J. Electroanal. Chem., 102 (1979) 415--419
415
© Elsevier Sequoia S.A., L a u s a n n e - Printed in The Netherlands
Short communication
O N T H E M E A N I N G O F T H E I M P E D A N C E C O N C E P T IN T H E C A S E O F A N
O B J E C T T H A T V A R I E S W I T H T I M E A N D IN T H E C A S E O F A S W E P T F
FREQUENCY
M. SLUYTERS-REHBACH and J.H. SLUYTERS
Van ' H o f f Laboratories, State University, Utrecht (The Netherlands)
(Received 5th February 1979)
ABSTRACT
Expressions are derived that describe the behaviour of a condenser the capacity of which
varies linearly with time under potentiostatic and galvanostatic a.c. conditions. The "impedances" are found to be different. Application to the dropping mercury electrode is indicated.
Also the behaviour of a constant capacitor subject to a frequency swept a.c. potential or
current is calculated. The admittance of the capacitor is found to have increased in the latter
two cases.
INTRODUCTION
R e c e n t l y we c o m m u n i c a t e d o n t h e i n s t r u m e n t a l a r t e f a c t s t h a t as a conseq u e n c e of slow r e s p o n s e of a d e t e c t o r can m a r i m p e d a n c e m e a s u r e m e n t s perf o r m e d o n t i m e - d e p e n d e n t objects [ 1 ]. H o w e v e r , also t h e c o n c e p t of i m p e d a n c e
itself s o m e t i m e s b e c o m e s m o r e c o m p l e x in such a case. We wish to s h o w this
c o m p l i c a t i o n on a p u r e c a p a c i t o r t h e c a p a c i t a n c e of w h i c h varies l i n e a r l y with
times"
(1)
C = at
By d e f i n i t i o n the charge o n the c o n d e n s e r is r e l a t e d to t h e voltage across it by
(2)
O : CV
In case of a sinusoidally v a r y i n g voltage
V = Vn~ sin w t
(3)
(1), (2), a n d (3) lead t o
Q = a t V ~ sin ~ t
(4)
w h e n c e , for t h e c u r r e n t i = d Q / d t o n e o b t a i n s
i = a V ~ sin cot + ~ a t V m cos cot
(5)
E v i d e n t l y in a d d i t i o n t o t h e t e r m to be e x p e c t e d , a t c o V ~ cos cot, an in-phase
c o m p o n e n t a V m sin cot is o b t a i n e d . In o t h e r w o r d s , in case of this t i m e - d e p e n d e n t c a p a c i t o r the o b s e r v e d a d m i t t a n c e Y is c o m p l e x i n s t e a d of being imaginary"
Y= a + j~at
= a + jcoC(t)
(6a)
416
and its reciprocal, the impedance Z is:
1
cot
Z = a(1 + co2t2) -- j a(1 + co2t2)
(6b)
instead of Z = - - j / c o C ( t ) .
If similar reasonings are applied to a time d e p e n d e n t resistor no complication
is met, because only in cases where the derivation involves differentiation with
respect to t an extra t e r m arises.
Thus far the e x p e r i m e n t was supposed to be c o n d u c t e d potentiostatically
(cf. eqn. 3) and one may w o n d e r h o w a t i m e - d e p e n d e n t condenser behaves under
galvanostatic conditions, i.e.
i = im sin co t
(7)
C o m b i n a t i o n of eqns. (1), (2), and (7) leads to
Q
.
r i m. sin. cot . d t
im cos
cot = at V
09
Whence
V-
im
coat
cos w t
(8)
Comparing eqns. (7) and (8) it is evident t h a t under galvanostatic conditions the
condenser presents the characteristics expected f r o m its impedance
Z =
1
jc0C
(9)
irrespective w h e t h e r C is a function of time or not.
F r o m the foregoing it must be concluded that the c o n c e p t of impedance
should be applied with care to t i m e - d e p e n d e n t objects: the impedance measured
will be d e p e n d e n t on the way the e x p e r i m e n t is c o n d u c t e d .
Because in a bridge the object c o n d u c t s electric current neither u n d e r potentiostatic nor under galvanostatic conditions its behaviour c a n n o t easily be predicted.
In case of a t i m e - d e p e n d e n t self inductance it can be shown in an analogous
way t h a t the object behaves normal u n d e r potentiostatic conditions and presents
a real c o m p o n e n t u n d e r galvanostatic conditions.
APPLICATION TO A DROPPING MERCURY ELECTRODE
It easily can be shown that also for this object the galvanostatic e x p e r i m e n t
gives no special effect. Therefore we confine to the potentiostatic e x p e r i m e n t
and for the sake of simplicity in this Section we restrict to a theoretical electrode
w i t h o u t faradaic process and w i t h o u t series resistance. The double layer capacity
of this electrode will be proportional to the electrode surface area, so on the
assumption of c o n s t a n t mercury flow
C = a t 2/3
(10)
Then with (3) following the same reasoning as before we find an expression for
417
the cell current
(11)
i = a V m ( ~ t -~/3 sin cot + cot 2t3 cos cot)
Again, in addition to the " n o r m a l " response VmcoC(t) cos cot, the current contains an in-phase c o m p o n e n t , 2 / 3 a V m t-1/3 sin cot. Evidently, in this case of the
d r o p p i n g mercury electrode, this term extinguishes slowly. It is interesting to calculate on which c o n d i t i o n the " e r r o r " is negligible.
An acceptable error in the phase angle is 0.2 degree, which corresponds with
tg~ < O.OO35.
F r o m eqn. (11) it follows tg~ = 2/3cot, whence cot > 200. For a drop life of
three seconds the lowest allowed frequency is henceforth f o u n d to be about 10
Hz. However, if one intends to apply the so-called rapid a.c. polarography technique as proposed by Zatka [2], Bond et al. [3,4] and Canterford [5], the small
time values involved (t = 0.16 s), require a frequency limit f > 200 Hz.
APPLICATION TO MORE COMPLEX EQUIVALENT CIRCUITS
Calculation of the current response of an R C series c o m b i n a t i o n where b o t h
R and C are functions of time requires solving the differential e q u a t i o n
~+
dt
E
dt
+
R(t)C(t
)1
i--
R-~
dt
=0
(12
)
in which for a practical dropping mercury electrode C(t) = at 2/3 and R ( t ) =
Rhom + bt -1/3. As y e t no analytical solution of this differential e q u a t i o n could
be found.
SWEPT FREQUENCY IMPEDANCE MEASUREMENTS
N o w that voltage controlled oscillators and phase locked detectors are available it is t e m p t i n g to c o n s t r u c t a n e t w o r k analysing system that automatically
measures the cell impedance or a d m i t t a n c e at a c o n t i n u o u s l y varying frequency.
Again the general feeling will be t h a t the rate of change of the frequency should
be low c o m p a r e d to the frequency applied. We believe it is interesting to k n o w
more quantitatively the errors involved and also to k n o w n the nature of the
error, viz. w h e t h e r s o m e t h i n g happens to the in-phase c o m p o n e n t or the quadrature. Again we treat the problem i n d e p e n d e n t l y of the effects of non-zero time
constants discussed in ref. 1.
For the sake of simplicity we restrict this discussion to a capacitor with a
t i m e - i n d e p e n d e n t capacity.
If the frequency is swept linearly with time
¢ o = a + bt
a potentiostatic p e r t u r b a t i o n applied (cf. eqn. 3) becomes
V = Vm sin(at + bt 2)
(13)
whence with i = Cd V / d t
i = (a + 2 b t ) C V m cos(at + bt 2)
(14)
41.8
Comparison of eqns. (13) and (14) leads to the conclusion that the " i m p e d a n c e "
of the capacitor is
Z(t)
1
j(a + 2bt)C
(15)
while neglect of the fact that the frequency is being swept leads to
1
Z-jcoC
1
j(a + bt)C
(16)
Comparing eqns. (15) and (16) we can conclude t h a t the admittance Y - 1/Z observed in a swept frequency e x p e r i m e n t is larger than the a d m i t t a n c e that would
have been observed at a certain constant frequency, the difference being equal
to jbtC = j(¢o -- o~t=o)C. Of course b also can be chosen negative, in which case
the admittance will be found smaller. An analogous calculation can be made
with a linearly frequency swept a.c. current
i = im
sin(at + bt 2)
(17)
with the intent to see what happens to the a d m i t t a n c e of a constant capacitor in
that case. Evaluation of the voltage across the capacitor requires solution of the
integral
im
t
V(t) = -~ / sin(at + bt2)dt
(18)
0
An alternative formulation of this integral is [6]
g ( t ) - im
7r
C 2 b cos ~-bC~
1
(a+2bt
--sin ~C
1
(a+2bt
(19)
in which c~(x) and e ( x ) stand for the sine and cosine Fresnel integral, respectively. Numerical values of these integrals have been tabulated [7]. For x =
(2~b)-1/2(2bt + a) :> 5 they can be a p p r o x i m a t e d by analytical expressions,
yielding for V(t)"
0.3181rim
V(t) = ~im ~/~_~ ICos a(~_b)-- sin a(~b)1 -- C(a
+ 2bt) cos(at + b t2 )
(20)
As 0.318~ equals 0.99997 the a.c. term in eqn. (20) leads to nearly the same
conclusion as in the potentiostatic case, i.e. that towards a linearly frequency
swept current a capacitor exhibits a changed admittance, the difference as compared to a constant frequency e x p e r i m e n t being equal to jbtC = j(co -- cOt=o)C.
The condition for (20) to be valid appears to be fulfilled in realistic practical
cases. For example it is met already at t = 0 for a swept frequency starting at 26
Hz swept with a rate of 26 Hz s -~.
CONCLUSIONS
F r o m the foregoing discussion it can be concluded that under normal practical
conditions the determination of the double layer at a dropping mercury electrode
Willnot be complicated by its time dependency. The in-phase c o m p o n e n t of the
419
capacitance of the double layer can be calculated from eqn. (10) with
a = 47rCd (4 7r13.6
3rn )2~3
always to be much smaller than the ohmic resistance of the cell.
In the study of electrode kinetics, however, the ohmic resistance is subtracted
before the analysis and moreover the analysis is most sensitive to errors in the
phase angle. Therefore under unfavourable conditions like small drop time and
very reversible electrode reaction the effect described could influence the result.
Unfortunately a quantitiative study based on an equivalent circuit comprising a
faradaic process looks impossible due to mathematical complexity.
Measurements with frequency swept a.c. voltage or current lead to an error in
the double layer capacitance that can be calculated easily. The allowed sweep
rate can be calculated and its value will also serve as a guide in more complex
equivalent circuits.
REFERENCES
1 M. Sluyters-Rel~bach, J.S.M.C. Breukel a n d J . H . Sluyters, J. E l e c t r o a n a l . Chem., 1 0 2 ( 1 9 7 9 ) 303 (this
issue).
2 A. Z a t k a , J. E l e c t r o a n a l . Chem., 27 ( 1 9 7 0 ) 164.
3 A.M. B o n d a n d D.R. C a n t e r f o r d , Anal. C h e m . , 44 ( 1 9 7 2 ) 1 8 0 3 .
4 A.M. B o n d a n d R.J. O ' H a l l o r a n , J. Phys. C h e m . , 77 ( 1 9 7 3 ) 915.
5 D . R . C a n t e r f o r d , J. E l e c t r o a n a l . Chem., 77 ( 1 9 7 7 ) 113.
6 M. A b r a m o w i t z a n d I.A. S t e g u n , H a n d b o o k of M a t h e m a t i c a l F u n c t i o n s , Dover P u b l i c a t i o n s , N e w Y o r k ,
1 9 7 0 , eqn. 7.4.39.
7 Ref. 6, p. 322.