Controlled-PotentiaI DifferentiaI DC Polarography Determinate and Statistical Errors in Comparative Polarography, Theory and Experiment W. D. Shults Analytical Chemistry Division, Oak Ridge National Laboratory, Oak Ridge, Tenn. 37830
W. B. Schaap Department of Chemistry, Indiana University, Bloomington, Ind.
Expressions for the determinate error and the statistical error associated with comparative polarographic determinations are derived, and theoretically predicted errors are compared with experimentally observed errors. An equation for the optimum compensation fraction, f = CR/C,, is also suggested; it may be used as a guide in t h e design of comparative polarographic procedures.
COMPARATIVE POLAROGRAPHY is a differential polarographic technique that can be used to obtain very accurate and precise analytical results. The technique involves the measurement of the small difference between two diffusion currents, one due to the electroactive species of interest in an unknown solution and one due to the same electroactive species (present in accurately known concentration) in a similar reference solution. This comparative technique has been studied by cathode-ray differential polarography utilizing a dual-cell apparatus that has two DME‘s with synchronized drop times (1-9). Comparative polarography has also been evaluated (IO) with a controlled-potential differential directcurrent polarograph ( I 1) that does not require drop-time control, and that makes possible a single-cell mode of operation as well as the dual-cell technique. In both casesoscilloscopic and direct-current comparative polarographyanalytical results of accuracy and precision of 0.1 have been obtained under optimum experimental conditions. Although comparative polarography has been evaluated thoroughly, and is being used to an increasing extent, no attention has been given specifically to the determinate and statistical (1) P. Valenta and J. Vogel, Collection Czech. Chem. Commun., 21, 502 (1956). (2) J. Vogel, Dissertation, Karlsuniversitet, Prag, 1952. (3) J. Vogel and P. Valenta, Chem. Listy, 49, 361 (1955). (4) H. M. Davis and J. E. Seaborn, “Advances in Polarography,” I , I. S . Longmuir, Ed., Pergamon Press, Oxford, 1960, pp. 239-50. U. K. At. Energy Authority Rept., RJ3472, September 1960. ( 5 ) H. M. Davis, Chem.-Zng.-Tech.,37, 715 (1965). (6) H. M. Davis and R. C . Rooney, J. Polarog. SOC.,8 (2), 25 (1962). (7) H. M. Davis and H. I. Shalgosky, “Advances in Polarography,” IZ, I. S. Longmuir, Ed., Pergamon Press, Oxford, 1960, pp. 618-27. U.K. At. Energy Authority Rept., R13413, September 1960. (8) H. I. Shalgosky, R. C . Smart, and J. Watling, U. K. At. Energy Authority Rept., RJ4270, June 1964. (9) H. I. Shalgosky and J. Watling, Anal. Chim. Acta, 26, 66 (1962). U. K. At. Energy Authority Rept., RJ3590, February 1961. (IO) W. D. Shults, D. J. Fisher, and W. B. Schaap, ANAL.CHEM., 39, 1379 (1967). (11) W. D. Shults, D. J. Fisher, H. C . Jones, M. T. Kelley, and W. B. Schaap,2.Anal. Chem., 224, l(1967).
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ANALYTICAL CHEMISTRY
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errors appropriate to the technique. Furthermore, only Shalgosky ( 9 ) has attempted to use a mathematical expression as an aid in the selection of the optimum experimental condition-Le., the optimum reference concentration. These are the items to which the present paper is addressed. We present here an analysis of the determinate error and the statistical error associated with comparative polarographic analysis, and some data to support the derived expressions. We also derive an expression for the optimum compensation fraction, f (E CR/C,), an expression that may serve as a guide to the optimization of comparative polarographic procedures. Our results are general and hence apply to either cathode-ray or direct-current comparative polarography. The expressions presented in the following sections are stated in terms that are empirical in nature. This is necessarily so if the expressions are to be useful in various laboratories and with various apparatus. Also, it is assumed for mathematical purposes that the various empirical parameters are independent of each other, an assumption that is not true in fact. The errors introduced by the invalidity of this assumption are small, however, as evidenced by the good agreement that is obtained between theory and experiment. EXPERIMENTAL DETAILS
The equipment, reagents, and procedures used to obtain the experimental results reported in this paper have been described fully in previous papers in this series (IO, 11). THEORETICAL CONSIDERATIONS
Analysis of Determinate Errors. The unknown concentration is computed according to
after experimentally determining Aid and (id)R.In this equation, Cu is the unknown concentration, C R is the reference concentration, (id)Eis the polarographic diffusion current corresponding to the reference solution, and Aid is the differential diffusion current corresponding to the difference between the diffusion current due to the unknown and that due t o the reference, i.e., (i& - ( i d ) R . One may write the following expression for small changes in
CC.:
The partial derivatives can be evaluated from Equation 1 and substituted in Equation 2 to yield
(3)
tration is zero and the comparative technique has become the subtractive technique (compensation is for the residual current only). Equation 5 may also be written
An expression for the relative determinate error is now obtained by dividing Equation 3 by Cu--i.e,, by Equation 1and simplifying the quotient; the result is
(id)R
+ Aid
(4)
It is desirable to express the relative determinate error in Cu in terms of,f, the compensation fraction. The following identities may be invoked:
In Equation 6, dcR/cR is the relative determinate error in CR, dAid/Aid is the relative determinate error in the differential diffusion current measurement, and d ( i d ) R / ( i d ) R is the relative determinate error in the reference diffusion current measurement. This expression is useful for estimating approximately the maximum relative determinate error in Cu from experimentally obtained information, as shown in a following section. Analysis of Random Errors. The fundamental equation of comparative polarography is
CL,= CR -k Ac
and for which
nu2 =
Substitution of these identities into Equation 4 yields dC0 - dcR + f d A i d CV CR
- (1 - f ) d(id)R (id)R
which shows that the relative determinate error in Ct7comprises two factors, one being the relative error in C R and the other being the measurement error. Only the maximum measurement error can be specified since both positive and negative errors in the measured variables are likely. This is achieved by employing the absolute values of the terms in the measurement error. Hence,
UR2
+
(7 )
(TAC'
In this equation, uLr2, uR2,and U& refer to the variances of the values of CU,CR,and AC (E Crr - C R ) , respectively. Equation 7 also may be written in terms of relative standard deviations, which for convenience is given the symbol D . [ D x = 100 u X / X E 100 sxlX; sx is the standard deviation associated with the measurement of X.]Thus, DRCR
or Uu2 =
+
1 0 - 4 [ D ~ 2 c ~ DAC'(CL~ 2 - CR)']
Substitution for CRin terms off yields Uu2 =
Equation 5 says much about the effect of experimental conditions upon the accuracy of comparative polarographic analysis. First of all, the relative determinate error in the unknown concentration can never be less than the relative determinate error in the reference concentration. Second, the measurement error decreases with increasing ( Q R ; the use of a large reference concentration is implied. Third, increasing the compensation fraction, f , places increased emphasis upon the effect of the error in Aid upon the measurement error. At f 0.9, the measurement error (and hence the relative determinate error of C,) is closely approximated by dAid/(id)R. In the limit, as f approaches unity, the measurement error approaches the balance error because at f = 1 , the unknown and reference solutions are of equal concentration. In single-cell comparative polarography, the balance error is the inherent uncertainty involved in compensating ( i d ) R , rinsing and refilling the cell with the same reference solution, and deoxygenating the solution. In dualcell comparative dc polarography, the balance error is the inherent uncertainty in the channel 2 gain adjustment after proper gain adjustment has been made and the cells are emptied, rinsed and refilled with the same reference solution, and then deoxygenated. The fourth and final point to be made from Equation 5 is that decreasing compensation fraction, f,places increased emphasis upon the error in (id)R. In the limit, as f approaches zero, the measurement error approaches the relative error in the diffusion current (id)R. This is to be expected since at f = 0, the reference concen-
>
+
1 0 - 4 c v ' [ D ~ 2 f 2 D A C ' ( ~- f ) ' ]
].e., Dc'
=
DRY'
+ Dac'(1
-f)'
(8)
In any comparative technique, the concentration of the reference solution is assumed to be known exactly. Determinate errors are possible, but statistical variation is not allowed, in the reference concentration. This means that randomness in the analytical result is due solely to variations in the quantity that is determined differentially (e.g., in AC). Hence, DR = 0 and Equation 8 becomes Du2 = D A ~ ('1
- f)'
(9)
The same result is obtained by setting uR = 0 in Equation 7. The differential concentration, AC, is computed from experimental quantities according to
The precision of AC can be expressed on a relative basis, namely
Converting to relative standard deviations and setting uR = 0 yields an expression for DAC, DAC'= Datd2
+ D(id)R2
VOL. 39, NO. 12, OCTOBER 1967
1385
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,
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I
I
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the precision of the analytical result is determined to an increasing extent by the precision of the measurement of (QR. The optimum value o f f will be determined by the relative magnitudes of U A and ~ ~ a ( i d ) R . Once these two parameters are determined experimentally, it is possible to estimate what f-value is optimum for the particular procedure and operating conditions. This is achieved graphically by plotting 104f2u~idZ as.fand lo4(1 - f ) 2 u ( i d ) R cs.2j'on the same axes and then summing the two curves at eachf-value. The resulting curve depicts the variation in the bracketed term of Equation 12 with J; which is analogous to the variation of D C 2 with f. Exemplary curves are shown in Figure 1 for the assumed values a~~~ = 0.005 pa and a ( i d ) R= 0.01 pa. The optimum value off corresponds to the minimum in the curve, about 0.8 in the example in Figure 1 . Similar curves can be constructed with BAid and u ( t d ) having R various values. Inspection of such curves reveals that, for equal U A and ~ ~ acid),, the minimum value of Dv occurs at f = 0.5. As U A decreases ~ ~ with respect to u ( t d ) the , minimum in the Du us. f curve shifts to larger f-values and the value of DLIbecomes smaller (improves). An expression for the optimum value (fo) of f can be obtained by differentiating Do (Equation 12) with respect tof, setting the derivative equal to zero, and solving for f . The expression is
I
0.9
0.8
\
0.7 n k
I
cu a..
0.6
b-
N
7
0.5
+ .P 0.4
ba N
2 2 0.3 0.2
0.4 0 COMPENSATION FRACTION, f
or
Figure 1. Graphical interpretation of Equation 12 which may be combined with Equation 9 to obtain
+ D(td),'I
- f)'[ D ~ t d '
Du' = (1
(10)
An expression for Datd in terms o f f and ( i d ) R is desired for substitution in Equation 10. By definition,
and
Optimum f-values can be selected more conveniently by utilization of Equation 13 than by the graphical technique metioned above. The foregoing considerations indicate that comparative polarographic analytical procedures should be designed with two variables in mind. First, the procedure and experimental conditions should be such that
>> UAid
a(td)~
so that
Substitution of Equation 11 into Equation 10 and simplification yields an expression for the relative standard deviation of the analytical result, CO,in terms of variables that can be assessed experimentally : Du' =
104 (ZdR
If'
Q~td'
+ (1 -f)*
(12)
Equation 12 gives some insight into the effect of procedural design upon the precision (Le., random error) of the analytical result. The relative standard deviation of Cv decreases (improves) with increasing (i&; hence it is preferable to use a reference solution of relatively large concentration. As f approaches unity (CR+C,), the precision of the analytical result is determined increasingly by the precision with which the balance can be adjusted (see discussion in preceding section). On the other hand, as f approaches zero (CR-.O), 1386
ANALYTICAL CHEMISTRY
and, second, an appropriately largef-value should be utilized. When these conditions are met, utmost precision is achieved (see Equation 12) and utmost accuracy is obtained (see Equation 6). It is to be emphasized, however, that all these parameters-j; ( i d ) R , and A i d , and hence g ( i d l Rand uAid-are interrelated. The assumption of constant a's is not strictly valid (see Tables I and 11). Consequently, while the foregoing treatment can prove quite useful, it must not be regarded as anything other than a guide to the design of comparative polarographic procedures. EXPERIMENTAL RESULTS
Equations 6 and 12 express the maximum determinate error and the statistical error, respectively, in terms of quantities that can be estimated or at least approximated experimentally. For example, the relative error in CR, dcR/cR, can be estimated from knowledge of the uncertainties involved in preparing the reference solution. The relative errors in Aid and (id)E can be estimated from statistical information by assuming that the absolute error in a measured variable is represented by twice the random error associated with that measured variable.
Summary of Results Obtained When Various Compensation Fractions Were Used in Single-Cell Comparative DC Polarographic Determination of 10-3M Cd(I1) (Medium: 1.OF KC1-0.001F HC1-0.001% Triton X-100, oxygen-free. f = CR/CU) Compensation CU,mM Standard deviation No. of runs fraction, f Taken Found (id)R, A i d , pa SL,, mM s(%~)R pa, SAld, l a 0.0017 6.07 0.1213 0.022 0.009 1 ,0225 5 0.983 1.0207 0.3388 0.0012 0.017 0.009 1.0034 5.75 5 0.945 1.0031 0.452 0.0010 0.013 0.006 1.0221 5.74 6 0.928 1 ,0207 1.705 0.0027 1.0417 4.64 0.007 0.020 6 0.732 1 ,0420 3.78 2.551 0.0028 0.010 0 016 1.0420 1.0429 6 0.597 Table I.
Table 11. Summary of Results Obtained When Various Compensation Fractions Were Used in Dual-Cell Comparative DC Polarographic Determination of 10-3M Cd(1J) (Medium: 1.OFKC1-0.001F HC1-0.001% Triton X-100, oxygen-free. f = C&U) Compensation CU m M Standard deviation No. of runs fraction, f Taken Found ( i d ) R , pa Aid, l a SLT, mM S(id)R, SAid, l a 0.015 0.014 0.1098 0.0023 6.10 1.0207 1.0212 9 0.983 0.0020 0.016 0.011 0.3316 1.0024 5.76 9 0.945 1.0031 0.016 0.005 0.4429 0.0008 1.0207 5.76 9 0.928 1 ,0207 0,023 0.0053 0.017 4.65 1.696 1 .0420 1.0413 6 0.732 0.012 0.0029 0.010 3.78 2.552 1.0431 5 0.597 1,0420 Table 111. Comparison of Computed and Experimentally Observed Values of Determinate and Random Errors (Determination of -1O-3M Cd(1I) i n oxygen-free 1.OFKC1-0.001F HC1-0.001% Triton X-100 solution) Determinate error, Random error, (relative error) (relative standard deviation) Compensation Computed by Computed by Technique fraction,f Observed Eq. 6 Observed Eq. 12 Single-cell 0.983 + O . 18 0.28 0.17 0.15 0,945 +0.04 0.34 0.12 0.15 0.928 $0.14 0.24 0.10 0.10 0.732 -0.03 0.73 0.26 0.32 0.597 $0.09 0.74 0.27 0.27 Dual-cell 0.983 +O. 04 0.46 0.22 0.22 0.945 -0.07 0.42 0.20 0.18 0.928 0.00 0.22 0.08 0.08 0.732 -0.07 0.94 0.53 0.38 0.597 f O . 10 0.61 0.28 0.22
Thus
and
The maximum relative error in Cc, (dCLr/Cu)max, can be estimated from this experimental information by means of Equation 6. Similarly, the relative standard deviation, Dc, appropriate to a given procedure can be computed by Equation 12 using experimental estimates (s2) of the required variances (u*). Tables I and I1 present summaries of the results that were obtained when lO-3M cadmium(I1) solutions were analyzed by single- and dual-cell comparative dc polarography with various values of compensation fraction, f. The medium was oxygen-free 1.OF KC1-0.001F HC1-0.001 Triton X-100. The data presented in Tables I and I1 were used to construct Table 111. Table I11 allows comparison of the experimentally observed determinate and random errors with the respective computed values, at different compensation fractions. For
z
computation of the maximum relative error according to Equation 6, a value of d c n / c Rof 0.0002 (i.e., 0 . 0 2 z )was estimated and the assumptions mentioned above were invoked. The contribution of the measurement error (the second term in Equation 6) far outweighs the error in CR as evidenced by the large computed values for maximum relative error compared to 0 . 0 2 z . Because it is the maximum relative error that is computed, the experimental determinate errors are expected to be smaller than the corresponding computed values. This was observed to be true (determinate error columns of Table 111). The agreement between experimental and computed values for the random error (relative standard deviation) of CV also is considered to be excellent at allf-values, especially in view of the fact that the experimental values are estimated from statistical samples of only moderate size. It is, of course, irrelevant to be able to compute the precision from experimentally observed variables when it can be estimated from the results themselves. The comparison in Table I11 is made in order to lend support to the treatment given in the preceding section. The real value of Equations 6 and 12 is that they may be used as guides for the optimum design of comparative polarographic procedures. VOL. 39, NO. 12, OCTOBER 1967
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In conclusion, we comment upon the statistical error relationship derived by Shalgosky and Watling (9). Their Equation 1 in reference 9, using our symbology, relates the ratio of the standard deviation of the unknown and differential concentrations as follows:
In other words, Dr(CR
+ A C ) = DcCr = D AcA C
This says in effect that the standard deviation of a polarographic (or differential polarographic) determination is inversely and linearly related to the concentration (or differential concentration) being determined. This is untenable in practice. Polarographic analysis is characterized by virtually constant precision over the concentration range of approximately lo-* tu 10-4M, but at smaller concentrations
the precision increases markedly (see, for example, the data in Table I of reference 12). Accordingly, one would expect the Shalgosky-Watling expression to be a useful guide to the selection of a reference solution for a comparative measurement only when the concentration and differential concentration being determined exceed lO-4M or so. Comparative polarographic procedures generally specify unknown concentrations of a t least this magnitude (9), but a t large values off the differential concentration may be significantly smaller than 10-4M. Hence, the Shalgosky-Watling equation must be considered only as a very approximate guide to the design of comparative polarographic procedures. RECEIVED for review June 1, 1967. Accepted August 1, 1967. Research sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corp. (12) W. D. Shults, D. J. Fisher, and W. B. Schaap, C/zem. h s f r . , i n press.
Binomial Distribution Statistics Applied to Minimizing Activation Analysis Counting Errors P. C. Jurs and T. L. Isenhour Department of Chemistry, Unicersity of Washington, Seattle, Wash. 98105
To investigate the statistical nature of the minimum theoretical error in activation analysis, the binomial distribution has been applied to radioactive decay. Equations a r e derived for radioactive counting cases involving constant background, decaying interferences, known and unknown decaying mixtures, and simultaneous and sequential comparative activation analysis. T h e iterative computer programs required to solve these equations a r e presented and discussed along with examples of their application to the resolution of complex radioactive mixtures and the evaluation of the statistical feasibility of analyses dependent o n radioactive decay.
MINIMIZING ERROR is a goal in any analytical determination. In the particular case of radioactive counting, the statistical nature of nuclear decay processes requires proper mathematical treatment if the maximum accuracy is to be attained. This work evaluates the validity of the statistical assumptions commonly employed in radioactive counting experiments, and presents mathematical expressions useful for minimizing error in activation analysis. The binomial (or Bernoulli) distribution which describes radioactive counting statistics in general (1-5) can be expressed as the general term of the binomial expansion ( p 4)” in the form
+
(1) P. C. Stevenson, “Processing of Counting Data,” NAS-NS3109, (1966). (2) G. Friedlander. J. W. Kennedy. and J. M. Miller, “Nuclear and Radiochemistry,” 2nd ed., Wiley, New York, 1964. (3) L. J. Rainwater and C. S. Wu, h’ircleonics, 1, No. 2, 60 (1947) and 2, No. 1, 42 (1948). (4) A. S . Goldin, “Radioactivity Statistics,” U. S. At. Energy CO117117. WIN-120, 1960. ( 5 ) A. H. Jaffey, Nucleonics, 18, No. 11, 180 (1960). .
p
=
n! -pa$ a!b!
where p is the probability that an event will occur, q is the probability that an event will not occur, and P is the overall probability that an event will occur a times, and fail to occur b times in n trials, where n = a b. The mean or expected value of a is
+
E(a)
=
(2)
np
and the variance, which is the square of the standard deviation, is var(a)
= ua2 =
npq
(3)
The binomial distribution, Equation 1, can be approximated by the Poisson distribution when n is large, p is small, and c1 is much less than n. For cases where the Poisson distribution is a valid approximation t o the binomial distribution, and a is quite large, the Poisson distribution may be well approximated by the Gaussian, or normal, distribution. For the Poisson and the Gaussian distributions, the expected value of a is the same as for the binomial distribution (Equation 2), but the variance of a is var(a)
= ua2 =
np = E(a)
(4)
To use any of these three distribution functions, the probability for an individual event must be formulated analytically. The basic law expressing the decay of radioactive substances is
I
1388
ANALYTICAL CHEMISTRY
where X = In 2/tiiz and N = the number of radioactive nuclei present. Equation 5 contains the assumption that the prob-
