I,
net intensity of element before addition Is+# = net intensity of element after addition R = ratio of net intensities of element before and after addition Composite C = yl yz yr+. .. =
+ +
Care must be taken to add an appropriate amount of the element of interest. If too much is added, the relationship loses its linearity and erroneous results are obtained. If too little is added, experimental errors may be large. For optimum results, the amount of added element was chosen so that the value of R in the equation was between 0.7 and 0.8. Although the correct amount can be determined only by trial, a general rule is to add about 25% of the expected amount present. For most accurate results, the line intensities from the elements of interest are measured on the same sample before (10 grams) and after addition of C grams), and a the composite (10 dilution factor is applied in the calculations. However, this method involves a time lapse of possibly several hours between measurements on the sample and its standard, during which the instrument may not remain stable. Therefore, because of instrumental variables and for increased simplicity of operation, it is more convenient to assume that the weight of the sample k
+
equal to the combined weight of the sample and the composite mixture, and to measure the intensity of the sample and its standard at the same time. To justify this approach, the powder sample and ita standard must be packed into the Lucite rings in such a manner that the packing densities of each are about the same. This is accomplished by adding C grams of composite to 10- C grams of sample. If this is done carefully, the error introduced by the equation is small, because the two rings contain nearly identical amounta of sample, one of which contains known excess amounts of the elements of interest. LITERATURE CITED
(1) Anderman, G.,ARL Spedrographer’s News Le& 13, No. 3, 1-3 (1960); Proc. 9th Ann. Cmf. A p lications of X-Ray Analysis, Denver, 8olo., 1960 (2) Applied Research Laboratones, AR’L Spectrographer’s News Letter 7, No. 3, 1-3 (1954). (3) Baird, A. K., MacCall, R. S., McIntyre, D. B., 10th Ann. Cod. Applications of X-Ray Analysis, Denver, 1961; A&. X-Ray A d . (to be published). (4) Berth, E.P., Longobucco, R. J., 10th Annual C o d Apphcations of X-Ray Analysis, Denver, 1961; A&. X-Ray Anal. (to be ublished). ( 5 ) Campbell, J., Carl, H. F., ANAL. CHEM.26, 800-5 (1954). (6) Carl, H. F., Campbell, W. J., ASTM Spec. Tech. Publ. 157, 63-8 (1953). (7) Claisse, F., Norelco Replr. 4, 3-7, 17,
%.
x Figure 1.
x+y
Standard addition method
19, 20 (1957); Province of Quebec, Canada, Dept. Mines, Rept. 327 (1956). (8)Liebhafsky, H. A., Pfeiffer, H. G., Winslow, E. H., Zemany, P. D., “X-ray Absorption and Emission in Analytical Chemlstry,” p. 218, Wdey, New York, 1960. (9) Maneval, D. R.,Lovell, H. L., ANAL. CHEM. 32, 1289-92 (1960). (10)Meyer, J. W., Zbid., 33,692-6 (1961). (11) Mitchell, B. J., Ibid., 33, 917-21 (1961). (12) Neff, H., Togei, K., Siemens and Halske AG, Karlsruhe, Germany, Rept. SH8203 (1959). (13) Scott, R. K., Fall Meeting, Refractories Div., Am. Ceramic SOC.,Bedford Springs, Pa., 1956; Pittaburgh Conference on Analytical Chemistry and A plied Spectroscopy, Pittaburgh, Pa., d r c h 1957. RECEIVEDfor review February 5, 1962. Accepted July 5, 1962.
Funda menta Is of Coulostatic Ana lysis PAUL DELAHAY Coates Chemical Laboratory, Louisiana State University, Baton Rouge, La.
b Fundamentals are discussed for an electroanalytical method in which POtential-time variations are determined at open circuit after abrupt change of the charge on the electrode. The most favorable range is approximately 10-6 to TO-’ mole per liter, and the method appears applicable to any substance which is reduced or oxidized under polarographic or voltammetric conditions. Methodology is briefly covered.
A
ELECTROANALYTICAL method is offered, based on a principle reported in a recent communication ( 1 ) . (However, for priority of idea, see this issue, page 1344.) The principle is as follows: The charge on the working electrode o & m electro-chemical cell is changed-in a short time (a few tenths of a microsecond to a few milliseconds) with an instrument which allows flow of the charging curN
rent but prevents %ow of the reverse current. Charging is achieved, for instance, by discharge of a capacitor, initially charged a t a known voltage, across the electrochemical cell (cf. Methodology). The potential, E, of the working electrode, before charging, is a t its equilibrium value or is set a t a value for which the faradaic current is negligible. The charge supplied to the working electrode is such that it brings E in a range in which there is an appreciable faradaic current. The cell is a t open circuit, for all practical purposes, after charging, and the faradaic current is entirely supplied by discharge of the double layer capacity. The potential tends to return to its initial value before charging as the double layer capacity is progressively discharged. Potential-time variations depend on the double layer capacity and the magnitude of the faradaic current. Since the faradaic current generally
depends on the concentrations of the substances involved in the electrode reaction, these concentrations can be determined, in principle, from the potential-time variations. This method was originally developed for the study of adsorption kinetics (6, 6) and electrode (I,3) processes. The expression “coulostatic method” was coined (6) for application to adsorption kinetics because the charge on the electrode remains constant during recording of potential-time curves. The double layer is discharged by the electrode reaction in analytical applications, and the expression “chargepulse method” would be more accurate. However, there hardly seems a need for two different expressions according to the applications of the method. The coulostatic method is related to the classical interrupter method for the study of electrode kinetics (7) in which potential-time variations are deterVOL 34, NO. 10, SEPTEMBER 1962
* 1267
mined at open circuit after interruption of the electrolysis current. Extrapolation of the potential to time zero in the interrupter method gives the potential which prevailed just before current interruption. The correction for ohmic drop in solution in the measurement of potential is thus avoided. The interrupter method has generally been applied to electrolysis a t steady-state current, though this condition need not necessarily be fulfilled. The coulostatic method is an interrupter method in which the duration of electrolysis, before interruption, is negligible in comparison with the time interval during the measurement of the potentialtime variations. Under ideal coulostatic conditions, the time interval during electrolysis before interruption would be equal to zero. METHODOLOGY
Discussion here is limited to the principle of a coulostat for supplying known quantities of electricity to the working electrode, since techniques, otherwise, are conventional. Coulostats based on two different principles have been developed in this laboratory: The cell, in series with ft rectifier (or equivalent transistor circuit), is connected to a pulse generator; and the charge is supplied by a capacitor charged at a relatively large voltage (10 to 20 volts) in comparison with the cell voltage. Some details are given about an instrument using t h e h t t e r principle, simple, since equipment is p&Br& inespensive, and entirely satisfactory for analytical applicatbg, [Application to electrode kinetics and a circuit analysis are given by Delahay and Aramata (f?).] Capacitor cc (FiguTe I), which is charged a t a known voltage, on potentiometer Pi3 i%. WRXE*& to the cell, C L , , b switdhlng relay RLl from positpoh 1 to 2. Relay RLn closes a
Yusted
Table 1.
&gq-y7? e.
w
p2
PI
Figure 1. Coulostat based on discharge of capacitor across electrochemical cell PI, Pt, Pa. Potentiometers
-4ssume that the circuit of an electrochemical cell is open at time t,,, after the beginning of electrolysis and that the potential of the working electrode at interruption is E,,,. The charge on the electrode a t time t, is
short time (perhaps 0.01 second) after contact 2 of RLI has closed. The charging current has then died out, and a potential-time curve is recorded by means of REC. The potential change resulting from discharge across the cell is compensated by means of potentiometer PJ at t = 0, and high sensitivity can be selected in the recording of potential-time curves. The potential of the working electrode before charging is adjusted by means of potentiometer PI.
c(Em)A(trn)[Em- E,]
Rl1, Rlz.
In general, capacity c, is very small in comparison with the double layer capacity, and the time constant of the cell-capacitor (c,) circuit is primarily determined by cc. This time constant therefore can be made very short in comparison with the interval for recording of potential-time curves. The quantity of electricity corresponding to leakage to REC during recording must be negligible in comparison with the quantity of electricity Aq consumed by the faradaic process. A high sensitivity amplifier with low input currente.g., 10-*2 ampere-and high-resistance voltage divider may be necessary, especially with electrodes of small area (hanging drop, wireelectrode), h t o matic determination of AE for a given t should offer no problem, and such an
Potential-+e Variations for Constant Double Layer Capacity and Fumdaic Current Independent of potential
- (Coulostatic conditions)
E
- t Variations4
Equation
z-It
Limiting current with steady-state mass transfer
fAE
Diffusioncurrent at planeelectrode
iAE =
Diffusion current at spherioal electrode
t l / z I nFDCo f A E = 2nFD1I2Co 1yC rc
Diffusion current at cylindrical electrode
f A E =
(2)
2nFD1I2C0t l l z
(3)
1y1,2c
2nFD1/2C0t"2 Xl'ZC
- E,. Plus sign corresponds to
(4)
I nFDCO 2rc
1 nFD3IZCo t 3 , 2 67r1Ie r2c
...
(5)
a net cathodic process, minus sign to a net anodic process. Concentrations in moles per cc.
AE
1268
=
E
ANALYTICAL CHEMISTRY
GENERAL EQUATION FOR POTENTIAL TIME VARIATIONS
Low leakage capacitor Relays CL. Electrochemicol cell eA, eR, ow. Auxiliary, reference, and working electrode, respectively REC. Recording instrument CC.
Current
a
instrument is now being developed in this laboratory. The reference electrode can be combined with the auxiliary electrode, but one must ascertain that its potential is not affected by the current pulse during charging.
where c(E,) is the integral capacity of the double layer per unit area at potential E,, A@,) is the electrode area a t time t,, and E , is the potential at the point of zero charge. The double layer capacity is discharged by the charge transfer reaction after interruption, and the electrode area may also vary. The potential thus varies, and the charge on the electrode a t time t is c ( E ) A ( t ) [ E- E,]
where c ( E ) is the integral capacity of the double layer per unit area a t potential E, A(t) is the electrode area at time t , and t is the time since the beginning of electrolysis (not since tm). The quantity of electricity consumed by the faradaic process since t, is
where i is the faradaic current. The general equation for the potential-time variations is c ( E ) A ( t ) [ E- E,] c(Em)A(tm)[Em- Es1 =
+
This equation, with t, 0 or t,, = 0, respectively, is applicable te the interrupter and coulostatic methods. The signs in Equation 1 are: E, Earopean (IUPAC or Steckholm) convention; i 3 0 for net cathodic or anodic processes, respectively. Forbs of Equation 1 corresponding to the followixig conditions can be distinguished:
I. The eleetpde mea, A , is constant I. The current, i, is constant a The capacity, c, is constant b. The capacity, c(E), is a function of E 2. The cum@, i(t), is a function of time only; cases a and b 3. The current, i ( E ) ,is a function of potential only; cases a and b
t“2
advantage over direct measurement of the limiting current. ELECTRODE OF CONSTANT AREA AND CURRENT INDEPENDENT OF POTENTIAL
t
(sec.)
Figure 2. Potential-time variations for plane electrode and coulostatic conditions Potential varying in diffusion current range and double layer capacity independent of potential
4. The current, i(E,t), is a function of potential and time; cases a and b 11. The electrode area, A ( t ) ,varies with time; dropping mercury electrode 1. The current, i ( t ) ,is a function of time only; cases a and b 2. The current, i(E,t), is a function of potential and time; cases a and b Cases with potential analytical applications are treated below by the following method : The explicit form of the faradaic current for the particular electrode being considered and the prevailing electrolysis conditions is introduced in Equation 1, and integration is carried out. The necessary equations for the current are available from the theory of polarography and voltammetry. ELECTRODE OF CONSTANT AREA AND CURRENT INDEPENDENT OF POTENTIAL AND TIME
This case corresponds, for instance, to variations of potential in the limiting current range for an electrode with steady-state mass transfer-e.g., a rotating electrode. If the double layer capacity is independent of potential, the potential for coulostatic conditions varies linearly with time (Equation 2, Table I) and the slope of the E-t line is proportional to the limiting current density, ZI. This is characteristic of discharge of a capacitor a t constant current. Since 12 is generally proportional to the concentration of the species being reduced or oxidized a t the electrode, this concentration can be determined. The capacity, c, is measured in a separate experiment or, more simply, is determined by caIibration with a known concentration of analyzed substance. There seems to be no
This case corresponds to variations of potential in the diffusion or limiting current range for an electrode with mass transfer controlled by diffusion. The potential-time variations will be analyzed for the plane, spherical, and cylindrical electrodes for a simple charge transfer reaction, and the influence of variation of the double layer capacity with potential will he considered for the plane electrode. Plane Electrode. The potentialtime variations are derived by introducing in Equation 1 the diffusion current density for semi-infinite linear diffusion. Equation 3 in Table I shows that E varies linearly with t 1 / 2 for coulostatic conditions. The shift of potential, AE = E-E,,,, is positive for a net cathodic process and negative for a net anodic process. This is to be expected, because a reduction process consumes electrons and thus causes the electrode to become less negative a t open circuit. Conversely, oxidation causes the electrode to become less positive a t open circuit. The concentration of analyzed substance is directly computed from the slope of the E us. t1’* plot. The double layer capacity can be computed from the shift of potential a t t = 0 from E , before charging to E,,, just after charging. Thus Aq = c ( E ,
- E,)
(6)
where Aq is the known charge increment supplied to the electrode a t t = 0, as expressed per unit area of the electrode. In view of the order of magnitude of the double layer capacity (approximately 20 to 50 pf. cm.-2 for a smooth surface), analytical application is most promising in the range of 10-5 t o IO-’ mole per liter (Figure 2). Concentrations below 10-7 mole per liter could hardly be determined because AE is too small and cannot reliably be measured for t > 30 to 60 seconds a t room temperature as a result of interference by convection. The shift of potential for a rough electrode is somewhat smaller than for a smooth electrode having the same area as the projected area of the rough electrode, all other conditions being the same. The double layer capacity is determined by the actual area a t the metal-electrolyte interface, whereas the diffusion current, after a sufficient time, depends more nearly on the projected area than the actual area. Roughness thus increases the double layer capacity without increasing the diffusion current in the same proportion, and AE a t a given
0
0.5
I
Figure 3. Potential-time variations for plane electrode and current interruption after different times t, of electrolysis Potential varying in diffusion current range and double layer capacity independent of potential
t is smaller than for a smooth electrode of the same projected area. Simultaneous analysis of several substances is also possible. The procedure for two substances is as fol1ows:hE is determined for the more easily reducible or oxidizable substance, after charging of the electrode to a potential in the diffusion current range for this substance; the electrode is charged anew without interruption of electrolysis to bring its potential in the diffusion current range for the other substance, and A E is determined. The slope of the first AE us. t * ’ 2 line is proportional to CIo, whereas this slope for the second line is proportional to (Clo Czo). Concentrations C1” and Czothus can be determined. The same method applies to more than two substances, hut errors are cumulative. More simply, one can obtain the A E us. t ” 2 plots in separate experiments for the successive plateaus of the currentpotential curve. The potential-time variation for the interrupter method is
+
and the shift of potential, after interruption, decreases with increasing time of electrolysis (Figure 3). This is to he expected, since the diffusion current density for a plane electrode varies with l/tl/*, and consequently the current for discharge of the double layer capacity after interruption decreases when t, increases. The coulostatic method therefore is preferable to the interrupter method in trace analysis in this particular case. Variations of the double layer capacity with potential complicate the potential-time relationship but, in practice, the AE us. t 1 / 2 plot is nearly VOL 34, NO. 10, SEPTEMBER 1962
1269
0
.01
.05
.2
.I
Spherical Electrode. We consider diffusion from solution toward a spherical electrode, and introduce in Equation 1 the corresponding diffusion current density. The potential shift (Equation 4, Table I) is larger than for the plane electrode for identical conditions, because the diffusion current density for a spherical electrode is larger than for a plane electrode of the same area. One deduces from Equations 3 and 4 t h a t
t0 Y2-
0.5
0
.5
t(sec.1
’
Figure 4. Potential-time variaiions for plane electrode and coulostatic conditions for integral capacity of double layer varying linearly with potential ( p = dc/dE) Potential varying in diffusion current range
linear for variations of ilE not exceeding 50 mv. even for very large variations of capacity with potentiali.e., 20 pf. over a 1-volt interval (Figure 4). The potential-time variations are derived from Equation 1, which takes the form
t l / l + constant * 2nFD1/2co p
(8)
Integration is performed after introduction of c as a known function of E , and the integration constant is determined from the condition, E = E, a t t = 0. For instance, E is solution of PE’
+ [can - p(E. + E,)IE
,“ .*
=
tkZ
and consequently experimental All’s for a spherical electrode can be reduced to the corresponding AE’s for a plane electrode if D and r are known. The concentration is obtained from the plot of the corrected shift of potential us. W’ as for the plane electrode. The correction of Equation 11 is significant for a hanging drop electrode of the usual size ( 7 = 0.05 cm.) and for t exceeding a few tenths of a second (Figure 5). Cylindric Electrode. The shift of potential in Equation 5 (Table I) was derived for diffusion toward a cylindric electrode with the field of diffusion limited by two planes perpendicular to the axis of the cylinder. Higher terms than those in Equation 5 are generally not needed for usual cylindric electrodes and the values of t for which convection is negligible. Shifts of potential can be corrected as for the spherical electrode. One has
] --I Or o
>
E
C
w
a I oc
2 00 0
3
2
I
4
t (sec.) Figure 6. Influence of difference of potential E. E, (volts) on potentialtime variations for dropping mercury electrode
-
Potential varying in diffusion current range and current interruption a t 1 second
growth of the drop. One can also assume that the electrode area is constant for polarographic electrodes with drop time of 3 to 5 seconds under coulostatic conditions but the time interval for recording of potential-time curves is then too short for adequate sensitivity a t low concentrations. Analysis of potential-time curves for intervals comparable with the drop time will be limited to the interrupter method. The shift of potential a t time t of the drop life a t 25’ C. is AE
=
( E , - E,)
-
[ ( E , - E,)t,2’3
*
(12) Charge Transfer Reaction with Coupled Chemical Reaction. Charge
transfer processes with various types of coupled chemical reactions can be treated by introduction of the corresponding expressions for the limiting current in Equation 1. Rate constants for the chemical reaction could be obtained from the variations of the shift of potential with time. Details are not given here since analytical application is remote.
I
4
I
2
0
I
I n A D-
IO”
cm’. sec-‘
t (sec 1
Figure 5. Correction of shift potential for spherical and cyclindrical electrodes according to Equations 1 1 and 12
1270
ANALYTICAL CHEMISTRY
DROPPING MERCURY ELECTRODE AND CURRENT INDEPENDENT OF POTENTIAL
Two types of dropping mercury electrodes are considered : electrodes with long drop time (perhaps 20 to 40 seconds) and usual electrodes of polarography. Potential-time variations in the coulostatic method with eIectrodes of long drop time are the same as for an electrode of constant area, provided the charging pulse is applied toward the end of drop life and potentials are measured over a sufficiently short interval in comparison with drop life. A correction can be made for the small increase of capacity resulting from the
t
3 (sec.)
Figure 7. Influence of concentration (moles per liter) of reducible or oxidizable substance on potential-time variations for dropping mercury electrode Potential varying in diffusion current range and current interruption at 1 second
where Co is espressed in moles per cubic centimeter and the f sign corresponds to cathodic and anodic processes, respectively. Equation 13 was derived by application of the IlkoviE equation, and a more rigorous equation could be derived for corrected forms of this equation. The shift of potential is caused by discharge of the double layer capacity by the faradaic process and by the change in capacity resulting from expansion of the drop. The AE us. t curve may exhibit a maximum under certain conditions (Figure 6). The conditions corresponding to this maximum and the value of t at the maximum are rcadily deduced by setting dAE/dt = 0. The shift of potential resulting solely from espansion of the drop is large, and the sensitivity is lower than for coulostatic analysis with an electrode of constant area (Figure 7). The difference between the shifts of potential with faradaic process and without is proportional to the bulk concentration of analyzed substance. Thus
out or with mass transfer control (diffusion). Shifts of potential do not exceed E to 50 mv. in kinetic studies. The analysis of AE os. t curves for larger variations of potential is very simple when mass transfer is controlled by steady-state convection. One has, for instance, for a reversible process with steady-state mass transfer and a double layer capacity independent of potential
ACKNOWLEDGMENT
(15)
When - ( E >> RT/nF, Z = Zr and Equation 15 reduces to Equation 2 (Table I). Variations of E with t become less rapid when E drifts to values at which the current is smaller than the limiting current, and I d A E / d t 1 decreases continuously with increasing t. Reversible processes for a process ne = involving soluble species (0 R) at a plane electrode of constant area can be analyzed as follows for diffusion control:
Although the interrupter method appears of little value for analytical applications of polarography, it may be useful in the determination of currentpotential curves in conventional polarography nith solutions of low conductivity. The potential of the dropping mercury electrode should then be measured a very short time (perhaps a few microseconds) after interruptionLe., before any appreciable variation of potential. Techniques applied in electrode kinetics could readily be transposed to the dropping mercury electrode. Further investigation would be needed to compare this technique with present methods for compensation of the ohmic drop. CURRENT DEPENDENT O N POTENTIAL
Conditions in which the faradaic current varies with potential are of interest in the study of electrode processes, and the essential theory has been worked out (8) for processes with-
The author is indebted to his colleague, L. W. Morris, Physics Department, Louisiana State University, for the suggestion of using capacity discharge in the design of a coulostat. NOMENCLATURE
A
= electrode area
c
= integral capacity of double layer
C
= concentration of
per unit area
Two boundary conditions are prescribed at the electrode surface (z = 0) for t > 0 : conservation of flux for substances 0 and R ; and Equation 1 in which the current is written in terms of the flux of 0 at x = 0 and E is expressed by the Nernst equation as a function of the concentrations of 0 and R a t z = 0. These two conditions are transformed into an integral equation in ( ~ C & Z ) ~= 0 and .(CO), = o by application of the convolution theorem. This equation could be solved after introduction of dimensionless variables. This is primarily a mathematical exercise, since the problem has no direct analytical application and seems t o be of no consequence for electrochemical studies. Irreversible processes can be analyzed by a similw method.
reducible or oxidizable substance Co = bulk concentration of reducible or oxidizable substance D = diffusion coefficient E = electrode potential Et = potential before charging E , = potential immediately after charging = half-wave potential F = faraday i = faradaic current I = faradaic current density Zl = faradaic limiting current density n = number of electrons in over-all charge transfer reaction q = quantity of electricity r = r d i u s of spherical or cylindrical electrode R = gasconstant t = time since beginning of electrolysis tm = time at which electrolysis is interrupted T = absolute temperature z = distance from electrode
CONCLUSION
LITERATURE CITED
+
Variations of AE with t become less rapid when t , increases, as one can ascertain by deriving the value of d A E / d t a t t = tm from Equation 13. The effect of variation of the double layer capacity with potential could be considered, but this seems hardly worthwhile.
problems. Instrumentation is much simpler than in some other electroanalytical methods for trace determination such as square-wave polarography, pulse polarography, and faradaic rectification. Possibilities of application of the coulostatic method w e much greater than for stripping methods, which, however, are more sensitive in some instances. Esperimental results will be reported (4).
Coulostatic analysis with an electrode of constant area appears promising in the range of 10-8 to mole per liter for substances which are reducible or oxidizable under polarographic conditions. The difficulty resulting from the double layer, which prevents application of polarography and some related methods to trace analysis, is entirely avoided and, in fact, the double layer is used advantageously. Purification of the supporting electrolyte-e.g., by pre-electrolysis-and oxygen removal remain serious
(1) Delahay, P., Anal. Chim. Acta 27, 90 (1962). (2) Delahay, P., J. Phys. Chem., in press.
(3) Delahay, P., Aramata, A., Ibid., in
press.
(4) Delahay, P., Ide, Y., unpublished in-
vestigation.
( 5 ) Delahay, P., Mohilner, D. M., J. Am. Chem. SOC.,in press. (6). Delahay, P., Takemori, Y., unpub-
llshed investigation.
( 7 ) Hickling, A., Trans. Faraday SOC. 33, 1510 (1937).
RECEIVEDfor review May 17, 1962. Accepted July 5, 1962. Research supported by grant G-10006 from the National Science Foundation.
VOL 34, NO. 10, SEPTEMBER 1962
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