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# Practical applications of sampling theory

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of the materials could be improved. The producers of the RMs usually carry out homogeneity tests that provide data that could be reported in a compressed form as sampling constants, but unfortunately at the moment, these data are not fully utilized. Below, some examples are given on how Gy\u2019s funda- mental sampling error model can be used in practice to design and audit analytical procedures. Some further examples can be found in Ref. [8]. As mentioned above, P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u201394 87 If the material to be sampled consists of particles having different shapes and size distributions, it is difficult to estimate the number of critical particles in the sample. Gy has derived an equation that can also be used in this case to estimate the relative variance of the fundamental sampling error: r2r ¼ Cd3 1 MS \ufffd 1 ML \ufffd \ufffd ð3Þ Here rr ¼ ra aL ¼ Relative standard deviation of the fundamental sampling error ð4Þ where ra, absolute standard deviation (in concentration units); aL, average concentration of the lot; d, characteristic particle size = 95% limit of the size distribution; MS, sample size; ML, lot size; and C is the sampling constant that depends on the properties of the material sampled. C is the product of four parameters: C ¼ fgbc ð5Þ where f is the shape factor (see Fig. 2). Shape factor is the Sampling Expert: Well, let\u2019s consider the problem. We could use the Poisson distribution to estimate the required sample size. Let\u2019s see: The maximum relative standard deviation sr = 20%= 0.2. From Eq. (2), we can estimate how many coarse particles there should be in the sample to have this standard deviation n ¼ 1 s2r ¼ 1 0:22 ¼ 25 If 1 tonne contains 5 coarse particles, this result means that the primary sample should be 5 tonne. This is a good example of an impossible sampling problem. Although you could take a 5-tonne sample, there is no feasible technology to separate and count the coarse particles from it. You should not try the traditional analytical approach in controlling the quality of your product. Instead, if the specification is really sensible, you forget the particle size analyzers and maintain the quality of your product by process technological means; that is, you take care that all the equipment are regularly serviced and their high performance maintained so that the product quality is always maintained. Plant Manager: Thank you. In light of what you said, it seems that the expensive laser diffraction particle size analyzer recommended to us will not solve our problem. 3.2. Applications of Gy\u2019s fundamental sampling error equation for designing sample preparation procedures ratio of the volume of the sampled particles having the characteristic dimension d to the volume of the cube having the same dimension. For spheroidal particles fc 0.5, which often can be used as the default value for this parameter. g is the size distribution factor ( g = 0.25 for wide-size distribution, and g = 1 for uniform particle sizes), and b is the liberation factor (see Fig. 2). Liberation factor is an empirical correction for materials, where the critical particles are found as inclusions in the matrix particles. Liberation size L is defined as the size of the opening of a screen below which 95% of the material has to be crushed in order to liberate at least 85% of the critical particles; bmax = 1 (liberated materials and materials ground below the liberation size L), bmin = 0.03 (materials where the critical particles are very small in comparison to d; note that because b is dependent on the particle size d for a given material, the sampling constant C changes when the material is ground or crushed). c is the constitution factor and can be estimated by using Eq. (6) if the necessary material properties are available. c ¼ 1\ufffd aL a \ufffd \ufffd2 aL a qc þ 1\ufffd aL a \ufffd \ufffd qm ð6Þ Here, aL is the average concentration of the lot; a, the concentration of the analyte in the critical particles; qc, the density of the critical particles; and qm, the density of the matrix or diluent particles. If the material properties are not available and they are difficult to estimate, the sampling constant C can always be estimated experimentally. International reference mate- rials (RMs), for example, are a special group of materials for which the sampling constant should always be esti- mated and reported. Unfortunately, this is seldom done. If the particle size distribution and sampling constants were Fig. 2. Estimation of particle shape factor and liberation factor for unliberated and liberated critical particles. L is the particle size of the critical particles. the fundamental sampling error is the minimum theoretical error achievable in a sampling step. Therefore, the funda- mental sampling error calculations give realistic estimates for the global sampling error, only if the material is well- mixed before sampling, and all sampling and subsampling contains as an average 0.05% of an enzyme powder that has a density of 1.08 g/cm3. The size distribution of the enzyme was available, and from it was estimated that the characteristic particle size d = 1.00 mm and the size factor g = 0.5. Estimate the fundamental sampling error for the following analytical procedure. The actual concentration of a 25-kg bag is estimated by taking first a 500-g sample from it. This material is ground to particle size \ufffd 0.5 mm. Then, the enzyme is extracted from a 2-g sample by using a proper solvent, and the concentration is determined by using liquid chromatogra- phy. The relative standard deviation of the chromatographic measurement is 5%. To estimate the errors of the two sampling steps, we have the following material properties: The total relative standard deviation can now be estimated by applying the rule of propagation of errors: st ¼ \ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03\ufb03X s2ri q ¼ 0:143 ¼ 14:3% The largest error is generated in preparing the 2-g sample for the extraction of the enzyme. To improve the overall precision, this step should first be modified. The P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u20139488 These values give for constitution factor (Eq. (6)) the value c = 2160 g/cm3, and for the sampling constants (Eq. (5)), C values C1 ¼ 540 g=cm3 and C2 ¼ 270 g=cm3 Eq. (3) gives now the standard deviation estimates for sampling steps. M1 = 500 g ML1 = 25000 g d1 = 0.1 cm g1 = 0.5 aL= 0.05% a= 100% qc = 1.08 g/cm 3 qm= 0.67 g/cm 3 f = 0.5 . . .default value for spheroidal particles b= 1 . . .liberated particles M2 = 2.0 g . . .sample sizes ML2 = 500 g . . .lot sizes d2 = 0.05 cm . . .particle sizes g2 = 0.25 . . .estimated size distribution factors sr1 = 0.033 = 3.3% . . .primary sample sr2 = 0.13 = 13% . . .secondary sample procedures are carried out with equipment and methods that follow the rules of sampling correctness defined in Gy\u2019s sampling theory. Therefore, if large lots are sampled, the uncertainty of the primary samples has to be estimated in a different way, e.g., by using Gy\u2019s variographic method. Example 2. A certain cattle feed (density = 0.67 g/cm3) sr3 = 0.05 = 5% . . .analytical determination recommendation from this exercise is that either a larger sample should be used for the extraction, or the primary sample should be pulverized to a finer particle size before secondary sampling, whichever is more economic in practice. Example 3. Evaluate the feasibility of the following procedure for calibrating an IR spectrometer for the determination of quartz in mineral mixtures. To prepare the calibration standards, pure minerals (d= 1 mm) were ground individually for 2 min in a swing mill. Then 30 mg\u20132.95 g of each mineral was carefully weighed to obtain the designed composition. The material was carefully mixed for 3 min in a Retsch Spectro Mill, and 20 mg of the mineral mixture was carefully weighed into 4.98 g of KBr and mixed for 3