POLAROGRAPHY IN A BINARY SALTSOLUTION
24 1
Polarography in a Binary Salt Solution
by Thomas W. Chapman and John Newman Inorganic Materials Research Division, Lawrence Radiation Laboratory, and Department of Chemical Engineering, University of California, Berkeley, California (Received March 8, 1966)
The instantaneous current and the average current to a dropping mercury electrode in a binary salt solution are calculated. At very small times the instantaneous current depends only on the ohmic drop and varies as (I8, whereas at sufficiently large times and voltages a limiting current is reached which varies as t1l6. The average current over a drop lifetime is not well represented by the limiting-current approximation of the Ilkovi6 equation if either the drop life or the applied voltage is too small. However, the current is always directly proportional to the bulk concentration of the reactant for a given voltage and drop life.
Introduction Polarographic analysis with a dropping mercury electrode is usually carried out in the presence of a large excess of indifferent, nonreacting electrolyte. This serves both to reduce the ohmic potential drop in the solution and to reduce the effect of the electric field on the movement of reacting ionic species. For a sufficiently large applied potential, the current to the drop is limited by the rate of diffusion and convection and corresponds to a zero concentration of the reactant at the surface. For the total cathodic current in this case IlkoviPs2 and Mac Gillavry and Rideala have obtained the expression
I D = 4nFy2c,-((/')
(1)
where 4 n y s / 3 is the constant volumetric flow rate of mercury through the capillary, F is Faraday's constant, cm is the bulk concentration, Di is the diffusion coefficient of the reactant, and n is the number of electrons consumed when one reactant ion or molecule reacts. The radius TO of the drop grows with the cube root of time
ro = rt =/a
where t+ is the cation transference number, D = D+D-(z+ - z-)/(z+D+ - z D - ) is the diffusion coefficient of the salt, and x+ and x- are the charge numbers of the cation and anion. Ionic migration in this case of cation discharge enhances the limiting current so that IL > I D . For example, for a symmetric electrolyte and a transference number of 0.5, IL= 21D. For solutions of intermediate compositions, the limiting current follows a similar time dependence and has been calculated for several cases by N e ~ m a n . The ~ effect of ionic migration on limiting currents has been discussed qualitatively by the early polarographers.lt6 If there is insufficient indifferent electrolyte, the current due to one discharging species will produce an electric field which enhances the limiting current for a second discharging species. The low electrical conductivity associated with the absence of supporting electrolyte can also lead to an ohmic potential drop in the solution which prevents the attainment of a limiting current, at least in the initial stage of the growth of the drop. The ohmic resistance for flow of current to a sphere of radius TO in a medium of uniform conductivity K is ~ / ~ T K T O .
(2)
Volume 71, Number 8 JaltUaTY 1987
THOMAS W. CHAPMAN AND JOHNNEWMAN
242
Hence, for an applied potential V , the total current can be no larger than I = 4TKTov = 47K”fVt/a (4) For small values of t, this value is less than the limiting diff usion-migration current, and the effect becomes particularly important for solutions of low conductivity. For polarographic analysis the applied potential is essentially constant over the drop life, and the current will at first follow eq 4. As the drop grows, the current and the concentration must adjust themselves so that the various overpotentials, including concentration overpotential, surface overpotential, and ohmic drop, balance the applied voltage. Levich7has already treated this problem for a binary electrolyte. However, he assumed a constant concentration at the surface of the drop and thus missed the ohmic limitation at short time represented by eq 4. His work also contains a number of mathematical errors which obscure the analysis. With some generosity, his result can be expressed as
IL(c,- co)/cm
I
(5)
where co is the assumed constant concentration at the surface and depends upon the applied potential. This current is proportional to tf/’ and represents an asymptote for large values of 2. Equation 5 does reproduce the limiting current of Lingane and Kolthoff for a large applied potential. Physical Basis for the Analysis. It is assumed that the solution can be treated as a dilute electrolytic solution; i.e., the diffusion coefficients and activity coefficients are taken to be constant and the molar flux of an ionic species is expressed as Ni = -zi(Di/RT)FciV@
- DiVci + V C ~
(6)
where v is the fluid velocity and CP is the electrostatic potential. The concentration of the reactant in a solution of a single salt then obeys the equation of convective diffusion bC bt
+ V - V C= DV2c
Z+
- Z- t- In co- - R T cSo In - +
-z+z-
C,
z+F
cB
lom !! dr (8)
K
where i, is the radial current density, co is the concentration in the solution near the surface, and csois the concentration in the mercury amalgam near the surface. Equation 8 is the potential of a Concentration cell with transference plus the ohmic potential drop in the solution. The surface overpotential associated with the electrode reaction has been ignored. The concentration in the solution differs from the bulk value c, only in a thin region near the surface. If in this region the concentration is approximated by a linear profile c = co
+ (c, c
- co)y/6 for 0 < y < 6 = cmfor y > 6
(9)
where
y=r-ro (10) then the integral in eq 8 can be evaluated, and the equation simplifies to (cf. ref 8)
(Levich? uses the crude approximation of taking c = See also editor’s note to his section 52.) The analysis of IIkoviE1*2applies to a drop growing radially with no tangential velocity and to a thin diffusion layer which becomes thicker with time but is thin compared to the radius of the drop. The same approximation can be introduced in the present analysis so that the equation of convective diffusion, eq 7, becomes
co in a region of thickness 6.
ac _
at
_ 2y _ -dro- ac - - D-uc ro dt a y
ay*
(12)
or
(7)
where D is again the diffusion coefficient of the salt. For simplicity and clarity, let the growing mercury drop be an amalgam with an initial concentration cs of the discharged reactant, and let the applied potential V refer to the potential of the growing drop relative to an amalgam reference electrode also of concentrac tion cs and located at a large distance from the capillary electrode. The applied potential V can then be expressed as The Journal of Physical Chemistry
RT
V=-F
ac = D-3% -at_ - 2y -3t ay ay2 dc
(13)
A similar equation applies inside the sphere, but with the diffusion coefficient D, of the discharged reactant in mercury. (7) V. G. Levich, “Physicochemical Hydrodynamics,” PrenticeHall, Inc., Englewood Cliffs, N. J., 1962, Section 110. (8) J. Newman, “The Effect of Migration in Laminar Diffusion Layers,” UCRL-16666, Lawrence Radiation Laboratory, Berkeley, Calif., Feb 1966.
POLAROGRAPHY IN A BINARY SALTSOLUTION
243
Finally the derivatives of concentration at the surface, both inside and outside the drop, are related to the total current by the expression
I
-
47rr02
=
1nFD - t+ by,=o+ b!
nFD,%l b y *=o-
RT -ln(1+ z+F
(14)
Derivation of Concentration at the Surface. I n this section we obtain an integro-differential equation from which the surface concentration can be determined. Equation 13 can be reduced to the transient heat conduction equation
Z+
- Z-
T
= (3/7)t”3
(16)
The solution to eq 15 satisfying the conditions
is9 c = c,
2 +4 ~
~
FV RT
-
co InCm
+ t+(l - ): +
-z+znF2yt z+ - z- R T K , ( ~- t+)
by means of the variables
c
=
c,at7 = 0
c
=
c,atx =
c
>+
c, - co 1 - t+ c,
Solution for the Concentration and the Current. We shall obtain the solution when the concentration variation inside the mercury drop can be neglected and eq 22 can be written -Z+L
z = t’Iay and
~
= c ~ ( T at )
(23)
The form of this equation suggests that most of the parameters can be eliminated by the use of appropriate dimensionless variables. For the potential and the concentration the choice is clear
(17)
a
z =0
sm [ s2) co( 7 -
- c,]e-”
Note that E is now positive while V is negative. Equation 23 then becomes dz
Zid401
(18) The concentration 4 7 ) at the surface is still to be determined. The derivative of the concentration a t the surface can be evaluated from this solution and substituted into eq 14 to yield
I n a similar manner, the solution for the amalgam inside the drop gives
By equating expressions 19 and 20 and integrating, one can relate cso to co
where
The dimensionless parameter N is related to the relative importance of concentration overpotential and ohmic drop in balancing the applied voltage. For a dimensionless time variable we choose
e
=
2/3aD/7t‘/‘//Ny
With this variable, eq 25 can be written
(28) where C is now a function of 8. From eq 11 it- now becomes apparent that an appropriate choice for the dimensionless current would be
J = An integro-diff erential equation for the surface concentration can now be obtained by substituting eq 19 and 21 into the potential relation 11. The result is
(27)
3(1 - t+)I 28nF y k, N
(9) H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” Clarendon Press, Oxford, 1959, pp 62, 63.
Volume 71, Number 2 Janwrg 1967
THOMAS W. CHAPMAN AND JOHN NEWMAN
244
that the value of the transference number has only a small effect on the current, particularly at large voltages, in which case the effect vanishes in both the small and large time limits. The current varies with O2 or tl’a at small times and then changes over to a e or tl” dependence at large times, the transition being earlier the greater the applied potential. The significant quantity for comparison with experiment is usually not the instantaneous current but rather the current averaged over the lifetime of a drop. This average current is defined as 1
I
1
=
m 1
0.01
0.04 0.06 0.1
0.02
0.2
0.4 0.6
1.0
8
Figure 1. The reduced instantaneous current to the drop as a function of dimensionless time, dimensionless applied voltage, and cation transference number.
With this choice, eq 11 becomes
J
=
+ In C + t+(l - C ) ]
@[E
(30)
Now C can be obtained as a function of 0 from eq 28, and the dimensionless current can then be obtained from eq 30. For small times, C can be expanded as a power series :m 0
c
:= 1 -
Ee
+
&e2
+ a,ea + . . .
(31)
with then
J = EO2
+ e303 + e4e4 + . . .
The first term of these equations corresponds to eq 4, and the applied voltage is opposed only by the ohmic potential drop in the solution. The coefficients ai and ei are functions of the parameters E and t+. At very large times, C will be a constant, given implicitly by the equation
-E
=
In C
+ t+(l - C)
Idt
I
o.6k 7
I
/ I /
t+=o.o
0.4
(33)
and the current will be given by eq 5, where co is determined from eq 33. For a large applied potential the limiting dimensionless current is J = 8
(34)
corresponding to eq 3. Using the small time expansion introduced in eq 31 we have calculated the dimensionless current to the drop for various values of E and for values of 0 up to 1. The results are shown in Figure 1. It is seen The Journal of Physical Chemistry
(35)
where Tis the drop life. The solution for the instantaneous current has been integrated to obtain the average current to the drop. The results are shown in Figure 2 where the dimensionless average current j, defined analogously to J , is plotted us. dimensionless drop life 8 = t9final with the dimensionless voltage and t+ as parameters. For a typical experiment 0 is on the order of 0.1-0.5. Also, for a 1-1 electrolyte an applied voltage of 1 v corresponds to E = 19.5. The line j = 68/7 is the limiting current result corresponding to eq 3 and represents the large-time, large-voltage asymptote. The value of the transference number has little effect on the average current particularly for reasonably large voltages. Curves for values of t+ between 0 and 1 fall between the two curves. 1.0
(32)
nT -
I JO
a Figure 2. The reduced average current to the drop as a function of dimensionless drop life, dimensionless applied voltage, and cation transference number.
POLAROGRAPHY IN A BINARY SALTSOLUTION
245
This analysis has been developed for a two-electrode polarographic technique where the reference electrode is far from the working electrode. If the applied voltage is controlled by a potentiostat which employs a probe reference electrode very close to the drop, the upper limit of the integral in eq 8 would be no longer infinity but rather the radial position of the reference electrode. The probe would still be outside the diffusion layer, which is at most to cm thick in these experiments, so that the approximation used to evaluate the integral would not be invalidated. The only change in the analysis necessitated by the use of a probe is the introduction of a factor of {/({ ro) in the last term in eq 23 where { is the distance of the tip of the probe from the surface of the drop. The introduction of this time-dependent factor in eq 23 complicates the solution of the problem, However, since the factor would normally vary from unity down to 0.7-0.6 over the lifetime of a drop, it should not greatly change the results of the analysis as stated. Its effect on the measured average current would be essentially the same as that of an increase in the conductivity of the bulk solution. We have neglected the effects on the transient current behavior of certain electrode phenomena. If the electrode is not reversible as we have assumed, it is necessary to include the overpotential involved with the electrode reaction in the boundary condition formulated in eq 8. I n addition to the Faradaic current calculated, a capacitive current must also flow as the drop grows in order to charge the double layer on the newly formed mercury-solution interface. The magnitude of this capacitive current can be estimated by considering the charging of a spherical capacitor of double-layer capacity K . The total voltage is made up of the double layer voltage and the ohmic drop
+
v=
vdl
+
(36)
Vohm
Since the total voltage is constant i _ dvohm _ - dvdi ~ -- I dt dt K 4aro2K ~
~
(37)
Differentiation of the expression for the ohmic loss to a sphere V o h m = I/4sKmr0 (38) yields dVohm ___
dt
-(!z1
4TK,rO
dt -
;)
(39)
Substitution of eq 37 into eq 39, integration, and application of eq 38 as an initial condition for I with Vohm = Vyield eq 40.
The ratio of this ohmically limited charging current to the total ohmically limited current at small times is eXp( - 3 K m t 2 ’ ‘ / / 2 ~ K ) For the typical values of the parameters y = 0.1 cm/secl’*, K = 30 pf/cm2, and K, = O.Ol(ohm cm)-l, this ratio is e-5X10at2”, where t is in seconds. Since we need not concern ourselves with extremely short times, the capacitive current may be considered negligible compared to the Faradaic current. A discussion of the effect of the solution resistance on the capacitive current in electrochemical kinetics studies, where it is not negligible, has been given recently by Oldham.lo Also Delahay has made a more thorough investigation of the contributions of Faradaic and capacitive currents in transient electrode processes.”
Conclusions For E > 20 and 8
> 0.07 the limiting-current approximation provides a good representation of the average current. However, for E = 10 the approximation is not very accurate unless 8 > 0.2, and it becomes much less accurate for smaller voltages and shorter times. Therefore, the correct interpretation of experimental data obtained under circumstances where values of drop life or applied voltage may be necessarily small or where instantaneous current is measured directly requires consideration of the detailed results of this analysis. A further conclusion of this work follows from eq 29 and 30. The right side of eq 30 is dimensionless and independent of e,. Therefore, since N is independent of c, for dilute solutions, the current I is always proportional to the bulk concentration of the reactant, even during the transient concentration behavior. This is the characteristic of all polarographic situations which gives the method its great usefulness as a quantitative analytical tool. The phenomena considered here for a binary electrolyte also occur in a supported electrolyte. However, because of the higher conductivity, the transient effects are of much shorter duration and therefore insignificant in that case. Acknowledgment. This work was supported by the U. S. Atomic Energy Commission. (10)
K.B. Oldham, J . Electroanal. Chem.,
11, 171 (1966).
(11) P. Delahay, J . Phua. Chem., 70, 2373 (1966).
Volume ‘71,Number 2 January 196’7
