03.1 Experimental Error

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1 03.1 Experimental Error Problems: 15, 18, 20 Dr. Fred Omega Garces Chemistry 251 Miramar College 1

2 Making a measurement In general, the uncertainty of a measurement is determined by the precision of the measuring device. A 1-ml pipette with a graduation of 0.01mL with give an uncertainty of , or 10% of the smallest graduation (10% 0.01 =.001). A 100-ml graduated cylinder with 1-ml graduation will have an uncertainty of +0.1mL. For a 25-ml graduated cylinder with graduation of 0.2 ml, the uncertainty is +.02-ml (10% of 0.2 =.02). A 500 ml graduated cylinder with a 5mL graduation will have an uncertainty of +.5 ml. 2

3 Significant Digits and the different type of Uncertainties Number of Significant figures Implied uncertainty Relative uncertainty % Relative uncertainty % % % % 3

4 Significant Figures A measurement will always have some uncertainty based on the measuring device used. One of the best ways of indicating reliability is to give a confident interval of 90% or better. Another method is to report the absolute standard deviation or the coefficient of variation. A less satisfactory but common indicator of quality of the data is the significant figure convention. Significant figures in a number contains all the certain digits in the measurement and the first uncertain digit. The uncertainty in a measurement is the estimate reading between the graduation in a measuring device. An easy method to remember significant figures rule is as follows: Place the number in the center of the map. The presence (Pacific) or absence (Atlantic) of a decimal point determines which strategy is used Decimal Present (Pacific) For numbers with a decimal point present, draw a line starting from the Pacific to the first non-zero number, all digits shown including the nonzero number are significant. i.e., Place Number here Decimal Absent (Absence) For numbers with a decimal point absent, draw a line starting from the Atlantic (right) to the first non-zero number, all digits shown including the non-zero number are significant. i.e., Significant Figures 3 Significant Figures

5 Significant Digits and the different type of Uncertainties Number of Significant figures Implied uncertainty Relative uncertainty % Relative uncertainty % % % % 5

6 What is the difference between Precision & Significant Figures What is the difference between significant figures and precision? In the examples shown, which number has the fewest significant figures and which is the least precise?. Fewest Significant figures Least precise a) 123 vs b) 1.23 vs c) vs d) 30 vs

7 Significant Figures in Arithmetic Math operation & significant figures Addition and Subtraction: Uncertainty of answer (Significant figures of answer) is limited to the value with the least precise value (number with fewest digit after decimal place Least precise Answ: = 105. Multiplication and division: Uncertainty of answer (Significant. Figure) is limited to the value with the fewest significant figures. Fewest number of significant figures S. F. 2 S. F. Logarithms and Antilogarithms: Generally when a number expressed in exponential notation, i.e.., , 10 is the base (base-10) and 6 is the exponent. This number contains 3 significant figures. The log of this number is is the characteristic and it correspond to the exponent of the number, is the mantissa and it represents the precision of the number. Since the original number contains 3-significant figures, the final logarithm answer should have the mantissa rounded to three digits S. F. Answer with 2 s.f. Log 10 ( ) = characteristic mantissa 7

8 Significant Figures in Graphs Graph format is based on information the graph is to portray. Qualitative information. Quantitative: These graph show trend and Graph needs to show precise value that may be used to make prediction. must be read within specific significant figures. At minimum, the graph must have tick mark on both axis it would be best to display graph on a grid. Calibration Curve for 50-ml Buret 8

9 Systematic Errors Errors that have definite values and have assignable cause. These errors lead to bias in measurement techniques. Types of systematic errors- Instrument errors - poor calibration of instrument, glassware at extreme temperatures, voltage instability. Method errors - Incomplete conversion of sample for analysis Personal errors - poor experimental techniques, prejudice Result of systematic error lead to constant or proportional errors in each measurment. 9

10 Random Errors Random or indeterminate errors lead to repeated inconsistency even though systematic errors have been minimized. The randomness indicates that the errors are unpredictable and have null expected value, that is measurements are scattered about the true value, and tend to have null arithmetic mean when a measurement is repeated several times with the same instrument. All measurements are prone to random error. Origin of random errors are caused by unpredictable fluctuations in the readings of a measurement apparatus, or in the experimenter's interpretation of the instrumental reading; these fluctuations may be in part due to interference of the environment with the measurement process. These errors cannot be eliminated, but can be reduce by careful measurements and better experimental design. 10

11 Terms Mean, ; the sum of the replicated measurement divided by the x number of measurements in the set. Also known as arithmetic mean or average Medium - The middle result when the data is arranged in order of increasing values or decreasing values. Precision - reproducibility of measurement, closeness of results. Terms indicative of precision (function of deviation from mean)- standard deviation variance coefficient of variation 11

12 Precision versus accuracy Errors in Experiments measurements Analysis1; High precision and high accuracy Analysis2; Poor precision but high accuracy Analysis3: High precision but poor accuracy Analysis4: Poor precision and poor accuracy 12

13 Absolute and Relative Error Absolute Error (a.u.); E = x i - x t, This error is the uncertainty associated with a measurement. For example if a measurement is taken to be mol, then the absolute error is This error is also referred to as the difference between measured value (x i ) from the true value (x t ). This can be positive or negative. Relative Error (r.u.); E = [ (x i x t ) /x t ] * 100 This is the absolute error compared to the measurement. In general the relative uncertainty is dimensionless but it can be express in terms of percent relative error (% or pph) or in ppt (parts per thousand) 13

14 Detecting Errors Systematic error are found and corrected by calibration. Personal error minimized by care and self-discipline (do not rush the procedure). Use reliable standard reference materials. Use independent analytical method Run blank determination Vary sample size 14

15 Significant Digits and the different type of Uncertainties Number of Significant figures Implied uncertainty Relative uncertainty % Relative uncertainty % % % % 15

16 Propagation of Uncertainty from Random Errors Addition and Subtraction Calc. A.U then R.U ( ± 0.03) e ( ± 0.02) e ( ± 0.02) e ( ± e 4 ) The absolute uncertainty (AU), e 4 can be determined by the following: e 4 = e 1 ( ) 2 + ( e 2 ) 2 + ( e 3 ) 2 e 4 = ( 0.03) 2 + ( 0.02) 2 + ( 0.02) 2 = Relative uncertainty (RU) e x 100 = x 100 = 1. 3 % Multiplication and Division Calc. R.U then A.U ( ± 0.03) x 1.89 ( ± 0.02) 0.59 ( ± 0.02) = ± e 4 convert a.u. to percent r.u ( ± 1. 7 %) x 1.89 ( ± 1. 1 %) 0.59 ( ± 3. 4 %) = ± e 4 take % r.u. and determine root sum square to find e 4 %e 4 = %e 1 ( ) 2 + (%e 2 ) 2 + (%e 3 ) 2 (%e 4 is the percent r.u. of the answer) %e 4 = 1. 7 % ( ) 2 + ( 1. 1 %) 2 + ( 3. 4 %) 2 = 4. 0 % AU 100 = RU reults Mixed Operation 1.Carry out Addition- Subtraction w/ AU 2. Convert error to RU [1.76 ( ± 0.03) ( ± 0.02)] = ±? 1.89 ( ± 0.02) Addition Subraction first, and determine a.u. [1.76 ( ± 0.03) ( ± 0.02)] = e a.u. e a.u. = (.03) 2 + (.02) 2 = ± Simplifying and determining e r.u. for each measurment 1.17 ( ± 0.036) 1.89 ( ± 0.02) = ± e 0 r.u. convert a.u. to r.u. for answer, e a.u. e r.u ( ± 3. 1 %) = ± 3. 3 % 1.89 ( ± 1. 1 %) solving for AU: AU = result RU 100 convert e r.u. to e a.u. for answer AU = 3. 3 x = AU = Answer: ± AU ± 4. 0 % RU = Answer ± AU ± 3. 3 % RU 16

17 Summary of Rules for Propagation of Uncertainty Propagation of Random Errors using various math operation. 17

18 Real Rules of Significant Figures The first digit of the absolute uncertainty, a.u. is the last digit in the answer. In multiplication and division Add an extra digit when the answer is between 1 & 2 Consider 82/80 = 1.0 if rules of S.F. is followed. It is better to write an answer of 1.02 showing that the uncertainty is in the order of 1%. A value of 1.0 suggest uncertainty is in order of 10% which is larger than actual uncertainty in the original data. 18

19 Calculation Molarity Example 3.20 The concentration of HCl can be precisely determined by reaction with sodium carbonate. 2 HCl (aq) + Na 2 (s) g 2Na + (aq) + 2Cl - (aq) + H 2 O (s) + CO 2 (g) If g of Na 2 (FM ) required ml of HCl, what is the molarity of HCl and the a.u.? 1 mol Na ( ± ) g Na mol HCl 1 = M HCl ± a.u. ( ± ) g Na 2 1 mol Na 2 ( ± ) L convert a.u. to percent r.u. convert measured a.u. to measured percent r.u. 1 mol Na ( ± % ) g Na mol HCl 1 = M HCl ± %e ( ± %) g Na 2 1 mol Na 2 ( ± % )ml determine % r.u. of each measurement and determine root sum square to find answer e r.u. %e 4 = (%e 1 ) 2 + (%e 2 ) 2 + (%e 3 ) 2 = ( %) 2 + ( %) 2 + ( %) 2 = % (This is the percent r.u. of the answer) ( ± % ) g Na 2 1 mol Na 2 2 mol HCl 1 ( ± %) g Na 2 1 mol Na 2 ( ± % )ml = HCl ± 0.17 % 6 3 Converte e r.u. of answer and convert to e a.u. of anaswer A.U. = Results (R.U.) 100 = = Molarity HCl = with r.u. = % 19

20 Calculation Molarity Example 3.20 The concentration of HCl can be precisely determined by reaction with sodium carbonate. 2 HCl (aq) + Na 2 (s) g 2Na + (aq) + 2Cl - (aq) + H 2 O (s) + CO 2 (g) If g of Na 2 (FM ) required ml of HCl, what is the molarity of HCl and the a.u.? 1 mol Na ( ± ) g Na mol HCl 1 = M HCl ± a.u. ( ± ) g Na 2 1 mol Na 2 ( ± ) L convert a.u. to percent r.u. convert measured a.u. to measured percent r.u. 1 mol Na ( ± % ) g Na mol HCl 1 = M HCl ± %e ( ± %) g Na 2 1 mol Na 2 ( ± % )ml determine % r.u. of each measurement and determine root sum square to find answer e r.u. %e 4 = (%e 1 ) 2 + (%e 2 ) 2 + (%e 3 ) 2 = ( %) 2 + ( %) 2 + ( %) 2 = % (This is the percent r.u. of the answer) ( ± % ) g Na 2 1 mol Na 2 2 mol HCl 1 ( ± %) g Na 2 1 mol Na 2 ( ± % )ml = HCl ± 0.17 % 6 3 Converte e r.u. of answer and convert to e a.u. of anaswer A.U. = Results (R.U.) 100 = = Molarity HCl = with r.u. = % 20

21 Propagation of Systematic Uncertainty Uncertainty in Atomic Mass- The error in the atomic mass is not mainly from random error but from isotopic variation in samples of elements from various source samples which is found to be / g/mol Consider the atomic weight of O, the error in the mass approximates a rectangular distribution. There is an equal probability of fining the atomic mass of oxygen between and The standard deviation for this distribution is called the standard uncertainty, + a/ 3. For oxygen the standard uncertainty is / 3 =

22 Propagation of Systematic Uncertainty Uncertainty in Molar MassWhat is the uncertainty of the molar mass? Consider the molar mass of O2, the error in the molar mass is for two oxygen, the lower limit would be *2 = The upper limit would be *2 = Therefore for O2, the mass would be ( )/2 = ± The uncertainty of mass therefore for one atom is /2 = , which means for n atoms, the standard uncertainty is n * ± Thus for n = 2, (2* ± ) = For systematic uncertainty, the uncertainty is added per each term. Consider the molar mass of C2H6O, ethanolatomic mass of C = ± / 3 = ± Atomic mass of H = ± / 3 = ± Atomic mass of O = ± / 3 = ± C: 2( ± ) = ± (2* ) = ± H: 6( ± ) = ± (6* ) = ± O: 1( ± ) = ± (1* ) = ± ± e ± e4, e4 = ( )½ = Molar mass C2H5OH = ± g/mol 22

23 Propagation of Systematic Uncertainty Multiple deliveries from Pipette- Consider a 25-mol Class-A volumetric pipette which is certified by manufacturer to deliver mol. The range therefore for a reading is to ml. For volumetric glassware the Triangular distribution is followed for the standard deviation, or + a/ 6 The standard uncertainty (std dev) for the pipet is therefor a/ 6 = +.03/ 6 = ml. From the triangular distribution, there is zero probability that a volume will fall out side of ml. If the pipette is un-calibrated and used four times to deliver 100mL, then the standard uncertainty (std dev) = + 4 x = ml and not + [ ] 1/2 or std dev = ml. This is due to systematic error Calibration of the pipette on the other hand, eliminates systematic errors. Calibration improves certainty by removing systematic error. For example consider a calibrated pipet that delivers a mean value of with a standard deviation ml (0.006 ml is random error). Using the pipet to deliver 4 times gives an uncertainty + [ ] 1/2 or ml. An un-calibrated pipet will give an uncertainty of 4 aliquots as 4 x ml = ml. Calibrated pipet volume: ml Un-calibrated pipet volume: mL 23

24 Summary Every measurement is influence by many uncertainties that produce a scattering of results. It is not easy to estimate the reliability of the experimental data, but sources of error must be identified whenever possible so that the results can be stated with confidence. The true value of a measurement is never really known exactly. 24

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