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ANALYTICAL PROPERTIES PART III ERT 207 ANALYTICAL CHEMISTRY SEMESTER 1, ACADEMIC SESSION 2015/16 OVERVIEW 2 ANALYTICAL SAMPLES AND METHODS STANDARDIZATION AND CALIBRATION EXTERNAL STANDARD CALIBRATION INTERPRETATION OF LEAST-SQUARE RESULTS TRANSFORMED VARIABLES QUALITY ASSURANCE OF ANALYTICAL RESULTS [email protected] ANALYTICAL SAMPLES AND METHODS 3 Macro analysis: Is used for samples whose masses are > 0.1 g. Semimicro analysis: Is performed on samples in the range of 0.01 to 0.1g. Micro analysis: Is used for samples whose mass is 10-4 – 10-2g. Ultramicro analysis: Is used for samples whose mass is > 10-4g. [email protected] ANALYTICAL SAMPLES AND METHODS 4 Major constituents: Those present in 1 – 100% by mass. Minor constituents: Species present in 0.01-1% by mass. Trace constituents: Those present in amount between 100 ppm (0.01%) and 1 ppb. Ultratrace constituents: Those present in amounts < 1 ppb. [email protected] ANALYTICAL SAMPLES AND METHODS 5 Fig 1: Classification of analytes by sample size Fig 2: Classification of constituent types by analyte level [email protected] ANALYTICAL SAMPLES AND METHODS 6 ↓ ↓ Fig 3: Inter-laboratory error as a function of [email protected] analyte concentration. ANALYTICAL SAMPLES AND METHODS 7 Note: Samples are analysed, but species or concentrations are determined. Determination of glucose in blood serum analysis or the analysis of blood serum for glucose concentration determination [email protected] ANALYTICAL SAMPLES AND METHODS 8 The composition of the gross sample and the laboratory sample must closely resemble the average composition of the total mass of material to be analyzed. The items chosen for analysis are often called as sampling units or sampling increments. In sampling, a sample population is reduced in size to an amount of homogeneous material that can be conveniently handled in the laboratory and whose composition is representative of the population. [email protected] ANALYTICAL SAMPLES AND METHODS 9 Statistically, the goals of the sampling process are: 1. To obtain a mean analyte concentration that is an unbiased estimate of the population mean. It can be realized if all members of the population have an equal probability of being included in the sample. 2. To obtain variance in the measured analyte concentration that is an unbiased estimate of the population variance so that valid confidence limits can be found for the mean, and various hypothesis tests can be applied. It can be realized if every possible sample is equally likely to be drawn. [email protected] ANALYTICAL SAMPLES AND METHODS The number of particles required in a gross sample ranges from a few particles to 1012 particles. Based on the Bernoulli equation: The standard deviation of the number of A particles drawn: σ N 1 p Probability of 10 A p randomly drawing an A type particles The relative standard deviation of drawing A type particles: σA 1 p σr [email protected] Np Np ANALYTICAL SAMPLES AND METHODS 11 The number of particles needed to achieve a given relative standard deviation: 1 p N pσ 2 t If 80% of the particles are type A (p=0.8) and the desired relative standard deviation is 1% (σr=0.01), the number of particle making up the gross sample should be: 1 0.8 N 2500 2 0.80.01 [email protected] ANALYTICAL SAMPLES AND METHODS 12 In reality: The type A particles contain a higher percentage of analyte, PA and the type B particles a lesser amount, PB. The average density d of the particles differs from the densities dA and dB of these components. d Ad B N p1 p 2 d 2 p A pB σr P 2 Overall average percent of active ingredient (%) [email protected] ANALYTICAL SAMPLES AND METHODS Rearrange the equation: The relative standard deviation: PA PB d Ad B p1 p σr x 2 P d N If we make the assumption that the sample mass m is proportional to the number of particles and the other quantities in the equation are constant, the product of m and σ, should be a constant. 13 K s m x σ r x100 2 σr x 100% = % relative standard [email protected] ANALYTICAL SAMPLES AND METHODS 14 When σr = 0.01, σr x 100% = 1%, Ks = m. Ks is the minimum sample mass required to reduce the sampling uncertainty to 1%. [email protected] ANALYTICAL SAMPLES AND METHODS 15 EXAMPLE 1 A column packing material for chromatography consists of a mixture of two types of particles. Assume that the average particle in the batch being sampled is approximately spherical with a radius of about 0.5 mm. Roughly 20% of the particles appear to be pink in color and are know to have about 30% by mass of a polymeric stationary phase attached (analyte). [email protected] ANALYTICAL SAMPLES AND METHODS 16 The pink particles have a density of 0.48 g/cm3. The remaining particles have a density of about 0.24 g/cm3 and contain little or no polymeric stationary phase. What mass of the material should the gross sample contain if the sampling uncertainty is to be kept below 0.5% relative? [email protected] ANALYTICAL SAMPLES AND METHODS 17 Number of laboratory samples: How many samples should be taken for the analysis? 2 2 t ss The number of samples N, N 2 xσ 2 r [email protected] ANALYTICAL SAMPLES AND METHODS 18 EXAMPLE 2. The determination of copper in a seawater sample gave a mean value of 77.81 μg/L and a standard deviation ss of 1.74 μ g/L. How many samples must be analysed to obtain a relative standard deviation of 1.7% in the results at the 95% confidence level? [email protected] STANDARDIZATION AND CALIBRATION 19 Calibration: It determines the relationship between the analytical response and the analyte concentration. The relationship is usually determined by the use of chemical standards. Interference could be reduced from other constituents in the sample matrix, called concomitants by using standards added to the analyte solution or by matrix matching or modifications. [email protected] STANDARDIZATION AND CALIBRATION The absolute method (e.g. gravimetric method) do not rely on calibration with chemical standards. Comparison with standards: 2 types of comparison methods: i. Direct comparison techniques ii. Titration procedures Null camparison or isomation methods: Comparison a property of the analyte with standards such that the property being tested matches or nearly matches that of the standard. [email protected] 20 STANDARDIZATION AND CALIBRATION 21 With some modern instruments, a variation of this procedure is used to determine if an analytes concentration exceeds or is less than some threshold level. Comparator can be used to indicate that the threshold has been exceeded. Titration is a type of chemical comparison. The amount of the standardized reagent needed to achieve chemical equivalence can be related to the amount of analyte present by stoichiometry. [email protected] EXTERNAL STANDARD CALIBRATION 22 A series of standard solutions is prepared separately from the sample. The standards are used to establish the instrument calibration function. It is obtained from analysis of the instrument response as a function of the known analyte concentration. Ideally, 3 or more standard solutions are used in the calibration process, although in some routine determinations, 2 point calibration can be reliable. [email protected] EXTERNAL STANDARD CALIBRATION 23 The calibration function can be obtained graphically or in mathematical form. A plot of instrument response versus known analyte concentrations is used to produce a calibration curve, sometimes called a working curve. It is often desirable that the calibration curve be linear in at least the range of the analyte concentrations. [email protected] EXTERNAL STANDARD CALIBRATION 24 Fig. 1: Calibration curve of absorbance versus analyte concentration for a series of standards. [email protected] EXTERNAL STANDARD CALIBRATION 25 The linear relationship is then used to predict the concentration of an unknown analyte solution. (1) The Least-squares Method The investigator must try to draw the ‘best’ straight line among the data points. Regression analysis provides the means for objectively obtaining such a line and also for specifying the uncertainties associated with its subsequent use. [email protected] EXTERNAL STANDARD CALIBRATION 26 Assumptions of the leastsquare method: 1)There is actually a linear relationship between the measured response y and the standard analyte concentration x. The mathematical relationship that describes this assumption is called the regression model (y = mx + c). Fig. 2: The slopeintercept form of a straight line. [email protected] EXTERNAL STANDARD CALIBRATION 27 2) We assume the y deviation from the individual points from the straight line arises from error in the measurement. There is no error in x values of the points (concentrations). When the uncertainties in the y values vary significantly with x, basic least-squares analysis may not give the best stright line. Instead, a correlation analysis (or weighted least squares analysis) should be used. [email protected] INTERPRETATION OF LEAST-SQUARE RESULTS 28 The closer the data points are to the line predicted by a least-square analysis, the smaller are the residuals. The sum of the squares of the residuals, SSresid, measures the variation in the observed values of the dependent variables (y values) that are not explained by the presumed linear relationship 2 between x and y: N SS resid yi c mxi i 1 [email protected] INTERPRETATION OF LEAST-SQUARE RESULTS A total sum of the squares, y y y y N 2 29 SStot S yy 2 i 2 i i SStot is a measure of the total variation in the observed values of y since the deviations are measured from the mean value of y. The coefficient of determination (R2) measures the fraction of the observed variation in y that is explained by the linear relationship: SS resid R 1 SStot 2 [email protected] INTERPRETATION OF LEAST-SQUARE RESULTS 30 The closer R2 is to unity, the better the linear model explains the y variations. The difference between SStot and SSresid is the sum of the squares due to regression, SSreg. In contrast to SSresid, SSreg is a measure of the explained variation. SS regr SStot SS resid & R 2 SS regr SStot [email protected] INTERPRETATION OF LEAST-SQUARE RESULTS EXAMPLE 3 Find the coefficient of determination for the chromatographic data below. 31 Mole percent isooctane, xi 0.352 0.803 Peak area, yi 1.09 1.78 1.08 1.38 1.75 2.60 3.03 4.01 The least-square line: SStot = 5.07748. y 2.09 x 0.26 [email protected] TRANSFORMED VARIABLES 32 Transformations to linearize function: Linear least squares gives best estimates of the transformed variables, but these may not be optimal when transformed back to obtain estimates of the original parameters. Nonlinear regression methods may give better [email protected] estimates. QUALITY ASSURANCE OF ANALYTICAL RESULTS 33 Control chart: A sequential plot of some characteristic that is a criterion of quality (quality assurance). It shows the statistical limits of variation that are permissible for the characteristic being measured. 3σ μ = population UCL μ mean Upper control limit: N 3σ Lower control limit: UCL μ N σ = population standard deviation [email protected] QUALITY ASSURANCE OF ANALYTICAL RESULTS 34 Fig 3: A control chart for a modern analytical balance. For example, from independent experiments, estimates of the population mean and standard deviation were found to be μ = 20.000 g and σ = 0.00012 g, respectively. [email protected] QUALITY ASSURANCE OF ANALYTICAL RESULTS 35 The mean of 5 measurement, 0.00012 = 0.00016 3x 5 UCL = 20.00016 g & LCL = 19.99984 g. (see Fig 3). As long as the mean mass remains between the LCL and the UCL, the process is said to be in statistical control. [email protected] QUALITY ASSURANCE OF ANALYTICAL RESULTS Fig. 4 shows the results of 89 production runs of a cream containing a nominal 10% benzoyl peroxide measured on consecutive days. Each sample is represented by the mean percent benzoyl peroxide determined from the results of five titrations of different analytical samples of the cream. The chart shows that, until day 83, the manufacturing process was in statistical control with normal random fluctuations in the amount of benzoyl peroxide. 36 [email protected] QUALITY ASSURANCE OF ANALYTICAL RESULTS Fig 4: A control 37 chart for monitoring the concentration of benzoyl peroxide in a commercial acne preparation On day 83, the system went out of control with a dramatic systematic increase above the UCL. [email protected] QUALITY ASSURANCE OF ANALYTICAL RESULTS 38 This increase caused considerable concern at the manufacturing facility until its source was discovered and corrected. These examples show how control charts are effective for presenting quality control data in real world problems. [email protected] EXAMPLE 1 The values for the average density and percent polymer: d = 0.20 x 0.48 + 0.80 x 0.24 = 0.288 g/cm3 0.20 x0.48 x0.30g polymer / cm P 0.288 g sample / cm = 0.10% Then, 3 3 x100% 2 0.48 x0.24 30 0 N 0.201 0.20 2 0.288 0.005 x0.10 = 1.11 x 105 particles required 39 [email protected] 2 EXAMPLE 1 Mass of sample = 3 4 cm 0.288g 3 1.11x10 particles x π 0.05 x 3 3 particle cm 5 = 16.7 g 40 [email protected] EXAMPLE 2 Assume we have infinite number of samples, t = 1.96 at 95% confidence level. Since σr = 0.017, ss = 1.74 and Ẋ=77.81, 2 2 1.96 x1.74 N 2 2 0.017 x77.81 = 6.65 We round the result to 7 samples, the value for t for 6 degrees of freedom is 2.45. Using the t value (2.45), we calculate N = 10.38. 41 [email protected] EXAMPLE 2 Now we use 9 degree of freedom, t = 2.26, N = approximately 9. Note that it would be good strategy to reduce the sampling uncertainty so that fewer samples would be needed. 42 [email protected] EXAMPLE 3 Finding the sum of the squares of the residuals: xi yi ẏi yi-ẏi (yi-ẏi)2 0.352 1.09 0.99326 0.09674 0.00936 0.803 1.78 1.93698 -0.15698 0.02464 1.08 2.60 2.51660 0.08340 0.00696 1.38 3.03 3.14435 -0.11435 0.01308 1.75 4.01 3.91857 0.09143 0.00836 5.365 12.51 SStot 0.06240 = 5.07748, SS resid 0.0624 R 1 1 = 0.9877 SStot 5.07748 2 43 [email protected]