Byron Kratochvil' R. Stephen Reid and Walter E. Harris University of Alberta Edmonton, Alberta Canada T6G 2G2
Sampling Error in a Particulate Mixture An analflical chemistry experiment
Adequate sampling of materials prior t o chemical analysis is universally acknowledged to be as important as the suhsequent steps. While instructors in introductory analytical courses mav?orovide some lecture discussion of the samoline . . process, such coverage is not generally extended to the laboratory because of the difficulty in isolating sampling error. This shortcoming in student instruction carries over to many real situations where too little attention is paid to proper sampling or time and money wasted through unnecessary replicate analyses. An experiment involving a particulate mixture has been devised that allows measurement of sampling variance after separate estimation of, and correction for, other experimental error. The variances observed depend primarily on particle size, sample size, and composition of the sample. The variances are compared with theoretical values calculated for perfectly random mixing. The theory of statistically-based sampling error is well developed (1,2), and an application to the design of reference samples for trace analysis has been made (3).However, undergraduate experiments clearly demonstrating sampling error appear to be nonexistent, mainly because of the difficulties in devising appropriate systems t o be sampled, in adequately separating sampling and measurement errors, and in obtaining sufficient information within a limited time t o allow meaningful statistical treatment. These difficulties have been circumvented here by: selecting as the sampling system a mixture of potassium hydrogen phthalate and sucrose, a nonacidic crystalline material of similar density; using a self-zeroing, automatically refillable buret to minimize the time required for the titration of multiple samples; and employing a dilute back-titrant to obtain high end-point precision. The statistical background required of the student is minimal, no more than that covered in most introductory analytical chemistry texts (41, and the calculations take little time with modern calculators. Since the experiment entails several weighings and titrations, i t is best assigned only after the student has attained facility in these techniques. Assignment later in the course also has the advantage that it allows more flexibility in scheduling balance and self-zeroing buret use. Background Samples provided in most introductory courses in chemical analysis are usually assumed to be sufficiently homogeneous
' To whom correspondence should he addressed.
518 / Journal of Chemical Education
that the results obtained can be used to evaluate the accuracy and precision of the separation and measurement operations without concern for sampling error. Real materials, however, are often difficult to s a m ~ l ein such a wav that a ort ti on for sampling analysis is representati;e of the whole. problems arise from inhomogeneities in the material under study; such inhomogeneities can occur in almost any mixture, including gases and liquids. As examples, the level of nitrogen oxides in the atmosphere is greatly affected by distance from a highway, and concentrations of soluble substances in a lake or stream bv location or d e ~ t h . For mixtires of solids of moderate particle size, the problem of samoline . can be oarticularlv severe. Samoles taken from the most carefully mixed materials may var; in composition because of statistical variations in the distribution of particles richer or leaner in the sought after substance. For this situation Benedetti-Pichler ( 1 , 2) related the percent relative standard deviation in the sample composition usto the total number n of two types of uniformly sized, spherical particles in a samole. With some rearrangement the relevant equation
any
where d l and dn are the densities of the two types of particles, d is the average overall density, p and (1- p ) are the fractions Approximate Mesh Size ,499
TW
l?,
, , 5p ,
4:
,
3p 2i
I
I 0
0.3
Particle Dnmeter,
0.6 mm
FigUTe 1. Relation between total percent relative standard deviation . o and particle diameter. Values calculated using eqns. (1) and (2).A, typical sh&nt measurement emn u, = 0.4%; B. overall enor u . k a mixhre containing 80% KHP: C, as above tor 5 0 % ; and D, as above for 20%.
(hv numher) of each t w e of o a r t i d e i n t h e mixture. PI and P4 are t h e percentages d f i h e s;hstance t o h e determined i n t h e richer a n d leaner of t h e two t w e s of ~ a r t i c l e sa, n d P,,, is t h e average percentage of t h e suithtance'in t h e overall i i t e r i a l . T h e i m ~ l i c a t i o nof i this relations hi^^ involving t h e expected sampling error have been discussedin detail elsewhere (2,3, 51. By judicious choice of parameters the theoretical sampling error us can be constrained t o any desired value. Experimentallv. " . the overall variance in a n analvsis . s .-, 2 is the sum of t h e variances associated with t h e sampling procedure s.2 and of t h e subsequent analytical operations sm2 so2= ss2+ a,=
(2)
T h e standard deviation2 of t h e sampling procedure is then .s = "'LFi? (3) Hence, measurement a s described later of s, a n d 3, permits direct estimation of t h e sampling standard deviations,. An experiment suitable for~instructionaluse should include t h e following: (1) Availabilrty of asimple, precise analytical proredwe, so that the
standard deviation of meawrement in independently measurable and preterahly smaller than the standard deviation of aampling. (2) Absence of systematic sampling error. For example, thoraughly mixed mixtures of particulates of different densities display anomalouslv " lame " samoline errors ( 6 ) . (3) Adequate mechanical sirength of the sample components so that fragmentation does not occur during mixing. (4) Features desirable to all undergraduate experiments-safety, ease of preparation and storage of samples, and Low cost. An acid-base titration of a mixture of pure potassium hydrogen phthalate (KHP: PI = 100%, dl = 1.63) and sucrose ( P z = 0%, dz = 1.59) meets t h e above criteria. Both materials are inex~ensiveand are available in pure form as well-shaped. . . robust crystals. A mesh ranee of 30-40 was selected for t h e studies o n t h e basis of Figure 1.For small particles, sampling error is insignificant compared with t h e error i n t h e analytical measurement; this is t h e objective in preparing samples for real analysis. For instruction in sampling error, i t should he recognized t h a t for particle diameters around 0.25 m m (60 mesh) asa n d om are comparable. W e recommend t h e use of particles of diameter about 0.4 mm, where u8 dominates. Experlrnental Apparatus The titration npparatusemploysa~elf.zcruinghuret with a two.way s t ~ p w kWith . practirr, reptrtive titrotiumran br perfumed quickly with this apparatus. Because burets should not he left either drained or full of sodium hydroxide, which slowly etches glassware, a distilled water reservoir is connected to the buret by way of a second two-way stopcock. After use the huret is filled with distilled water and drained several times, and is finally left full. The apparatus for mixing the sample components is shown in Figure 2. A low-speed (15 rpm) motor is directly coupled to a 9-in. diameter metal plate on which the mixing containers are mounted radially. Metal sample containers are preferable to glass in that they minimize the effects of buildup of static charge on the particles during mixing. Since this buildup is more serious on the particles of KHP than those of sugar, segregation results. Metal cans formerly used for 35-mm film cartridges make convenient mixing containers. Glass containers are stored a t a humidity of 50% or so before use, as described below for KHP. The apparatus is grounded. Chemicals Potassium hydrogen phthalate (BDH AnalaR grade) and e m mercial granulated sugar were sieved, and the 30- to 40-mesh fraction 10.42- to0.60-mm oarticle diameter) was used. Ifdesired. other oartick sizes can be taken to eive a ranee of samoline . .. uncertainties. Initmlly, rhr KHI'wnsd;ird at 1 I&"' for itvrral hours twfure use. Howwer, huildup of stntrc charge un the dry KHI' rrystali, even i n metal containers, makes quantitative transfer without segregation difficult. The problem is reduced by storing theKHPin a desiccator
Figwe 2. Mixing apparatus for uptoeight samples.Rotation speed 15rpm.Note use of metal sample container to reduce static charge buildup.
with a beaker containing a saturated aqueous solution of Na2Cr207. 2H20, which provides a constant relative humidity of 54% (7). Sodium hydroxide titrant, approximately 0.04 M, was prepared by dissolvine solid oellets in distilled. freshlv demineralized water. A dilute 8rld hnck ritrant solutim w n i prepared by diluting concentrated HC1 wth diitilled wnrer until 1 dmp walirquknimt to001 ml of the wdium hyururide stork r h t i o n . 'ilk *olut~onwas stored in and dispensed from a dropping bottle. ~~~~
~~
Procedure 1) Weigh the twocommnents of the mixture direetlv into the mixing container in the relative proportions assigned by the instructor: (25 Ahout will he ~- recommenddl -~~~~~~~~ - 15 -.~ e of ~ material ~ ~ . to 70% KHP is needed. Mount the container on the mixing apparatus and rotate end-over-end for at least 5 hr. Carefully remove the container, open, and pour the mixture onto a clean sheet of white paper. Do not dislodge particles adhering to the walls or cap of the container, as these may not be representative of the well mixed sample. Take twelve 0.5 LtO.02)-g portions from the circumference of the pile and weigh accurately lf0.1 mg) directly into 200-ml conical flasks. Dissolve each portion in 20 ml of distilled water, and then titrate with sodium hydroxide solution to a definite phenolphthalein pink. Back-titrate dropwise with the dilute acid solution to the first complete disappearance of pink to obtain a highly precise end point. Record the end-point volume as the huret reading in milliliters minus the number of drops of backtitrant required (in units of 0.01 ml). (2) To obtain an estimate of the error in the measurement .~PXzdure. weigh into flasks, and titrateasnho\r,sixportionsol KHI'rorrespmding tu the weight present in the u.5-g sarnplrp of the mixlure t i m n W'; KHI' r~,ixturr.11.2-r: purtiun.;). hut not i r s than 0.2 g i n any ease. ~
Calculations A typical set of results is shown in the example that follows. From the weights of KHP and the titrationvolumes taken in Part (2), calculate six values for the nominal molarity of the solution of sodium hydroxide. The mean of these is used in the example; their percent relative standard deviation gives ,s directly. From the weights and volume in Part (I),and the mean molarity of the titrant, calculate the twelve values for percentage of KHP in the mixture. Their relative standard deviation gives 3,. Compute the sampling standard deviation s, from eqn. (3). .. The ZRieorouslv.the auantitvs, is defined as the estirnote of c~. validity of the estimate will depend on the numher uf sampler examined, the sampling procedure wed, and tht- erperimmtal suund"err wrth which it i~ cnrried our.
Volume 57, Number 7, July 1980 / 519
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100 200 300 Mixing Time, Minutes fa system described Figue 4. Effect d mixing time u p relative sampling in example. Dashed tics indicates value expected fa mmpletely mixed samples under these conditions. 0
Fraction of KHP in Sample Figure 3. Relation between percent rslal'we stan&ddsviationin sample am position a. an0 pacent potarsidm hydrogsn phlhalate. Sold line, calculated using eqn. (1); A, experimental values. If desired for compaxison,at this stage the instruetor can advise the student of the mesh size of his mixture and ask him to calculate the statistical sampling error a, and compare it with the actual sampling error obtained. Alternatively, the student may be asked to suhmit the results as an "unknown determination." If s. and us differ significantly, there was a significant contribution to sampling standard deviation by the mixing and sampling technique used. A grade can be assigned on the basis of how dose s, is to the expected u, for the samole. Hrcause the assumption made here that the sample particles are sphrres is only approximately valid, cautron is recommended in setting up a grading scale on the basis of theory alone. However, we fmd close correspondence between theoretical prediction and experimental reality.
11
Example: Typical Set of Data and Calculations (a) Molarity of NaOH from titrations of pure KHP: 0.03361 0.03338 0.03345 0.03364 0.03340 0.03322 Mean, 0.03345; s.d.,0.00014;sm,0.42% (b) Percentage of KHP found in sample portions: 17.79 18.78 18.56 19.15 18.96 17.44 18.75 17.07 18.95 19.60 19.27 19.45 Mean, 18.65%;s.d.,0.74;s0, 4.0% From eqn. (3):s, = [(4.0)2- (0.42)2]0.6= (15.8)0.6= 4.W (c) Calculation of ai For a 30-mesh sieve. assume all particles are 0.6 mm in diameter. (This is valid to within a few percent provided the distrihution of particle sizes is not too highly skewed toward the small end.) The number of particles in 0.5 g of a 30-mesh sample of density 1.60 will be
0.5X-X 1.60 Thus from eqn. (1)
-X--
(0.06P
- 2.76 X
lo3
Results and Conclusions An experienced analyst can complete the experimental work (exclusive of mixing time and calculations) in under 2 hr. obtaining 8, value; on the order of 1 p p t . ~ y p i c a l under: graduates require about 3 hr, obtaining s, values on the order of 1.5-5 ppt. When correct sampling technique is used, both groups obtain values for s, that lie close t o the theoretical a* Figure 3 shows the theoretical relation between a* and %KHP in a mixture of 30- to 40-mesh particles, along with some experimental values. When incorrect sampling technique is used, the results have been found to lie above the line; experimental ooints sienificantlv below the line are imorohabk. Fieure 4 shows the effect of different mixing times on sampling standard deviation. For a range in values of sample standard deviation, the ~roportionsof K H P and sucrose can be varied. or samples of &ing mesh sizecan he issued. The validity of the &dents9 sampling . - techniaue can be assessed bv com~arisonwith expected values.and that of their other experimental technique hv examinina the data for titration of pure KHP. As noted above, grading scales for such experiments should he developed by experience on the basis of student performance. The experiment described illustrates that it is not always the method of measurement or the s e ~ a r a t i o ntechnique emnloved that limits the orecision of aianalvsis. Its incirpor'atibn into analytical iaboratory training"shou1d make students more aware of the importance of the sampling operation.
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Acknowledgment Thanks are due to Mary Lau, Dan Dean, and Peter Wilson for assistance with portions of the experimental work.
Literature Cited 11) Benedetti.Piehlu. A. A,, in "Phyaieal Msthodr of Chemical Analysis," (Eddiloc Bell. W. M.,), Vol 3,Academic P m ,New York,1956,~. 183.
(2) hitinen, H. A., and Hanis, W. E., in "Chemieal Anaimis,"2nd E d MfCraw-Hill. New York. 1975. p. 565. (3) Harris, W. E., and Kratoeh4.B.. Anol Chsm. 46,313 (1974). 14) Harris, W. E.. and Kratoehvii, B., in '"Chomieal Separations and Msasurementa," Saundera, Philadelphia, 1974. p. 33. 15) Harris, W.E.,Amor.Lob. 10.31 1197.3). (6l Wiiwn, P. D., and Kretahvil, 8.. unpublished data, 1977. 17) Lange, N. A,. (Editor). "Handhmk ofChomi8try." 10th Ed..McCraw-Hill.NewYork. 1967. p. 1433.
520 1 Journal of Chemical Education
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