Chapter 3 Experimental Error

1. Chapter 3 Experimental Error

2. - Significant figures - Precision (Reproducibility) - Accuracy (Error) - Uncertainty

3. 3-1 Significant Figures • Significant figures: minimum number of digits required to express a value in scientific notation without loss of • accuracy (with the appropriate accuracy). Guidelines for Sig. Figs., when looking at a number 1) All nonzero digits are significant figures • 142.7 = 1.427 x 102 • 0.000006302 = 6.302 x10-6 4 significant figures (sf) Zeros are simple place holders (3 sf) 9.25 x 104 9.250 x 104 9.2500 x 104 (4 sf) 92500 (5 sf) • 2) Zeros are counted as significant figures only if: • i) occur between other digits in the number • 106 0.0106 0.106 • ii) occur at the end of a number on the right-hand side of the • decimal point • 0.1060 (3 sf) (4 sf)

4. The number of significant figures is the minimum number of digits needed to write a given value in scientific notation without loss of precision. 92, 500  9.25× 104  3significant figures 9.250 × 104  4significant figures 9.2500 × 104  5significant figures Zeros are significant 1) in the middle of a number, 2) at the end of a number on the right-hand side of a decimal point.

5. 3-2 Significant Figures in Arithmetic How many digits to retain in the answer after you have performed arithmetic operations with your data? Addition and Subtraction:the same decimal place in the final answer 1.362 × 10 -4 + 3.111 × 10 -4 4.473 × 10 -4 5.345 + 6.728 12.073 7.26 × 10 14 -6.69 × 10 14 0.57 × 10 14 18.9984032 (F) + 18.9984032 (F) + 83.789 (Kr) 121.794 8064 1.632 × 105 +4.107 × 103 +0.984 × 106 1.632 × 105 +0.04107 × 105 +9.84 × 105 11.51 × 105

6. - Multiplication and Division In multiplication and division, we are normally limited to the number of digits contained in the number with the fewest significant figures. 4.3179 × 1012 × 3.6 × 10-19 1.6 × 10-6 34.60 ÷2.46287 14.05 3.26 × 10 -5 ×1.78 5.80 × 10-5 - Logarithms and antilogarithms IF n = 10a, then we say that a is the base 10 logarithm of n :

7. A logarithm is composed of a characteristic and a mantissa. The characteristic is the integer part and the mantissa is the decimal part log 339=2.530 log 3.39×10-5= -4.470 CharacteristicMantissa Characteristic Mantissa = 2 =0.530 =-4 =0.470 10 2.531 =340(339.6) 10 2.530 =339(338.8) 10 2.529 =338(338.1) antilog (-3.42) = 10-3.42 = 3.8 × 10 -4 log0.001237= -2.9076 antilog4.37=2.3 × 10 4 Log1 237= 3.0924 10 4.37=2.3 × 10 4 Log3.2=0.51 10-2.600=2.51 × 10 -3

8. Significant Figures and Graphs

9. 3-3 Types of Error Experimental error from uncertainties of measurements - SYSTEMATIC ERROR: Systematic error , also called determinate error arises from a flaw in equipment or the design of experiment. - RANDOM ERROR: called indeterminate error arises from uncontrolled (and maybe uncontrollable) (GROSS ERRORS) ⇒mainly originated by person ⇒statistical calibration

10. Systematic errors: determinate error ⇒ definite value, assignable cause. ⇒ bias-> 모든 결과에 같은 크기의 영향을 미치며, 부호를 가짐. Box 3-1. Case study : systematic error in Ozone Measurement

11. Sources of systematic errors Three types of  systematic errors 1) instrumental errors          2) method errors         3) personal errors  

12. ■ Instrumental errors ⇒ measuring device error Ex) ① Pipet, burette, mass flask, etc. Reasons : The calibration temp.is nonequal to the measuring temp. Dry or heating cause the distortion of glass Contamination inside surface of vassel ②Electronic instrument error Reasons: potential change by dry cell life time Wrong calibration electric devise error by temperature change Noise from AC power source ③ Instrument operation in error conditions 이유: pH meter in strong acid solutions -> acid error ⇒ vibration → detectable, correctable  

13. ■ Method errors ⇒ From nonideal behaviors of reactions and reagents. ⇒Slow reaction, incomplete reaction, using unstable chemical, nonselectiveity of reagents, side reactions, etc.. ⇒The most difficult to remove it. Ex) using excess reagents - Due to chemical properties of nicotinic acid – degradation using hot and conc. H2SO4 - pyridine ring bearing nicotinic acid -> incomplete degradation. - Addition of potassium sulfateand elevate the boilng temp. -> complete degradation.

14. ■ Personal errors ⇒ personal decision needs in many measurements. ⇒error, toward one direction 예) • Reading in position of pointer between two points. • Color at the end point • liquid level of burret • Color sensitivity ⇒personal bias 예) • 정밀도를 증가시키는 방향으로 눈금을 읽을 때 • 측정의 참값을 미리 마음 속으로 정해 놓을 때 • 숫자에 대한 편견이 있을 때 - 눈금 위의 바늘의 위치를 읽을 때 숫자 0과 5를 선호, - 큰 수보다 작은 수, - 홀수보다 짝수 선호

15. The effect of the systematic error on analytical results ⇒ Systematic errorsisconstant orproportional ⇒constant errors size • 측정되는 양의 크기에 따라 달라지지 않음 • 절대오차는 시료크기에 대하여 일정 • 상대오차는 시료의 크기에 따라 변함 ⇒proportional errors size •분석에 사용된 시료의 크기에 따라 증가 또는 감소 • 절대오차는 시료크기에 따라 변함 • 상대오차는 시료의 크기에 대하여 일정

16. Detection of systematic instrumental & personal errors ⇒ Instrumental error • can be founded and corrected by calibration • periodical calibration • Instrumental error by interferences in samples → 단순한 검정으로 영향제거 불가능 ⇒Personal error • It can be minimize by precaution, excise, etc. • Check instrument reading, notebook entries & calculations • chose the adequate method -- errorminimizing

17. Detection of systematic method errors ■ Method errors Analysis of standard samples ⇒The best way of estimating the bias of analytical method is by the analysis of standard reference materials(SRMs) ⇒SRMs : 정확하게 잘 알려진 농도를 가지고 있는 analytes를 하나 또는 그 이상 포함한 물질 ⇒합성하여 사용 • 순수한 성분들을 혼합하여 조성을 알 수 있는 균일시료 제조 • 합성 표준물질의 조성은 분석시료의 조성과 거의 같아야 함 • 표준시료의 합성이 불가능하거나, 쉽지 않고 시간이 많이 걸리는 경우 가 있음 → 실제적이지 못할 수 있음 ⇒• 미국 정부기관인 NIST(National Institute od Standards & Technology)에서 1300 종 이상의 SRMs 공급 •몇몇 시판 공급회사에서도 공급

19. http://www.nist.gov/

20. http://ts.nist.gov/measurementservices/referencematerials/index.cfmhttp://ts.nist.gov/measurementservices/referencematerials/index.cfm

21. 105.4 Toxic Substances in Urine (powder form) SRMs 2670a, 2671a and 2672a are for determining toxic substances in human urine. They consist of freeze-dried urine and are provided in sets of four 30 mL bottles -- two each at low and elevated levels. NOTE:The values listed for these SRMs apply only to reconstituted urine.

22. ■ Independent analysis ⇒ 표준시료를 사용할 수 없을 경우 ⇒ 같은 시료를 또 다른 독립적이고 신뢰성 있는 분석법으로 분석 (parallel analysis) ⇒두 방법은 가능한 한 많이 달라야 함 → 두 방법에 모두 영향을 줄 수 있는 공통요인을 최소화 하기 위함 ⇒ 두 방법간의 차이가 random error 또는 방법에서 오는 bias 때문인지를 평가하기 위해 반드시 통계적 test를 실시 (7B-2)

23. ■ Blank determinations ⇒ blank • reagents and solvents without analyte. • Similar condition of analyte environment (sample matrix): - >addition of sample constituents. ⇒blank determination • every step in analysis : blank material analysis needs.

24. Precision and Accuracy Precision describes the reproducibility of a result, if you measure A quantity several time and the values agree closely with on another your measurement is precise. Accrue describes how close a measured value is to the “true” value If a known stands is available , accuracy hoe close your value is to the known value.

25. Precision ⇒ 측정의 재현성(reproducibility of measurements) ⇒ 정확히 똑같은 방법을 이용하여 얻은 측정값들 사이의 일치성 ⇒반복시료를 사용하여 반복적인 측정을 함 • Three terms to describe the precision • Standard deviation • Variance • Coefficient of variance (분산계수) ⇒ deviation from the mean (di)의 함수.

26. Absolute and Relative uncertainty Absolute uncertainty expresses the margin of uncertainty associated with a measurement. Relative uncertainty compares the size of the absolute uncertainty with the size of its associated measurement.

27. 3-4 Propagation of Uncertainty from Random Error We can usually estimate or measure the random error associated with measurement, such as the length of an object or the temperature of a solution. - Addition and Subtraction 1.76 (± 0.03) + 1.89 (±0.02) -0.59(±0.02) 3.06(±e4) e1 e2 e3 Percent relative uncertainty=0.041 / 3.06 × 100= 1.3% 3.06 (± 0.04): absolute uncertainty, 3.06 (± 1 %): relative uncertainty

28. (EXAMPLE) Uncertainty in a Buret Reading

29. -Multiplication and Division

30. - Mixed Operations

34. Exponents and logarithms Uncertainty for powers and roots: (3-7) Uncertainty for logarithm: (3-8) (3-9) Uncertainty for Natural logarihm (3-10) Uncertainty For 10x: Uncertainty for ex (3-11)

36. 3-5 Propagation of Uncertainty from Systematic Error The standard deviation ( defined in section 4-1 ) for this distribution, Also called the standard uncertainty , is ±a√3= ±0.0003/ √3= ±0.00017

37. Uncertainty in molecular mass

38. - Uncertainty in Molecular Mass

39. MultipleDeliveries From One Pipet: The Triangular Distribution The standard uncertainty ( standard deviation) in the triangular Distribution is ± a √6 = ±0.03 √6= ±0.012ml The standard uncertainty is ±4 × 0.012= ±0.048 ml , not ± √0.012 2 + 0.012 2 +0.012 2 + 0.012 2= ±0.024ml Calibration improves certainty by removing the systematic error