J. Braun steinand G. D. Robbins

Reactor Chemistry DivisionOak Ridge National Laboratory

Oak Ridge, Tennessee 3/830

Electrolytic Conductance Measurements

and Capacitive Balance

Reliable and meaningful measurementsof electrolytic conductance require attention to designof cells, electrodes, and measuring circuitry.

Extraction of an Ohmic resistance from ac bridgemeasurements is a non-trivial procedure that requires arealistic model for the equivalent electrical circuit of a

conductance cell. This entails an understanding of theelectrochemical processes occurring when current ispassed through a system consisting of electrolyte, elec-trodes, and bridge (1). However, electrode processesare seldom dealt with in the undergraduate chemistrycurriculum—almost never in connection with conduc-tance measurements. Furthermore, the principles ofconductance measurements are unevenly covered inmany standard physical chemistry textbooks, referencebooks, and laboratory manuals. For example, an as-

pect often dealt with either incompletely or erroneouslyis the physicochemical basis for capacitive balance andfrequency dependence in conductance measurementswith alternating current. Also, since the elements ofbridge circuits and balance conditions tend to be com-

partmentalized into the physics curriculum, a unifiedtreatment is rare.

Detailed consideration of all of electrochemistry un-

doubtedly is impossible in an undergraduate course, butthere is no reason not to introduce some of the basicconcepts in a qualitative manner; it is poor pedagogy toachieve simplicity by introducing erroneous concepts(it is not necessary, after all, to derive statistical me-chanical equations for virial coefficients to discuss inter-molecular forces in gases).

In this paper we review the principles of electrolyticconductance measurements, citing applications tomolten salts and to solid electrolytes as well as toaqueous electrolytes. We discuss capacitive balanceand the frequency dependence of ac measurements in thelight of processes occurring in the bulk of the electrolyteand at the electrode surfaces in a conductance cell, aswell as relevant properties of the Wheatstone bridge.

Research sponsored by the U.S. Atomic Energy Commissionunder contract with the Union Carbide Corporation.

Presented at 21st Southeastern Regional ACS Meeting, Rich-mond, Va., Nov. 5-8, 1969, Division of Chemical Education,Paper No. 110.

1 The term ionic constituent refers to the ion-forming portion ofan electrolyte, without specification of the extent of dissociation,association, solvation, etc. See, e.g., reference (4), p. 60. Thusthe ionic constituents in an aqueous solution of CdR are Cd! +

and I-, regardless of the fact that, the actual species in solutionmay include hydrated cadmium ions, anionic complexes such as

Cdl4!_, etc. Measurements of macroscopic properties yield in-formation only on the components (Cdl-j and water) or the ionicconstituents. Unusual values of these properties may suggestthe microscopic nature of the species (e.g., ions). More detailedinformation on the nature of the species requires measurementssensitive to the species properties and distribution (e.g., spectra).

Alternative methods of conductance measurement are

pointed out, as are some aspects pertinent to the mea-surement of dielectric constant. The extent of cov-

erage can easily be modified to suit the level of thecourse and the time available.

Phenomenology: Ohm's Law Applied to Electrolytes

Electrolytes conduct current by transport of ions, incontrast to electronic conductors or n-type semiconduc-tors in which electrons are the charge carriers, orp-typesemiconductors in which positive holes predominantlycarry the current; both electronic and ionic conductionoccur simultaneously in some substances {2). Sincethe mass of an ion is of the order of 104 times the elec-tronic mass, quantum mechanical effects such as ex-

change, delocalization, and tunneling are much smallerfor ions than for electrons. Electrolytic conductionoften is accompanied by transfer of mass as well as

charge across the electrode-electrolyte interface withthe formation or discharge of ions at (usually metallic)electrodes. Massive ions, rather than delocalized elec-trons, drift through the bulk medium under the in-fluence of an electric field, generally at velocities fiveorders of magnitude lower than electron velocities.These distinctions dictate differences in the methods ofmeasurement. Thus the low inertia of electrons in ametal permits measurement of resistance with eitherdirect or alternating current and ready demonstrationof Ohm’s law, V — IR. Extraction of an Ohmic resis-tance (reciprocal of conductance) for an electrolyte ismore involved, since the variety of chemical as well as

physical phenomena occurring at the electrodes must beseparated from the voltage drop associated with themigration of ions through the bulk electrolyte (3). Al-though no physical principle requires the feasibility ofextracting an Ohmic resistance, and, hence, conduc-tivity of an electrolyte, it is experimentally possible todo so over an appreciable frequency range for a widevariety of electrolytes (4)-

The conductivity (or specific conductance), k, of amaterial may be thought of as the conductance (re-ciprocal of resistance) of a cube of 1 cm edge, assumingthe current to be perpendicular to opposite faces of thecube. Since it. depends partly on the number ofcharge carriers, a more useful indication of the natureof interactions of the constituents with the medium isthe conductance per unit concentration of chargecarrying constituent.,* 1 *or equivalent conductance, A =

k/c; c is the concentration in equivalents per cm3(normality times 1000) (4)  Combined with densities,molecular weights, and transference numbers (fractionsof the current carried by the various constituents), theconductivity thus yields the relative velocities of theionic constituents under the influence of an electric field.

52 / Journal of Chemical Education

Dow

nloa

ded

via

UN

IVE

RSI

DA

DE

DO

PO

RT

O 0

1100

on

May

1, 2

020

at 1

7:02

:39

(UT

C).

See

http

s://p

ubs.

acs.

org/

shar

ingg

uide

lines

for

opt

ions

on

how

to le

gitim

atel

y sh

are

publ

ishe

d ar

ticle

s.

The mobilities (velocity per unit electric field, cm2 sec-1volt-1) vary with the nature of the constituents, theirconcentrations in mixtures, temperature, medium, andother factors. In dilute aqueous solutions of dissociatedelectrolytes, ionic mobilities vary little with concenlra-tion, and the equivalent conductance at vanishingelectrolyte concentration may be expressed as the sum

of independent equivalent conductances of the con-

stituent. ions. The limiting equivalent, ionic conduc-tance may be considered as the conductance of ions farremoved from other ions so that solute-solute interac-tions are negligible. At finite concentrations, interionicforces generally lower the mobilities, and the concentra-tion dependence of equivalent conductance has been a

classical probe of the strength of ionic interactions (1).In dissociated solvents such as molten salts, the concept,of independent ionic mobilities may be significant fordilute solute ions not common to the solvent, but notfor solvent ions because of their mutual proximity.The equivalent conductance and (in mixtures) mobili-ties, and their temperature dependence, are importantin testing theories of transport in liquids (5).

Some of the more important phenomena associatedwith the application of a voltage between electrodesimmersed in a liquid electrolyte are indicated below,with reference to Figure 1, for an idealized system inwhich it is presumed possible to isolate the variousphenomena.

DOUBLE LAYER /CAPACITANCE ^

ELECTROLYTERESISTANCE

V- DOUBLE LAYERCAPACITANCE

Figure 1. Electrolytic conductance cell—an oversimplified schematicrepresentation of the double-layer at the electrodes, Faradaic processes,and migration of ions through bulk electrolyte.

Double-Layer CapacitanceA positively charged electrode will preferentially at-

tract a layer of negative ions (and a negative electrode a

layer of positive ions).2 With increasing distance froma charged electrode, its orienting effect on ions in so-lution becomes rapidly more diffuse owing to thermaldisordering (Brownian motion) (6). The double-layer,consisting of charges on the electrode and oppositelycharged ions adjacent to it in solution (or separatedfrom it by a layer of solvent molecules), constitutes anelectrical capacitor, capable of storing charge. If a

sufficiently low steady voltage is applied to the elec-trodes, virtually no current will flow beyond that re-

quired initially to charge the double-layer. The struc-ture of the double-layer is considerably more compli-cated than the idealized version in Figure 1, and isgreatly influenced also by the interactions of polar sol-vent molecules with the ions and with the electrode sur-faces.

Electrolysis

As the voltage applied to the electrodes is increased,the charge accumulated in the double-layer increasesuntil a decomposition voltage analogous to the break-down voltage of a capacitor is exceeded;3 current thenflows across the electrode-solution interface, accom-

panied by oxidation at the positive electrode, reductionat the negative. This Faradaic (electrolytic) processpartially short circuits the double-layer, behaving elec-trically somewhat like a resistor (although a voltage-dependent one) shunting a capacitor. Slow electrontransfer processes at the electrodes may contribute a

polarization resistance (1,7).Ohmic Resistance

The current is carried though the bulk electrolyte bycations moving toward the cathode, anions toward theanode. Velocity-dependent retarding forces opposethe accelerating force of the electric field, and result in a

constant drift, velocity at a given field. Current flow isaccompanied by dissipation of energy as heat, since theions must overcome frictional forces in their motionthrough the medium, whereas the charging of thedouble-layer corresponds to storage rather than dis-sipation of electrical energy.

Concentration Polarization

On further increase of voltage, Faradaic removal ofelectroactive ions (i.e., ions reducible or oxidizablein therange of voltages applied to the electrodes) near theelectrode may proceed faster than diffusion from thebulk electrolyte can replenish their supply. This can

establish a concentration gradient between bulk elec-trolyte and the electrode surface, such that the currentmay reach a diffusion-limited value (8). The phenom-enon of concentration polarization is exploited for ana-

lytical purposes in polarography (9).Calculation of the Ohmic resistance of the electrolyte

requires not only the (easily measured) current but alsothe voltage accelerating the ions—not the voltage ap-plied at the electrodes. The latter voltage must alsosupply the driving force for the electrical work neededfor the Faradaic processes, double-layer charging andconcentration polarization. The problem of resolvingelectrolytic conductance (reciprocal of resistance) isthat of eliminating the effects of the processes asso-ciated with the electrodes, and is generally accomplishedby measurements with alternating current.

Alternating Current Effects

Reversal of the sign of the applied voltage reversesthe direction of motion of the ions. With an alter-

2 Even without an external applied voltage, the surface of ametal immersed in an electrolyte will tend to become charged,owing to factors such as preferential adsorption of ions of one sign.Application of a voltage will modify the ionic distribution at thesurface and the structure of the double-layer ((?).

3 In a real system it may not always be possible to separateexactly the double-layer and Faradaic processes (3). For exam-ple, the presence of traces of reducible or oxidizable species maypermit current to flow at low voltages, before the double-layer iscompletely charged. The term decomposition voltage is usedhere in the phenomenological sense to indicate an experimentallyobserved voltage near which a reasonably sharp increase of cur-rent occurs whether the system be liquid, solid, or gaseous, singlecomponent or multicomponent. (See reference (47)).

Volume 48, Number 1, January 1971 j 53

nating (ac) rather than a steady (dc) voltage applied tothe electrodes, the above processes reverse themselveswith the period of the alternating voltage. But,having differing specific rates (or relaxation times),their relative contributions to energy dissipation varywith frequency. As the frequency is increased, con-

centration polarization can be greatly reduced or elim-inated. The Faradaic effect can be eliminated in prin-ciple either by sufficiently reducing the applied voltageor by employing reversible (nonpolarizable) electrodes.

High-Frequency Effects

Although increased frequency of the applied voltage(to several thousand hertz) reduces many of the effectshindering accurate conductance measurements, in-definite increase of frequency leads to other problems.For example, the effects of stray capacitance in the mea-

suring circuits increase. Also, the rearrangement ofions and solvent molecules in the vicinity of a movingion may not be fast enough to keep up with the periodicreversal of the field. A moving ion will then essentially“see” a different environment than at a lower fre-quency. At frequencies comparable with the time re-

quired for an ion’s oppositely charged ionic atmosphereto readjust to the reversal of motion of the ion, the con-

ductivity increases because of the reduction of the dis-symmetry of the ionic atmosphere. This is the Debye-Falkenhagen effect (10). The critical frequency is~107 Hz, in a 0.001 M solution, and increases with in-creasing concentration. With still further increase offrequency, translational motion of the ions graduallyceases to respond to the rapid reversal of the electricfield. Since dipole reorientation can still occur, di-electric properties of the material are measured; at stillhigher frequencies, e.g., optical, only electrons wouldrespond to the oscillating electric field.

In spite of the necessity for sorting out many super-imposed phenomena, conductance measurements havebeen developed to a very high art, primarily during thedecades between 1920 and 1950, with precision of theorder of 0.01% attainable (1). The techniques de-veloped for dilute electrolytes were based to a consid-erable extent on the thorough investigations of Grin-ned Jones and co-workers, who discussed Wheatstonebridge design (11), conductance cells and electricalleads (12), the electrode polarization resistance and itsfrequency dependence (13), the effect of high electricfields in increasing the conductance (the Wien effect)(14) , and errors arising from the use of water rather thanoil in constant temperature baths (11). It has becomecustomary to adopt the techniques developed specificallyfor dilute aqueous electrolytes by Jones and co-workerseven for studies of other media, e.g., molten salts, wheremodifications required for measurements of high ac-

curacy follow from the principles enunciated by Jones(15) .'

Alternating Current Conductance Measurements:Equivalent Circuits

If should be recognized that, just as there are no per-fect gases or ideal solutions, there are no perfect, (i.c.,frequency-independent) electrical components—pureresistances, pure capacitances, or pure inductances (16).Even if there were, it would not be possible to representmost electrochemical processes exactly in terms of such

linear electrical components. Nevertheless, even an

approximate representation facilitates analysis of elec-trochemical measurements in terms of ionic models (1).Electrical components representing electrochemicalprocesses and stray current paths in a conductance cellare indicated schematically in the equivalent circuit inFigure 2 and are discussed below. Some of the high-

PARALLELCAPACITANCE

Figure 2. Electrical equivalent circuit of a conductance cell showing re-

sistive and capacitive elements corresponding to circuit components andelectrochemical processes.

frequency paths are primarily nuisances, to be elim-inated when possible. The double-layer capacitanceand Faradaic impedance, on the other hand, althoughdifficult to evaluate, contain valuable information on

surface properties and on rates of electrode processes.

Conductance Measurement

The evaluation of electrolytic conductivity generallyis a comparative procedure. A conductance cell iscalibrated by determining its cell constant, l/A, usinga solution of known conductivity. l/A is the ratio ofeffective length and cross sectional area of the conductingpath (rather than the geometric value). The con-

ductivity, or specific conductance, is then calculated as k= (l/A) (1 /R) where R is the measured resistance. Theconductivity standards are aqueous KC1 solutions mea-sured by Jones and Bradshaw (17) and traceable back tothe measured conductance of a mercury thread in a cellfor which the geometric length and cross-sectional area

give the cell constant. The Kohlrausch method em-

ploys a Wheatstone bridge, alternating current, andplatinum electrodes which have been coated electro-lytically with a deposit, of platinum black to increase thesurface area and reduce the polarization resistance (18).(In very dilute solutions, platinization may introduceerrors through adsorption of solute.) The bridgecircuit, generally must contain not only resistance, butalso capacitance (or inductance), to balance the capaci-tive effects in the conductance cell.

Capacitive Balance

A cursory examination of several widely used andhighly regarded textbooks and laboratory manuals ofphysical chemistry reveals the following incomplete or

inaccurate expositions of the origin of capacitive effects

54 / Journal of Chemical Education

in the cell and the need for capacitance in the bridgeX. No discussion.2. The capacitance is required simply to sharpen detection of the

minimum in the imbalance potential but has no effect on themeasured resistance.

3. The balancing capacitance is required to compensate thecapacitance of a dielectric medium between the electrodes inparallel with the resistance of the electrolyte in the con-ductance cell.

4. The capacitance is required to balance stray capacitance in thebridge circuit components, connections, water bath, etc.

The third and fourth factors certainly introduce ca-

pacitance in the circuit, but the major capacitive contri-butions and source of frequency dependence under theusual conditions in conductance measurements arisefrom the electric double-layer at the electrode-electro-lyte interface at applied voltages below the decomposi-tion voltage,3 and from the frequency-dependent, resis-tance (impedance) associated with the Faradaic pro-cesses at voltages above the decomposition voltage.

Parallel Capacitance

The parallel capacitance of the medium between theelectrodes is given by

10«Z>

4 *c*(l/A)where D is the dielectric constant of the medium, c isthe velocity of light (3 X 10'° cm/sec), and l/A is thecell constant (19). For a dilute aqueous electrolyte thedielectric constant may be taken as that of water, ~78.A cell constant of order 1 would thus yield a parallelcapacitance of only C' = 7 picofarads. The corre-

sponding magnitude of impedance (ac analog of resis-tance, i.e., ratio of voltage to current), \/2ir$C’, is of theorder of 2 X 107 ohms at 1000 Hz, in parallel with theelectrolyte resistance, and therefore has little effect on

measurements of electrolyte resistances below 104 ohms.

Series Capacitance: The Ideal Polarized Electrode

It has been pointed out above that the accumulationand dissipation of an excess of positive or negative ionsat. the electrode surfaces constitutes the charging anddischarging of a condenser. This model of the elec-trolyte-electrode interface has been supported by exten-sive investigation, e.g., by Grahame (20) with mercuryelectrodes and highly purified electrolyte solutions.He showed that the current at dc voltages below thedecomposition voltage is negligibly small (as expectedfor a “true” capacitor), and that with alternating cur-

rent, the double-layer capacitance is essentially inde-pendent of frequency, behaving as a capacitance inseries, not in parallel, with the solution resistance. Anelectrode such that no charge crosses the phaseboundary when the potential is altered slightly is termedan ideal polarized electrode (21). It. requires the ab-sence of species capable of being oxidized or reduced atthe voltage applied to the electrodes. Absence of fre-quency dependence of the capacitance indicates that theionic distribution in the double-layer readjusts rapidlyto changes of potential (at frequencies up to those cor-

responding to the Debye-Falkenhagen effect (22)).Although electrodes in conductance cells generally do

not satisfy all the criteria of the ideal polarized elec-trode, the model of a resistance in series with a capaci-tance appeal's to be a reasonable approximation. Thus

extensive measurements with conductance cells havedemonstrated that the series capacitance varies littlewith electrode separation but increases with increasingsurface area (obtained by increased platinization of theelectrodes) (13). An important physical implication ofthe scries capacitance is that it, derives from interactionsat the electrode surfaces rather than from bulk effects,such as relaxation processes in the electrolyte, whichwould be manifested as a parallel capacitance (23).

The magnitude of the double-layer capacitance formany electrolyte-electrode systems including aqueoussolutions and molten salts is of the order of 10-100microfarads per cm2 of electrode surface (1, 13, 15).This leads to a magnitude of impedance at 1000 Hz ofthe order of

dfC=

6283C“ 16 “ 16 0hmS

at a 1 cm2 electrode. The impedance in scries with theelectrolyte decreases with increasing capacitance or in-creasing frequency, and a high double-layer capacitancegreatly reduces errors in ac conductance measurements.

Hence for resistances of less than 104 ohms the modelof a conductance cell as consisting of the electrolyteresistance in series with the double-layer capacitance,though not exact, is a more reasonable physical ap-proximation than that of a parallel resistance-capaci-tance combination.

Faradaic Impedance

Conductance measurements are sometimes made withac voltages as high as several volts, i.e., above the de-composition potential, and with elect rodes platinized toreduce polarization (13). The use of such high volt-ages probably stems from the formerly low sensitivityof detectors, which is no longer a problem. At ap-plied voltages above the decomposition voltage,3 thedouble-layer capacity, Cs, in scries with the electrolyteresistance, is shunted by an impedance, Zr, associatedwith the transfer of charge through the electrolyte-electrode interface, as indicated in Figure 2. There isno simple representation of this Faradaic impedance interms of frequency-independent resistances and ca-

pacitances. In conductance measurements, this im-pedance leads to high apparent values of the measuredresistance, but decreases with increasing frequency.It is considered to have a resistive component at-tributed to a finite rate of electron transfer at the elec-trode surface (7). Warburg attributed the observedfrequency dependence in conductance measurements toconcentration polarization—the depletion of electro-active species at the electrode surface so that the cur-

rent is limited by diffusion of such species to the elec-trode—resulting in an impedance proportional to thereciprocal of the square root, of frequency, Ali = k(8, 13). Linear extrapolation of measured resistancesto infinite frequency by plotting against elim-inates this error. Grahame also found this form of fre-quency dependence for impedances associated with anelectrochemical reaction, but under the restrictive as-

sumption of vanishing sine wave amplitude, whichlimits its validity for conductance measurements (20).Many cases of frequency dependence other thanhave been reported, however, and extrapolation to in-finite frequency is best carried out with functions

Volume 48, Number 1, January 1971 / 55

found empirically to be nearly linear unless a frequencyindependent range can be found {2/f).

The part of the polarization resistance resulting fromelectron transfer processes of finite rate is not elim-inated by varying the frequency, and the extrapolatedresistance then includes both the electrolyte resistanceand a residual polarization resistance related to thespecific rate of the electrochemical process (7, IS).Detection and elimination of this source of error, whichgenerally is only serious with very low electrolyte re-

sistances, i.e., several ohms or less, would requiremeasurements in cells with differing conductive pathlengths, and hence electrolyte resistances, but withidentical electrodes.

Parasitic Effects

High-frequency effects resulting from stray capaci-tances in leads, connections, and improper ground con-

nections can be minimized by avoiding close placementof electrode leads and by the use of the Wagner ground{1, 16). The “Parker effect” (Fig. 2) in high-resistancesolutions may result when there is a resistive paththrough the electrolyte in series with, and capacitivclycoupled to, parallel (insulated) electrode leads passingthrough the solution; it can be eliminated with appro-priate cell design {12,25).

Mysels et al. have analyzed an effect important withvery high resistances, e.g., 10s ohms, resulting fromcapacitive coupling of the electrolyte in the conductancecell to ground through the high resistance in the cell andthermostat {26). This is difficult to avoid in measuringthe conductance of solvent (necessary for obtainingelectrolyte conductances in very dilute solutions).The effect leads to an apparent increase of resistancewith increasing frequency, in contrast to the effects ofpolarization and capacitive shunts. The authors showthat the anomalous excess resistance varies as the squareof the frequency at low frequencies, and extrapolationto zero frequency is required to extract the Ohmic resis-tance.

Bridge Balance for Series or Parallel RC Circuits

At any one ac frequency it is possible to represent a

resistance-capacitance combination in a “black box”equally well as a series or parallel combination, since ac

measurements with sinusoidal applied voltages pro-vide only an absolute impedance and a phase shift4 {16,27). It is thus necessary to vary the frequency to dis-tinguish between the two possibilities (or to use wave-forms other than sinusoidal). Although many kinds ofdetectors may be used as null indicators for the Wheat-stone bridge, an oscilloscope is recommended since itprovides a visual display of both the phase shift and im-balance potential.

Balance of Pure Resistances

Figure 3 shows the circuit of a Wheatstone bridge andthe conditions of bridge balance for the idealized casewhere all components are pure resistances. The un-

known resistance is indicated by Rs, and Rb is a vari-able resistance for bringing the bridge to balance by ad-justing the current, 7, in the lower arm until the differ-ence of potential between points C and D is zero. Ifthe vertical deflection plates of an oscilloscope arc used

as the voltage null detector, and asine-wave generator is connected tothe horizontal plates, bridge imbal-ance is indicated by a sloping oscil-loscope trace whose vertical projec-tion is proportional to the voltagedifference between C and D, andwhose horizontal projection is pro-portional to the peak-to-peak volt-age applied between A and B. Ahorizontal line   indicates balance{Vc — VD = 0, Rs = RB), and thiscondition is independent of the fre-quency of the applied voltage. Ifthe unknown resistance is a conduc-

Figure 3. Balanceconditions for a

Wheatstone bridgewith pure (frequencyindependent) resis-tances and either ac

or dc voltage source.

tance cell, the oscilloscope tracegenerally is an ellipse rather than a

straight line, because the cell is nota pure resistance. It is then im-possible to bring the imbalance po-tential to zero at any frequency

simply by adjusting RH because the current is notin phase with the applied voltage. Introductionof a variable capacitance (or inductance) into the bridgepermits compensation of the phase shift (time lag) andbalance of the bridge.

4 The phase shift or time lag between current and voltage,caused by the capacitance follows from consideration of the ap-plication of a sinusoidal voltage, Vo sin 2x ft = Vo sin oil, across a

capacitor, C. The charge stored by the capacitor at any instantis q = C'V and the current is i = dq/At — CAV/At = otC Vocos oil. Hence the current leads the voltage by 90°, reaching itspeak at t = 0, 2ir/oi, . .. while the voltage peak occurs at t =

ir/oi, 5w/2oi . . . For a series resistance-capacitance circuit (likethat between points A and D in Fig. 4) the voltage drop vc acrossthe capacitor is the difference between the voltage v applied tothe circuit and the ill drop through the resistor

Vc = V — iR

The current is given by the rate of change of charge of thecapacitor

 _

Aq_ n d(» - ill)

__ n dc_ pp di

1 “

d(- C '

dt~

At1

d<

For sinusoidally varying voltage and current whose peak valuesare shifted in phase (displaced in time), the solutions are combina-tions of trigonometric functions. The algebra is much simpler ifcomplex functions replace the trignometric functions, recallingthat e>x = cos x + j sin x, where j — \/ — I. Substitutingti = Vf/at and i = 7eJ“! in the above differential equation, notingthat V and/ are independent of time but may be complex (e.g.,/ = ]/|V»r', V — |F| ei°,T'',oi (r' — r") being the phase shift)yields

juiCRIet+ IeJ“‘ — jo}CVe‘<*‘

(joiCR + 1)/ = joiCV

V = 11 4- joiCR~joiC~ “'(K + jz)

= 1Z

|F| = |/| |Z| = |/| VR> + ^= |/| R Vi + \/(wCRY

Z is the total impedance, \Z\ its absolute value. 1 /juC is thecapacitive reactance. See, e.g., reference (28).

56 / Journal of Chemical Education

Balance of Series RC Circuit

For cases where the conductance cell may be ap-proximated as a series combination of a pure resistanceand pure capacitance, the balance conditions for twobridge circuits are shown in Figure 4. (The resis-

Figure 4. Balance conditions for a series RC circuit in bridges with seriesor parallel RC balancing arms.

tances in the upper arms of the bridge have been setequal {Ri) to simplify the equations, but the equationscan be derived similarly for unequal resistances.) Theconditions arc exact if the electrical components arethose represented in Figure 4, i.e., if the cell is a trueRC circuit. If it is not, the equivalent circuit must beknown (or assumed) if a physically significant resistanceis to be calculated from the resistance and capacitancereadings of the balanced bridge (1).

The condition for zero potential difference betweenthe vertical plates of the oscilloscope is, as in Figure 3,that the voltage drop between points A and C be equalto that between A and D; similarly, equality of thevoltage drop from B to C and that from B to D is re-

quired. Now however, the phase shift, or lag betweenvoltage and current peaks, resulting from the capaci-tance in the circuit, requires expression of the voltagedrop in terms of the impedance, comprising a reactivecomponent (here a capacitance) in addition to a resistivecomponent.* ***5 The algebra describing the componentsof the impedance is handled most simply with complexnotation {27, 28). Thus the impedance of a seriesresistor and capacitor is the complex sum of imped-ances of the resistance, R, and of the capacitance,l/jo>C, where/ = V — 1, and w = 2irf.

z,„i„ - * + (sc) - s + (f) - R -

(/)

Parallel impedances add like parallel resistances, butthe complex notation must be retained, resulting in theequations for a parallel resistance-capacitance com-

bination6

_1_ _i , 1_ J_ ,   r

Zp»„n«i“

R + l/j»C R

(1 + juCR)R

„ R R{ 1 - jo,CR)/Wuel -

J + .aCR j + ("CRYR . wCR*

~

l + (o,CRY J 1 + (uCRY

Equating impedances at balance in Figure 4, it is seenthat the balance condition for a scries combination byanother series combination (Fig. 4a) at any frequency isequality of both resistances and of both capacitances,since equality of two complex numbers requires equalityof the real parts and of the imaginary parts indepen-dently. A similar result would obtain for balance of a

parallel combination by another parallel combination.(It should be noted that complex numbers are notvectors, as is sometimes erroneously stated. Al-though they add vectorially, their products are not dotor cross products; i.e.,/2 = — 1, not +1 orO.)

In balancing a series RC combination with a parallelcombination (Fig. 4b), however, the resistances in thetwo lower arms of the bridge (and the capacitances)will not be equal, and the balancing resistance (andcapacity) will vary with frequency, i.e., show frequencydispersion. The balance conditions are those of theWien bridge (a circuit formerly used to measure fre-quency with calibrated resistors and capacitors in thebridge circuit) {16). Thus the resistance reading on a

parallel RC component bridge, when balancing a seriesRC circuit, will vary with frequency, and the true re-sistance must be calculated with the equations in Figure4b. Often, in conductance measurements, this cal-culation is omitted. Rather it is assumed that RB' —

Rs, an approximation valid at small values of oiCb'Rb'(or, correspondingly, at high values of oiRsCs). Thelatter condition generally holds quite well for mod-erately dilute aqueous electrolytes under the usual con-

ditions of measurement of resistances of the order of103 ohms and greater. It is more difficult to attain formolten salts, owing to their much higher conductivities,unless capillary cells are used. Thus (Fig. 4)

Rb' = R> [l +

It may be seen that the factor in brackets differs littlefrom unity for, say Rs = 100012,/ = 1000 Hz, and Cs =

100 (i.e., 1 + 3 X 10-6), and the computation ofRs from RB'

p -

Rb'1 + {uCb’Rb’Y

is unnecessary. However for measurements in moltensalts, with high conductivities, or in cells with lowelectrode capacitance, the term

l(wR,C,Y (wRb'Cb'Y

6 The balance conditions for Figures 4a and 46 are derived as

follows

Similarly

Vac = Vad; iRi = IZBVbc — VBd; iRi — iZ,

JZ, = I Zb; z, = Zb

Zb' = Z,Also see footnote 4.

6 These electrical equations are of analogous form to those formany mechanical, acoustic, or chemical systems in which theresponse to a sinusoidally varying perturbation is shifted inphase as a result of relaxation processes that can store (capaci-tive) or dissipate (resistive) energy (48, 49).

Volume 48, Number 1, January 1971 / 57

may become quite large relative to unity, requiring thecalculation of Rs from RBr, CB> and u> (7, 15, 29). Fora 10-ohm resistance, the error of assuming R, = RB>would be 3%. The error (Rbi — Rs) could exceed thevalue of Rs at still lower resistances, that might, be ob-served in, e.g., crucible cells with small electrode sep-arations. It may be convenient in some cases to elim-inate the calculation by employing a bridge with a seriesRC balancing arm as in Figure 4a. The parallel com-

bination is more frequently used because, as may beseen from the equations of Figure 4, the parallel ca-

pacitance required to compensate a series capacitance of100 /rF at 1000 Hz with a resistance /?, = lOOOfi isonly CB' 3 X 10 ~4 /jF = 300 pF, and small capaci-tances can be obtained with higher accuracy and lessfrequency dependence than large ones.

With a conductance cell that Is not. a pure series RCcombination, some frequency dependence of measuredor calculated Rs still may be observed, and extrapola-tion to infinite frequency as a function of w or a more

appropriate empirical function of frequency still maybe required (15, 24)   This extrapolation should not,however be based on an incorrect equivalent circuit ofthe cell. The fact that one does not know the exactequivalent circuit of the cell does not eliminate thenecessity of a choice if a resistance is to be calculatedfrom the measurable impedance and phase angle, andthe series RC combination is the best approximation inmany cases.

Application to Special SystemsMolten Salts

There has been increasing interest in molten salts, asmodels of ionic liquids including concentrated aqueouselectrolytes (30), and for practical application inmolten salt nuclear reactors (31), fuel cells (82), andmetallurgical processes (33). Many molten salt elec-trochemical experiments are suitable for undergraduatelaboratories (34,85).

In molten salt conductance measurements the ap-proximation Rb< = Rs in Figure 5b frequently is notvalid because of the low resistance, especially for cor-rosive melts which cannot be contained in capillarycells, but must be measured in crucible-type cells withlow cell constants (29). Failure to make the appro-priate calculation of Rs from RB> and CB' (or an erro-

neous correction resulting from flaws in the bridge) maylead to spurious frequency dispersion. Measurementwith a series component bridge (Fig. 4a) has been em-

ployed to eliminate this error, and has yielded nearlyfrequency independent results at frequencies up to 20khz, confirming the usefulness of the series RC repre-sentation of the conductance cell for molten salts (15,36). Any bridge should of course be checked by mea-suring standard electrical components in the range ofanticipated resistance and capacitance values. Itwould appear desirable to reduce the amplitude of thesinusoidal voltage below the decomposition potential,3if possible, to approach the frequency independent, con-ditions at an ideal polarized electrode (15) and avoidelectrolysis (37). Some very careful studies of moltenzinc chloride by Bloom and Weeks, however, showed no

frequency dispersion greater than 0.1-0.2% at fre-quencies between 0.5 and 100 khz and at voltages be-tween 0.1 and 1.0 V (38). The polarization resistance

can be further reduced by platinizing the electrodes(13), thereby increasing the surface area and reducingthe current density.

Special Methods

Difficulties associated with frequency dispersionhave encouraged attempts at other moans of measuringconductance. King and Duke (39) measured dc con-ductance in molten alkali nitrates using silver electrodesin thin-walled glass compartments to sense the iR dropthrough the solution. Their cell contained four elec-trodes, two to carry current and two to measure the po-tential drop without carrying current. This method,however, requires reversible electrodes, frequently notavailable for systems of interest. A four-elect rode ac

potentiometric method also has been reported (40).One way to eliminate electrode polarization is to elim-inate electrodes. Electrodeless conductance measure-ments have been accomplished by coupling a loop ofelectrolyte in an appropriate tube to a transformer cir-cuit or a heterodyne circuit (41). These special meth-ods, while useful in particular cases, lack the generalityof the ac bridge method, and have not been widelyadopted.

Solid Electrolytes

Interpretation of polarization effects at electrodes insolid ionic conductors is greatly simplified by recog-nizing the series representation of the resistance ca-

pacitance combinat ion (23). Double-layer capacities insolid electrolytes are comparable to those of liquidelectrolytes. A value of 200 ^iF/cm2 has been re-

ported for solid silver bromide.

Dielectric Constant

A realistic equivalent circuit must be employed alsofor the evaluation of dielectric constant from capaci-tance measurements in the presence of an ionic leakagecurrent (42). Here it is the capacitance which is re-

quired; the parts of the impedance resulting from dis-sipation of energy, i.e., dielectric loss, inherent elec-trical conductivity, and impurity conduction, must beeliminated in order to obtain the part of the impedanceassociated with the storage of energy.

SummaryConductance measurements are among the most ac-

curate of the classical techniques available in physicalchemistry, but attainment of high accuracy for sys-tems of diverse properties requires an understanding ofthe electrochemical processes accompanying currentflow through a conductance cell and its electrodes.Many commercial instruments operate at a fixed fre-quency and fixed voltage which make it difficult toassess possible errors associated with electrode polar-ization, bridge circuitry, parasitic effects, or unrealisticequivalent circuit. (The black box syndrome has beencommented on also with regard to other areas of chem-ical instrumentation (43).)

With increasing interest in unusual materials andconditions (5, 15, 44)* accompanied by the extension ofconductance measurements to systems of very high(molten salts) or very low (solid electrolytes, organicsolvents) conductance, it becomes necessary to modify

58 / Journo/ of Chemical Education

standard techniques developed for aqueous electrolytesin accord with fundamental principles if serious error isto be avoided.

Conductance experiments in undergraduate labora-tories, or as special projects, might well include studiesof the effects of varying frequency, voltage, and elec-trode surfaces in various media. These could serve asan introduction to ideas on electrode processes impor-tant not only to conductance measurements, but towide areas of physical chemistry (45) and electroana-lytical chemistry (46, 47).Literature Cited

(1) Robinson, R. A., and Stokes, R. H., “Electrolyte Solutions" (2ndEd. Rev.) Butterworths, London, 1968; Harmed, IT. S., and Owen,B. B., "The Physical Chemistry of Electrolyte Solutions” (2nd Ed.),Reinhold, New York, 1950.

(2) Inokuchi, H., "Electrical Conduction in Solids," Dover Publications,New York, 1965; Weidner, R. T., and Sells, R. L., "ElementaryModern Physics," Allyn and Bacon, Boston, 1968.

(3) Delahay, P., J. Phys. Chem., 70, 2373 (1966).(4) MacInneb, D. A., "The Principles of Electrochemistry,” Reinhold

Publishing Corp., New York, 1939.(5) Moynihan, C. T., J. Chem. Edtjc., 44, 631 (1967).(6) Conway, B. E., "Theory and Principles of Electrode Processes," The

Ronald Press Co., New York, 1965, p. 25.(7) Hills, G. J., and Djordjevic, S., Electrochim. Acta., 13, 1721 (1968).(8) (a) Warburg, E., Wied. Ann., 67, 493 (1899). (b) Warburg, E.,

Ann. Phys., 6, 125 (1901).(9) Heyrovsky, J., and Kuta, J., "The Principles of Polarography,"

Academic Press, New York, 1966.(10) (a) Prock, A., and McConkey, G., "Topics in Chemical Physics,"

Elsevier Publishing Co., Amsterdam, 1962, p. 246. (b) Debye, P.,and Falkenhagen, H., Physifc. Z., 29, 121, 401 (1928).

(11) Joneb, G., and Josephs, R. C., .7. Amer. Chem. Soc., 50, 1049 (1928).(12) Jones, G.t and Bollinger, G. M., .7. Amer. Chem. Soc., 53, 411 (1931).(13) (a) Jones, G.t and Christian, S. M., J. Amer. Chem. Soc., 57, 272

(1935). (b) Jones, G., and Bollinger, D. M., J. Amer. Chem.Soc.. 57, 280(1935).

(14) Jones, G., and Bollinger, G. M., .7. Amer. Chem. Soc., 53, 1207 (1931);Eckbtrom, H. C., and Schmelzer, C., Chem. Rev., 24, 367 (1939).

(15) Robbins, G. D., \nd Braunstein, J., in “Molten Salts: Characteriza-tion and Analysis," (Editor: Mamantov, G.), Marcel Dekker, NewYork. 1969, p.443.

(16) Stout, M. B., "Basic Electrical Measurements," Prentice Hall, Inc.,Englewood Cliffs, N. J. (2nd Ed.) 1960.

(17) Jones, G., and Bradshaw, B. C., J. Amer. Chem. Soc., 55, 1780 (1933).(18) Kohlrausch, F., Wied. Ann., 60, 315 (1897).(19) Siskind, C. S., "Electrical Circuits” (2nd Ed.), McGraw Hill, New York,

1965, p. 235.(20) Grahame, D. C., J. Amer. Chem. Soc., 68, 301 (1946).(21) Grahame. D. C., and Whitney, R. B., J. Amer. Chem. Soc., 64, 1548

(1942).(22) Ferry, J. D., J. Chem. Phys., 16, 737 (1948).(23) Raleigh, D. O., J. Phys. Chem. Solids, 29, 261 (1968).(24) Nichol, J. C., and P'uobs, R. M., J. Amer. Ckem. Soc., 77, 198 (1955);

Buckle, E. R., and Tbaoubboglou, P. E., J. Chem. Soc. (London),667 (1961).

(25) Siiedlovbky, T., J. Amer. Chem. Soc., 54, 1411 (1932).(26) Mysels, E. K., Sholten, P. C., and Mysels, K. J., J. Phys. Ckem., 74,

1147 (1970).(27) Page, L., and Adams, N. I.. "Principles of Electricity," D. Van Nos-

trand, Princeton, N. J. 1969.(28) Page, C. H., "The Algebra of Electronics,” D. Van Nostrand Co.,

Princeton, N. J. 1958.(29) Robbins, G. D., .7. Electrochem. Soc., 116, 813 (1969).(30) Braunstein, J., in "Ionic Interactions: Dilute Solutions to Fused

Salts” (Editor: Petrucci, S.), Academic Press, New York, inpress.

(31) Grimes, W. R., Nuclear Application and Technology, 8, 137 (1970).(32) Swinkels, D., “Molten Salt Fuel Cells and Batteries," in "Advances in

Molten Salt Chemistry” (Editor: Braunstein, J., Mamantov, G.,and Smith, G. P.), Plenum Pub. Corp., New York, Vol. 1, in press.

‘

(33) Grjotheim, K., and Matiasovsky, K., Tidsskr. Kjemi Berov. Metal-lurgi., 26, 226 (1966).

(34) Braunstein, J., J. Ciiem. Educ., 44, 223 (1967).(35) Clark, P. V., J. Chem. Educ., 46, 329 (1969).(36) Robbins, G. D., and Braunstein, J.. J. Electrochem. Soc., 116, 1218

(1969).(37) Christe, K. O., J. Phys. Chem., 74, 2039 (1970).(38) Bloom, H., and Weeks, I. A., J. Chem. Soc., A, 2028 (1969).(39) King, L. A., and Duke, F. R., J. Electrochem. Soc., Ill, 712 (1964).(40) Anderson, F. P., Brookes, H. C., Hotz, M. C. B., and Spong, A. H.,

J. Sci. Instr., 2, 499 (1969).(41a) Yosim, S., Grantham, L. F., Luchsinger, E. B., and Wike, R.,

Rev. Sci. Instr., 34, 994 (1963).(41b) Bertram, R.t Z. f. Analyt. Chemie., 222, 189 (1966).(42) Metcalfe, W. S., J. Sci. Instrum., 42, 742 (1965).(43) Hesse, G., Chromatographia, 2, 183 (1969).(44) Marshall, W. L., Rev. Pure and Appl. Chem., 18, 167 (1968).(45) Conway, B. E., and Salomon, M., J. Chem. Educ., 44, 554 (1967)

Laidler, K. J. Chf.m. Educ., 47, 600 (1970).(46) Anderson, L. B., and Reilley, C. N., J. Chem. Educ., 44, 9 (1967).(47) Lingane, J. J., "Eleetroanalytical Chemistry" (2nd Ed.), Interscience,

New York, 1958, p. 202.(48) Caldin, E. F., "Fast Reactions in Solution," Blackwell Scientific Pub-

lications, Oxford, 1964, p. 81.(49) Olson, H. F., "Solutions of Engineering Problems by Dynamic

Analogies,” D. Van Nostrand, Princeton, New Jersey, 1966.

+

Volume 48, Number 1, January 1971 / 59