate and potassium bromide. After separation of the precipitated lead bromide, it was redissolved in hydrobromic acid, recrystallized, and washed several times with dilute hydrobromic acid solution and finally with water. The lead electrodes were Fisher certified reagent lead metal.
Results and Discussion The formulas, which have been derived in the previous section, are easily applied to AgN03 and PbBrz. After eliminating one of the transport numbers by eq. 2, one gets for AgN03
Mt
vt + v,= Mt -
P(vb
+
TA~MA~N -OMNO* , voAg
(19)
- (1 - T A g ) v A g N 0 8 (20)
v e ) = 1WAg
-
PAgNOsvoAg
(21)
for PbBrz
Mt
vt + ve Mt -
P(vt
=
' / 2 ( T P b M P h B r , - 2MBr)
= '/2[VoPb -
+ ve)
-=
(1 - T P b ) v P b B r , ]
1/2(AvPb -
PPbBr,
voPb)
(22) (23) (24)
Equations 21 and 24 clearly show the physical nieaning of the asymptotic slope. When the steady state is reached, the level of the melt in the electrode compartments is constant; and for each faraday of electricity, one equivalent of metal is transported from anode to cathode, while a corresponding volume of molten salt flows in the other direction. R e c o r d i n g of W e i g h t C h a n g e us. Time. Figures 2 and 3 represent, for AgN03 and PbBrz, respectively, the weight changes recorded for periods of time larger than the hydraulic time constant l/a of the cells. The asymptotes are determined theoretically from eq. 21 and 24. The densities of Ag and P b are taken from ref. 12 and the densities of AgrO3 and PbBrz from ref. 13 and 14, respectively. Volume 68, Number 7
J u l y , 1964
PAUL DUBYASD HERBERT H. KELLOGG
1760
0.30
I
I
I
I
30
40
50
60
I
I
0.25
0.20 M
al
8
5 0.15 c .c
.-
; 0.10
0.05
0
0
10
20
Time, min.
Figure 2. Weight change as a function of time. AgN03 a t 229", I = 50 ma. The dashed line represents the theoretical asymptote. 0.100
ences of behavior between the two salts. They were chosen to test this new technique, and they are presented in the same paper because of these differences. The equivalent mass transport of silver nitrate is positive and rather large because of the high value of the silver transport number, but the volume change is comparatively small. Accordingly, there is little curvature on the weight us. time curve. For PbBrz, on the contrary, the equivalent mass transport and the volume change are both negative because of the low value of the lead transport number. The mass transport is rather small but the volume change is quite large. This explains the shape of the curve. Porosity of the Jfembrane and Heat Dissipation. The two different fritted disks used for hgSOa and the one used for PbBrp are all in the medium porosity range (pore diameter 10-13 p ) . Their electrical resistance, when filled with the molten salt, is only a few ohms. Their porosity defined as the ratio of the effective electrical cross section to the apparent cross section is roughly 10%. Table I gives the electric and hydraulic characteristics of the different cells for a temperature of the melt in the middle of the investigated range. The resistance values are derived from a measurement with an aqueous solution of potassium chloride a t room temperature. They agree with measurements of current and voltage across the cells in operation. The values of the hydraulic constant result from the observation of water flowing through the cell a t room temperature. They are in agreement with rough measurements of the flow rate a t high temperature. The characteristics of the cell with a capillary are given in the same table. The hydraulic resistance of the capillary is slightly lower than the one of the fritted disks, but its electrical resistance is much higher than the others.
7
Table I : Characteristics of the Transport Cells Cell Type Salt Temp., "C. Resist. cell, ohms Resist. disk, capillary l/a, hr. -0.050
L
I
I
0
10
20
I
30 Time, min.
I
I
40
50
60
Figure 3. Weight, change as a function of time. PbBrz a t 504", I = 50 ma. The dashed arrow represents theoretical asymptotic slope.
These curves can be easily fitted to eq. 15 or 16. The comparison of Fig. 2 and 3 emphasizes the differThe Journal of Physical Chemistry
1 Disk AgNOa 250 88 2 7 2 27
2 Disk AgN03 250 66 2 0 79
3 Capillary AgNOs 250 10,700 10,700 0 70
4 Disk PbBrz 450 47 3 1 16
No effect of the current intensity on the mass transport was observed. The investigated range was ~~~
(12) "Metals Handbook," 8th Ed., American Society for Metala, Novelty, Ohio, 1961, pp. 1063, 1181. (13) R. C. Spooner and F. E. Wetmore, Can. J . Chem., 29,777 (1961). (14) N. K. Boardman, F. H. Dorman, and E. Heymann, J . Phys. Chem., 53, 375 (1949).
TRANSPORT NUMBERS
IN PURE SILVER NITRATE AND
LEADBROMIDE
from 20 to 60 ma. With an apparent cross section of 0.785 and an estimated porosity of lo%, the actual current density ranged from 0.25 to 0.75 amp./ These values are much lower (10-100 times) than the current densities used by previous investigators. l5,I6 KO change of the electrical resistance was observed when passing the current or when changing the intensity. This indicates that the temperature increase of the fritted disks remains very small during electrolysis. A comparison with the measurements of Duke and Wolf,16who observed an elevation of temperature of 10' per watt, confirms our findings since the power dissipated within our fritted disks is only to watt. Any temperature change is definitely smaller than the precision of the temperature measurement, namely 1'. Because of the much higher resistance of the capillary, however, a larger amount of heat' was dissipatemd even for small currents and the temperature rose sig;nificantly. The temperature increase cannot be measured accurately; and although t,he mass transport wajs measured reliably by the balance, the resulting transport numbers are less reliable because of the uncertainty about the true temperature. The temperature increase withiin the capillary js estimated by measurement of the change of the cell resistance. The results agree with a rough heat transfer calculation. For currents from 10 t'o 20 ma., the temperature rises, respectively, 15-40', For current's of 25 ma. and up, the temperature within the capillary rises abobve 300°, and bubbles are formed by decomposj.tion or volatilization of Agn'Os. Precision and Accuracy of the Measurements. Figures 2 and 3 show that the initial slope was easily determined from the weight recording vs, time obtained b y a long experiment. .A more precise method, however, was to record the weight change for a short period (IO min.) and then to reverse the current and record weight change for a n equal period. The average slopes of the recordings so obtained were then corrected foE the small amount of hydraulic flow by means of the theoretica,l equation of the curve (eq. 15). Because of the rather small period of time At compared to the time constant l / a of the cell, this correction is smaller. than 1% on the mass transport and still smaller on the value of the transport number. The main obstacle to precision is not the hydraulic leakage through the plug, but results from the practical problem of suspending t'he cell under the balance. If it hangs as shown in Fig. 1, if the distance bet)ween the two suspension points is equal to the beam length, and if the two suspension wires are equal, then the cell
1761
rotates around an axis equidistant from the two points, when the beam rotates around the center knife edge. It is easily shown that the complex system consisting of the balance and the cell has the same sensitivity as the balance alone when the center of gravity of the cell coincides with the rotation axis. The sensitivity decreases when the center of gravity is lower, and the decrease is greater the greater. the weight of the cell. The preliminary nieasurements which have been reported previously,j were made with cell No. 1 suspended from ita top as shown in Fig. 1. The sensitivity of the balance was appreciably decreased by such a suspension, and the resulting transport number values have been accordingly discarded. The following measurenients were made with cells SO. 2, 3, and 4, the suspension point being inside a small tubing going through the electrode cups or attached to them. It is, however, very difficult to fill the cell in such a way that the position of the center of gravity is exactly on the SUBpension line. Practically, the sensitivity was limited to reading about 0.5 mg. Some other practical problems were encountered : either friction of the hanger assembly or excessive oscillation of the cell under the balance. They are responsible for a scattering of the data, amounting to 0.01-0.02 on the transport number. This is as good or better than any previously published result. It is, however, possible to improve further by changing the design of the cell and its suspension. The main source of inaccuracy lies in the geometry of the cell and its suspension, since the Pyrex cells cannot be made to meet rigid size specifications. The distance between the center of gravity of the two electrode compartments does not need to be equal to the length of the beam, but in any case, it must be measured. The accuracy of the weight change determination depends directly on the measurement of the distance between the center of the two cups. The error on the latter is estimated smaller than 1% and this leads to an error of 0.04 on T A ~and , 0.007 on TPb. The good reproducibility of the results obtained with lead bromide seems to indicate that the salt layer was an effective protection of the molten lead from oxidation by air. The effect of a slight solubility of lead in lead bromide is presently being studied by the pressuree.m.f. technique." Results obtained so far under an atmosphere of argon are in good agreement with the present ones. l8 ~
~
(15) A. Lbnden, J . Eleelrochem. Soc., 109, 260 (1962). (16) E. D. Wolf and F. R. Duke, ibid.,110, 311 (1963).
(17) P. Duby and H. H. Kellogg, ibid., 110, 349 (1963).
Volume 68, niumber 7
J u l y , 19C4
PAULDUBYAND HERBERT H. KELLOGG~
1762
Temperature Dependence. The transport numbers
of Ag and P b computed by means of eq. 19 and 22 from average values of M t are shown on Fig. 4 and 5 as functions of temperature. The vertical lines indicate the corresponding estimated error. The best lines obtained by a least-square method are drawn on the graph. They are represented by the following equations: for AgN03
Tho = 0.800
-
1
3.5 X 10p4(8 - 200)
0
call
0
L B O . (ref.?)
(std. dev. = 0.005)
NO. 4
for PbBrz
T P ~= , 0.370
-
1.66 X lO-l(O
I
- 370) (std. dev, = 0.006)
The results obtained with a capillary are indicated on the graph with the horizontal lines showing the uncertainty of the temperature. They prove that the balance technique makes possible measurements with cells of low hydraulic resistance. The agreement with the results obtained with porous plugs is quite significant and it shows that the value of the transport number does not depend on the presence of a porous membrane,
I
T
I
I
I
I
I
T
T
d
0.70
1I
200
I
I
240
220
I
260
I
I
I
-II
280
TemDerature, ‘ C .
Figure 4. Transport number in molten AgN03 as a function of temperature.
For each salt, the values of the transport number published by Laity and Duke7 are shown on the graph. The one for PbBr?, agrees very well, while the ones for
The Journal of Physical Chemistry
I 400
I
I
I
I
I 450
I
I
J
I 500
I
I
I
~
I
340
Temperature. ‘C. Figure 5 . Transport number in molten PbBrz as a function of temperature.
AgNOa are only within limits of error. The agreement between our results and those of Laity and Duke is quite significant since the two techniques are completely different. Because of a higher sensitivity, it is possible with the balance technique to use coarser porosity membranes. This shows that the porosity has no effect on the transport number within the limits of precision. Also, this minimizes any temperature riae during the experiment, and it is possible to find a significant temperature dependence, No assumption is made in this paper on the actual ionic carrier of electricity in either AgSOs or PbBrz. The experimental results are interpreted only as the net flow of silver or lead element due to the passage of electricity. This quantity is quite reproducible with different cells and different techniques. Further, it agrees with the interpretation of pressure-e.m.f. results obtained by the authors.17 Accordingly, this transport of matter is a characteristic property of the investigated salt. Acknowledgments. The authors wish to acknowledge the generous support of the Kational Science Foundation under contract NSF-G-20879. P. D. also wishes to thank the Belgian American Educational Foundation for financial support during the preliminary stage of this research. (18) P. Duby, L. J. Howell, and H. H. Kellogg, Abstract No. 162, Electrochemical Society Meeting, April, 1963, to be published.
