1. ANALYTICAL PROPERTIES PART III ERT 207 ANALYTICAL CHEMISTRY SEMESTER 1, ACADEMIC SESSION 2015/16

2. OVERVIEW  ANALYTICAL SAMPLES AND METHODS  STANDARDIZATION AND CALIBRATION  EXTERNAL STANDARD CALIBRATION  INTERPRETATION OF LEAST-SQUARE RESULTS  TRANSFORMED VARIABLES  QUALITY ASSURANCE OF ANALYTICAL RESULTS 2 bblee@unimap

3. ANALYTICAL SAMPLES AND METHODS bblee@unimap 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 -2 g.  Ultramicro analysis:  Is used for samples whose mass is > 10 -4 g.

4. ANALYTICAL SAMPLES AND METHODS bblee@unimap 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.

5. ANALYTICAL SAMPLES AND METHODS bblee@unimap 5 Fig 1: Classification of analytes by sample size Fig 2: Classification of constituent types by analyte level

6. ANALYTICAL SAMPLES AND METHODS bblee@unimap 6  Fig 3: Inter-laboratory error as a function of analyte concentration. ↓ ↓

7. ANALYTICAL SAMPLES AND METHODS bblee@unimap 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

8. ANALYTICAL SAMPLES AND METHODS bblee@unimap 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.

9. ANALYTICAL SAMPLES AND METHODS bblee@unimap 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.

10. ANALYTICAL SAMPLES AND METHODS bblee@unimap 10  The number of particles required in a gross sample ranges from a few particles to 10 12 particles.  Based on the Bernoulli equation:  The standard deviation of the number of A particles drawn:  The relative standard deviation of drawing A type particles: Probability of randomly drawing an A type particles

11. ANALYTICAL SAMPLES AND METHODS bblee@unimap 11  The number of particles needed to achieve a given relative standard deviation:  If 80% of the particles are type A (p=0.8) and the desired relative standard deviation is 1% ( σ r =0.01 ), the number of particle making up the gross sample should be:

12. ANALYTICAL SAMPLES AND METHODS bblee@unimap 12  In reality:  The type A particles contain a higher percentage of analyte, P A and the type B particles a lesser amount, P B.  The average density d of the particles differs from the densities d A and d B of these components. Overall average percent of active ingredient (%)

13. ANALYTICAL SAMPLES AND METHODS bblee@unimap 13  Rearrange the equation:  The relative standard deviation:  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. σ r x 100% = % relative standard deviation

14. ANALYTICAL SAMPLES AND METHODS bblee@unimap 14  When σ r = 0.01, σ r x 100% = 1%, K s = m.  Ks is the minimum sample mass required to reduce the sampling uncertainty to 1%.

15. ANALYTICAL SAMPLES AND METHODS bblee@unimap 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).

16. ANALYTICAL SAMPLES AND METHODS bblee@unimap 16  The pink particles have a density of 0.48 g/cm 3.  The remaining particles have a density of about 0.24 g/cm 3 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?

17. ANALYTICAL SAMPLES AND METHODS bblee@unimap 17  Number of laboratory samples:  How many samples should be taken for the analysis?  The number of samples N,

18. ANALYTICAL SAMPLES AND METHODS bblee@unimap 18 EXAMPLE 2.  The determination of copper in a seawater sample gave a mean value of 77.81 μ g/L and a standard deviation s s 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?

19. STANDARDIZATION AND CALIBRATION bblee@unimap 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.

20. STANDARDIZATION AND CALIBRATION bblee@unimap 20  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.

21. STANDARDIZATION AND CALIBRATION bblee@unimap 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.

22. EXTERNAL STANDARD CALIBRATION bblee@unimap 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.

23. EXTERNAL STANDARD CALIBRATION bblee@unimap 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.

24. EXTERNAL STANDARD CALIBRATION bblee@unimap 24 Fig. 1: Calibration curve of absorbance versus analyte concentration for a series of standards.

25. EXTERNAL STANDARD CALIBRATION bblee@unimap 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.

26. EXTERNAL STANDARD CALIBRATION bblee@unimap 26  Assumptions of the least- square 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 slope- intercept form of a straight line.

27. EXTERNAL STANDARD CALIBRATION bblee@unimap 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.

28. INTERPRETATION OF LEAST-SQUARE RESULTS bblee@unimap 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, SS resid, measures the variation in the observed values of the dependent variables (y values) that are not explained by the presumed linear relationship between x and y:

29. INTERPRETATION OF LEAST-SQUARE RESULTS bblee@unimap 29  A total sum of the squares,  SS tot 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 (R 2 ) measures the fraction of the observed variation in y that is explained by the linear relationship:

30. INTERPRETATION OF LEAST-SQUARE RESULTS bblee@unimap 30  The closer R 2 is to unity, the better the linear model explains the y variations.  The difference between SS tot and SS resid is the sum of the squares due to regression, SS reg.  In contrast to SS resid, SS reg is a measure of the explained variation. &

31. INTERPRETATION OF LEAST-SQUARE RESULTS bblee@unimap 31 EXAMPLE 3  Find the coefficient of determination for the chromatographic data below.  The least-square line:  SS tot = 5.07748. Mole percent isooctane, x i Peak area, y i 0.3521.09 0.8031.78 1.082.60 1.383.03 1.754.01

32. TRANSFORMED VARIABLES bblee@unimap 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 estimates.

33. QUALITY ASSURANCE OF ANALYTICAL RESULTS bblee@unimap 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.  Upper control limit:  Lower control limit: μ = population mean σ = population standard deviation

34. QUALITY ASSURANCE OF ANALYTICAL RESULTS bblee@unimap 34  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. Fig 3: A control chart for a modern analytical balance.

35. QUALITY ASSURANCE OF ANALYTICAL RESULTS bblee@unimap 35  The mean of 5 measurement, = 0.00016  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.

36. QUALITY ASSURANCE OF ANALYTICAL RESULTS bblee@unimap 36  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.

37. QUALITY ASSURANCE OF ANALYTICAL RESULTS bblee@unimap 37  On day 83, the system went out of control with a dramatic systematic increase above the UCL. Fig 4: A control chart for monitoring the concentration of benzoyl peroxide in a commercial acne preparation

38. QUALITY ASSURANCE OF ANALYTICAL RESULTS bblee@unimap 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.

39. 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/cm 3 = 0.10% Then, bblee@unimap 39 = 1.11 x 10 5 particles required

40. EXAMPLE 1 Mass of sample = = 16.7 g bblee@unimap 40

41. EXAMPLE 2 Assume we have infinite number of samples, t = 1.96 at 95% confidence level. Since σ r = 0.017, s s = 1.74 and Ẋ =77.81, = 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. bblee@unimap 41

42. 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. bblee@unimap 42

43. EXAMPLE 3 Finding the sum of the squares of the residuals: SS tot = 5.07748, = 0.9877 bblee@unimap 43 xixi yiyi ẏiẏi yi-ẏiyi-ẏi (y i - ẏ i ) 2 0.3521.090.993260.096740.00936 0.8031.781.93698-0.156980.02464 1.082.602.516600.083400.00696 1.383.033.14435-0.114350.01308 1.754.013.918570.091430.00836 5.36512.510.06240