T
is improved. In addition, it should be stressed that the present analog data processing system is a simple, convenient, and inexpensive adaptation of a commercial instrument. Improvement in sensitivity may be achieved by employing any of the following pieces of equipment and/or techniques: (i) higher intensity continuum source, (ii) high resolution monochromator, (iii) programmable rapid scan or slew scan spectrophotometer, and (iv) wavelength modulation or spectral-line modulation. Such studies are in progress.
Absorbance
0.025
i
I
I
,
210
,
,
250
,
,
,
,
,
n
I
300
350
LITERATURE CITED
300 nm
400nm
Figure 2. Sequential linear scan AAC spectrum of elements in an air-acetylene flame. Concentrations of all the elements (Zn, Cd, Ni, Co, Fe, Mn, Mg, Cu, and Cr) at 10 pg mL-’
10 pg mL-’) into an air-acetylene flame. The experimental conditions were the same as those described above. The scan speed of wavelength was 10 nm min-’. The spectrum in Figure 2 was obtained with the aid of the following analog data treatment. The background spectrum (upper curve in Figure 1)and total absorption spectrum (background + sample) were observed separately and stored in two magnetic tapes in the analog data processing system (SBA-1). As a consequence, the corrected absorption spectrum in Figure 2 was obtained from the output of the analog data processing system, which can electronically subtract the background spectrum from the total absorption spectrum in order to measure only the sample signals. The absorption signals for all nine elements can clearly be seen in Figure 2, although the baseline is not linear below 250 nm probably because the intensity of the light source is too weak in this wavelength region. Especially, magnesium atomic absorption a t 285.2 nm can be observed without any interference of OH background absorptions, while the background absorptions of OH around 306 nm could not be corrected completely because of very narrow lines in the bands. Although the present work is only preliminary, the sequential linear scan absorption spectrum of atomic absorptions for nine elements shown in Figure 2 suggests that the present instrumental arrangement might be applied to multielement analysis by AAC, if the atomic absorption sensitivity of AAC
(1) K. W. Busch and G. H. Morrison, Anal. Chem., 45, 712A (1973). (2) D. G. Mitchell, K. W. Jackson, and K. M. Aldous, Anal. Chem., 45, 1215A (1973). (3) V. A. Fassel and R. N. Kniseiey, Anal. Chem., 46, l l l O A (1974). (4) J. D. Winefordner, J. J. Fitzgeraid, and N. Omenetto. Appl. Spectrosc., 29, 369 (1975). (5) P. N. Keliher and C. C. Wohlers, Anal. Chem., 48, 333A (1976). (6) N. G. Howeii and G. H. Morrison, Anal. Chem., 49, 106 (1977). (7) D. J. Johnson, F. W. Piankev. and J. D. Winefordner, Anal. Chem.. 47, 1739 (1975). (8) J. J. Duika and T. H. Risby, Anal. Chem., 48, 640A (1976). (9) “Trace Elements in Human and Animal Nutrition”, E. J. Underwood, Ed., Academic Press, New York, 1971. (10) K. W. Jackson, K. M. Aidous, and D. G. Mitchell, Appl. Spectrosc., 28, 569 (1974). (1 1) K. M. Aidous, D. G. Mitchell, and K. W. Jackson, Anal. Chem., 47, 1034 (1975). (12) A. T. Zander, T. C. O’Haver, and P. N. Keliher, Anal. Chem., 48, 1166 (1976). (13) R. L. Cochran and G. M. Hieftje, Anal. Chem., 49, 98 (1977). (14) V . A. Fassel, V. G. Mossotti, W. E. Grossman, and R. N. Kniseiey, Spectrochim. Acta, 22, 347 (1966). (15) W. W. McGee and J. D. Winefordner, Anal. Chim. Acta, 37, 429 (1967). (16) R. C. Eiser and J. D. Winefordner, Anal. Chem., 44, 698 (1972). (17) H. Haraguchi, N. Furuta, E. Yoshimura, and K. Fuwa, BunsekiKagaku, 24. 733 11975). (18) H. Haraauchi. N. Furuta, E. Yoshimura, and K. Fuwa. Anal. Chem.. 46, 2066 (1376). (19) N. Furuta, E. Yoshimura, Y. Nemoto, H. Haraguchi, and K. Fuwa, Chem. Lett., 1976, 539. (20) E Yoshimura, N. Furuta, H. Haraguchi, and K. Fuwa, unpublished work.
Naoki Furuta Hiroki Haraguchi* Department of Chemistry and Physics National Institute for Environmental Studies Yatabe, Ibaraki 300-21, Japan Keiichiro Fuwa Department of Chemistry Faculty of Science The University of Tokyo Hongo, Bunkyo-ku, Tokyo 113, Japan
RECEIVED for review February 17,1977. Accepted April 29, 1977.
Information Theory as Applied to Chemical Analysis Sir: Chemical analysis is defined ( I ) as a process of obtaining revelant information on the composition of matter. In order to evaluate the information properties of analytical results and methods, some authors (2-5) have adopted relations derived for the needs of communication technique, namely Shannon’s measure of information content (6). The dependence of analytical studies on communication theory in considering the information properties of analytical results and methods is testified to by the very terminology used in these studies (e.g., signal, signal detection, signal-to-noise ratio, etc). According to the Shannon’s measure, the average information contained in the received item of information, on the
assumption of the continuous distribution, may be defined as
H
= -Sf:p(x)
log, p ( x ) d x
(la)
where p(x) is the probability density of random variable x . T h e information content of the analytical result is given by the relation
I(H,H,) = Ho - H I(H,Ho) = Sf,“P ( X ) log, p(x)dx - ST: P O ( ~log, ) Po(xc)dx
(1b)
i.e., as the difference of the average information before H,, and ANALYTICAL CHEMISTRY, VOL. 49, NO. 8, JULY 1977
1265
after H reception of the information item. The average information H is often called “information entropy” and used as a measure of the information disorder of a random variable whose distribution is characterized by probability density p ( x ) . This measure, however, does not allow differentiation of the information content of the analytical result according to whether it did or did not confirm the preliminary assumptions. This differentiation is of particular importance in the analytical quality control which is carried out in order to differentiate whether the product has or has not the presupposed content of the determined component. Besides, it evaluates the information content only according to the accuracy of results. It can be demonstrated (7),however, that in order to express the information content of analytical results, it proves more advantageous to utilize the so-called divergence measure (8)
following two types of variables, derived from the information content. (1)The amount of information obtained in a time period can be expressed as the so-called information flow J=
cu
(3)
dt
For the purpose of evaluation of analytical methods, it is more convenient, however, to use the so-called time-information performance, which was primarily defined by Danzer (11)as 1
and which can be rewritten by means of the divergence measure as 1
which may be more appropriately written as difference
T h e divergence measure assesses the information content according to the nonsimilarity of the probability density of distribution of the results of an analysis p ( x ) and probability density po(x) which characterizes our preliminary knowledge of the composition of the analyzed sample. The advantages of the use of the divergence measure in evaluation of the information content of analytical results as against the use of Shannon’s measure were discussed in detail before (9, 10). The divergence follows Shannon’s measure into which it is transformed for the case p&) = const for xc(x1,x2) and x , ~ x p p ( x ) d x = 1. In practice, these conditions are fulfilled when the only information about the content of the assessed component, available before the analysis is carried out, is that it lies within the interval (x1,x2) and the results of the analysis confirm this assumption. Accordingly, everything stated in studies (2-5), in which these conditions are fulfilled, holds also when the divergence measure is used. The divergence measure permits the assessment of the information content even when p o ( x )is the probability density of any continuous distribution, as stated in (7), and not only the probability density of rectangular distribution, including the case when the analysis does not confirm the original assumption. Then, however, the information content I ( p , p o )expressed by means of the divergence measure depends not only on the accuracy of results, but also on the difference between the presupposed and the determined content of the assessed component. This can be demonstrated on an example discussed in greater detail in an earlier study (7). For case in which p ( x ) and p o ( x ) are probability densities of normal distribution and uo 2 u , and if we denote a = u / u o , b = (h-go)/uo,than I(H,Ho) = - log, a and I(p,po) = - log, a + (a2 + b2 - 1). This case corresponds to the higher accuracy results of the preliminary analysis. The information content is then assessed by means of the Shannon’s measure according to the attained higher accuracy ( a ) and by means of the divergence measure according to the higher accuracy ( a ) but also according to the correction, the new analysis adds to the original result ( b ) . Since the divergence measure allows, as has been shown ( I O ) , deriving relations relevant in the assessment of the information and wholly specific for different cases of analytical processes, it is convenient to use as a basis for the variables which make possible optimization of analytical methods. Then the evaluation and optimization of the individual analytical methods may be carried out with the help of the 1200
ANALYTICAL CHEMISTRY, VOL. 49,
NO. 8,
JULY 1977
where t A stands for the time needed for carrying out the analysis, that is the analysis itself and preparation of the equipment for the analysis of another sample. The time t A can be divided into permanent t o and that depending on the number of performed parallel determinations t,. Then the time needed for carrying out n parallel determinations is t A = t o nt,. The information performance is expressed, e.g., in binary or natural units-s-’. (2) The amount of information obtained for the expended financial costs can be expressed by means of the relation
+
R
1
(5a)
= - M(P,Po) 7
where M(p,po) = C,,lkI(p,p& for i = 1,2...k simultaneously determined components of the analyzed sample. A more useful variable proves to be a reciprocal value that can be referred to as the specific price of information
where the costs, T , can also be divided into permanent T O and those depending on the number of performed determinations T,. The specific price of information is of more general importance than the time-information performance, for i t is frequently possible to find the economic equivalent of time, e.g., for the amortization of equipment, the costs of analysts’ salaries, and of economic loss due to waiting for the results of the analysis, etc. The specific price of information is expressed in $, binary or natural unit-’. Practical usefulness of the information performance according to the relations 4a and 4b and of the specific price of information according to the relation 5b, e.g., for the optimization of the analytical process, is quite obvious. It can be demonstrated on a simple case of the optimal number of parallel determinations. The information content of the quantitative assessment of one component, carried out with the precision characterized by the standard deviation u, is, according to ( I O ) given by relation
where n is the number of parallel determinations and x1 and x2 represent the lower and upper limits of the interval in which the content of the determined component is proposed. Then the time-information performance
L=
1
to + nt, loge
(x2 - X I ) +
shows maximum in dependence on n, as a rule for n
5.5,.
according to the relation of toand t,. The minimum number of parallel determinations has to be n = 2, since it is impossible to carry out the statistical control of the reliability of the analysis for a single determination. The time-information performance may, naturally, acquire relatively high values and the specific price of information very low values, respectively, even at a low information content, provided the determinations are performed rapidly or at low costs. Therefore the minimum permissible information content should be determined also. In the case of the analytical control of quality, x 1 and x q are then the lower and upper tolerance limits. I t is useful to choose the tolerance limits (12)so that (xq- xl) = 3.29 oF,where OF characterizes the total analytically found variability of the content of the assessed component in different lots of product, i.e., the variability due to production and inaccuracy of the analytical determination of the assessed component. Then it will always be CTF 2 o,so that it has to be ( x 2- x l ) / u 2 6.58, from which it follows that the minimum information content for the analytical method used for quality control in two parallel determinations is Z ( ~ , J J & ~ = ~ log, (6.58 &/&I = 0.81 natural unit. Besides, the relations 5a and 5b may be used in making the decision about the most profitable equipment of the analytical laboratory which performs the analytical control of the quality
of a product of the chemical industry. ACKNOWLEDGMENT The author is indebted to I. Vajda, Institute of Automation and Information Theory, Czechoslovak Academy of Science, Prague, and K. Danzer, Technical University, Karl-MarxStadt, GDR, for their helpful discussions. LITERATURE C I T E D (1) G. Gottschalk, Fresenius’ 2. Anal. Cbem., 258, 1 (1972). (2) H. Kaiser, Anal. Cbem., 42 (2),24A; (4), 26A (1970). (3) B. Grieping and W. Krijgsman, Fresenius’ Z. Anal. Cbem., 265, 241 (1973). (4) H. Malissa and J. Rendel, Fresenius’ Z . Anal. Cbem., 272, 1 (1974). (5) F. Dupuis and A. Dijkstra, Anal. Cbem., 47, 379 (1975). (6) C . E. Shannon, Bell Syst. Tech. J . , 27, 379, 623 (1948). (7) K. Eckschlager and I. Vajda, Collect. Czech. Cbem. Commun., 39, 3076 (1974). (8) S. Kullback, “Information Theory and Statistics”, J. Wiley, New York, 1959. (9) K. Eckschlager, Collect. Czech. Cbem. Commun., 41, 2527 (1976). (IO) K. Eckschlager, Fresenius’ Z. Anal. Cbem., 227, 1 (1975). (11) K. Danzer, Z. Cbem., 15, 326 (1975). (1 2) A. Hald, “Statistical Theory with Engineering Applications”, Wiley, New York, 1952, p 315.
K a r e l Eckschlager Czechoslovak Academy of Science Institute of Inorganic Chemistry 250 68 Rei, Czechoslovakia Received for review July 30, 1976. Accepted March 22,1977.
AIDS FOR ANALYTICAL CHEMISTS Needle Guide for Liquid or Gas Syringe Arye Tishbee Department of Organic Chemistry, The Weizmann Institute of Science, Rehovot. Israel
The septum material and hardness differ according to the special requirement of the system, and in most cases especially in LC. The septums used are hard to penetrate and very often while injecting the needle will bend. A small diameter needle though is required for long septum and column life. The needle guide described below protects the needle from bending without interfering with the regular functions of the syringe. It permits a very fast and easy penetration through the hardest septums. As much as 4-mm plywood could be penetrated by a 10-pL syringe with a 0.0185 in. X 2 in. needle equipped with the needle guide, and thus new types of inert and thermally stable septum materials may be used now.
i d a e
f DESCRIPTION
b The guide consists of three flat disks a, b, and c, Figure 1, 24 mm in diameter and 7 mm thick made from AKULON (Nylon 6) (Verenigd Plastic Verkoopkantoor n.v., Holland). The two upper disks a and b are made to slide freely on two stainless steel rods (d), 80 mm X 1.5 mm in diameter, tightly fitted into disk c, 16 mm apart. Disk a is drilled 6.6 mm in diameter in the center and is fitted with a Nylon tightening screw 6 ) for syringe mounting. Disks b and c are drilled 0.5 mm in diameter in their centers to allow needle travel (m) (Figure 3). The position of disk 2 is controlled by a polypropylene line (f) 60 mm X 0.4 mm in diameter (any Nylon 0.3-0.5 mm diameter fishing line can be used) attached with screws (e) to the disk.
C
Figure 1. Prior to injection position: (a, b, c) disks: (d) stainless steel rods: (e) line attaching screw: (f) polypropylene line: (9) syringe: (h) 0.0185 in. needle; (i) Nylon screw ANALYTICAL CHEMISTRY, VOL. 49, NO. 8, JULY 1977
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