Sampling Variance in Analysis for Trace Components in Solids Preparation of Reference Samples W. E. Harris and Byron Kratochvil Department of Chemistry, University of Alberta, Edmonfon, Alberta, Canada
Sampling errors may be classified as determinate or indeterminate. We consider here a statistical approach to the question of variance of sampling of particulate matter, as distinct from the variance of analysis, with special emphasis on problems in trace analysis. In addition, the related topic of preparation of reference samples for trace measurements is considered. Sampling Variance in Solids. When traces of material in solids are to be determined, minimization of sampling errors can in some cases impose extreme demands on the size and preparation of samples. In real situations, solid samples are often complex mixtures of several components with varying particle size, each particle containing the constituent of interest a t a different concentration. The magnitude of sampling variance, or error, is then a function of particle size, heterogeneity of the material sampled, relative density of the compounds, and the desired precision of analysis. One question faced by an analyst is predicting the minimum amount of sample to be taken if the sampling error is to be less than a predetermined value. Although rigorous statistical evaluation of indeterminate ( I ) error (fluctuations arising as a result of the application of a random sampling procedure and denoted by the appropriate sampling error) in the sampling of complex mixtures is impractical, with several simplifying assumptions some guiding conclusions can be drawn. Consider a simplified system in which the material to be sampled consists of a mixture of small spherical particles of the same size and of uniform density, but of two types-one, A, containing some quantity of the component of interest, the other, B, none of it. Also assume that each type of particle is uniform in composition, and that the distribution of the two types is random. For such a system, the properties of a binomial distribution ( 2 , 3 ) permit the standard deviation (square root of the variance) of A in a sample to be calculated from the relation
where n is the number of particles, p the fraction of the desired component A, and (1 - p ) the fraction of diluent B. The relative standard deviation (in per cent) for component A is u, = 100 i
w
p
(2)
We will call the relative standard deviation the sampling error; fractions or multiples of us also could be chosen as circumstances warrant. Figure 1 indicates the relative standard deviation, or sampling error, for values of p (1) I. M. Kolthoff, E. B. Sandell, E. J. Meehan, and S. Bruckenstein. "Quantitative Chemical Analysis," 4th ed, Macmillan. New York. N.Y., 1969, p 385. (2) M. R. Spiegel. "Statistics," McGraw-Hill. New York. N.Y.. 1961, p 122. (3) W. J. Dixon and F. J. Massey. "Introduction to Statistical Analysis," 3rd ed, McGraw-Hill, New York. N.Y., 1969, p 413.
ranging from O.OOO1 to 0.999, ( i e . , 0.01 to 99.9%), for particles containing either 100 or 0% of the component A. When the constituent being analyzed for is present as only a small fraction of the total, the number of units or particles required in the sample becomes enormous if the sampling error is to be kept small. In practice, the factor of interest is most often the weight of sample to be taken for analysis. Figure 2 gives the approximate relation between the mesh size of spherical particles of varying density and the number of particles per gram of material. For example, if IO6 particles are needed to hold the sampling error to a predetermined level, material of density equal to 3 should be ground to pass a 200-mesh sieve. For high-precision analysis of traces in heterogeneous materials, it is a formidable task to grind the material sufficiently fine that samples of reasonably small weight can be taken. In Figure 1, the component of interest is assumed to be present only as pure (100%) discrete particles. Often this is not the case, and single particles may contain both component A and inert material. Consider now a situation where two kinds of uniform particles are present as before, but have compositions less divergent than 0 and 100% in the component of interest. Benedetti-Pichler ( 4 ) developed the following formula for the number of units required in this case to obtain a random sample:
where d l and dz are the densities of the two types of units making up the sample; d is the overall sample density; PI and Pz are the percentages of A in the richer and leaner of the two types of particles; Pa, is the percentage of A in the overall sample; and us is the per cent relative standard deviation (Equation 2). With the simplifying assumption that d l = dz = 1, Figure 3 shows the relation between the number of units required to keep sampling error within predetermined limits and the percentage of the component of interest in a sample that contains two kinds of particles with a relatiL'e difference in composition of from 100 to 10%. The per cent relative difference in composition, 100 ( P I - P z ) P I , is used to label the curves so that any range of absolute. values on the horizontal axis can be chosen: Figure 3 indicates that sampling error decreases sharply as the relative dzfference in percentage of the sought-for substance in the two kinds of particles becomes smaller. On the other hand, when the relative difference in composition is 100% (topmost curve in Figure 3), it is difficult to provide a sufficient number of particles per sample as the concentration Pa, of the constituent of interest decreases. (4) A. A. Benedetti-Pichler in "Physical Methods of Chemical Analysis." W. M. Berl. Ed., Vol. 3, Academic Press, New York. N.Y., 1956. p 183.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 2, FEBRUARY 1974
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10
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T 10'
2
2 L
&
a,
b"
e
L
1
L
W
01
0.01
io4
io5
lo6
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lo8
Particles in Sample Figure 1. Relation between the sampling error us in percentage and the total number of particles n for samples in which p rang-
0 0 0
es from 0.0001 to 0.999
10 0.1
20 0.2
0.5x Component of interest Present, %
X
Figure 3. Relation between the minimum number of units in a sample required for sampling errors (relative standard deviations in percentage) of 0.1 and 1% (y-axis) and the overall composition of a sample (x-axis), for mixtures having two types of particles with a relative difference in composition ranging from 100 to 10%
10'
io4
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1o7
Particles per Gram
Figure 2. Approximate relation between the number of spherical particles per gram of sample and mesh size (U.S. Sieve Series, ASTM E-1 1-61) or particle diameter, for densities from 1 to 10
In pracuce, samples consisting of particles whose compositions diverge strongly on a relative basis (left side of Figure 3) should be avoided unless they contain more than about 106 particles if the error is to be kept to an acceptable level. Often a larger sampling variance may have to be accepted because the sampling operation becomes too difficult. In regions near the left edge of Figure 3, the amount of sample required becomes excessive, even with extensive particle size reduction through grinding. I t is thus apparent that in the analysis of, for example, airborne particulate matter for a specific trace element, small particles are essential if gross sampling errors are to be avoided (unless either large samples or a large number of samples are analyzed). The effect of the difference in density of the two types of particles can be deduced directly from Equation 3. The effect of the more complicated factor of variable particle size has been treated by Benedetti-Pichler; in essence, there is little advantage to fine grinding only a fraction, even a substantial fraction, of a sample. Preparation of Reference Samples for Trace Components. An extension of the basic problem of sampling error is the preparation of reference samples of materials in which the component of interest is present a t trace levels. Here an attempt to reduce sampling error by simply increasing sample size may be unacceptable because of the procedures used. The alternative of reducing the material 314
to small particle size would require an unreasonable amount of grinding. The finer a material is ground, the more trouble it is to sieve, the more likely changes in composition will occur, and the more likely problems with caking and static charges during mixing will arise. Therefore, the largest particle size consistent with acceptable sampling error is the most desirable. A third technique is to prepare samples from two uniform materials, each containing the component of interest. For example, if a series of samples for chloride analysis are to be prepared, there is great advantage in using mixtures of NaCl and KC1 instead of mixtures of NaCl with an inert diluent such as NazS04. When a sample contains only NaCl (60.6670 chloride) or only KC1 (47.55% chloride), sampling error is zero and is not a function of particle size. Maximum sampling error occurs with an approximately 40:60 NaC1-KC1 mixture; if in this case a sampling error of 0.5 ppt is acceptable, about 60,000 particles must be taken for analysis. For a 1-gram sample, this corresponds to grinding the mixture to pass a 50-mesh sieve. When the mixture composition varies either side of 40% NaC1, a larger particle size can be tolerated. Thus techniques for reduction to modest size are required throughout. If Na2S04 is used as diluent, sampling error exceeds 1 ppt for 50-mesh material when the Na2S04 level is greater than 5% of the total composition of the sample. Extending this approach to the preparation of reference materials for traces by mixing a small amount of pure substance with a large quantity of diluent presents a virtually impossible sampling problem. With 0.1% or less sampling error and a reasonable sample size, the lowest percentage that can be prepared in this way without grinding to finer than 400 mesh is about 1070. Circumvention of this problem may be illustrated by the following example. A series of reference samples containing from 0.1 to 0.2% copper were required for atomic absorption analysis. First, a quantity of small (200 mesh) pure-nickel beads of uniform diameter were prepared by a
ANALYTICAL CHEMISTRY, VOL. 46, NO. 2, F E B R U A R Y 1974
hydrometallurgical technique. A thin layer of copper was deposited on the surface of the beads, followed by a layer of nickel. Two batches, one containing approximately 0.1% and the other 0.2% copper, were each prepared and analyzed carefully by two independent methods. The two materials then were mixed in varying proportions to give a series of 10 to 20 samples. From Figure 3, the relative sampling error for these samples should be no greater than 0.1% if lo5 particles are taken (the most unfavorable mixture has about 1 part of 0.1% to 2 parts of 0.2%). For ZOO-mesh materials, a sampling error below 0.1% is easily achieved with samples of 0.2 gram (Figure 2). If these samples were prepared from pure copper and a diluent such as pure nickel, 1O1O particles (about 20 kg) would be required for each sample analyzed. In summary, the sampling of heterogeneous solids may be a major source of error if the sought-for component is
present in small quantity in discrete particles. Poor precision in trace analysis may be the result of variation in sample composition. In the preparation of reference materials for trace components, the use of inert material plus pure component is not recommended. Other approaches to their preparation, particularly use of homogeneous material or of two materials close together in percentage of the sought-for component, should be used.
ACKNOWLEDGMENT The assistance and cooperation of D. R. Weir and Sherritt-Gordon Mines, Ltd., Fort Saskatchewan, Alberta, in the preparation of the copper-in-nickel reference materials is gratefully acknowledged. Received for review July 31, 1973. Accepted October 4, 1973.
New Method for Calibration of Permeation Wafer and Diffusion Devices Russell N. Dietz, Edgar A. Cote, and James D. Smith Department of Applied Science, Brookhaven National Laboratory, Associated Universities, Inc., Upton, N. Y. 7 1973
Standard reference materials for calibrated sources of gaseous pollutants at ppm and ppb levels are needed to provide improved calibration of current air monitoring instruments. Permeation Teflon (Du Pont) tubes containing liquified gases were developed as such sources and have recently' been standardized for SOz(l a ) . Permeation tubes for condensible gases, originally suggested by O'Keeffe and Ortman ( 2 ) and evaluated for SO2 by Scaringelli, Frey, and Saltzman ( 3 ) , have been calibrated by their weight loss with time (4, 5) and more recently by volumetric displacement (6, 7) and by pressure differential measurements with a capacitance manometer (8). However, the rates are inconveniently high for ambient air applications (1500 ng/min per cm of length), the tubes last only a few months, and the permeation rates for NO2 decline with time because of interaction with moisture. Permeation bottles capped with Teflon disks reduce the rates by about an order of magnitude, but calibration by conventional weight loss measurements is extraordinarily long-about 1 month. Utilization of a recording electrobalance (9) reduces the calibration time, but the total weight of the permeation device is limited. (1) J. R. McNesby and R. Byerly, Jr., "Measures for Air Quality, Annual Report-FY 1971." Nat. Bur. Stand. (U.S.) Tech. Note, 711, January 1972: a. The Sulfur Dioxide Permeation Tube, p 68; b. The Nitrogen Dioxide Permeation Tube, p 70; c. Molecular Complexes of Gaseous Pollutants, pp 29-32. (2) A. E. O'Keeffe and G. C. Ortman, Anal. Cnem., 38, 760 (1966). (3) F. P. Scaringelli. S. A. Frey, and 8. E. Saltzman, Amer. Ind. Hyg. Ass., 28, 260 (1967). (4) F. P. Scaringelli, A. E. O'Keeffe, E. Rosenberg, and J. P. Bell, Anal. Chem., 42, 871 (1970). (5) F. P. Scaringelli, E. Rosenberg. and K. A. Rehme, Environ. Sci. 924 11970). Techno/.. 4.,~~ (6) 8. E. Saitzman. C. R;'Feidmann, and A. E. O'Keeffe, Environ. Sci. Techno/., 3,1275 (1969). (7) B. E. Saltzman, W. R. Burg, and G. Ramaswamy, Environ. Sci. Techno/., 5, 1121 (1971). (8) J. J. McKinley, "A Calibration System for Trace Analyzers," 16th National Symposium, Instrument Society of America, Pittsburgh, Pa.. May 1970. (9) L. J. Purdue and R . J. Thompson, Anal. Chem., 44, 1034 (1972).
Noncondensible pollutant gases such as nitric oxide, methane, and carbon monoxide can permeate through Teflon, but permeation tubes have not been made since the gases cannot be liquified a t ordinary temperatures. Other suitable NO source materials are being investigated including encapsulated gas bubbles, molecular complexing in conjunction with the permeation tube principle, certification of NO-Nz mixtures in cylinders, and quantitative catalytic conversion of NO2 from permeation tubes to NO ( I C ) . Monitoring instruments for the determination of CH4 and CO in the environment ( I O ) are presently calibrated by preparing mixtures in cylinders at about 10 ppm (11); however, there is no way to check the accuracy of these preparations or to determine the loss of calibration with time. McKinley (8) measured permeation rates of noncondensible gases through polymeric membranes (e.g., F E P Teflon tubing) by a pressure differential technique, although many of the experimental details were not given. This paper describes a new pressure differential method for the calibration of permeation wafer and diffusion devices for both noncondensible and condensible gases. Permeation rates less than 5 ng/min can be accurately determined in hours instead of weeks.
EXPERIMENTAL T h e permeation wafer device consisted of a T e f l o n disk h e l d b y compression in a Swagelok tee such t h a t the leg containing the disk was connected t o the p o l l u t a n t gas source a n d the other connections provided t h e diluent gas input a n d o u t p u t flow ports (cf. Figure 1). For calibration, t h e diluent side was connected t o a
~
(10) R. K. Stevens, T. A. Clark, C. E. Decker, and L. F. Ballard. "Field Performance Characteristics of Advanced Monitors for Oxides of Nitrogen, SOz, CO, CH4, and Non-Methane HC." 68th APCA Meeting, Miami, Fla., June 1972. (11) E. E. Hughes and J. K. Taylor, "Standard Reference Materials For Air Pollution and Gas Analysis." 164th National Meeting AEC. New York, N.Y..; Amer. Chem. Soc., Div. Water, Air, Waste Chem., Gen. Pap., 12(2), 238 (1972).
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