232
ANALYTICAL CHEMISTRY, VOL. 51, NO. 2, FEBRUARY 1979
expect their representations to be accurately described in the file. This problem is a n inadequacy in the data rather than in the search strategy and as such it is the responsibility of those in charge of maintaining the integrity of the data base. One approach that will accommodate "incomplete" spectra is to separate them all from the main file and create a second file ordered on a mass range for which those spectra are defined. Both files would then need to be searched with their respective representations of the query spectra. In most instances presumably the latter file would be relatively small. In discussing the performance of the system the preliminary searches of the dictionary have been ignored because of their very s m d overhead relative to spectrum matching operations. Further, the number of comparisons for retrieval from both a hash file and an ordered file are known and have been completely characterized by theoretical arguments (9).
and S. R.Heller and G. W. Milne for providing the mass spectral data base.
ACKNOWLEDGMENT T h e author thanks Ann Titus for typing the manuscript
RECEIVED for review July 24, 1978. Accepted November 6, 1978.
LITERATURE CITED (1) F. W. McLafferty, R. H. Hertel, and R. D. Viilnock, Org. Mass Spectrom., 9, 690 (1974). (2) S. Grotch, Anal. Cbem., 43, 1362 (1971). (3) H.S. Hertz, R. A. Hites. and K. Biemann, Anal. Cbem.. 43, 681 (1971). (4) S. R. Heller, Anal. Cbem., 44, 1951 (1972). (5) L. R. Crawford and J. D. Morrison, Anal. Cbem., 40. 1469 (1968). (6) D. H. Smith, Anal. Cbem., 44, 536 (1972). (7) R. G. Dromey. Anal. Cbem., 48, 1464 (1976). (8) R. G. Dromey, J . Cbem. I n f . Cornput. Scl., (November 1978). (9) D.Knuth, "The Art of Computer Programming", Vol. 3, Addison-Wesley, Reading, Mass., 1973. (IO) S. L. Grotch, Anal. Cbem., 42, 1214 (1970). (1 1) B. H. Kennett, K. E. Murray, F. B. Whitfieid, G. Stanley, J. Shipton, and P. A. Bannister, "Mass Spectra of Organic Compounds", CSIRO report, (1977). (12) K. J. McDonell and A. Y. Montgomery, Aust. Comput. J . , 5, 115 (1973). (13) D. Severance and R. Duane, Comm. ACM, 19, 409 (1976).
Optimizing Precision in Standard Addition Measurement Kenneth
L. Ratzlaff
Department of Chemistty, The Michael Faraday Laboratories, Northern Illinois University, DeKalb, Illinois 60 1 15
Equations are presented and plotted which describe the effect of the increment size on the precision of a standard addition or standard subtraction measurement. Linear, exponential, and logarithmic transfer functions are considered in which the uncertainty may take several forms.
The standard addition (SA) technique (also called known addition, standard increment, or known increment) is well known to be of value when solution conditions are not readily reproducible in reference media (1). In this technique, a measurement is made on the analyte followed by the addition of a known quantity of standard and a second measurement; the original quantity of analyte is then computed without reference to a calibration curve. The increment size must be chosen such that good precision can be maintained in the measurement and the solution conditions are not disturbed ( I ) . In this paper, the choice of increment size will be discussed as it relates to precision. Although SA is in common use, the increment size has received little or no attention except in dc polarography where, for example, Lingane suggests that the increment should double concentration (2). There are several criteria which although seldom mentioned, must be satisfied for SA utilization. (1)The instrumental technique must either have a linear response with concentration or must have a known transfer function over the range of concentrations encountered. (2) T h e chemical system must be free of deviations from linearity over the range of concentrations observed; these deviations could be caused by effects related to complexation equilibria, ionic strength, pH, free electron concentration in flames, etc. A large increment may shift these equilibria and produce nonlinearity. (3) The intercept must be known; generally it is considered to be zero. 0003-2700/79/035 1-0232$01.OO/O
The increment may be either positive or negative; a negative addition, or standard subtraction, occurs when the added reagent removes analyte from availability to the measurement system. This is in many ways similar to an incremental or Gran plot titration ( 3 ) . The advent of computer-controlled instrumentation has made possible automated SA measurements ( 4 , 5 ) . It is now feasible, after computing a first approximation of the concentration from the initial measurement, to determine an optimum addition based on the initial value. Consequently, a better understanding is needed of how precision relates to increment size. Because the increment size is to be determined after the first measurement, standard addition measurements in which two samples are taken and one is incremented before chemical workup are outside the scope of this study as are all errors and uncertainties associated with the sample preparation. Only the measurement step is included.
PRECISION OF S A MEASUREMENT Factors affecting the choice of increment size are examined herein by considering the contribution of various types of uncertainty for each of several transfer functions. For any measurement the overall uncertainty, expressed as the relative standard deviation, is equal to the square root of the sum of the squares of the relative standard deviation due to each source of uncertainty. The relative standard deviation is determined using propagation of errors mathematics. This approach has been used on several occasions (6-8) for single measurement techniques so that certain types of uncertainty are well defined. The following transfer functions will be considered. (a) Linear. Examples include atomic or molecular fluorescence spectroscopy, flame emission spectroscopy, and voltammetric techniques. (b) Exponential. An example is potentiometric measurement. (c) Logarithmic. Molecular and atomic absorption spectroscopies are included.
D 1979 American
Chemical Society
233
ANALYTICAL CHEMISTRY, VOL. 51, NO. 2, FEBRUARY 1979
For spectroscopic measurements, the fundamental nature of the uncertainties has been discussed previously, and these uncertainties have been grouped into three categories (6-8). (a) Independent. The uncertainty in the measured parameter is independent of the magnitude of the value. (b) Square root. T h e uncertainty is proportional to the square root of the magnitude of the measured parameter. (c) Proportional. T h e uncertainty is proportional to the measured parameter. For electrochemical measurements and most others, there appear to be no fundamental sources of noise except for independent uncertainty generated by noise in electronic circuitry and either digitization or meterlrecorder reading uncertainty. Linear Transfer Function. A quantity of analyte, ml moles, in volume, V , is related to the readout, X I by a proportionality factor, k .
X I = kml/V
(1)
After addition of m2 moles of standard, the second measurement is made. The increasingly common use of micropipets which reproducibly dispense volumes under 10 pL makes reasonable the assumption of negligible dilution. Then,
X z = K(m, + m 2 ) / V
(2)
ml = m2Xl/(X2 - Xl)
(3)
Solving for m,, Since the variance in ml will be the result of the variances in the measured parameters, the variance in ml can be computed using propagation of error mathematics (9) if the contributing variances can be defined. The variance in ml ( u ~ )can ~ , be written as
(4) where uXl2 and uXz2are the variances in X I and X 2 respectively. The percent relative standard deviation, 70ul = (ul/m,) X 100% is then written in terms of ml and m2using Equations 1 and 2,
+
+
z
Q
c4 5
n
D
Independent Uncertainty. Where the uncertainty is independent of the measured parameter, the standard deviation of the measurement can be defined as a constant, ux,so that ux = uxl = ux2. Substituting into Equation 5 , the percent relative standard deviation due to independent uncertainty, %qL, is % Ul,‘ =
uxV(mZ2+ 2m,mz km1m2
+ 2m12)1/2 X
100% (6)
It is useful to present % q ias a function of the ratio of increment to analyte, R , where R = m 2 / m l . Then
9c u/,i =
uxV(1
+ 2R-1 + 2R-2)1/2x 100% hml
(7)
Equation 7 is plotted as curve A in Figure 1 vs. R for ux km,/V. It should be noted that R may assume negative values when a standard subtraction is carried out. Square Root Uncertainty. When the standard deviation is proportional to the square root of the signal level, ox, = ( X 1 / P ) 1 / and 2 ux2 = ( X 2 / P ) 1 / 2 .The conversion factors grouped in P are discussed in references 6 and 8; X I P effectively represents the number of discrete events (photons or radioactive decay counts) included in the measurement, =2 X
01
10 0
10
INCREMENT RATIO
Figure 1. Percent relative standard deviation as a function of increment-to-analyte ratio. (A) Linear transfer function and independent uncertainty (Equation 7). uxVlkrnl = 2 X (B) Linear transfer function, square root uncertainty (Equation 9). Pkml/ V = 2.5 X lo5. (C) (i) Linear transfer function, proportional uncertainty (Equation 11). 5=2X (ii) Also represents exponential transfer function, independent uncertainty (Equation 15). u , / S = 1.4 X (D) Exponential transfer function, independent uncertainty in slope (Equation
17). u t / s= 1.4x
and is often termed “shot noise”. Substituting into Equation 5 , the relative standard deviation is
+
%ul,s = ((2m12 3mlm2
+ m22)V/Pkmlm22)1/2 X 100% (8)
In terms of R , % q s= ( ( 1
+
3;;r12R-’)V )
1’2
x 100%
(9)
Equation 9 is plotted as curve B in Figure 1 vs. R for kml/ V = 2.5 X 105/P. Proportional Uncertainty. Where the standard deviation terms in Equation 5, uxl and ux2, are proportional to XI and X 2 , respectively, uxl = [XI and ux2 = [ X 2 where E is the proportionality or “flicker” factor. The relative standard deviation becomes
% u i = ((uX12(ml mz)2 u x z 2 m , 2 ) ~ / k 2 m ~ 2 m x~2)1~2
100% (5)
01
IO
% u l , p= f i t ( ( m l
+ m 2 ) / m 2 x)
100%
(10)
In terms of R
Equation 11 is plotted as curve C in Figure 1 for 6 = 0.002. Exponential Transfer Function. For the common example of this function, potentiometric measurement, the measured value is proportional to the logarithm of the concentration, Le., for the analyte, X I = X o S In ( m l / V ) (12)
+
and after the addition
X 2 = X o + S In ( ( m , + m 2 ) / V )
(13)
In these equations X,is a constant, and S is the slope of the plot of potential against the logarithm of concentration; the theoretical value of S is the well-known R T / n F where R is the gas constant, T i s the temperature, F is the value of the Faraday, and n is either the number of electrons transfered at a Redox electrode or the charge of the selected ion a t an ion-selective electrode. Solving Equations 12 and 13 for ml, m , = m Z / ( e ( X 2- X d / S - 1) (14)
Independent Uncertainty. Again applying propagation of error mathematics, the variance in ml for exponential transfer function, ue2,is
234
ANALYTICAL CHEMISTRY, VOL. 51, NO. 2, FEBRUARY 1979
ue2
=
+
10 0
(uX12 ux22)m22e2(xz X1)IS SZ(e(X2- xd/s - 1 ) 4
z
0
The only type of uncertainty in the signal is independent, contributed by Johnson noise in the amplifiers and noise pickup in the electrodes. Consequently, both standard deviations can be written as ux. Including Equations 12 and 13, the percent relative standard deviation is then
5> a
B 5:
5
: w
J
U W
+
0’
W
In terms of R
U
e
2ax(l + R) 70 ue,i = x 100% SR Equation 15 has a dependence on R identical to that of Equation 10 so that it may be represented by Figure 1,curve C, where ux = 1.4 X S. Slope Uncertainty. A second independent uncertainty to be considered in a potentiometric measurement is the standard deviation, ut, in the slope, S; S must be determined before Equation 14 can be applied, but is dependent on temperature and the condition of the electrode. The uncertainty in the ~ , to slope uncertainty is found entire measurement, D ~ , due from application of propagation of error mathematics to Equation 14. ue,t = a,(X2 - X1)m2e(xz - x1)/s/(S2(e(x2- x l ) / s -
measurement of Io, I,, and I2respectively. Substituting for Io, 11, 1 2
Combining with Equations 12 and 13 and dividing by ml, the relative standard deviation is uAm1 + m2) In ((ml + m2)/mJ % ue,, = X 100% (16)
where k = cb/V. Independent Uncertainty. If the uncertainty in I is independent of Z, all standard deviations can be set equal, Le., ur = q0= uz1 = uz2 Then the relative standard deviation, 70uA, is
Sm2
.10
01
01
10 0
10
INCREMENT RATIO
Figure 2. Percent relative standard deviation as a function of increment-to-analyte ratio for logarithmic transfer function, independent uncertainty (Equation 23) where u J I 0 = 4 X
In terms of R , u,(l Ya u,,, =
+ R) In ((1+ R ) / R ) SR
X
100%
(17)
Equation 17 is represented in Figure 1 as curve D where 1.4 X S; i.e., for a slope of 59.2 mV per decade, ot is about 0.1 mV. Logarithmic Transfer Function. For absorption measurements, the measured values, transmitted and incident intensities, I and Io,are logarithmically related to concentration. Thus, according to Beer’s law, for the analyte of concentration of ml/ V, 0, =
In (Io/Il) = 2.303ebm1/V
Substituting A = kml, R = m 2 / m 1
(18)
where t is the molar absorptivity and b is the path length. After the increment, (19) In (10/12) = 2.303eb(ml + m 2 ) / V Solving for m,,
This equation is plotted in Figure 2 for various values of Al where 01 = 4 X Io. Square Root Uncertainty. If the uncertainty in Z is due , 012 = ( ~ z / P ) ” ~ . to shot noise, oro = ( Z O / P ) ~ ~=~ U1/P)l”, Substituting into 2 1 and solving for the relative standard deviation, %“A,s
=
+
At least three measurements must be made, I*, 11, and Io. The latter may be made either once or twice depending on whether a single or double beam instrument is used; L will represent that number and may be 0 or 1. T h e variance in ml, uA2is u p 2
In (Iz/IO)
Il(ln (Iz/IJ)2
where ul,,, uIl and
012
>’.
are the standard deviations in the
In terms of R and Al
ANALYTICAL CHEMISTRY, VOL. 5 1 , NO. 2, FEBRUARY 1979 I”
n
235
The relative standard deviation is
z 0 W
100% (26) In terms of A I and R Y G (TA,p
=
~
E
2.303A1
(L
+
(R l + R ) + ($>’)”’ ~
(27)
c
Equation 27 is plotted in Figure 4 for [ = 2
w
8
L
-1 0
-01
01
10
10 0
INCREMENT RATIO
Figure 3.
Percent relative standard deviation as a function of increment-to-analyte ratio for logarithmic transfer function, square root uncertainty (Equation 25), where PI, = 4 X
lo6
4 > 8
n
04
s W
2
U W
X
DISCUSSION Aside from the equations which have been presented, there are several noteworthy points concerning the choice of increment size. (1)When R is significantly less than unity, significant loss of precision results. (2) The standard subtraction method is quite attractive although care must be taken to assure that R is less than one. (3) In absorption spectroscopy, the uncertainty will always go through a minimum since the variances from each source are additive. Although for some techniques, the uncertainty monotonically decreases with R , it is important to note that a large addition may change the solution conditions and consequently the slope of the transfer function ( I ) . Consequently the equations developed herein should be used primarily to determine minimum values of R.
I-
w U
a W 13
10
-01
01
10 0
10
INCREMENT RATIO
Figure 4. Percent relative standard deviation as a function of increment-to-analyte ratio for logarithmic transfer function, proportional uncertainty (Equation 27) for [ = 2 X
Equation 25 is plotted in Figure 3 for various values of A l where Io = 4 X 106/P. Proportional Uncertainty. Finally, if the measurement is “flicker” limited, Le., the standard deviation is proportional to the signal by the factor [, cTro = [Io, “II = [I1,ur2 = (I2. Then the variance in m, is
(
2.3;im2)
ACKNOWLEDGMENT The author gratefully acknowledges the helpful comments of Arnold M. Hartley of the Institute for Environmental Studies, University of Illinois. LITERATURE CITED (1) Robert Klein, Jr., and Clifford Hach, A m . Lab., 9 (7),21 (1977). (2) James J. Llngane, “Electroanalytical Chemistry”, 2nd ed., Interscience, New York. 1958. (3) Lester P. Rigdon, Gwilym J. Moody, and Jack W. Frazer, Anal. Chem., 50, 465 (1978). (4) Axel Johansson and Sten Johansson, Analyst(London), 103,305 (1978). (5) Christian Stahii, John H. Wharton, and Hans Noll, Anal. Biochem., 88, l(1978). (6) J. D. Ingle and S . R. Crouch, Anal. Chem., 44, 1375 (1972). (7) H. L. Pardue. T. E. Hewitt, and M. J. Milano, Clin. Chem. ( Wlnsfon-Salem, N.C.), 24. 1028 (1974). (8) K. L. Ratziaff and, D. F. S. Natusch, Anal. Chem., 49, 2170 (1977). (9) P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences”, McGraw Hill, New York, 1969.
RECEIVEDfor review August 8, 1978. Accepted November 6, 1978.
