110
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
Although the surface-active agents generated on less than exhaustive ozonolysis of samples high in organics forced us to modify the ASV standard additions method somewhat, it is likely that this method will be completely acceptable to those using solvent extraction or chelating resin preconcentration techniques prior to the analysis of metals by other instrumental methods. As an outgrowth of the foregoing work, we are now building a 5-tube ozonizer which will permit a much higher sample through-put and will be powered with a 60-kV transformer. T h e tubes will be wired in series so the voltage drop across each will be 1 2 kV. The next logical extension of the method is to the digestion of solid or liquid biological samples. Nitric acid rarely has sufficient oxidative capacity to be used alone; the presence of H2S04and/or HC104 is required. However, calcium is precipitated in the presence of the former and potassium in the presence of the latter. I t seems likely that if the sample were dissolved in warm concentrated nitric acid, then subjected to ozonolysis, complete digestion would occur. If the sample were not being prepared for analysis with ASV, then oxidation catalysts such as Ag or Hg could be included. The problems associated with the nonvolatile acids could thus be circumvented.
ACKNOWLEDGMENT T h e authors thank G. V. Shalimoff for making the spectroscopic measurements and D. Mack for supporting this effort. LITERATURE CITED (1) R. G. Clem and A. F. Sciamanna, Anal. Cbem., 47, 276 (1975). (2) J. P. Riley and D. Taylor, Anal. Cbim. Acta. 40, 479 (1968).
(3) W. G. Cox in "Proceedings of the Specialty Conference on Dredging and Its EnvironmentalEffects", American Society of Civil Engineers, wblisher, New York. 1976, pp 242-252. (4) W. R . Matson. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 1968. (5) F. A. J. Armstong, P.M. Williams. and J. D. H. Strickland. Nature(London), 211, 481 (1966). (6) R. G. Clem, Anal. Cbem., 47, 1778 (1975). (7) C. C. Patterson and D. M. Settle, "7th Materials Research Symposium", National Bureau of Standards, Gaithersburg. Md., October 7-1 1. 1974. (8) H. H. Willard and L. L. Merritt, Id.Eng. Chem., Anal. Ed., 14, 486, 489 (1942). (9) J. G. Calvert and J. N. Pitts, "Photochemistry", Wiley, New York, N.Y., 1967. (IO) F. L. Evans, "Ozone in Water and Wastewater Treatment", Ann Arbor Science, Ann Arbor, Mich., 1972. (11) H. R. Eisenhauer, J . Water follut. Control Fed., 40, 1887 (1968). (12) E. Bernatele and C. Frengen, Acta Cbem. Scand., 15, 471 (1961). (13) P. S. Bailey, Cbem. Rev., 58, 925 (1958). (14) P. S. Bailey and J. E. Keller, J . Org. Cbem., 33, 2680 (1968). (15) P. S. Bailey, J. E. Keller, D. A. Mitchard, and H. M. Wite, Adv. Chem. Ser., 77, 58 (1968). (16) P. S. Bailey, J. E. Keller, and T. P. Carter, J. Org. Chem., 35, 2777 (1970). (17) P. S. Bailey, D. A. Mtchard, and A. Y. Khashab, J . Org. Chem., 33, 2675 (1968). (18) L. Horner, H. Schaefer, and W. Ludwig, Chem. Ber., 91, 75 (1958). (19) F. J. Welcher, Ed., "Standard Methods of Chemical Analysis", Vol. 2, part 8. Van Nostrand, New York, N.Y., 1962, p 2461. (20) J. J. Alberts, J. E. Schindler, D. E. Nutter, and E. Davis, Geocbim. Cosmochim. Acta, 40, 369 (1976). (21) A. Nissenbaum and D. J. Swaine, Geocbim. Cosmochim. Acta, 40, 809 (1976). (22) M. D. Ahmed and C. R. Kinney, J . A m . Cbem. SOC.,7 2 , 559 (1950). (23) F. Dobinson and G. J. Lawson, Fuel, 38, 79 (1959). (24) R. Ernst, H. E. Alien, and K. H. Mancy, Wafer Res., 9, 969 (1975). (25) R. L. Malcolm, J . Res. U . S . Geol. Surv., 4, 37 (1976). (26) J. L. Kassner and E. E. Kassner, Ind. Eng. Cbem. Anal. Ed., 12, 655 (1940). (27) A. Zirino and S. H. Lieberman, Adv. Cbem. Ser., 147, 82 (1975).
RECEIVED for review June 28, 1977. Accepted October 10, 1977. Work supported by the U S . Energy Research and Development Administration.
Numerical Differentiation by Fourier Transformation as Applied to Electrochemical Interfacial Tension Data Robert de Levie, * Srinivasan Sarangapani, and Philip Czekaj Department of Chemistty, Georgetown University, Washington, D.C. 20057
George Benke Department of Mathematics, Georgetown University, Washington, D.C. 20057
Numerical differentlatlon of experlmental data must be accompanied by data smoothing, since differentiation greatly enhances the effect of random experimental error. Here we describe a numerical method which performs both the differentiation and the data smoothlng on an arbitrary set of equidistant data by Fourier transformation after subtraction of a polynomial chosen to avoid truncation errors. In the examples given, an effectlve amount of smoothing can easily be selected with a simple criterion based on the power spectrum of the original data set.
The numerical differentiation of experimental data presents some interesting problems. By their very nature, experimental data sets are finite collections of discrete numbers. This may be self-evident, as with a tabulation of experimental obser0003-2700/78/0350-0110$01.OO/O
vations, or hidden, as with the strip-chart recorder output of a spectrophotometer with non-zero monochromator slit-width. Clearly, a derivative is not defined for a finite collection of points. If we assume that the experimental data are more or less representative samples of a n underlying, continuous function, then the problem is to reconstruct that continuous function. This is usually achieved by fitting the available data to a continuous analytical function, such as a polynomial, which can then be differentiated algebraically. The two operations can be combined into a single, moving convolution (1). algorithm In fitting a n analytical function through a set of discrete data, a compromise must be made. The experimental data will contain errors, due to their discrete nature (truncation or round-off errors) if not also due to experimental artifacts. An analytical function fitting the data set exactly would include such errors, which will affect the derivative. On the 1977 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
other hand, systematic distortion may result if the “noise” in the experimental data set is filtered out by too constrained an analytical function. The trade-off involved, of distortion vs. noise, is not always transparent. For example, if the data are fitted with a moving least-squares polynomial ( I ) , the order of the polynomial and the length of the moving patch fitted to the polynomial strongly influence the result, yet there does not seem to be a general scheme by which a rational choice of polynomial order or patch length can be made. In the present communication we will use a method in which the amount of noise rejection can be chosen on a rather rational basis, and in which no arbitrary choice of fitting function (parabola, cubic etc.) need be made, except a t the extremes of the data set. In order to understand the principle of the method, let us first consider what happens when one “eyeballs” a curve through a set of data points. When we make a judgment of what the underlying, assumedly smooth curve should be, we tend to ignore the short-period fluctuations in the data set but to retain its long-term trends. Such a procedure is akin t o a Fourier transformation of the data set and rejection of its high-frequency components. T h e Fourier transform of an integrable function f ( x ) is defined as
and the Fourier transform of derivatives, when they are themselves integrable functions, is given by (2)
F
1%\
= ( 2 ~ j sF){~f ( x ) }
where j = v q . Consequently, once the Fourier transform of a function has been obtained, m-fold differentiation can be achieved merely through multiplication by (2.irjsjm. Subsequent inverse transformation will then yield dmf(x)/dxm. Thus, when Fourier transformation is used for the purpose of smoothing, there is the additional convenience that differentiation of the original function can be replaced by a simple multiplication in the Fourier-transformed domain (3). Fourier transformation of a finite set of discrete data is performed most efficiently using one of the “fast” discrete Fourier transform algorithms ( 4 4 ) ,which can be made to be a close approximation of the continuous Fourier transform (7). One of the main sources of discrepancy lies in the fact that the fast Fourier transform algorithm treats a finite data set as if it were a repeat unit of a periodic function extending from x = -a to x = f a . The repetition of an arbitrary data set will, in general, exhibit discontinuities at the junctions of the repeat units, which correspond with high-frequency components and may cause large truncation errors in the Fourier transform. This problem can be avoided in special cases, such as with absorption and emission spectra ( 3 , 8 ) ,by choosing the data set such as to begin and end in a stretch of baseline. I n general, however, data sets do not start and terminate with zero values and zero-valued derivatives, so that Equation 2 cannot be used directly. However, since both Fourier transformation and differentiation are associative, one can subtract from the original data set, S(x), an analytical function, P ( x ) ,such that the remainder, R ( x ) ,is zero-valued and has zero-valued derivatives a t its beginning and ending (9). The analytical function, P ( x ) ,can then be differentiated analytically, the remainder, R ( x ) , by Fourier transformation, and the two derivatives added to yield the derivative of the original function:
d m S ( x )dmP(x) dx” dxm
+
dmR(x) dxm
(3)
Table I. Coefficients aiof the Polynomial u Used to Represent Synthetic Surface Tension Data, in p J cm- *
=
c
:=
111
aiE’
a, = 42.0000000 a, = 0.00000000 a , = -4.75155549 a3 = - 0.796049021 a4 = -12.0782515 a i = -4.67943267 a6 = t 15.3273189 a7 = +8.35398806 a 8 = -3.07552677
when S(x) = P ( x ) + R ( x ) . A requirement that R(xj and its first m derivatives are zero at its first and last points imposes 2m + 2 independent restraints on the analytical function, P ( x ) , which can be met, e.g., by a polynomial of order 2m + 1:
P ( x )=
2m+l
c aid
i=O
(4)
The coefficients, a,, can be determined from, e.g., least-squares polynomial fits to the first and last few data points in the set,
S(x). When m is much smaller than the number of data in the set, virtually all of the experimental noise will remain in R ( x ) , and can be filtered out; see below. Application to S y n t h e t i c S u r f a c e Tension Data. We will illustrate the method with electrocapillary data, Le., surface tension data as a function of applied electrode potential. At the interface between an ideally polarizable liquid metal electrode and an electrolyte solution, the surface tension, u , can be related to the electronic charge density, Q , and the capacitance (per unit area), C, by the thermodynamic relations
where E denotes the potential of the metal electrode as measured against a suitable reference electrode, the differentiation being made at constant temperature, pressure and chemical composition. The parameters u , Q , and C are all directly measurable quantities and especially CT and C can be measured with high precision. In order to test whether the method will indeed yield the correct first and second derivatives of numerical data, and to have an objective measure of any distortion it may introduce, we will first use a set of artificial surface tension data. For the purpose of differentiation, electrocapillary curves have been represented in the past by polynomial expressions of fixed (10) or variable (11) order, and we will do likewise by using an eighth-order polynomial, see Table I, selected such that it has a realistic magnitude and (near-parabolic) shape, and has a second derivative exhibiting considerable detail. The polynomial was used to generate 128 equidistant points, spaced 10 mV apart. This synthetic data set was differentiated by using Equations 2 and 3, and the results were compared with those obtained by direct, analytical differentiation of the original polynomial used to generate the data set, S ( x ) . The differences found for the first and second derivatives, A Q and X, respectively, were listed, and their standard deviations, {Z(SQ)2/N)1/2 and { Z ( X ) 2 / N 1 ’ ’ 2calculated. , T h e beneficial effect of the subtraction of P ( x ) is most dramatically seen in the capacitance: direct double differentiation of S(x) leads to a standard deviation of almost 4000 pF cm-2, whereas subtraction of a polynomial, P ( x ) , with m = 0, 1,and 2 respectively, leads to standard deviations of 147, 0.3, and 0.01 pF cm-* respectively. In terms of relative errors, 1 pF cm-’ corresponds with an error of about 4%. Direct
112
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
7
i
i
61 lob P~
5t
! I t9
3
t* I *
-I
t
-2
t
-3t
1 0
-1
j
IO
20
30
40 + 5 0
60
Figure l a . Power spectrum of the synthetic surface tension data ( 0 ) and of the same rounded to four significant figures (x). The power, p , is in units of V2 pJ2 ~ m - the ~ , "frequency", s, in V-'
experimental determination of C is seldom more accurate than f O . l pF cm-2. Inspection of the individual values of AQ and AC shows, that by far the largest errors occur at the extremes of the data set. Such edge errors can be reduced greatly by very minor modifications in the coefficients used in P ( x ) ,and thus reflect errors in the estimates (using a five-point unweighted least-squares fit to a parabola) of value, slope, and second derivative at the first and last points of the data set. If only the central 116 (out of the total of 128) points are considered, the standard deviation in C is reduced to 0.0004 pF ern-'. Clearly, the polynomial subtraction, especially for m > 0, is very effective in reducing the distortion introduced by truncation. Rejection of Round-Off Errors. The real test of any practical differentiation procedure lies in its ability to reject a maximum of experimental "noise" with a minimum of resultant distortion. We must therefore introduce realistic noise into our synthetic data set. T h e determination of surface tension is based, either directly or through calibration, on the measurement of mercury height, which is usually done by cathetometer. This limits the resolution to about hO.01 cm. Since the total pressure used is typically of the order of 40 cm of mercury, this results in a round-off error of about 1 in 4000. Such an error can be simulated in our synthetic surface tension data by rounding the numbers in the data set to four significant figures. I n this case, subtraction of a polynomial, P ( x ) , though somewhat beneficial, fails to yield acceptable results: even with m = 3, standard deviations in charge density and capacitance of 0.7 pC cm-' and 155 p F ern-', respectively, are obtained. Clearly, the round-off noise interferes with the differentiation and needs t o be filtered out. At this point, i t is useful to consider the power spectrum, which depicts the sum of the squares of the amplitudes of the real (in-phase) and imaginary (quadrature) components of the Fourier transform, and therefore represents the squares of the absolute magnitudes of its frequency components. Figures l a , b, and c show the power spectra of S ( x ) , Figure l a in the absence of a subtracting polynomial, P ( x ) , b and c after subtraction of P ( x ) with m = 0 and 1, respectively. Comparison of the unrounded sets illustrates that the high-frequency content is progressively reduced by our polynomial
-4
c I
...
-51 0
IO
20
30
40 5-50
1
60
Figure l b . The same as in Figure l a after subtraction of a straight line connecting the first and last data points, Le., P ( x ) = xI=oa$
i
I
1
Figure IC. The same as in figure l a after subtraction of a third-order polynomial, P ( x ) = x:==oa&'fitted using the first and last five points of the data set. Note that the contribution of the r o u n d 4 noise is hardly distinguishable in Figure l a , but becomes clearly separated from the power spectrum of the noise-free set in Figure IC
subtraction procedure, thus removing the effect of truncation. T h e essentially white round-off noise then becomes increasingly more distinct and, in Figure k, this noise clearly dominates the power spectrum a t all but the lowest few frequencies. The same applies to data obtained with m = 2.
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
The above observation immediately suggests a strategy for efficient noise rejection. All contributions with s > 3 in the rounded data set of Figure ICcontain more noise than signal. Thus, after the polynomial subtraction method has, as it were, uncovered t h e noise, the latter can be removed almost completely by multiplying all contributions for s > 3 by zero, followed by processing of the data array (now containing a number of zeroes) as before. This procedure eliminates more noise than signal, and is therefore beneficial. Calculation of the standard deviations in Q and C verifies that the best fit is obtained when the cut-off frequency of the sharp low-pass filter used is indeed selected a t the point where the initial downward trend in the power spectrum is halted by the near-horizontal noise band. (If the noise were nonwhite, the noise band would not be horizontal, but might follow another trend. The cut-off frequency can still be determined, however, as long as the parts of the power spectrum dominated by signal and noise, respectively, are clearly distinguishable, so that their intersection can be determined.) For m = 2, we obtain the lowest standard = 3, where we find them to deviations for Q and C with,,s be 0.03 (0.02) p C cm-2 and 1.1 (0.23) pF cm-*, respectively, in our example, the bracketed numbers referring to the central 116 data points. Such a result is totally acceptable for the charge density, and marginally so for the capacitance. Comparison with the corresponding standard deviations for t h e unrounded data set shows that, in our example, the resulting errors in Q and C are almost exclusively due to loss in signal rather than to remaining noise. T h e combination of polynomial subtraction, reducing truncation and thus uncovering the noise, with digital filtering of the latter, thus appears to be a viable strategy. In an experimental situation, the uncorrupted, accurate result is, of course, not available for comparison, so that the standard deviation cannot be used as a criterion. However, the cut-off frequency can still be determined from the power spectrum of the data at hand, rather than by some a priori choice. Thus, if the signal-to-noise ratio of the experimental data set is larger, more frequencies are automatically retained in the differentiation procedure. For example, if the synthetic data set is rounded to five (rather than four) figures, the power spectrum suggests,s = 10, which choice of cut-off frequency indeed yields the smallest standard deviations: 0.005 FC cm-* and 0.20 pF cm-2 for m = 2 and the central 116 points. Differentiation of Integrated Capacitance Data. One can criticize the foregoing analysis on the grounds that simulated rather than actual experimental data have been used. This objection can be avoided, while still retaining the possibility of monitoring distortion, by using doubly integrated capacitance data. We have ascertained, by double numerical integration of the analytical second derivative of our eighth-order polynomial, t h a t the errors introduced by the numerical integration are entirely negligible compared with those observed in Figure 2. We have used 128 contiguous data from Grahame's meaM aqueous NaF at 25 "C, since this surements (12,131 of set shows considerable detail, especially around the potential of zero charge, and consequently provides an exacting test case. Double integration of a set of 128 data points was performed in double precision, using Simpson's rule, and subsequent double differentiation followed the procedure outlined before. T h e power spectrum of the doubly integrated capacitance data, after subtraction of the fifth-order polynomial fitting the end points, is shown in Figure 2. Clearly, the contribution of noise becomes dominant beyond about the first 12 frequencies, and elimination of all contributions at higher frequencies yields capacitance data which are compared in Figure 3 with Grahame's original ones. Again, the minimum
2 I.
113
7
'
i
-51
..
-6 -71
.....*' ..,.* *.,..".. .. .*."
a .
-81 0
20
40
.
.*
e: . .
2 : '
s-+
60
Figure 2. Power spectrum of the surface tension calculated from Grahame's capacitance measurements at the Hg/aqueous 1 mM NaF after subtraction of the fifth-order polynomial fitting the interface ( 72), ~ in , V-' extremes of the data set. Dimensions: p in V 2 h J 2 ~ r n - s
t C
Act 0 O
1 ,
0 -
IL-, c5
1
1
1
C
€-Ez-
. I -
-05
1
Figure 3. The difference between Grahame's original data and those reconstructed by double integration followed by double differentiation, as a function of "rational" potential E - € z , where E, denotes the potential of zero charge. LC in pF cm-', E - €, in V
standard deviation (of 0.031 pF cm-:' for the central 116 data points) is obtained with, ,s = 12 as suggested by Figure 2. Differentiation of Experimental Surface Tension Data. As a final test, we have used a set of surface tension data reported by Nakadomari et al. ( 2 4 ) which was obtained with a computer-controlled bubble pressure instrument ( 2 5 ) . The raw surface tension data are reproduced in Table 11, as are the charge densities calculated from these measurements by Nakadomari et al., using a sophisticated moving polynomial fit (16, 17), and by the present method. Although the correct values of the charge density are not known, a measure of their reliability can be obtained by considering their increments, 1Q. These are divided by the corresponding increments in potential, LE = 0.025 V, to provide an estimate of the differential capacitance, C. A comparison of the values of AQ/ S so obtained from the smoothed charge density data of Nakadomari et al. ( 2 4 ) with those of the present study is shown in Figure 4, and clearly shows the advantages of the present method. The corresponding power spectrum of R ( x ) is shown in Figure 5.
DISCUSSION The method used here is conceptually simple. and is readily implemented. I t is based on the premise that, after sub-
114
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
Table 11. Comparison of Moving Least Squares and Present Methoda Nakadomari et al. E,
v
- 1.300
-1.275 -1.250 - 1.225 - 1.200 - 1.175 - 1.150 - 1.125 -1.100 - 1.075 - 1.050 - 1.025 - 1.000 - 0.975 - 0.950 - 0.925 -0.900 - 0.875 -0.850 - 0.825 -0,800 -0.775 -0.750 -0,725 - 0.700 - 0.675 -0,650 - 0.625 -0.600 -0.575 -0.550
OexpyJ
cm-
35.408 35.771 36.171 36.534 36.884 37.220 37.567 37.891 38.190 38.506 38.812 39.094 39.369 39.630 39.880 40.136 40.371 40.598 40.808 41.007 41.206 41.373 41.557 41.714 41.858 41.997 42.119 42.233 42.323 42.408 42.482
Q, IL? cm-15.32 -15.26 -14.92 -14.38 -13.84 -13.70 -13.25 -12.54 -12.41 -12.34 -11.77 -11.16 -10.64 -10.31 - 10.14 -9.76 -9.26 -8.70 -8.27 -7.87 -7.35 -7.07 -6.71 -6.11 -5.66 -5.22 -4.69 -4.05 -3.55 -3.17 -2.63
This study
Nakadomari et al.
cm-2
Q,PC cm-*
C,uF
35.413 35.796 36.169 36.531 36.883 37.226 37.560 37.884 38.200 38.506 38.803 39.089 39.366 39.633 39.889 40.135 40.370 40.595 40.809 41.012 41.204 41.384 41.554 41.712 41.859 41.993 42.116 42.226 42,324 42.410 42.482
-15.60 -15.11 -14.68 -14.28 -13.90 -13.54 -13.17 -12.81 -12.44 -12.06 -11.67 -11.27 -10.87 -10.46 -10.04 -9.62 -9.20 -8.77 -8.34 -7.90 -7.46 -7.01 -6.56 -6.09 -5.62 -5.15 -4.66 -4.17 -3.67 -3.16 -2.64
20.9 18.4 16.6 15.4 14.8 14.5 14.5 14.7 15.0 15.4 15.7 16.0 16.3 16.5 16.7 16.9 17.0 17.2 17.4 17.6 17.8 18.1 18.3 18.6 18.9 19.2 19.6 19.9 20.2 20.6 21.0
0,
PJ
cm-2
E,
v
- 0.525 - 0.500 - 0.475 - 0.450 - 0.425 -0.400 - 0.375 -0.350 - 0.325 -0.300 -0.275 -0.250 -0.225 - 0.200 -0.175 - 0.150 -0.125 -0.100 - 0.075 - 0.050 -~0.025 0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200
0expj PJ
Q, P C
42.541 42.582 42.619 42.637 42.642 42.623 42.594 42.552 42.489 42.401 42.298 42.179 42.032 41.863 41.677 41.449 41.220 40.962 40.681 40.367 40.036 39.681 39.301 38.898 38.484 38.026 37.548 37.050 36.523 35.971
-2.03 - 1.57 - 1.09 - 0.46 0.28 0.92 1.42 2.12 3.01 3.78 4.45 5.33 6.23 7.19 8.28 9.09 9.79 10.78 11.88 12.87 13.77 14.74 15.58 16.36 17.49 18.63 19.54 20.51 2 1.64 2 3.07
cm-’
cm-’
This study 0,
IJ:
cm-
42.542 42.587 42.619 42.637 42.639 42.626 42.597 42.551 42.486 42.403 42.299 42.175 42.030 41.862 41.672 41.458 41.221 40.960 40.676 40.368 40.037 39.682 39.304 38.904 38.479 38.030 37.555 37.053 36.521 35.958
Q, P C cm-2
C, P F
-2.10 - 1.56 -0.99 -0.41 0.20 0.84 1.51 2.21 2.95 3.73 4.54 5.38 6.26 7.16 8.08 9.02 9.96 10.90 11.85 12.78 13.71 14.64 15.57 16.51 17.47 18.47 19.52 20.65 21.89 23.25
21.6 22.2 23.0 23.9 24.9 26.1 27.4 28.8 30.3 31.7 33.1 34.4 35.6 36.5 37.2 37.6 37.8 37.8 37.6 37.4 37.1 31.1 37.3 37.9 39.1 40.9 43.6 47.1 51.7 57.2
cm-’
a Experimental surface tension uexp as a function of applied potential E as reported by Nakadomari et al. f o r the interface between mercury and aqueous 0.1 M Na,SO,. The third column lists smoothed values for the corresponding charge density, Q, as calculated b y Nakadomari et al. ( 1 4 ) . The fourth, fifth, and sixth columns show the smoothed, rounded values of surface tension, charge density, and capacitance, C, derived from the measurements of Nakadomari et al. ( 1 4 ) by the Fourier transform method described in the text, using no, single, and double differentiation, respectively. For convenience of Fourier transformation, the data set shown (consisting of 61 measurements) was extended by extrapolation with the following data: 35.012, 34.606, and 34.192 p J cm-- at - 1.325, - 1.350, and - 1 . 3 7 5 V, respectively. Errors in these extrapolated points as well as errors in the polynomial fitted to the curve extremes degrade the accuracy of the derived data in columns 4 through 6 at potentials positive of about 0 V and negative of about - 1 V.
-_
Figure 4. The capacitance, C , in WF cm-’, of Hglaqueous 0.1 M Na2S04,estimated as A Q l A E f r o m the data listed in Table I1 (0):from the smoothed charge densities listed by Nakadomari et al. ( 14);(0): from the smoothed charge densities calcuhted by the present technique
traction of a polynomial, the power spectrum of the signal in the remainder, R ( x ) ,is clearly distinct from that of the noise. In the examples used, the separation is simply one of frequency range, so that a sharp low-pass filter suffices to remove most of the noise and only a minor amount of the signal. The power spectrum provides a rather convenient criterion for the optimization of the unavoidable trade-off between noise
__
rejection and signal distortion. No absolutely objective criterion can be indicated unless the functional form of both signal and noise content of R ( x ) are known, which it is neither in general nor in our particular case. Our method of subtracting a polynomial fitting the extremes of the curve can be considered as an extension of the “translocation-rotation transformation” of Hayes et al. (9), which consists of subtracting a straight line connecting the end points of the curve. T h a t method, corresponding to m = 0, was sufficient in the absence of differentiation. For single and double differentiation of near-parabolic surface tension curves, we find that m = 1 and m = 2, respectively, gives satisfactory results. Other smoothing procedures, or different mathematical operations facilitated by Fourier transformation, such as data interpolation ( I @ , are readily implemented in the present method. In order to capitalize on the efficiency of fast Fourier transform algorithms, the method requires equally-spaced data. This is no major problem in the examples given, and only a minor inconvenience in, e.g., the differentiation of surface tension vs. the natural logarithm of solute activity as required for the determination of the relative surface excess of the solute. Such an inconvenience is more than compensated by the possibility of multidimensional differentiation-cum-smoothing (19). There is no strict requirement on the number of input data, although some of the available algorithms may be restricted to powers of two. However, even
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1 , JANUARY 1978
I -4
-
0
. ' 20
5-
49
Figure 5. Power spectrum of the surface tension data reported by Nakadomari et al. ( 7 4 ) for Hg in contact with aqueous 0.1 M Na2S04, after subtraction of a fifth-order polynomial fitting t h e extremes of the data set. Units: p in V2 KJ*~ r n - ~s ,in V-' the more flexible algorithms work more efficiently with a number of data points which is a multiple of 4 or 2 or, at least, highly composite rather than prime (20). Although rather large data sets have been used here, smaller sets still yield fairly reasonable results, even though the uncertainties in determining the coefficients of P ( x ) increase. For example, if seven out of every eight adjacent data points in Grahame's data set are left out, so that only 16 equidistant data points remain, the capacitance can still be recovered with a standard deviation of 0.6 FF cm-2. T h e method used here does not impose a preconceived structure onto the data set. The effect of an isolated, spurious data point is distributed over the entire array, but greatly attenuated because it introduces mostly high-frequency noise. For example, introduction of a clearly extraneous point in the data set of Nakadomari e t al., by adding 0.100 fiJ cm-2 to the surface tension reading at 4.925 V and subsequent processing, led to values of Q and C which were all within 0.06 pC cm-2 a n d 0.7 NF cm-2, respectively, of those calculated for the unadulterated set, without any single point showing visible irregular behavior. Note that a n incidental error of 0.100 ,uJ cm-2 is highly unlikely, since Nakadomari et al. (14) used a seven times more stringent criterion for rejection of outlying results. T h e present method does not accommodate physical restraints, such as a requirement that the capacitance always be positive, nor do we believe this to be a deficiency. When negative capacitance values are obtained, they clearly indicate a problem in the original data set (e.g., too wide a data spacing), and such a warning sign should be heeded rather than swept under the rug. In the present paper, we have emphasized double differentiation because it provides such a demanding quantitative test of our method. Single differentiation of surface tension data by the present technique appears to us to be probably more reliable than integration of capacitance data, since integration enhances minor systematic errors. We certainly d o not advocate surface tension measurements followed by double differentiation as a practical means for obtaining the differential capacitance: direct measurement is definitely more accurate and usually more convenient. (A possible exception
115
might be the determination of the zero-frequency capacitance in a desad peak, provided that the data spacing is taken sufficiently small.) However, in order to keep scientific progress in its proper perspective, we note that, as early as 1895 and fully 40 years before the first useful capacitance measurements were made (211, Gouy ( 2 2 , 2 3 )calculated the differential capacitance of aqueous HzS04 solutions by graphical differentiation of manually gathered surface tension data, and reported its detailed structure including the existence of the capacitance "hump"! In the introduction we alluded to the difference between continuous and discrete Fourier transform. The two will be essentially equivalent (7) when the function R ( x ) is periodic, is sampled over exactly one period or an integer multiple thereof, and if R ( x ) is band-limited and sampled with a rate at least twice the largest frequency component of R ( x ) . T h e method outlined in the present paper satisfies all the above requirements. If applied to curves exhibiting rather abrupt bends, such as electrocapillary curves in the presence of organic adsorption (24), sufficiently narrow data spacing must of course be used. Finally, it might be useful to point out that integration can be performed as the inverse operation of differentiation, merely by changing the sign of m in Equation 2 . Thus, for m-fold integration of a function, one divides its Fourier ) ~ by inverse transformation. transform by (2 ~ j s followed Such a division will leave the zero-frequency component undefined, as one would expect since integration requires an additional integration constant. When the data are available already in a convenient format as, e.g., in Fourier-transform NMR, integration by division in the Fourier-transformed domain may well be an efficient route, especially since, in that case, neither noise reduction by frequency-selective filtering nor subtraction of a polynomial fitting the curve extremes will usually be necessary.
ACKNOWLEDGMENT Helpful comments by G. Horlick are appreciated.
LITERATURE CITED A. Savitzky and M. J. E. Golay, Anal. Chem., 36, 1627 (1964). e.g., R. Bracewell, "The Fourier Transform and Its Applications", McGraw-Hill, New York, N.Y., 1965, p 117. G. Horlick, Anal. Chem., 44, 943 (1972). G. C. Danielson and C. Lanczos, J . Franklin Inst.. 233, 365, 435 (1942). I . J. Good, J . R . Statist. Soc. 6 ,20, 361 (1958); 6 ,22, 372 (1960). J. W. Cooley and J. W. Tuckey, Math. Comput., 19. 297 (1965). e,g., E. 0.Brigham, "The Fast Fourier Transform". Rentice-Hall, Englewood Cliffs, N.J., 1974, p 99. P. D. Willson and T. H. Edwards, Appl. Spectrosc. Rev., 12, 1 (1976). J. W. Hayes, D. E. Glover, D. E. Smith, and M. W. Overton, Anal. Chem., 45, 277 (1973). R. G. Barradas, F. M. Kimmerle, and E. M. L. Valeriote, J . Polarogr. Soc., 13, 30 (1967). W. R. Fawcett and J. E. Kent, Can. J . Chem., 48, 47 (1970). D. C . Grahame, Tech. Rept., No. 14 to ONR, Feb. 18, 1954. D. C. Grahame, J . A m . Chem. Soc., 76, 4819 (1954). H. Nakadomari. D. M. Mohilner, and P. R. Mohilner, J. Phys. Chem., 80, 1761 (1976) and corresponding supplementary material available from the American Chemical Society. J. Lawrenceand D. M. Mohilner, J. Electrochem. Scc., 118, 1596 (1971). D. M. Mohilner and P. R. Mohilner, J. Electrochem. S c c , 115, 261 (1968). P. R. Mohilner and D. M. Mohilner, in "Computers in Chemistry and Instunentation", Vol. 2, J. S. Mattson, H. B. Mark, Jr., and H. C. Macdonald, Jr., Ed., Dekker, New York, N.Y., 1972. G. Horiick and W.K. Yuen, Anal. Chem., 48, 1643 (1976). e.g.. Ref. 2 , p 245. e.g., N. M. Brenner, Tech. Note, 1967-2. MIT Lincoln Lab.. July 28. 1967. M. Proskurnin and A. N. Frumkin. Trans. Faraday Soc , 31, 110 (1935). M. Gouy, Compt. Rend., 121, 765 (1895). M. Gouy, Ann. Chim. Phys., (7) 29, 145 (1903). M. Gouy, Ann. Phys., (8) 8, 291 (1906).
RECEIVED for review March 23, 1977. Accepted October 12, 1977. This work was supported by AFOSR Grant 76-3027 and N I H Grant GM 22296-02.
