Topic 2 Measurement and Calculations in Chemistry

Transcription

1 Topic Measurement and Calculations in Chemistry

2 Nature of Measurement Quantitative observation consisting of two parts. number scale (unit) Examples 0 grams joule seconds

3 The Fundamental SI Units

4 Prefixes Used in the SI System Prefixes are used to change the size of the unit.

5 Parts per million and parts per billion In IUPAC terminology, parts per billion, parts per million, and parts per thousand are mass concentrations. For very dilute solutions, parts per million (ppm) is a convenient way to express concentration: massofsolute c 6 10 ppm ppm massofsolution 10 9 ppb = 10 6 ppm

6 Significant figures & Calculations 1. Exact numbers have an infinite number of significant figures. 1 inch =.54 cm, exactly. 9 pencils (obtained by counting).. Nonzero integers always count as significant figures. 3456: 4 sig figs

7 Significant figures & Calculations 3. Three classes of zeros. a. zeros that precede all the nonzero digits: do not count :? sig figs. b. zeros between nonzero digits: always count : 4? sig figs. c. zeros at the right end of the number: significant only if the number contains a decimal point : 4? sig figs. 150:? sig figs.

8 Significant Figures in Mathematical Operations For multiplication or division, the number of significant figures in the result is the same as the number in the least precise measurement used in the calculation =

9 Significant Figures in Mathematical Operations For addition or subtraction, the result has the same number of decimal places as the least precise measurement used in the calculation Corrected 31.8

10 Significant Figures in Mathematical Operations Logarithms and Antilogarithms The base 10 logarithm of n is the number a, whose value is such that n=10 a : The number n is said to the antilogarithm of a.

11 Logarithms and Antilogarithms In converting a logarithm to its antilogarithm, the number of significant figures in the antilogarithm should equal the number of digits in the mantissa.

12 Logarithms and Antilogarithms [H + ]= ph=-log( ) = -( )=.70 antilogarithm of logarithm of log 339 = =.530 antilog (-3.4) = = = 3.8x10-4

13 Concept Check You have water in each graduated cylinder shown. You then add both samples to a beaker (assume that all of the liquid is transferred). How would you write the number describing the total volume? 3.1 ml What limits the precision of the total volume?

14 The beakers below have different precisions. You pour the water from these three beakers into one container. What is the volume in this container reported to the correct number of significant figures? a) ml b) 78.8 ml c) 78.8 ml d) 79 ml

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16 Errors in chemical Analyses 1. Errors are caused by faulty calibrations or standardizations or by random variations and uncertainties in results.. The term error has two slightly different meanings. - error refers to the difference between a measured value and the true or known value. - error often denotes the estimated uncertainty in a measurement or experiment. 3. We can only hope to minimize errors and estimate their size with acceptable accuracy.

17 Uncertainty A digit that must be estimated is called uncertain. A measurement always has some degree of uncertainty. Record the certain digits and the first uncertain digit (the estimated number).

18 Types of Errors Every measurement has some uncertainty experimental error. Experimental error is classified as either systematic or random. Maximum error v.s. time required

19 Systematic error Determinate error = consistent error - errors arise: instrument, method, & person - can be discovered & corrected - from fixed cause, & is either high (+) or low (-) every time. - Instrumental, method, and personal errors - ways to detect systematic error: examples (a) ph meter (b) buret

20 Random error Indeterminate error - Always present & cannot be corrected - Has an equal chance of being (+) or (-). - examples: (a) people reading the scale (b) random electrical noise in an instrument. (c) ph of blood (actual variation: time, or part)

21 Precision & Accuracy reproducibility confidence of nearness to the truth

22 The boiling point of a liquid was measured in the lab, with the following results: Trial Boiling Point 1.0 C ± C ± C ± 0.1 The actual boiling point of the liquid is 8.7 C. The results of the determination of the boiling point are: a) accurate and precise. b) precise but inaccurate. c) accurate but imprecise. d) inaccurate and imprecise.

23 Types of Errors Absolute & Relative uncertainty a) Absolute : the margin of uncertainty 0.0(the measured value - the true value) b) Relative absolute uncertainty magnitude of measurement (ex) ml %

24 Statistical treatment of random errors Statistical analysis only reveals information that is already present in a data set. Statistical methods, do allow us to categorize and characterize data and to make objective and intelligent decisions about data quality and interpretation.

25 Is my red blood cell count high today? "normal"days cells μl Today's count cells μl

26 Properties of Gaussian Curves 1) Nerve cells muscle cells (1991 Nobel Prize in Medicine & Physiology) Sakmann & Neher absence neurotransmitter present neurotransmitter

27 Properties of Gaussian Curves 9 ion channels response Typical lab measurements: Gaussian distribution y e ( x ) / Gaussian curves can be described by an equation that contains two parameters, the population mean m and the population standard deviations.

28 Properties of Gaussian Curves Gaussian distribution is characterized by 1)Mean: x N x1 i 1 N )Standard deviation: N i1 ( x ) i N

29 Properties of Gaussian Curves The two curves are for two populations of data that differ only in their The smaller the, the more precise the results. reproducible x,s μ, σ for for a finite set. an infinite set.

30 Areas under a Gaussian Curve σ & probability: regardless of its width, 68.3% of the area beneath a Gaussian curve for a population lies within of the mean m. 95.4% of all data points within of the mean and 99.7% within 3. Other terms: Median & Range

31 The Sample Standard Deviation A Measure of Precision - The sample standard deviation s is expressed as: s N i1 ( x i x) N 1 N i1 d i N 1 - This equation applies to small sets of data. - Number of degrees of freedom: (N - 1) is used instead of N. - When N is greater than about 0, s is usually a good estimator of, and these quantities can be assumed to be identical for most purposes. - The sample variance : s.

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33 Other terms Pooling Data to Improve the Reliability of s for several subsets of data Variance (s ) is the square of the standard deviation. The sample variance s is an estimate of the population variance and is given by: Relative Standard Deviation (RSD) and Coefficient of Variation (CV) t N i N j k k j i i pooled N N N N x x x x x x s ) ( ) ( ) ( ) ( 1 ) ( 1 1 N d N x x s N i i N i i x s s RSD r

34 s RSDinppt 1000 ppt x The relative standard deviation multiplied by 100% is called the coefficient of variation (CV). CV RSDinpercent s x 100% Spread or Range (w) is used to describe the precision of a set of replicate results. It is the difference between the largest value in the set and the smallest.

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37 Standard deviation of calculated results We must estimate the standard deviation of a result that has been calculated from two or more experimental data points, each of which has a known sample standard deviation.

38 1) Addition & Subtraction ) e ( 3.06 e 0.0) ( 0.59 e 0.0) ( 1.89 e 0.03) ( %) ( ) ( 3.06 % e e e e

39 ) Multiplication & Division use % relative uncertainties. (ex) e 1.76 ( 0.03) 1.89 ( 0.0) 5.64 e 0.59 ( 0.0) 1.76 ( 1. 4 %) 1.89 ( ( % 1. % 3. % % ( 0.) 5.6 ( 4%) 4 4 %) %) %

40 3) Mixed Operations 1.76 ( 0.03) 0.59 ( 0.0) 1.89 ( 0.0) ? ( 0.00 ) ( 3. 3%) 0.6 ( 0.0) 0.6 ( 3%)

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42 Homework ( ) (c) & (d)

43 Statistical Data Treatment & Evaluation Errors can be made? Judicial procedures? Statistical test?

44 Four Freedoms (Norman Rockwell) of Speech from Want of Worship from Fear

45 Student s t p. 18 Student s t is the statistical tool used to express confidence intervals & to compare results from different experiments. confidence interval: -In most quantitative chemical analyses, the true value of the mean m cannot be determined because a huge number of measurements (approaching infinity) would be required. -However, the interval surrounding the experimentally determined mean x can be determined within which the population mean m is expected to lie with a certain degree of probability. This interval is known as the confidence interval.

46 The confidence level (CL) The confidence level (CL) is the probability that the true mean lies within a certain interval and is often expressed as a percentage. CI for x z

47 The confidence level (CL) for the experimental mean x of N measurements CI for x z N

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50 Finding the confidence interval when s is unknown s calculated from a small set of data may be quite uncertain. Thus, confidence intervals are necessarily broader when we must use a small sample value of s as our estimate of s. To account for the variability of s, we use the important statistical parameter t, often called Student s t. t can be defined as t x s

51 For the mean of N measurements, t approaches z as the number of degrees of freedom becomes large. t s x / N

52 Calculating confidence intervals (ex) In replicate analyses, the carbohydrate content of a glycoprotein (a protein with sugars attached to it) is found to be 1.6, 11.9, 13.0, 1.7, and 1.5 g of carbohydrate per 100 g of protein. Find the 50 % and 90% confidence intervals for the carbohydrate content. P87

53 μ 50% x ts n μ 90% x ts n P86

54 Improving the reliability of your measurements Smaller confidence intervals Better measurement For 90% sure that a quantity lies in the range vs

55 Improving the reliability of your measurements ts μ x n (1) make more measurements 1 n n () improve expt. procedure S

56 The confidence interval for the mean x of N replicate measurements can be calculated from t as follows: CI for x ts N

57 T-test t test : used to compare one set of measurements with another to decide whether or not they are different.

58 Examples Case A : comparing a measured result with a known value Sample: 3.19 wt% (known value) a new analytical method : 3.9, 3., 3.30, 3.3 wt% X = S =

59 Does answer agree with the known answer? t calculate known value s x 4 3. n 41 95% confidence t calculate > t table result is different from the known value.

60 Case B significantly different comparing replicate measurements Nobel Prize by Lord Rayleigh. for discovering Inert gas argon :

61 Cu (s) 1 O CuO (s) N O NO NH4NO

62 t test for comparison of means : where n n 1 n s 1 n s s n n n n s x x t pooled 1 1 pooled 1 P86

63 Case C: Comparing individual differences Cholesterol content (g/l) Sample Method A Method B Different (d i ) d =

64 t calculate d s d n t s d calculate d i n 1 0 d t cal t table 0.1 two techniques are not significant different at the 95% confidence level

65 Q test for bad data help decide whether to retain or discard a datum Q test for discarding : Q gap range

66 Q calculate > Q t discard

67 Finding the Best straight line calibration methods prepare calibration curve.

68 Finding the Best straight line Mrthod of Least Least - - d d i y i least D mx y i (y i - square y mx n b i b) squares intercept x x i where the denominator, D, is given by y squares slope : - (mx b) m : b (postive only) n i x i x y x y i i i i x ( or i y i D -) D y i x i

69 Constructing a Calibration Curve 1) Blank standard soln Table 4-6 Spectrophotometer readings for protein analysis by the Lowry method Sample (μg) Absorbance of three independent samples Corrected absorbance ( after Range subtracting average blank ) blank Standard soln

70 Constructing a Calibration Curve ) Finding the protein in an unknown m b

71 Sample Size Techniques for handling very small samples are quite different from those for treating macro samples.

72 homework (a)