G. A. Lonergan and D. C. Pepper Trinity College University of Dublin, Ireland
Transport Numbers and Ionic Mobilities by the Moving Boundary Method
Few textbooks on physical chemistry treat the moving boundary method of determining transport numbers in the same detail that they devote to Hittorf's method. Texts on practical physical chemistry usually confine their account to the experiments of MacInnes, Longsworth ( I ) , and others (8, S), which yield transport numbers only, and are silent about the obvious pedagogic value of the method in illustrating the motion of ions during electrolysis and in yielding absolute mobilities. The reasons for this silence are plain. To derive ahsolute mobilities we need to know also the value of the potential gradient, and in the usnal experimental arrangement this is not usually uniform or measurable, nor are the changes in the over-all potential gradient as the boundary moves adequately discussed. The excellent experiment described by Longsworth (1) is an electrolysis of a solution of KC1 in a vertical graduated capillary pipet between an upper cathode (Ag/AgCI electrode) and a lower anode of cadmium
metal. A sharp, easily visible houndary is formed between the KC1 and a solution of CdCI2generated by the electrolysis a t the anode. If the electrolysis current (i) is kept constant the houndary travels upward a t constant rate, and the transport number of the potassium ion (T) is simply calculated from 3 dV T = e, (1) z dl where c = equivdents KCl/ml, 3 = the Farrtday, V = volume (ml) swept by the boundary in t seconds.
Longsworth's experiment stops a t this point, but the question arises: can the argument not also be applied to the Cd++ ions traveling behind the boundary and yield their transport number? The answer is yes; but only provided one realizes that the equivalent concentration of cadmium ions behind the boundary is less than that of the potassium ions in front of it, even though the cadmium ions are being generated in equivalent numbers a t the anode. The distribution of Cd++ from the anode to the boundary is in fact complex, consisting of a very concentrated region near the anode and a fairly uniform region extending from this to the boundary. This fact can be demonstrated, and the concentrat,ion ions and potential gradients in the various regions measured, by the modified form of experiment described in this paper. Electrical conductivities are also measured, so that from a simple experiment, lasting perhaps one hour, it is possible to derive transport numbers, absolute mobilities (by two arguments) and ionic conductances for cations both above and below the boundary (and also of course for the chloride anion). Two important relationships can then be verified, viz., and
T,/C, = Talc2 dE/dl = potential gradient p,
dE/dh
= p,
dE/&
p =
mobility
(Subscripts refer to the cations above and below the houndary.) The second relationship is a necessary condition for stability of the boundary. Apparatus
The modifications to Longsworth's experiment consist of:
u L
Flgure 1 . The apporotur. A, air inlet; B, cathode Ag/AgCl; C, Cathode compartment; 0, rubber sleeve; E, thermometer; F, 1-ml pipet graduated in 0.01-mli G, inner gloss envelope sontoining oil; H, glass water jocketi I , plalnum probe electrodes; I, cadmium cathode; K, piden wox r e d ; 1, brass connector; M, sonnectlonr to conductivity bridge; 0, voltmeter, 0 - 5 0 0 v; N, milliammeter, 0-5 ma; P, varioble 2 0 0 kilohm resistor; 0, power supply 0-500 v, 5 ma, dc.
82 / Journal of Chemical Education
(a) Alteration of the upper catbode compartment so that the po-
tential gradient is virtually confined to the cylindrical part of the electrolysis vessel, of known diameter and length. ( b ) Insertion of a pair of probe electrodes at a point in the tube so that the conductivity and hence the concentration of the solution may be determined before and after the boundary passes this point. (c) Connection of a voltmeter between one of the probe electrodes and the electrolysis cathode to permit measurement of the potential gradient. (d) Replacement of KC1 by HC1 containing an indicator (2).
A diagram of the apparatus is shown in Figure 1.
The electrolysis tube F consists of a 1-ml pipet graduated in 0.01-ml divisions, whose upper end projects into a wider tube carrying the Ag-AgCl cathode. Into the lower end fits a long cadmium rod (made by casting in a glass capillary of similar bore) adjustable so that its upper end (point J) may be a t any desired position, and secured by picien wax a t the bottom. The probe electrodes, a t point I, consist of fairly stout platinum wire (-0.3 mm diameter) sealed through the capillary waU and soldered to copper wires led out through the base of the apparatus. Some care is needed in sealing these electrodes to avoid excessive distortion of the capillary bore, and to ensure that they are a t the same level, otherwise the potential gradient along the tube causes a potential difference across the probes which upsets the conductivity measurement. The probe electrodes should be platinized before any experiments are made. The tube is mounted in a double-walled jacket, the inner space filled with paraffin oil to insulate the probe electrodes and the outer with circulating water for thennostating. The whole graduated length of the tube is within the jacket and in the present design the probe electrodes are fixed at approximately its midpoint, a t a distance 6 cm (0.36 ml) from the zero graduation mark and 18 cm from the upper end of the tube (virtual position of cathode). The current supply can consist of a simple full-wave rectifier unit giving about 500 v a t 5 ma. Its output is connected through a variable resistance (0-200 kilohm) and an accurate milliammeter (0-10 ma). This resistance must be adjusted a t frequent intervals during the experiment to keep the current constant as the total potential drop along the cell increases. Alternatively a constant current unit may be used (4, 6),which is more convenient, though more elaborate. A more accurate measurement of the current may of course be made by a potentiometer measurement of the voltage drop across a standard (-100 ohm) resistor connected in series with the electrolysis cell. However the simpler arrangement used here gives adequate precision for a class experiment. Finally a high impedance voltmeter (e.g., Heathkit Model V-7A valve voltmeter) is connected between the cathode and one of the probe electrodes. This serves to measure the potential gradient first in the HC1 solution and later, when the boundary passes, the increasing potential gradient as high-conductance HCl solution is replaced by low-conductance CdCI,. A low-impedance voltmeter is unsuitable since the resistance of the cell containing CdClz solution can reach 100 kilohm.
The conductivity bridge used in this design was a MuUard Model GM 4140/1 with Magic Eye detector, giving -2% precision over ranges from 0.1-10' ohm. An instrument with similar range but greater precision would be preferable. The cell is conveniently filled from a 5-ml medical syringe fitted with a polythene or nylon catheter tube long and narrow enough to reach to the bottom of the cell without disturbing the probe electrodes. Washing water is injected in the same way, and allowed to overflow into the cathode compartment from which it is removed by suction or siphoning. Experimental Procedure
The electrolysis tube is first calibrated, i.e., its volume to length ratio, dV/dl, and the distance of the probes from the anode and from the upper end, aremeasured. The apparatus is assembled, the cell filled to about 1 cm above the upper end of the tube with HC1 solution of known concentration containing about 0.05 g/l bromophenol blue indicator. Water is circulated through the outer jacket to bring the apparatus to constant temperature. The resistance across the probes is measured on the conductivity bridge. From literature values (6, 7) of the conductivity (K) of the HC1 solution the "cell constant" (B) of the probes is evaluated from K = B/R. The power supply is then connected, the current adjusted to the desired value, and a stopclock started. The boundary forms sharply within a few seconds and moves steadily upward. Its time of passing the graduation marks (e.g., every 0.02 ml) is noted. Readings of the conductivity a t the probes and of the voltage drop (E) are made at frequent intervals. Periodic adjustments of the variable resistor are made to keep the cnrrent constant. From these observations, graphs are drawn of V versus t, E versus t and R versus t, and if desired the corresponding plots of E and R against (1 - the anodeboundary distance). Provided the anode is set well below the probes (5-6 cm) a very simple behavior is observed. The volun~e (and length) traversed by the boundary remains accurately proportional to the time, except for a slight discontinuity as the boundary moves through the slightly distorted region of the tube a t the probes. The probe resistance (R) and the potential drop (E) remain constant as long as the boundary is below the probes. As it passes them, the resistance jumps sharply to a new constant value (R' = 10 R), and E starts a steady increase, linear with distance traveled beyond the probes. The concentration-distribution of the CdCI, solution behind the boundary can be studied by raising the anode closer (e.g., 1.5-2 cm) to the probes. The same initial behavior is observed, but the value of R' remains constant for only a short period after which it falls, a t first gradually, and then more sharply as the more concentrated CdC1, solution near the anode diffuses to the probes. Typical behavior is shown in the diagrams in Figure 2. Calculations
Figure 2. Variation of V, E ond R with time. (a1 Large di-tonse between probe electrodes and anode. Ib) Probe electrodes dose to anode.
From the linear V versus t plot the rate of movement Volume 42, Number 2, February 1965
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83
of the boundary with respect to volume dV/dt and length dl/&
.
=
dV/dt -dV/dl
aor;.,~a --. The transport number of the hydrogen ion Tn+ in the HC1 solution can then be calculated from equation (1) using dV/dt and the known value of c. For the cadmium ion T' is calculated in the same way hut the concentration c' must be derived from the measured conductivity K' = B/R' by interpolation in published data (8) relating conductivity to concentra..,+bL:",. The mobilities may be derived in two independent ways. They may be obtained from the transport numbers together with the conductivities, using the general relationships defining these quantities and the law of independent migration of ions, i.e., aro
m u . .
After the boundary has passed the probes, say by a length l', the measured potential drop has two components, that in the remaining length of HCl solution given by (L - 1') (dE/dl),, and a component from the length 1' of cadmium solution. In principle the latter may not have a uniform composition, and the potential gradient may also be nonuniform. However the constancy of conductivity, K' observed, and the linear increase of E with 1indicate that it is in fact effectively uniform. The value of the potential gradient in the Cd solution can then he deduced to be the sum of the eradient of the E venus 1 d o t d.u s the gradient in the lHcl Thus: E
Equivalent conductance A = Klc
= A+,+
A-
=
T
= A+/A
Fur
+ FL
dl
(Lo& inabilities) = Fp /A
(2) TAIF = T&FC They may also be obtained directly from the linear rate of movement of the ions (boundary) since
i'e.,
p =
-
-
- -
E
T Podtio. o l Prober
F ~ ~ U3.W
/
=
E
d
(") .d l . r--a
-
-
-
(.E dl .)H
since Eoconstant and since
1 Potllion el ~oundary
~erivotionof potential gradients.
T
TOP of pipet
acd
Journal o f Chemical Educofion
=
dl/dt d~ E (;it
+?)
Results and Conclusions
Table 1 shows some typical results. Better concorby more effective dance could no doubt he thermostating and more sensitive instrumentation, hut there are good reasons for restricting the experiments to the low concentrations and currents (where the results agreement with published values (9,10). in At the higher concentrations and currents a temporary disturbance of the boundary is noticed when the experiment is about half completed-the blue color of the HC1-bromophenol blue solution becomes bleached and an anomalous jump is observed in the potential drop E. This anomaly is probably to be associated with the diffusion of chlorine produced a t the anode under conditions of high current density and high accumulation of CdClpa t the anode (see later). Furthermore, a t the higher concentrations the formation of the various cadmium halide complexes, CdCl+, CdCla-, CdC14=, will become more important and this would introduce unnecessary complications into the con?putations. The value of the experiment lies, however, not so much in the precision of the results obtainable as in
Table 1
84
L(%)=
dE l,(&
and
by evaluating the potential gradient in the appropriate regions of the electrolysis tube. The method is best illustrated by a consideration of the plot of E against 1, which has the same form as that of E against t in Figure 2a since dl/& remains constant. The constant value Eo maintained as long as the boundary is below the probes, measures the potential drop in the uniform solution of HC1 between the probes and the outlet (neglecting the very small potential The potential in the wide gradient jn the HCl solution ( d E l d 0 ~is therefore simply Eo/L.
I-*
=
dE
p+ =
rate under unit potential gradient = -
.-
(L - 1 0
,., dE -=
(iorm conductmces)
(& + + 1,[(3ca (?)El] z , [ ( $ )G)=] ~~
=
correlating the various "laws" of electrolysis and in focusing attention on the conditions governing the migration of ions. For instance, it shows directly that if a stable boundary is to be formed between a solution of fast moving cations and one of more sluggish cations there must be a higher potential gradient in the latter. It shows, too, that since current must be "conserved" across the boundary, the concentration of the slower moving ions must be lower, i.e.,
The realization that, at the boundary, c' < c gives important insight into the conditions obtained in the cadmium solution. At the anode the cadmium ions are generated originally a t a rate equal to that of the departure of hydrogen ions. The concentration will then be originally the same (apart from a small effect of the different partial molar volumes of the cationsneglected here) but will build up to a still higher value because of the normal Hittorf effect of accumulation in the "anode compartment." How far does this anode compartment extend along the electrolysis tube? The experiment shows that it is confined to a narrow region (a few mm), and that most of the column of CdC1, solution in the cell has a uniform composition and
potential gradient. The experiment could be extended to study the concentration distribution in the anode compartment by raising the anode to a level close to the probes and changing the voltmeter connections to register the potential drop between anode and probe. The conditions in the anode region are however complex depending on the rate of generation and diffusion of Cd++ and hence on the temperature and the current, and on the anomalous production of Clz. Such a study is therefore more suited to a small research project than to a class experiment. Literature Cited (1) LONGSWORTH, L. G.,3. CHEM. EDUC.,11, 420 (1934). (2) DANIELS,F. ET AL., ''Experimental Physical Chemistry," 6th ed., MeGraw-Hill Book Ca., New York, 1962, pp. 165-9. (31 S. W.. J. CHEM.EDUC.. 38.516 (1961). ~,TOBEY. . (4) BENDER, P. AND LEWIS,D. R., J. CHEM.EDUC.24,454. (5) TOBEY, S. W., J. CHEM.EDUC.,38, 517 (1961). (6) SHEDLOVSICY, T. 411 (1932).
. .
.
2811 (1941). (8) . . NOYES,A. A. AND FALK,K. G., J. Am. Chem. Soc.,. 34,454 . (mi). (9) LUCA~SE, W. W., J. Am. C h a . Sot., 51, 2605 (1929). (10) MACINNES, D. A. AND LONGSWORTH, L. G., Chem. Reu., 11, 171 (1932).
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