Harris: Quantitative Chemical Analysis, Eight Edition CHAPTER 03: EXPERIMENTAL ERROR

Transcription

1 Harris: Quantitative Chemical Analysis, Eight Edition CHAPTER 03: EXPERIMENTAL ERROR

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3 Chapter 3. Experimental Error -There is error associated with every measurement. -There is no way to measure the true value of anything. - The best way we can do in a chemical analysis is to apply a technique that experience tells us is reliable. - Repetition of one method of measurement several times tells us the precision (reproducibility) of the measurement - If the results of measuring the same quantity by different methods agree with one another, then we become confident that the results are accurate, which means they are near the true value.

4 Chapter 3. Experimental Error 3-1. Significant Figures The number of significant figures is the minimum number of digits needed to write a given value in scientific notation without loss of accuarcy. Fig Scale of a spectrophotometer. Absorbance is a logarithmic scale. Absorbance = by interpolation. 2 and 3 are completely certain and the number 4 is an estimate. This number has three significant figures.

5 Chapter 3. Experimental Error 3-2. Significant Figures in Arithmetic ( rule of thumb ) 1) Addition & Substraction - In the addition or subtraction of numbers expressed in scientific notation, all numbers should first be expressed with the same exponent. - If the numbers being added do not have the same number of significant figures, we are limited by the least certain one. Rounding off - In the special case where the number is exactly halfway, round to the nearest even digit. ex) , , When rounding off, look at all the digits beyond the last place desired. ex)

6 Chapter 3. Experimental Error 3-2. Significant Figures in Arithmetic ( rule of thumb ) 2) Multiplication & Division We are normally limited to the number of digits contained in the number with the fewest significant figures. A rule of thumb : it suggests for multiplication and division that the answer should be rounded so that it contains the same number of significant digits as the original number with the smallest number of significant digits. * Rounding should only be done on the final answer (not intermediate results), to avoid accumulating round-off errors

7 Use the rule of thumb with Caution!! 2) Multiplication & Division A rule of thumb: it suggests for multiplication and division that the answer should be rounded so that it contains the same number of significant digits as the original number with the smallest number of significant digits. But use the rule of thumb with Caution!! Unfortunately, this procedure can lead to incorrect rounding. Example 1; consider the two calculations = 1.08 and = By the rule just described, the first answer would be rounded to 1.1 and the second to 0.96.

8 Use the rule of thumb with Caution!! However, if the last digit of each number making up the first quotient ( = 1.08 ) is uncertain by 1, the relative uncertainties associated with each of these numbers are 1/24, 1/452, and 1/1000. Because the first relative uncertainty is much larger than the other two, the relative uncertainty in the result is also 1/24; the absolute uncertainty is then /24 = = 0.04 By the same argument, the absolute uncertainty of the second answer is given by /24 = = 0.04 Therefore, the first result should be rounded to three significant figures, or 1.08, but the second should be rounded to only two; that is, 0.96

9 Use the rule of thumb with Caution!! In multiplication and division, keep an extra digit when the first digit of answer lies between 1 and 2 (p.59) Example 2: The quotient 82/80 is better written as 1.02 than 1.0. If the uncertainty in the last digit of each number, 82 and 80 is 1, the uncertainty is of the order of 1 %, which is the decimal place of If we write 1.0, you can surmise that the uncertainty is at least 1.0 ± 0.1 = ± 10 %, which is much larger than the actual uncertainty. Example 3: Even though the dividend and divisor each have three figures, the quotient is expressed with four figures ( ± 0.002) = ( ± 0.004) ( ± 0.002)

10 Use the rule of thumb with Caution!! Example 4: It is reasonable to keep an extra digit when the first digit of the answer is 1. ( x ) x (3.6 x ) = x 10-6 = 1.6 x 10-6 = 1.55 x 10-6 ( Better)

11 3-2. Significant Figures in Arithmetic 3) Logarithms and Antilogarithms * Number of digits in mantissa of log x = number of significant figures in x : log ( ) = digits character mantissa * Number of digits in the mantissa of x = Number of significant figures in the antilogarithm, antilog x (=10 x ) antilog = = mantissa, 3 digits 3 digits

12 Exam ) Convert p-function to molar concentration: Solution1) Number of digits in antilog x (= 10 x ) = number of significant figures in mantissa of x (P.54, L22) pmn = antilog ( ) = = 0.99 M Log 0.99 = (It s not reasonable!) Solution 2) Number of significant figures in antilog x (= 10 x ) = number of digits in mantissa of x (P.54, L20) pmn = antilog ( ) = = M (It s reasonable!) = M = M = M Log = Log =

13 3.3. Significant Figures and Graphs - When the graph is used to read data points, better is a fine grid superimposed on the graph - Rulings on graph should be compatible with the number of significant figures of the coordinates. a) Divisions are fine enough (0.53, 0.63), (1.08, 1.47)

14 3.3. Significant Figures and Graphs - When the graph is used to read data points, better is a fine grid superimposed on the graph -Rulings on graph should be compatible with the number of significant figures of the coordinates. b) Divisions are not fine enough (0.53, 0.63), (1.08, 1.47)

15 3.3. Significant Figures and Graphs When you are not expected to be able to read coordinates accurately on the graph, Qualitative behavior of data is OK (i.e., fine grid is not required)

16 3-4. Types of Errors in Experimental Data 1) Systematic errors (= Determinate Errors) - It is always in the same direction (unidirectional), and could be discovered and corrected. -It causes the mean of a set of data to differ from the accepted value ( Next slide, analyst 3 ) 2) Random Error ( = Indeterminate error) 3) Gross Errors

17 Systematic error (Analyst 3 & 4) Fig 5-3, p.94

18 3-4. Types of Errors in Experimental Data 1) Systematic errors (= Determinate Errors) i) Instrument Errors - imperfections in measuring devices. pipets, burets and volumetric flasks frequently deliver or contain volumes slightly different from those indicate by their graduations. The reasons are as following: Use of glassware at a temp. that differs significantly from the calibration temp. Distortions in container walls due to heating while drying. Errors in the original calibration. Contaminants on the inner surfaces of the container. Most systematic errors of this type are readily eliminated by calibration ( see next slide, textbook Fig. 3-3) - instruments powered by electricity. decreased voltage of battery-operated power supplies increased resistance in circuits because of dirty electrical contacts temp. effects on resistor

19 3.3. Significant Figures and Graphs Most systematic errors of this type are readily eliminated by calibration! Ex.) Buret reading: ml, Correction: ml Actual volume = =29.40 ml Fig. 3-3 Calibration Curve for a 50 ml buret

20 3-4. Types of Errors in Experimental Data ii) Method Errors Nonideal chemical or physical behavior of the reagents and reactions upon which an analysis is based. incompleteness of the reaction (ex: decomposition of pyridine ring) slowness of the reaction possible occurrence of side reactions that interfere with the measurement process in volumetric titration, the small excess of reagent required to cause an indicator to undergo the color change that signals completion of the reaction Errors inherent in a method are frequently difficult to detect, and are thus the most serious of the three types of determinate error.

21 3-4. Types of Errors in Experimental Data iii) Personal Errors Many measurements require personal judgments. - Prejudice (bias) Most of us have a natural tendency to estimate scale readings in a direction that improves the precision in a set of results or causes the results to fall closer to a preconceived notion of the true value. level of a liquid in buret color at the end point in titration the position of a pointer between two scales

22 3-4. Types of Errors in Experimental Data iii) Personal Errors Many measurements require personal judgments. - Number bias prefer 0 or 5 even number over odd number. * Detection of determinate errors. 1. Analysis of standard reference materials 2. a second reliable analytical method 3. Blank determinations 4. Variation in sample size in case of a constant error

23 3-4. Types of Errors in Experimental Data * Determinate Error may be either constant or proportional Constant Errors : magnitude is independent of size of the quantity measured. Ex). 500 mg of precipitate 0.5 mg wash out 50 mg of precipitate 0.5 mg wash out 0.5 relative error = = % relative error = = -1.0 % 50 Proportional Errors : increase or decrease in proportion to the size of the sample taken for analysis. Ex). the presence of interfering contaminants in the sample.

24 Absolute and Relative Uncertainty - Uncertainty is usually expressed as the standard deviation. Absolute Uncertainty : margin of uncertainty associated with a measurement. Ex). a buret reading : 12.35±0.02 ml Relative Uncertainty : comparing the size of the absolute uncertainty to the size of its associated measurement. R.U = absolute uncertainty magnitude of measurement 0.02 ml ex). R.U = = = 2 ppt = 0.2 % ml

25 3-4. Types of Errors in Experimental Data 2) Random Error ( = Indeterminate error) - It arises from natural limitation on our ability to make physical measurements. - It causes data to be scattered more or less systematically around a mean value. The precision of the data reflects the indeterminate errors in an analysis. - Sometimes positive, sometimes negative. - It is the ultimate limitation on the determination of a quantity. See next slide Ex). Electrical noise : small fluctuation resulting from electrical instability of the meter itself (voltameter)

26 3-4. Types of Errors in Experimental Data 3) Gross Errors - personal and arise from carelessness or ineptitude on the part of the experimenter. - Gross errors usually affect only a single result in a set of replicate data, causing it to differ significantly from the remaining results for that set. (outliers) Ex). Arithmetic mistakes Reading scale backward Reversing a sign Spilling a solution - They can be eliminated through self-discipline

27 3-4-1 Methods For Expressing Precision and Accuracy Precision : A measure of reproducibility of a result defined as the agreement between the numerical values of two or more measurements that have been made in an identical method. Accuracy : How close a measured value is to the true one - the nearness of a measurement to its accepted value - Accuracy is expressed in terms of error.

28 Fig 5-3, p.94 Precision and Accuracy

29 Accuracy (= nearness to the truth ) Methods for expressing accuracy 1) Absolute Error = x i x t x i : observed value x t : accepted value (It may itself be subject to considerable uncertainty) 2) Relative Error = x x x i t t 100 %

30 Precision (= reproducibility) Methods for expressing precision Mean or Average = x = Geometric Mean = n i x i i N x i 1) Standard Deviation For a very large set of data σ = N i= 1 ( x µ ) i N 2 σ: population S. D. µ: population mean ( = true value) σ 2 : variance

31 Methods for expressing precision For a small number of replicate measurements, S = N i= 1 ( x x) i N -1 2 S : sample standard deviation x : measured mean for the small set 2) Relative Standard Deviation (RSD) S RSD = x 1,000 (ppt) S CV (coefficient of variation) = 100 % x

32 Methods for expressing precision 3) Variance (S 2 ) S 2 = N i= 1 ( x x) i N ) Spread or Range (w) = highest lowest

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34 3-5. Propagation of Uncertainty from Random Error - We can easily estimate or measure the random error associated with a measurement. - Uncertainty is usually expressed as the standard deviation of calculated results or as a confidence interval. These parameters are based upon a series of replicate measurements. - In most experiments, it is necessary to perform successive arithmetic operations on several numbers, each of which has its associated random error. - The most likely uncertainty in the result is not the sum of individual errors, because some of these are likely positive and some negative. We expect some cancellation of errors. - The first uncertain figure should be the last significant figure : ex) (±0.0002)

35 3-5. Propagation of Uncertainty from Random Error 1) Addition and Subtraction 1.76 (±0.03) e (±0.02) e (±0.02) e (± e 4 ) e Uncertainty in addition and subtraction: = e1 + e2 + e3 = Absolute uncertainty e = ( 0.03) + (0.02) + (0.02) = Percent relative uncertainty = 100 = 1. 3% 3.06 Two expressions of final result : 3.06 (±0.04) (absolute uncertainty) 3.06 (±1%) (relative uncertainty)

36 3-5. Propagation of Uncertainty from Random Error 2) Multiplication and Division Uncertainty in multiplication and division: For example, 1.76( ± 0.03) 1.89( ± 0.02) = 0.59( ± 0.02) % e + e 5.64 ± e = (% e1 ) + (% e2 ) (% 3) First, convert absolute uncertainties to percent relative uncertainties ( ± 1. 7 %) 1.89( ± ( ± 3. 4 %) 1 %) = 5.64 ± e % e4 = (1. 7 ) + (1. 1) + (3. 4 ) = 4. 0% 4. = 0% 5.64 = Two expressions of final result : 5.6 (±0.2) (absolute uncertainty) 5.6 (±4%) (relative uncertainty)

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38 tip) Propagation of Uncertainty in the Product x x Table 3-1 says that the uncertainty in the function y = x a is % e y = a (%e x ). If y = X 2, then %e y = 2(%e x ). A 3% uncertainty in x leads to a (2)(3%) = 6% uncertainty in y. But what if we just apply the multiplication formula 3-6 to the product x x? 2 x ( ± e1 ) x( ± e2 ) = x ( ± e3) % e + e = (% e1 ) (% 2 ) = 2 2 ( 3%) + (3%) = 4. 2% Which uncertainty is correct, 6 % or 4.2 %?

39 tip) Propagation of Uncertainty in the Product x x

40 3-6. Propagation of Uncertainty from Systematic Error Systematic error occurs in some common situations and is treated differently from random error in arithmetic operation 1) The Rectangular Distribution : uncertainty in atomic mass - atomic mass of oxygen in the periodic table : ± ( Next slide) - The uncertainty is not mainly from random error in measuring atomic mass, but it is predominantly from isotopic variation in samples of oxygen from different sources. Therefore, the atomic mass of oxygen in a particular lot of reagent has a systematic uncertainty - There is approximately equal probability of finding any atomic mass between and and negligible probability of finding an atomic mass outside of the range. - Mean: , range : The standard uncertainty (standard deviation) = ± a (range)/ 3

41 End Papers: Periodic Table

42 3-6. Propagation of Uncertainty from Systematic Error -Oxygen atomic mass from different sources approximates a rectangular distribution. - There is approximately equal probability of finding any atomic mass between and and negligible probability of finding any atomic mass outside of the range (a).

43 3-6. Propagation of Uncertainty from Systematic Error - Systematic error occurs in some common situations, and is treated differently from random error in arithmetic operations 1) The Rectangular Distribution : uncertainty in atomic mass - The periodic table gives the atomic mass of oxygen as ± g/mol. The uncertainty is not mainly from random error in measuring the atomic mass, but it is predominantly from isotopic variation in sampled oxygen from different sources. - The atomic mass of oxygen in a particular lot of reagent has a systematic uncertainty.

44 3-6. Propagation of Uncertainty from Systematic Error - Propagation of systematic uncertainty : uncertainty of n identical atoms = n x (standard uncertainty in atomic mass) - The sum of atomic masses of different elements : Use the rules for propagation of random uncertainty Example: what is the standard uncertainty in molecular mass of C 2 H 4? From periodic table, Atomic mass of C = ± / 3 = ± Atomic mass of H = ± / = ±

45 3-6. Propagation of Uncertainty from Systematic Error -For the uncertainty in the sum of the masses of 2C +4H, we use equations for random error, because the uncertainties in the mass of C and H are independent of each other. One might be positive and one might be negative ± = ± = ±

46 3-6. Propagation of Uncertainty from Systematic Error 2) The Triangular Distribution : multiple deliveries from one pipet - A 25 ml pipet is certified to deliver ± 0.03 ml - The volume delivered by a pipet is reproducible, but can be in the range of ~ ml, depending on a pipet. - However, the manufacturer works hard to make the volume close to 25.00mL. In such case, we approximate the volumes of a large number of pipets by the triangular distribution.

47 3-6. Propagation of Uncertainty from Systematic Error - The probability falls off approximately in a linear manner as the volume deviates from ml because the manufacturer works hard to make the volume close to 25.00mL. - There is negligible probability that a volume outside of ± 0.03 ml will be delibered. - The standard uncertainty (standard deviation) = ± a (range)/ 6 3

48 3-6. Propagation of Uncertainty from Systematic Error Example: 1) What is the uncertainty in 100 ml if you use an uncalibrated 25 ml pipet four times to deliver of 100 ml? The uncertainty is a systematic error. So the standard uncertainty = ± 4 x 0.03/ 6 = ± 4 x = ± ml So uncalibrated pipet volume = ± 0.05 ml - The difference between ml and the actual volume delibered by a particular pipet is a systematic error. In other words, it is always the same, within a small random error - You could calibrate the pipet by weighing the water it delivers to eliminate a systematic error. In such case, the pipet always delivers, say, ± ml. The uncertainty (± ml) is random error.

49 3-6. Propagation of Uncertainty from Systematic Error Example: 1) What is the uncertainty in 100 ml if you use a calibrated pipet which delivers a mean volume of ml with a standard uncertainty of ± ml? You deliver four times, so the volume is 4 x ml = ml But the uncertainty = ± = ± ml We use equations for random error, because the uncertainties in four delivered aliquots are independent of each other. One might be positive and one might be negative because they are random error. So calibrated pipet volume = ± ml