Semiintegral Electroanalysis: Analog Implementation Keith B. Oldham Department of'chemistry,Trent University, Peterborough, Ontario, Canada Placed in the feedback loop of an operational amplifier, certain resistor-capacitor ladder networks provide an output accurately proportional to the semiintegral of the input current. This principle may be advantageously used for analytical purposes i n semiintegral electroanalyzers.
IL
0
IN A PREVIOUS ARTICLE (I), the principles of semiintegral electroanalysis were described and it was demonstrated experimentally that the theoretical predictions are well obeyed. Some of the advantages of the new technique over conventional electroanalytical methods were enunciated in an earlier note (2). One disadvantage of semiintegral electroanalysis as hitherto described is the data reduction stage of the experiment, which requires tedious transcription of current-time (or charge-time) data from a chart recording into a computer. The present article describes an analog technique for avoiding this stage, adding convenience to the other advantages of semiintegral electroanalysis. Experimental results obtained with the analog technique concur with results acquired digitally. PRINCIPLES
In the simplest embodiment of semiintegral electroanalysis, the solution to be analyzed for an electroreducible species Ox is placed within a two-electrode cell, as diagrammed in Figure 1. Initially, the cell potential is maintained a t a value E,, sufficiently positive just to inhibit the electroreduction reaction, Ox
+ Ne-
+
Rd
(1)
so that no current flows. The potential applied to the working dectrode cs. the counter electrode is then changed, as by turning the potentiometer shown in Figure 1 from one extreme position to the other, to a more negative value, E, - AE. If Reaction 1 is a reversible reduction, (240/N) millivolts would often be a suitable value for AE; a larger change would be needed for an irreversible electroreduction. Figure 2 shows the change from E, to E, - AE occurring linearly [a capped ramp ( 2 ) signal1 but a feature of semiintegral electroanalysis is its independence from the manner in which this change is made. In response to the more negative potential, electroreduction occurs at the working electrode, resulting in a time-dependent cathodic faradaic current i(t). Generally, the potential E(t) across the interface of the working electrode (referenced to the counter electrode) will be more positive than E, - AE by the product R*i(r),where the resistance R* includes the solution resistance, all resistors in the external circuit, and any polarization of the counter electrode. Because of progressive concentration polarization at the working electrode, i(r) will soon decrease with time, and E(r) will approach E, - AE, as shown in Figure 2. The magnitude of AE is so chosen that, after a time interval 7 of a few seconds, E(t) enters a potential region corresponding to virtually complete concentration polarization with respect to Ox. Theory shows that the
I REC
I
Figure 1. Schematic diagram of a simple electrochemical apparatus permitting the potential applied to the working electrode of a cell us. counter electrode to be changed from E, to E, - AE (resistance and polarization effects being ignored). Voltage-time recorder REC measures i( t ) by means of dropping resistor R. Effects of replacing R by a capacitor C and by a resistive capacitative line R,C are discussed in the text
equation m(t)
=
NAFV'DEC~- c,(t)l
(2)
holds. Here D and C, are the diffusion coefficient of Ox and its bulk concentration, A and C,(t) are the area of the working electrode and the Ox concentration at its surface, F is Faraday's constant and m(t) is the semiintegral of i(t). Because C,(t 3 T ) is virtually zero, Equation 2 may be rearranged to
(3) a relationship demonstrating the proportionality between Cb and m(t 3 7 ) which constitutes the basis of semiintegral electroanalysis. The veracity of Equation 3 has been clearly established ( I ) . This equation also implies that, subsequent to t = T, m will remain constant indefinitely. A practical limit, of order 10 seconds, is imposed by the onset of convection, a process ignored in the derivation of Equation 2. In previously published work ( I , 2), the preferred means of determining m(t)was, in effect, to place a resistor R , paralleled by a voltage-time recorder, in the current path as shown in Figure 1. The response (4)
e(t) = Ri(t)
of the recorder was directly proportional to i(r) and was used to provide evenly spaced i us. t data which were digitally semiintegrated to provide m(r). An alternative, less frequently adopted, method was to replace the resistor by a capacitor C. In this configuration the recorder response
(1) M.Grenness and K . €3. Oldham, ANAL.C H ~ M44, . , 1121 (1972). (2) K . B. Oldham, (bid., p 196.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 1, JANUARY 1973
0
39
potential
and leakage conductance between them (4). Here, however, we restrict consideration to an idealized transmission line in which resistance along one conductor and capacitance between the two conductors are the only significant elements. Such a line is symbolized in Figure 3. By p (ohm per centimeter) and y (farad per centimeter), we denote the resistance and capacitance of unit length of the line. The potential and current at the end of the resistiveSimcapacitative line will be represented by e(t) and i(r). ilarly, e(x,t) and i(x,t) will be used to represent the interconductor potential and intraconductor currents at a distance x from the end of the line at time t. These definitions will be clarified by reference to Figure 3 and by noting the identities e(r) = r(0,t)and i(t)E i(0,t). If the line is initially at rest, the potential and current are everywhere zero at t = 0, i.e.
t
e(x,O) Figure 2. Graph showing (-) total applied potential and (- -) potential across working electrode interface. Crosshatching indicates potential range within which concentration polarization is complete; the interfacial potential enters this region at time t = 7
-
...
w 1
(7)
(8)
was directly proportional to the charge q(r), Le. to the integral of i(t). To obtain m(t), the q DS. t data were digitally semidifferentiated. Inasmuch as m(t) possesses an intermediacy between i(t) and q(t), it might be supposed that the recorder in Figure 1 could be made to measure m(t) directly if the resistor were replaced by an element which is suitably intermediate between a resistor and a capacitor. Heaviside (3) postulated that a resistive-capacitative line is such an element, and the next section of this article is devoted to demonstrating this fact. If R and C are the total resistance and capacitance of such a line used to replace the resistor in Figure 1, then the response of the recorder is simply
-
d:
- m(t) 7,
to
RESISTIVECAPACITATIVE LINES A two-conductor transmission line, exemplified by a coaxial cable, generally has a number of impedances associated with it, including inductance along and between the conductors
e(x,t)
3X
-
(9)
+ pi(x,t) = 0
and
and combined into the single partial differential equation 32 e(x,t) 3x2 -
a
= py - e(x,t)
at
by elimination of the current. Solution of Equation 12 is accomplished by use of Laplace transformation and insertion of the initial Condition 7 a2
- B(x,s) = py[sa(x,s) - e(x,O)] = pysE(x,s)
dX2
(13)
to convert the problem to one involving an ordinary differential equation. Two arbitrary functions, P(s) and Pl(s), of the dummy variable s are present in the general solution a(x,s)
3
+ dx,t) = [ydx] ata e(x,t)
to the time derivative of the interconductor potential. These two equations may be rewritten as
a
Figure 3. This diagram defines some of the quantities employed in the theory of a resistive-capacitative line. Potential difference between the two conductors at distance x is e(x,t)
and is directly proportional to m(t) and, hence, for t the concentration C, of Ox.
i(x,O)
+ dx,t) = e(x,t) - [pdx]i(x,t)
i(x,t) - i ( x
=
=
to a section of line of length dx, and relating the interconductor reactance current, by means of Coulomb’s law
- - - UI d x y
e(t)
0
Subsequently a signal is applied to the end of the line, as a result of which a perturbation is transmitted down the line. The equations governing the transmission may be deduced by applying Ohm’s law e(x
0
=
=
P(s) expi - x d z f
+Pl(s)exp{x2/Z)
(14)
of Equation 13. This equation relates the Laplace transform e(x,s) of e(x,t) to the independent variables x and s. Let us temporarily treat the line as of infinite length and consider the x -.t m limit of Equation 14. This consideration shows that for E(x + m,s) to remain finite for nonzero s, Pl(s) must be zero. Hence Equation 14 reduces to a(x,s)
= ~ ( s exp )
{ --x~‘pySl
(1 3
-
(3) 0. Heaviside, “Electromagnetic Theory, Volume 11,” (1920) reprinted by Dover Publications, New York, N. Y., 1950, pp 35, 36 and 434-6. 40
(4) W. C . Johnson. “Transmission Lines & Networks,” McGrawHill Book Co., New York, N. Y., 1950.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 1, JANUARY 1973
which yields 0
(16) on partial differentiation with respect to x . Combination of the transform of Equation 10 with Equation 16 leads to I-
0-1
o--,
r.1
n
(d)
I-
kl5C
C
where use has been made of the rule (5) that the Laplace transform of the semiintegral of a function of t equals the It now requires transform of the function divided by only specialization to x = 0 [recall 1 (0,s) 3 Z (s), etc] and identification of ply with the ratio RIC of the total line resistance and capacitance to yield
di.
which generates Equation 6 on inversion. The derivation thus far has dealt with a line of infinite length. The next step is to show that this inconvenient requirement may be relaxed. Though the proof will be omitted, it may be shown rigorously and straightforwardly that the effect of curtailing a resistive-capacitative line to a finite length is to replace Equation 18 by
For large argument. the hyperbolic tangent approaches unity, so that $8) = dR/C f i ( s ) even for a finite line for sufficiently large s. If we quantify the meaning of the symbol = by using it to link quantities which differ by not more than 2 z , then 3-
Z(s) =
J ER f i ( s ) ,
because arctanh 10.9801 Relationship 20 to
1
-
s =
RC < (2.3)2 - = 0.19 RC
(20)
2.3. It is tempting to invert
but caution is necessary in the Laplace inversion of inequalities. However, as Appendix A demonstrates, Relationship 21 is almost identical to the result deduced by exact inversion being, in fact, slightly too conservative. In words, this relationship reveals that a truncated line accurately semiintegrates at short times, but that departures are to be expected for times in excess of about 20Z of the RC time constant. Relationship 21 was derived for a truncated line in which the distant end is left opencircuited, as in Figure 4(a). Exactly the same relationship is applicable to the shortcircuited line shown as Figure 4@). Judicious termination of the truncated line, however, may be shown to greatly lengthen the time range over which semiintegration occurs without appreciable distortion. Thus, for either of the circuits shown as Figures 4(c) and 4(d), the relationship -
e([)
=
dz R
m(t), t
< 1.3RC ~
~~
( 5 ) K . B. Oldham and J. Spanier, J . Electroand. C/iem., 26, 331 (1 970).
Figure 4. Resistive-capacitative lines of finite length. Respectively, (a) and (b) have open-circuit and short-circuit terminations. The other six diagrams show terminating components of optimally-chosen values
replaces Relationship 21, Likewise for the Figure 4(e) or 4(f) terminations,
and for 4(g) or 4(h) -
e(t) =
dg
rn(t),
t
< 52 RC
(24)
The virtue of careful termination in lengthening the time range of semiintegration is evident. However, it must be appreciated that the terminating components need to have their design values within close tolerance if departures from perfect semiintegration are to be maintained within the 1 2 % design margin. It may be noted in passing, that the effect of an inductiveconductive line, Le. a transmission line in which inductance along one conductor and conductance (ohmic leakage) between the two conductors are the only significant elements, is to semidifferentiate the input current: ,-
where L is the total inductance (henries) and G the total conductance (mhos) of the line. The proof of Equation 25 follows the same pattern that has been here used for resistivecapacitative lines. For the treatment of transmission lines in general, the interested reader is referred to monographs on that subject (e.g. Reference 4). LINE SIMULATION
Our electroanalytical application requires that semiintegration be carried out accurately for time intervals as long as ten seconds. This means that, even with excellent termination, a resistive-capacitative line with an RC product of order one second is needed. A typical coaxial cable has a so that a length of order py product of order lo-" sec
ANALYTICAL CHEMISTRY, VOL. 45, NO. 1, JANUARY 1973
41
(a 1
v
(b)
-
The lower time limit for accurate semiintegration by the ladder shown as Figure 5(b) is determined by n , the number of “rungs” in the ladder, and may be shown (7) to be
7‘ Terminotion
IP Termination
Termination
Figure 5. A truncated resistive-capacitative line is shown in (a), arbitrarily terminated. Ladder networks which simulate the line are shown in (b) and (c), (b) being an arithmetic ladder and (c) a geometric ladder. In the third diagram, g denotes a number greater than unity R, equals (g - l)R/(gn+l- g); and C , equals (g - l)C/(gn+*Notice that (c) duplicates (b) as g approaches 1.00 and that the total resistance and capacitance are, respectively, R and C in all
g).
three cases n
n
R/2n
R/2n
. I
-
Termination
n
L
< t
on the same basis used in the discussion of Relationships 20 and 21 (Le. no more than 2 departure from ideal behavior in transform space). Qualitatively, the reason for th,e existence of a lower limit is clear. Any signal applied to the ladder network of Figure 5(b) progresses through it from left to right. Once the “leading edge” of such a signal is in the interior of the ladder, it experiences an environment which resembles Figure 5(a) sufficiently that semiintegration ensues. Initially, however, the leading edge experiences only the series combination of the resistor R/2n and the capacitor C/n, which is not a semiintegrating environment. A time interval of some twelve times the (R/2n)(C/n)time constant must elapse before accurate semiintegration occurs. Just as the time range for accurate semiintegration by a truncated line may be extended upwards by judicious termination, so judicious initiation enables a ladder to semiintegrate accurately down to much shorter times. The initiating element depicted in Figure 6 , for example, may be shown to depress by almost a factor of ten the lower time limit for accurate semiintegration by the arithmetic ladder, the relationship ,-
replacing Relationship 26. The generation of the circuit in Figure 5(b)from that in 5(a) may be regarded as having two stages. First, the resistivecapacitative line is subdivided into n equal segments; and, second, each segment is replaced by a “T” element composed of two resistors and a capacitor having values equal to those of the line segment it replaces. The geometric ladder of Figure 5(c) is generated from that of Figure 5(a) by a similar twostage process, the only difference being that the initial subdivision is into unequal segments. The lengths of the line segments form a geometric sequence of ratio g, where g exceeds unity. Thus the capacitance of thejth segment is
Jn-1
gT,
Figure 6. An arithmetic ladder, well initiated
one-hundred kilometers would be needed to provide a onesecond time constant. Any similar device having conductors and a dielectric of macroscopic dimensions would also need to be inordinately large. Thin-film technology offers an obvious solution to this dilemma, but in the present study ladder networks (6) designed to simulate resistive-capacitative lines have been used rather than actual lints. Consider the arithmetic ladder shown as Figure 5(b). It seems reasonable to suppose that, as n + m , the behavior of this circuit will approach that of Figure 5(a). That this is indeed the case, except at very short times, may be demonstrated mathematically. The pertinent mathematics will be deferred to a later article (7) which will deal with ladder design in some detail. (6) M. F. Gardner and J. L. Barnes, “Transients in Linear Systems,” Volume I, John Wiley & Sons, New York, N.Y., 1961. (7) K. B. Oldham, unpublished, 1972. 42
.-
=
gj-’(g gn
- 1)C -1
the resistance of the jth segment being similarly related to the resistance R, of a fictitious zeroth segment and the total resistance R . The advantage which a geometric ladder possesses over an arithmetic ladder of the same total number of components lies in its having a lower short-time limit. This arises because smaller components are present a t the left-hand end of the ladder, the time constant experienced by a n entering signal being (g - 1)2RC/2(gn- 112. This time constant equals the RC/2n2of an arithmetic ladder for a g of unity, but is significantly less than RC/2n2for a g value of, say, 312. On the other hand, geometric ladders have the disadvantage of requiring components of a wide variety of resistance and capacitance values (but see Appendix B). The brief description just given of arithmetic and geometric ladders leaves many important issues unresolved. Thus, it does not touch upon the factors which dictate the optimal values of n or g. Nor is the question faced of the effect upon the semiintegrating properties of the ladders of departures of
ANALYTICAL CHEMISTRY, VOL. 45, NO. 1, JANUARY 1973
1 TI,,, 1 ‘1.’ Figure 8. A twentythree-rung arithmetic ladder constructed of T units
This is one of the three semiintegrating circuits actually constructed, the others being depicted in Figures 9 and 10. Component values in all three diagrams are given in kilohm and nanofarad units
1 1
(b)
C/2n
1
2860
Termination
L L
-In
L
Figure 7. Complementary arithmetic ladders. The ladder shown as (a) is; built from “T” units, that shown as (b) from “II” units. In practice, the central sections of both ladders are identical, only 1:hefirst and last rungs differing
component values from their nominal resistance or capacitance. These matters are among the ones discussed in the forthcoming article (7). The principle of complementarity is an important one in ladder design. In designing a ladder, one is trying (recall Equation 20) to obtain a circuit which gives g ( s ) / f i ( s ) equal to a constant d R / C , within close limits (such as the =k2 we previously adopted), for as wide a range of s values as possible. Let it be supposed that, for a particular s value, the circuit is imperfect by a fractional error term 6, such that
Figure 9. A geometric ladder, terminated and initiated. The design is based on the values n = 7, g = 101’6, and = lo6 ohm1’2farad-1’2 = 1 volt per microamplomb. Most components are of +1 % tolerance. Slightly different component values were used in some of the electrochemicalstudies
1/RiC
6.3
2
630
63
w
20
bvvv 4-50
7-500
200
2000
E30
Figure 10. Semiintegrating circuit based on the paralleling of an n = 4, g = 10, d R / C = 2.00 volt per microamplomb geometric ladder with its complement This equation gives
Now, it can be shown that there exists a complementary circuit such that
[-“I
iR(s)
comp
=p [-L] c 1 - S(s)
The circuits in Figure 4(a) and 4(b) are complementary in this sense, as are 4(c) and 4(4, 4(e) and 4(f), etc. Likewise the circuits shown as Figures 7(a) and 7(b) are complementary. Consider now what results from the paralleling of a circuit with its complement. One finds, using the familiar property of impedances in parallel, that
2 on reciprocation. Now compare Equations 29 and 32. Besides halving the effective resistance and doubling the effective capacitance, the effect of paralleling the ladder circuit with its complement has been to change the fractional error 1/263 . . . . For small 6 values, this from 6 to 1/262 represents a dramatic improvement in performance. Thus, if 6 = 10% for a particular value of s, the corresponding error in the paralleled ladder is only about 1/2%. Toward the end of the next section, we shall see practical proof of the value of paralleling a ladder with its complement.
+
+
*
ELECTRICAL EXPERIMENTS 1
=
$12
+ a2(s) + S3(s) + S4(s) + . . , ]
(31)
To test the ideas of the previous section, three ladders were constructed from components of = t l or =k2% tolerances, using the circuits displayed in Figures 8, 9, and 10. The Figure 8 circuit is a simple arithmetic ladder with a onecomponent termination. Such ladders have been described
ANALYTICAL CHEMISTRY, VOL. 45, NO. 1, JANUARY 1973
43
~
~
~
_
_
Table I. Response of the Figure 8 Ladder to a Ramp Signal t , sec 0 0.050 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.OO 1.20 1.40 1.60 1.80 2.00 2.20 2.40
i(0,P A 0 1.20 1.79 2.56 3.12 3.58 3.91 4.32 4.68 4.98 5.30 5.60 6.32 6.94 7.55 8.15 8.77 9.38 9.99
5.4 5.66) 5.721 5.701 5.66 5.6115.63 4c 0.05 5.581 5.59 5.571 5.58 5.601 5.78 5.87 5.97 6.07 6.20 6.32 6.45
I
Figure 11. Semiintegral analyzer employing an operational amplifier with an R,Cline in its feedback loop
in the literature (8, 9) and have been used to perform analog semiintegration, though the articles which report the circuits are not couched in terms of this operator. Figure 9 shows a ladder which is essentially geometric: its design was based partly on the considerations outlined in the previous section and partly on the local availability of components. No prior description of a geometric ladder is known to the author. The circuit in Figure 10 is a four unit, g = 10, geometric ladder which has been suitably terminated and initiated and then paralleled with its complement. Semiintegrating circuits exist which are not ladders in the sense of this article. For details of these, and for some interesting electrochemical applications of the fractional calculus, the reader is referred to Reference (IO). The Princeton Applied Research Model 170 instrument, used in our electrochemical studies, was also used to evaluate the performance of the three ladders. The instrument has the capability of applying voltage ramps to external circuitry and displaying on its pen-recorder a trace us. time of the current flowing in response to these signals. To test the ladder of Figure 8, the ramp signal
40
=
{
0, t < O
et, t
>0
1
(33)
was applied, with e = 2.00 volts per second. The expected response from a semiintegrating circuit is given by the following reasoning. For an ideal semiintegrator, from Equation 18 m(t
> 01 =
Ji
e(t
> 0) = c
Jf t
> 0)
= 21
t: -
(35)
results. The predicted square-root dependence of current on time was clearly shown on the pen recording and from it the (8) K. Holub and L. Nemec, J. Electroanal. Chenz., 11, 1 (1966). (9) R. F. Meyer, Aeronuuricul Rep., LR-279, National Research Council, Ottawa, 1960. (10) M. Ichise, Y . Nagayanagi, T. Kojima, J . Electroanal. Cliem., 33, 253 (1971). 44
data assembled in Table I were gathered. Notice that i/di is virtually constant in the 0.1 sec 6 t 6 1 sec range, the value of the constant being 5.63 microampere second-''*, with a root-mean-square deviation of 0.05 microampere second-1'2. The magnitude of this constant is to be compared with the theoretical value i(t
("""'"'N
> 0)
4;
100 nanofarad 15 kilohm
second
=
5.83 microampere second- u 2 (36) obtained by substituting into Equation 35 the numerical values o f t and of the C / Rratio. Table I demonstates the failure of the Figure 8 ladder to semiintegrate properly for times greater than one second. Beyond this time, the ladder is too "resistor-like." Theory predicts that the time limit of the one-component, terminated ladder should be close to 1.3 times the RC constant of the ladder (recall Equation 22). The time constant of the ladder is, in fact, 23 X 15 kilohms x 23 x 100 nanofarads, or 0.8 second. Multiplication by the 1.3 factor gives a 1.0-second limit, in perfect agreement with observation. The circuit of Figure 9 was tested in essentially the same fashion but with an applied ramp of 200 millivolts per second. For this ladder the C / R ratio was designed to be exactly 1.00 nanofarad per kilohm, i.e. 1.00 picosecond per ohm2. Hence application of Equation 35 leads to the prediction i(t
> 0)
2
200 mV
10-12 sec
-
z/i =z(Sec>dTG-0.226 pA sec-1'2 (37)
(34)
and on semidifferentiation (5) of each side of this relationship, i(t
w
The experimental results, presented in Table 11, concur extremely well with this prediction, over the 0.05 sec < t < 15 sec range. The properties of the Figure 10 circuit were evaluated in a more extensive study, involving a variety of ramp rates, e, and a number of different current and time scales on the i-t recorder. Table I11 summarizes the large amount of data SO produced. This circuit shares with the Figure 9 ladder the design ratio 4 T R = 1.00 second1'2 megohm-' and, hence, from Equation 35,
ANALYTICAL CHEMISTRY, VOL. 45, NO. 1, JANUARY 1973
= 1.13
microampere secl'z volt-'
(38)
is expected. Once again, agreement between experiment and prediction is seen to be very satisfactory. Notice that the semiintegrating properties of the Figure 10 circuit are quite as good as those of the geometric ladder of Figure 9 and far superior to those of the arithmetic ladder of Figure 8, despite the fact that Figure 10 has the fewest components of the three circuits. This reflects the value of exploiting the complementarity principle. The experiments just described show that, when subjected to a known voltage signal, each of the three ladder circuits tested passes a current accurately equal to the semiderivative constant of the ladder. of the signal multiplied by the It may therfore be inferred that the voltage that is generated by passing a current through each ladder will equal its 1 / r C constant multiplied by the semiintegral of the applied current. It is this property which is of electroanalytical value, as was demonstrated in connection with Equation 6.
Table 11. Response of the Figure 9 Ladder to a Ramp Signal t , sec
0,027 0.051 0.072 0.100 0.160 0.228 0.324 0.506 0.710 0.892
0.050
0.100 0.200 0.500 1.OO 2.00 5.00 10.00 15.00
-\/m
i(t)/di,PA sec-1’2
i(0, MA
0.020
0.19 0.228) 0.2281 0.2241 0.2261 0.228 0.227 0.229) 0.2261 0.2251 0.230)
0.002
Table 111. Response of the Figure 10 Circuit to Various Ramp Signals
;. volt sec-’ 2.000 1.Ooo
INSTRUMENTATION
1,000
Let us return now to Figure 1 (with the resistor replaced by the R,C line) and consider the impedance criteria which a semiintegrating circuit must meet in the configuration shown there. Substitution into Equation 3 of values C,, D,A , and N for a typical electroanalytical experiment lead to the conclusion that m(r 2 7)will usually lie within an order of magnitude of 10 microampere sec1j2[i.e. ten “microamplombs” ( 2 ) ] . The voltage generated across the R,C transmission line (or its simulant) by an rn value of this magnitude depends on the ratio RIC of the line resistance to capacitance and is given by
0.500 0.500
0.200 0.200
time range, sec 0.30-0.60 0.35-0.65 0.15-0.65 0.75-1.45 J 0.20-1.40 1.60-2.60 0.13-2.50 3.00-7.00 ( 0.25-2.00 3.00-7.00 8.00-14.0 3.00-7.00 9.00-1 3 . 0 [ 15. W 2 5.O
i 1
1
mean f(i)/d< i(t)/id/t, 9~ sec1’2 pA sec-1’2 volt -1 2.328 1.164) 1.160 1.1601 1.142 1.1421 1.149 1.149 0.5580 1.1161 0.5565 1.1131 0.5505 1.101 mean = 0.5700 1.140 1.136 0.2228 1.1141 0.2306 1.153 1 0.2304 1.1521 0.2298 0.2282 0.2218 1.109) ~
i
I;: :
(39) as Equation 6 shows. Because of the signal-independence feature of semiintegral electroanalysis, this technique is relatively insensitive to the effect of voltage-drop in the response-measuring segment of the cell circuit. Nevertheless, this voltage drop must not be too large, as otherwise one would need to know m(r 2 7) in order to select a suitable AE. A voltage drop in excess of, say, 100 millivolts could be considered intolerable. Putting this value as the upper limit in Equation 39, the result
are easily constructed: the ladders shown in Figures 8 through 10 are examples attesting to this fact. The convenient properties of operational amplifiers provide a ready solution to this paradox. Consider the circuitry of Figure 11. Without any significant loading of the cell circuit, the operational amplifier produces an output equal to 1/RT m(r). With m(r 3 7) in the range 1 to 100 microampere sec’/2,and with R/C having the convenient gigaohm farad-1 ratio given in Equation 42, then the amplifier output will be e(t
2
7) =
kilohm 1 nanofarad
(lO-5*1
ampere second”*) = lO’*t‘ volt (43)
is arrived at, whence Rlohm -__ Clfarad
- 106
This ratio expresses a suitable design criterion for the Figure 1 circuit. Capacitors are not normally encountered as close-tolerance components with values in excess of 1 microfarad. Accordingly, to satisfy Equation 41, resistors are needed with resistances of order 1 ohm, which is a value lower than is conveniently available as close-tolerance components. In fact it proves impossible to meet the criterion of Equation 41 with conveniently available components. Unfortunately, therefore, the simple circuit of Figure 1 is impractical. On the other hand, resistors in the kilohm range of values and capacitors of about nanofarad magnitude are freely available in a wide variety of close-tolerance values. Thus, ladders simulating R,Clines of ratio close to
an almost ideal range of voltages for actuating output devices such as meters, chart recorders, and oscilloscopes. Few operational amplifiers can provide outputs as large as the upper limit, 100 volts, of Equation 43. A paralleled-resistor current-divider in the amplifier input, however, easily solves this problem when larger-than-average electroactive concentrations are present. A circuit almost identical with that of Figure 11 has been constructed and operates as predicted. The results reported in the next section, however, were obtained by modifying the Princeton Applied Research Model 170 instrument. The output stage of this instrument incorporates an integrator of conventional design: an operational amplifier with a precision capacitor in its feedback loop. The modification consisted of simply replacing this capacitor by the Figure 9 ladder network. The resultant wmiintegrator was tested and calibrated by using a standard resistor as load.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 1, JANUARY 1973
45
Millivolts venu8 S . C . E . 0
-280
I
- 30
-20
m In
0 0
I
I
t I
I
I
1
0
0
*
0
0
-520
In
-520
I-
I
I
30
-
20
-
Figure 12. Neopolarogram of Pb2+electroreduction
-
Horizontal scales are linear in both time and (between -280 and -520 mV) potential. The ordinate is in microamplomb units (amplomb = ampere s e i 9 E coulomb sec-1’2)
* Y)
-E 0
i 0
‘E
-IO
10
-.E
0
0-
I
I
I
I
I
I
ELECTROCHEMICAL EXPERIMENTS
As previously explained (I), a capped ramp signal applied to a stationary mercury drop working electrode leads to an m cs. t curve which resembles a classical polarogram. Such “neopolarograms” as have been previously reported (I,2) were calculated digitally, using a computer program to generate m(t)from i(t)or q(r). The neopolarogram reproduced as Figure 12 is a direct tracing of the chart-recording produced by the Princeton Applied Research Model 170, modified as explained above. The system employed was 1.00 m M lead nitrate in 100 m M potassium nitrate solution, the aqueous solution being deoxygenated by passage of argon. The capped ramp signal commenced at -280 mV cs. SCE and proceeded to -520 mV at a ramp rate of 100 mV per second. The working electrode was a mercury sphere of 1.00-pl volume and all other experimental conditions were as in our previous studies (1). A three-electrode configuration was employed. Figure 12 clearly shows that m(t) approaches a constant, the magnitude being m(t
3
7)
=
25.8 microamplombs
is derived. This equation involves the m(t - T)/m(t)ratio, where 0 T 6 t. In our electroanalytical application ( I ) , m(t) always increases monotonically with time, so that this ratio invariably lies between zero and unity. Accordingly
Two duplicate runs with the same system, but with m(t r) determined digitally, gave 25.2 and 25.6 microamplombs. This agreement demonstrates the equivalence of the digital and analog techniques to within 2 %. Many other neopolarograms have been recorded, as well as m 6s. t curves with signals other than capped ramps. In all cases, agreement with digital semiintegration was observed. The analog technique is now our preferred means of determining m and all our recent results have been obtained by use of this convenient, rapid, and accurate method. APPENDIX A
Equation 19 may be rewritten
on integration. The bracketted term in Equation A4 acquires values in the range 0.980-1.000 for dwt values not less than 1.8. Therefore, recalling the quantification we have attached to the = symbol,
This inequality permits t to be as large as 30 of the RC time constant, which is slightly more liberal than the range permitted by the less exact Relationship 21. APPENDIX B
by expressing the hyperbolic tangent as its exponential equivalent. For large s, the series converges very rapidly. Inversion is possible by utilizing the well-known convolution theorem of Laplace transformation; thereby 46
In designing geometric ladders, advantage may be taken of the fact that the “preferred values” used as standards by component manufacturers form an approximately geometric sequence. Thus, for example, the commonly encountered values 1.0, 1.5, 2.2, 3.3, 4.7, 6.8, and 10 [and power-of-ten
ANALYTICAL CHEMISTRY, VOL. 45, NO. 1, JANUARY 1973
ACKNOWLEDGMENT
multiples and submultiples thereof] form a geometric sequence with ratio loll6. Similarly, ladders with g values of l o l l 3 , 101i4, erc. are readily constructed. Frustratingly, however, the preferred values do deviate significantly from being a perfect geometric sequence. Thus, to two significant figures, 1.0, 1.5, 2.2, 3.2, 4.6, 6.8, and 10 are the numbers which form a sequence with ratio
It is a pleasure to acknowledge the experimental assistance of Marten G~~~~~~~ and the financial assistance of the N ~ tional Research council of Canada. RECEIVED for review July 12, 1972. Accepted August 22, 1972.
Rapid Determination of Matrix Components by Neutron Activation Analysis: The Analysis of Gold Alloys Michel Heurtebise, Francois Montoloy, and J. A. Lubkowitz Centro de Incestigaciones Tecnoldgicas, Instituto Venezolano de Inoestigaciones Cientificas, Apartado 1827, Caracas, Venezuela The possibility of analyzing the macroconstituents of a matrix by using neutron activation without utilizing any standards has been studied, provided all major constituents can be activated and detected. Equations have been derived that directly relate the concentration of each of the constituents to the ratio variations of the accumulated counts due to the activation of these elements present. The general theory has been successfully applied to the determination of gold alloy constituents. A relative standard deviation of 0.92, 3.3, and 0.91% has been obtained for gold, silver, and copper determinations, respectively. Average deviation of the results obtained by this method and atomic absorption was 0.58%.
ACTIVATION ANALYSIS is most commonly used in trace analysis. In this case, standards are generally prepared and irradiated simultaneously with the samples being determined. Alternatively, it is common practice to incorporate a reference material in the sample. This technique presents the advantage over the former method in that errors due to flux inhomogeneity are avoided. Differences due to counting geometry are also minimized. Nevertheless, this method is dependent on standard preparation. In an earlier paper, it has been shown how to eliminate errors due to counting geometry, flux inhomogeneity, and standard preparation, by utilizing a matrix component as a standard ( I ) . This technique can be used only in case of trace analysis when the quantitative composition of the macroconstituents is constant. Thus, this method is inadequate for the determination of macroconstituents which vary from sample to sample. Other techniques, for the analysis of macroconstituents in alloys, based upon isotopic dilution after activation have been described (2). Although this technique does not require standards, it has the disadvantage of requiring radiochemical separations which may not be necessary using conventional methods. This paper describes a novel technique for the rapid determination of the macroconstituents of a matrix. If after an appropriate irradiation and decay period of a qualitatively known matrix, the y-activities due to activation of each and every constituent can be observed, then the ratio (1) M. Heurtebise and J. A. Lubkowitz, ANAL.CHEM.,43, 1218 ( 1971). (2) C. Capadona, “Modern Trends in Activation Analysis,” J. R. De Voe, Ed., Nat. Bur. Stand. (US.)Spec. Publ., 312, Volume I, 574 (1 968).
of the accumulated counts of the different isotopes will be a function of the matrix composition. A study of the variations of the ratios of accumulated counts due to the activation of the elements present in the sample has permitted the development of a mathematical model that relates this ratio variation with composition. The principles developed in this work have been successfully applied to the analysis of gold alloys employed in the manufacture of coins and ornamental jewelry. Das and Zonderhuis (3)have considered the problems that occur in analyzing samples containing gold, silver, and copper, such as high cross sections and high sample density which cause flux depressions in and around the sample. In such cases, the comparison of the induced activity with standard foils of the pure metals yields incorrect results and corrections have to be made. The work described here shows that this method is independent of flux depression and self neutron shielding effects at least for a homogeneous matrix. Mixtures of gold and copper have also been analyzed previously by activation analysis ( 4 ) although no quantitative data have been given. THEORY
The following reactions are considered during the irradiation of a matrix constituted by two elements:
‘E (n,y) ‘+lE where the first element has a cross section ul,decay constant XI, isotopic abundance el, number of atoms N I . The induced activity immediately after irradiation is Aol. For the second element present in the matrix, the following reaction is considered :
yF (n,y) y+lF where this second element has a cross section u2, decay constant X2, isotopic abundance &, number of atoms N 2 . The induced activity immediately after irradiation is A,>. The ratio of the activities of ‘+lE and Y f l F immediately after irradiation is given by: A,, _ = u1 _ . _a . -Nl. [1 - e-”’’] (11 1 A,, u2 0 N2 where ti is the irradiation time.
___
(3) H. A. Das and J. Zonderhuis, Rec. Trav. Chim., 85, 837 (1966). (4) M. Okada and Y.Kamemoto, Nature, 197, 278 (1963).
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