Lecture 3 SIGNIFICANT FIGURES e.g. - all digits that are certain plus one which contains some uncertainty are said to be significant figures 10.07 ml 0.1007 L 4 significant figures 0.10070 L 5 significant figures Addition & subtraction - keep only as many decimal places as in the number which has the least number of decimals. Multiplication & division e.g. - keep as many significant in the final answer as are found in the original number with the least number of significant figures 4.3179 x 10 1 3.6 x 10-19 1.6 x 10-6 Week -1
Logs & antilogs logs: keep as many digits to the right of the decimal point (mantissa) as there are significant figures in the original number e.g. log 97.1.4793 antilog: the number of digits to the right of the decimal point (mantissa) should equal the number of significant figures in the antilog e.g. antilog (-3.4) 10-3.4 3.8 x 10-4 ROUNDING i.e. for 5.555 do we use 5.55 or 5.56?? when rounding a 5, always round the preceding digit to the nearest even number i.e. 5.56 and 5.545 is rounded to 5.54 If the number after the last significant figure required is < 5 e.g. 5.554 no increment i.e. 5.55 If the number after the last significant figure required is > 5 e.g. 5.556 increase to the next higher number i.e. 5.56 Week -
ERROR ANALYSIS Types of errors Systematic - reproducible inaccuracy introduced by faulty equipment e.g. a leaking burette error in calibrating an analytical instrument etc This type of error can be detected and corrected Random - measure of fluctuation in results after repeated experimentation cannot be eliminated follows a normal distribution (Gaussian) - STATISTICS we will look at random errors Week -3
Absolute Uncertainty - associated with a measurement and has units e.g. weight ± 0.0001 g burette ± 0.0 ml Relative Uncertainty has no units absolute uncertainty magnitude of measurement e.g. weight 0.345 ± 0.0001g then relative uncertainty 0.0001 g 0.345 g 0.0004 0.4 ppt Accuracy - measure how close the result of the experimental value is to the true value (or mean value) Precision - measure how consistent the result is determined without any reference to any true value Week -4
Propagation of Errors (approximate method) - shows how uncertainty in the measurement of individual quantities translate into random variation in the final result of a calculation. Addition and subtraction The uncertainty of the measured values are additive in the determination of the uncertainty of the final result. If R (± e R ) A (± e A ) + B (± e B ) + C (± e C ), then e e + e + e R A B C Multiplication and division The uncertainty are transmitted i.e. for R AB/C e R R e A e B e C A + B + C Week -5
Week -6 Powers i.e. F X e F F ex X also if F (XY) ½ e F F 1 e XY XY and e XY XY 1 e X X e Y Y +
A student makes up a solution of NaCl in water by weighing the NaCl, dissolving it in water and diluting the solution to 1 L in a volumetric flask. Results: weight of empty beaker 5.183 g (e 0.000 g) weight of beaker + NaCl 5.911 g (e 0.000 g) volumetric flask 1.000 L (e 0.001 L) Calculate the analytical concentration and its standard deviation. 1. Calculate mass of NaCl: 5.911 ± 0.000-5.183 ± 0.000 0.1079 ± e ˆ mass NaCl 0.1079 ± 0.000 8 g (extra sig fig for later calculation) Week -7
. Calculate moles of NaCl: 0.1079 g 1.846 x 10-3 mol 58.44 g/mol e NaCl 1.846 x 10 3 0.0008 + 0.1079 0 e NaCl 4.79 x 10 6 3. Calculate concentration of NaCl C 1.846 x 10 3 mol 1.000 L 1.846 x 10 3 M ±?? ec 1.846 x 10 3 4.79 x 10 6 1.846 x 10 3 + 0.001 1.000 6.73 x 10-6 + 1.000 x 10-6 7.73 x 10-6 e c 5.13 x 10-6 C NaCl (1.846 ± 0.005) x 10-3 M Week -8
Lecture 4 STATISTICAL TREATMENT OF DATA Chemists generally repeat analyses of a given sample 3 to 6 times. Since the results are seldom the same, which data are selected as the best results to report?? How to report data 1. Arithmetic Mean or Average X_ N X i i N. Range (spread) - difference between the highest and lowest results. The spread or dispersion of this set of values is measured by the variance, v s : s N x _ i x i N 1 for N 0 and standard deviation s Week -9
In some disciplines and Quattro Pro, the variance is defined as v' N x i i N _ x for N s' v' As the number of values increases, s 6 s Week -10
Reporting Results in the Lab We use a very simple method for reporting results in the laboratory e.g. An analysis gave the results 10.06, 10.0, 10.10, 10.10 w/w % _ 10. 06 + 10. 0 + 1010. + 1010. mean x 101. wt% 4 variance v s (. 006) + (. 008) + (. 00) + (. 00) 4 1 00108. 3 000036. and s 00036. ± 006. The result would be reported as 10.1 ± 0.06 wt % Week -11
However, in the lab, we use an even simpler method of reporting the results: mean _ x 10.1 wt % average deviation from the mean N x _ i x i 0.04 N 5 ˆ relative uncertainty 0.04 10.1 and ppt 0.04 x 1000 4 ppt 10.1 the result is reported as: 10.1 wt % ± 4 ppt GAUSSIAN CURVE The variation in experimental data is normally distributed when replicate measurements follow a Gaussian distribution. Week -1
Q Test - Rejection of Outlying Data Point Sometimes a set of data points contains a result that differs significantly from the remaining data points must have at least 4 points could we ignore this point? let the Q test decide Q observed gap range *(diff between suspected results & its nearest neighbour)* *diff between lowest & highest values* At 90 % confidence level N Q critical 4 0.76 5 0.64 6 0.56 7 0.51 8 0.47 If Q observed > Q critical discard data point Week -13
e.g. 5 determinations of vitamin C content of a citrus drink gave the following results: 0.18, 0.19, 0.30, 0.15 and 0.0 mg/ml Apply the Q test to see if the 0.30 value can be rejected. Q obs 0.30-0.0 0.30-0.15 0.67 for 5 data points, Q crit 0.64 since Q obs 0.67 > Q crit then the 0.30 value can be rejected using the Q test rule Week -14
Student s t - is a statistical tool frequently used to express confidence intervals of the population mean, :, and for comparing results from different experiments. Confidence interval: µ _ x ± ts n where µ is population mean _ x is the measured mean Week -15
e.g. An analyst gave the results of an iron sample as: x _ 10.5, S 0.05, n 10 NRC gave a value of 10.60 % Fe. Are the results significantly different at 95 % probability level. _ µ x ± ts n 10.60 10.5 ± t x 0.05 10 t 5.06 From the tables ( table 4- pp 74): for degrees of freedom 9 and 95 % probability level t.6 since 5.06 >.6, the results are significantly different from the NRC result. Week -16
Using Student s t to: 1. Compare a measured result with a known value.. Compare replicate measurements. - different experiments 3. Compare individual differences - single measurements using two different methods on several different samples. Week -17
Using Spreadsheets Solving Problems Sample problems will be explained in class. Quattro Pro and Excel will be used in this course. You are free to use any spreadsheet program that you are familiar with. Harris also have similar type spreadsheets that you will encounter in this course, on his text web site. http://bcs.whfreeman.com/qca/ Week -18