Chapter 8 Sampling Standardization and Calibration TMHsiung2014 159

Chapter 8: Sampling, Standardization, and Calibration TMHsiung@2014 1/59

Contents in Chapter 08 1. 2. 3. 4. Analytical Samples and Methods Sampling The Least Squares and Calibration Curve Calibration Methods 1) External Standard Calibration - Matrix Effect 2) Standard Addition Method 3) Internal Standard 5. Figures of Merit for Analytical Methods TMHsiung@2014 2/59

1. Analytical Samples and Methods l Classification of analyses by sample size l Classification of constituent types by analyte level TMHsiung@2014 3/59

Continued l For example: Determining Hg in the ppb to ppm range in a 1 μL (≈ 1 mg) sample of river water would be a micro analysis of trace constituent. * Interlaboratory error as a function of analyte concentration: TMHsiung@2014 4/59

2. Sampling l Classification of sampling plan (Ref/Harvey) - Random sampling - Judgmental sampling - Systematic–judgmental sampling - Stratified sampling - Convenience sampling l Random sample: A sample collected at random from the target population. (provides and unbiased estimate of the target population) - using random number table - generating from spreadsheet TMHsiung@2014 5/59

l Steps in obtaining a laboratory sample TMHsiung@2014 6/59

l Statistics of sampling 1) Sampling variance versus overall variance so 2 = ss 2 + sm 2 so 2: overall variance ss 2 : variance of sampling process sm 2: variance of analytical method TMHsiung@2014 7/59

Example: A quantitative analysis for an analyte gives a mean concentration of 12. 6 mg/kg. The standard deviation for analytical method (sm) is found to be 1. 1 mg/kg, and that due to sampling process (ss) is 2. 1 mg/kg. (a) What is the overall variance for the analysis? (b) By how much does the overall variance change if sm is improved 10% to 0. 99 mg/kg? (c) By how much does the overall variance change if ss is improved 10% to 1. 9 mg/kg ? Solution: (a) so 2 = sa 2 + ss 2 = (1. 1)2 + (2. 1)2 = 5. 6 (b) so 2 = (0. 99)2 + (2. 1)2 = 5. 4 (c) so 2 = (1. 1)2 + (1. 9)2 = 4. 8 l In general, ss 2 is larger than sm 2 l More significant improvement in the so is focus on sampling problems. TMHsiung@2014 8/59

Example: The following data were particular drug concentration in a animal-feed formulations. The data on the left were obtained under conditions in which random errors in sampling and the analytical procedure contribute to the overall variance. The data on the right were obtained in a repeatedly analysing a single sample. Determine the overall variance and the contributions from sampling and the analytical procedure. so 2 sm 2 TMHsiung@2014 9/59

Solution: From left data: so 2 = 4. 71 x 10– 7 From right data: sm 2 = 7. 00 x 10– 8 Sampling variance: ss 2 = so 2 – sm 2 = 4. 71 x 10– 7 – 7. 00 x 10– 8 = 4. 01 x 10– 7 TMHsiung@2014 10/59

2) Drawing particles with two types particles (binomial distribution) Application example: n. A: analyte containing n. B: analyte free * TMHsiung@2014 11/59

Example 1: A mixture contains 1% KCl particles and 99% KNO 3 particles. What is the number of particle required for a sampling relative standard deviation of 1% for KCl ? Solution: TMHsiung@2014 12/59

Example 2: A mixture contains 1% KCl particle and 99% KNO 3 particle. The average density of the particles is 2. 108 g/cm 3, and the average diameter of the particle is 0. 152 mm. What is the sample mass equivalent to total 9. 9 x 105 particles of the mixture? Solution: For solid sample: Finer particles require less mass of sample to reach same RSD of sampling operation TMHsiung@2014 13/59

Continued For two types particles with different percentage of active ingredient and different density: d. A: density of type A particle d. B: density of type B particle d: the average density of the particles PA: the percentage of the particle contain a higher amount of active ingredient PB: the percentage of the particle contain a less amount of active ingredient P: the overall average percent of active ingredient TMHsiung@2014 14/59

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l Choosing a sample size (mount) Ks = m ( r 100)2 m: mass of the collected sample r : percent relative standard deviation of analyte measurement due to sampling Ks: Ingamells sampling constant TMHsiung@2014 16/59

Example: The following data were obtained for determination of the amount of inorganic ash in a breakfast cereal. (assume the Sm << Ss) (a) (b) (c) Determine sampling constant Ks. What is the amount of a single sample needed to give a relative standard deviation for sampling of ± 2. 0%. Predict the relative standard deviation and the absolute standard deviation if a single sample of 5 g are collected. (assume same average of percentage ash) TMHsiung@2014 17/59

Solution: (a) Mean of m = 1. 0007 Measurement mean = 1. 298 Ss = 0. 03194 r = 2. 46% Ks = m ( r 100)2 = (1. 0007)(2. 46)2 = 6. 06 g TMHsiung@2014 18/59

(b) m = Ks/ ( r 100)2 = 6. 06 g/(2. 0)2 = 1. 5 g TMHsiung@2014 19/59

l Number of replicate to be analyzed Assuming Ss >> Sa, the number of defined mass samples (ns) to reach a desired overall error (e): TMHsiung@2014 20/59

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3. The Least Squares and Calibration Curve 1) Method of Least Squares A mathematical optimization technique that attempts to find a "best fit" to a set of data by attempting to minimize the total squares of residual error, between the fitted function and the data. A linear least squares for example: Regression line (Calibration curve) ŷi Residual error = yi–ŷi yi yi: Experimental value ŷi: Calculated by y = mx + b Total squares of residual error = Σ(yi–ŷi)2 Assumptions in method of least squares: l error in y >> error in x l Uncertainties (standard deviation) in all y values are. TMHsiung@2014 same 22/59

2) Linear Calibration Curve The linear relationship between the amount of analyte and a method’s measuring signal: Smeas = k. CA + b (i. e. , y = mx + b) (established by the method of least squares) Smeas: Signal response by sample CA : Analyte concentration k: Slope between Smeas and CA b: y-intercept of the linear equation TMHsiung@2014 23/59

3) Establishing Calibration Curve Linear equation: y = mx + b TMHsiung@2014 24/59

Example: Establish a linear calibration curve by following data: x (concentration) y (signal) 1 2 3 3 4 4 6 5 Solution: TMHsiung@2014 25/59

Ans: Linear equation is y = 0. 61538 x + 1. 34615 TMHsiung@2014 26/59

4) Example of Applying Calibration Curve Following are results of spectrophotometric analysis of protein standard: 982_Ch 04_Calibration_C urve_Application_Demo. xls i) Reject datum of (0. 392) (it is an outlier) TMHsiung@2014 27/59

ii) Reject data of 25. 0 μg protein standard (out of the linear range) TMHsiung@2014 28/59

iii) 剩下 14 個數據建立 linear equation m = 0. 01628 b = 0. 10400 y = (0. 01628)x + 0. 10400 iv) 計算 unknown sample 中分析物含量 Example: 已知 calibration curve: y = (0. 01628)x + 0. 10400 樣品分析結果 吸收值為 0. 406，樣品中 protein 含量為多少？ Solution: TMHsiung@2014 29/59

4. Calibration Methods 1) External Standard Calibration l A calibration curve, y = ax + b, established by standard containing known amount of analyte. l External standard calibration is feasible if the standards and the sample’s matrix has no effect on the value of the slope of the calibration curve. TMHsiung@2014 30/59

Example: A spectrophotometric method for the quantitative determination of Pb 2+ levels in blood yields an absorbance of 0. 474 for a standard whose concentration of Pb 2+ is 1. 75 μg/L. How many μg/L of Pb 2+ occur in a sample of blood if its absorbance is 0. 361? Solution: Ans: 1. 33 μg. L– 1 TMHsiung@2014 31/59

Example: A spectrophotometric method for the quantitative determination of Pb 2+ levels in blood gives a linear normal calibration curve for which Sstand = (0. 296 μg – 1 L)CS + 0. 003. What is the Pb 2+ level (in ppb) in a sample of blood if Ssamp is 0. 397? Solution: Ans: 1. 33 μg. L-1 TMHsiung@2014 32/59

– Matrix Effect l Matrix: Everything in the unknown sample other than analyte l Matrix effect: The change in the analytical signal caused by matrix. l Value of slope versus matrix effect: Analyzing groundwater Cl. O 4– for example Signal of sample Incorrect Actual TMHsiung@2014 33/59

l Solutions for matrix effect: - Matrix matching: Adjusting the matrix of an external standard so that it is the same as the matrix of the samples to be analyzed. - Standard addition: By adding aliquots of standard solution into sample, and resolving analyte concentration by adequate algebra. Cautionary for Standard Addition Method: l Amount of spiked analyte (standard) should close to the amount of analyte in the sample l Volume of the spiked standard should be designed to avoid the matrix difference. l Drawback of standard addition: It can not avoid the signal from blank. TMHsiung@2014 34/59

3) Standard Addition Method i) Single-point standard addition: l Initial sample l Spiked sample Add Vs of [S]i Vo of [X]i: Vo : IX : [S]i: VS : IS+X: Analyte conc. of sample Vol. of sample Signal for [X]i Analyte conc. in standard Vol. of spiked standard Signal of spiked sample TMHsiung@2014 35/59

Example: A spectrophotometric method for measurong Pb 2+ in blood sample yields an absorbance f 0. 712. A 5. 00 m. L blood sample after spiking with 5. 00 μL of a 1560 μg. L– 1 Pb 2+ standard, an absorbance of 1. 546 is measured. Determine the concentration of Pb 2+ in the original sample of blood. Solution: Ans: 1. 33 μg. L– 1 TMHsiung@2014 36/59

ii) Multiple-point standard addition (Linear Graphic application): Experimental example: Place constant volume (Vo) of sample with conc. of [X]i Add different volume (VS) of standard with conc. of [S]i Dilute to a particular constant volume (Vf) TMHsiung@2014 37/59

[X]i: IX : V o: [S]i: V S: IS+X: V: Analyte conc. of sample Signal for [X]i Vol. of sample (constant) Analyte conc. in standard (constant) Vol. of added standard (variable) Signal of spiked sample (variable) Final vol. after diluting y= b + a x TMHsiung@2014 38/59

y= b + a x TMHsiung@2014 39/59

Example: Ascorbic acid in orange juice was determined by an electrochemical method. Following data and a linear graph were constructed by eight standard addition. Calculate the concentration of ascorbic acid in the sample. TMHsiung@2014 40/59

Solution: y= b + a x Ans: 2. 89 m. M TMHsiung@2014 41/59

4) Internal Standard i) Definition: l Adding known amount of a reference substance (internal standard) other than analyte is added to all blank, analyte standard and samples. l Establish a response factor based on the ratio of standard analyte signal/internal standard signal versus standard analyte conc. /internal standard conc. l Analyte concentration in unknown is then calculated according to the established response factor. - Standard addition: the “standard” is the same substance as the analyte. Internal standard: The internal “standard” is a different substance from the analyte. TMHsiung@2014 42/59

iii) Internal standard interpretation 以一個製備樣品其溶劑為揮發性者為例：。 Signal a) 溶劑未揮發 Analyte Standard 200 Internal Standard 300 Absorption wavelength (nm) b) 部分溶劑揮發 Internal Standard Signal Analyte Standard l Analyte b) 之訊 號值高於 a) l a) 和 b) Analyte standard 訊號值 /Internal standard 訊號值 之比例不變 200 300 Absorption wavelength (nm) TMHsiung@2014 44/59

iv) Establishing response factor (F) F: Response factor AX: Signal of analyte standard [X]: Conc. of analyte standard AS: Signal of internal standard [S]: Conc. of internal standard v) Calculate analyte in unknown F: Response factor AX: Signal of analyte in unknown [X]: Conc. of analyte in unknown AS: Signal of internal standard [S]: Conc. of internal standard TMHsiung@2014 45/59

Example: A spectrophotometric method for the quantitative determination of Pb 2+ levels in blood uses Cu 2+ as an internal standard. A standard containing 1. 75 μg. L– 1 Pb 2+ and 2. 25 μg. L– 1 Cu 2+ yields a ratio of AX/AS of 2. 37. A sample of blood is spiked with the same concentration of Cu 2+, giving a signal ratio of 1. 80. Determine the concentration of Pb 2+ in the sample of blood. Solution: Ans: 1. 33 μg. L– 1 TMHsiung@2014 46/59

5. Figures of merit for analytical methods l Linearity: A measure of how well data in a graph follow a straight line, e. g. , square of the correlation coefficient R 2 required close to 1. l Linear dynamic range: The concentration interval over which linearity, accuracy, and precision are acceptable. l (Calibration) Sensitivity: The change in the response signal per unit change in analytical concentration, thus, the slope of the calibration curve. TMHsiung@2014 47/59

l Limit of Detection (LOD): The smallest quantity of analyte that is “significant” different from blank, e. g. , 99% chance of being greater than the blank. l Only 1% of measurement for a blank are expected to exceed the detection limit. l But, 50% of measurement for a sample containing analyte at LOD will below LOD. TMHsiung@2014 48/59

l LOD estimation procedure: 1) Prepare sample whose analyte concentration is 1~5 times the LOD. 2) Measure 7 replicate samples. 3) Compute the signal standard deviation (s) of the 7 measurements. 4) Establish a calibration curve: y = mx +b 5) LOD (concentration) = 3 s/m TMHsiung@2014 49/59

Example: Signal from 7 replicate sample with a concentration 3 times the detection limit were 5. 0, 5. 2, 4. 6, 6. 0, and 4. 9 n. A. The slope of the calibration curve is m=0. 229 n. A/μM. What is the concentration of the limit of detection? Solution: Ans: 7. 3 μM TMHsiung@2014 50/59

l Limit of quantitation (LOQ): The minimum quantity of analyte that can be measured “accurately”, often calculated by 10 s/m. s: signal standard deviation (s) of replicate measurements. m: slope in the calibration curve l Instrument detection limit (IDL): Obtained by 7 replicate measurements of one sample. l Method detection limit (MDL): Obtained preparing 7 individual samples and analyzing each one once. Not detected (ND): Reported “ND” if the analytical result below the reporting limit of detection TMHsiung@2014 51/59

l Quality Assurance (QA) ü The protocol designed to demonstrate whether data is meeting criteria that has been established to ensure adequate data quality ü Process includes quality control and documentation of procedures and results. ü Quality control (QC): The measures taken to ensure that an analysis meets the required accuracy and precision. TMHsiung@2014 52/59

ü Duplicate analysis: (for checking precision) (Relative Percent Difference, RPD) 重複分析相對差異值百分比 Xl、X 2: 樣品重覆分析二次之個別分析值 TMHsiung@2014 53/59

ü Recovery of SRM analysis: (for checking accuracy) R： 回收率 (recovery) Ct： 已知參考值或確認值 Cm： 實際分析值 TMHsiung@2014 54/59

ü Recovery of Spike analysis: (for checking matrix effect/accuracy) R： T： ) S： X： 回收率 (recovery) 添加於樣品中之標準品的之濃度(已知值 添加樣品之測定濃度 未添樣品之測定濃度 TMHsiung@2014 55/59

l Control chart Example: A control chart for a modern analytical balance 失控時，需停止分析 作進行故障排除。 TMHsiung@2014 57/59

Example of QC application: Bear. Lab 石墨爐原子 吸光譜分析試算表 TMHsiung@2014 58/59

Homework (Due 2014/3/20) Skoog 9 th edition, Chapter 08 Questions and Problems 8 -4 8 -5 8 -9 8 -13 8 -22 End of Chapter 08 TMHsiung@2014 59/59