Solution to precision mixology challenge

Analytical and Bioanalytical Chemistry, Apr 2016

Juris Meija

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Solution to precision mixology challenge

Solution to precision mixology challenge Juris Meija 0 0 Measurement Science and Standards, National Research Council Canada , 1200 Montreal Road, Ottawa, ON K1A 0R6 , Canada - Solution Alice has a three-decimal-digit digital balance, which can weigh up to 100 g [1]. If she performs a single six-fold dilution of the stock solution with the mass fraction of cadmium w0 = 3000 μg/g, then the following equation can be written for the mass fraction of cadmium in the final solution, wCd: maliquot wCd ¼ w0 mfinal where maliquot ≈ mfinal/6. Best results are always obtained when mfinal is as large as possible, i.e., 100 g in this case. The uncertainty of wCd is estimated via uncertainty propagation as follows: u2ðwCd Þ w2Cd u2ðwCd Þ w2Cd ¼ u2 maliquot m2aliquot mf2inal ð1Þ ð2Þ ð3Þ If Alice performs serial dilution of the stock solution n-times, each time diluting the stock ki times where i = 1, 2,…,n and k1×k2×…×kn = K = 6, then the following equation can be written for the mass fraction of cadmium in the final solution, wCd: m1; aliquot wCd ¼ w0 m1; final m2; aliquot m2; final … mn; aliquot mn; final The relative uncertainty of wCd from Eq. 4 can be evaluated similar as it was done for Eq. 1. For n-step dilution, u2ðwCd Þ w2Cd ¼ X n i¼1 u2 mi; aliquot mi2; final=ki2 þ X n i¼1 u2 mi; final mi2; final Note that mi,final = (100 ± 0.002) g and mi,aliquot = (100 ± 0.002)/ki g. Given that all mass uncertainties are equal, u2ðwCd Þ w2Cd ¼ u2ðmÞ X n mf2inal i¼1 1 þ ki2 ∝X n The best results are those with lowest relative uncertainty of wCd and we can find out what values of n and ki in Eq. 6 satisfy this requirement. In other words, we are seeking the minimum value for ∑in¼1 1 þ ki2 as a function of n and ki given the constrain k1×k2×…×kn = K. This condition is met when all dilution factors are equal, which means ki = pnffiKffiffiffi. It remains now to find the optimal number of steps for serial dilution in order to achieve the desired dilution factor and guarantee the lowest relative uncertainty of wCd. This amounts to finding the minimum for function ∑in¼1 1 þ pnffiKffiffiffi2ffi ¼ n þ n pnffiKffiffiffi2ffi , which occurs at n = 2.8 ≈ 3 for K = 6. The minimum at n ≈ 3 is also shown in Fig. 1. The best strategy for Alice is to serially dilute the 3000 μg/g cadmium solution three times, each time using a ð4Þ ð5Þ ð6Þ J. Meija dilution factor of p3ffi6ffi . In this problem we have looked only at the mathematical side of the problem and one has to acknowledge other sources of errors in dilutions. For example, while a three-step serial dilution might be preferable from a mathematical point of view, each step can lead to potential contaminations from glassware and pipettes. Fig. 1 Uncertainty of wCd when serial dilution of 3000 μg/g solution is performed in n steps, each time with a dilution factor of ki = pnffi6ffi (i = 1,2, …,n) Meija J . Anal Bioanal Chem . 2016 ; 408 : 7 .

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Juris Meija. Solution to precision mixology challenge, Analytical and Bioanalytical Chemistry, 2016, 3055-3056, DOI: 10.1007/s00216-016-9413-3