1. Measurements Glenn V. Lo Department of Physical Sciences Nicholls State University

2. Measurement assigning a value to an observation For science and trade: value assigned should be in comparison to a reference standard. “reference” – basis for comparison “standard” – agreed upon Reported measurement generally has 2 parts: Number (indicates the value) Unit (indicates what it is compared to) Ex: 5.0 kg  (means 5.0 times the mass of a cylinder stored in a vault at the International Bureau of Weights and Measures in France; by agreement the cylinder is defined to have a mass of 1 kg)

3. Reliability of Measurements Accuracy: “correctness” Precision: “reproducibility” or “consistency” Precision is necessary, but not sufficient, for accuracy. It is possible to be consistently wrong (“precisely inaccurate”)

4. Accuracy To determine accuracy: calculate error Smaller error, more accurate Absolute error = (Measured value) – (Correct value) Relative error = (Absolute error)/(Correct value) Percent error = Relative error x 100%

5. Accuracy In a lab practical (exam), a student was given a tablet with 250 mg of ascorbic acid. The student determined the amount to be 225 mg. Calculate absolute error and relative error.

6. Correct values Defined or standard value Ex. Speed of light = 2.99792458 x10 8 m/s (by definition) Standard Reference Material: certified by manufacturer to have a particular value for a given property Literature value – from a reputable publication Theoretical value --- predicted based on a well- established theory

7. Why measure what is known? To test the competency of person doing measurements To calibrate or standardize instrument Use “standard reference materials” Use “standard procedures” Validity of “standards” must be ultimately traceable to an authorized agency. In U.S., the NIST. To validate a method to be used in an experiment

8. Systematic errors Main cause of inaccuracies Tend to be “one-sided” (consistently low or consistently high) Avoidable Examples: improperly calibrated instrument, limitations of method used

9. Precision Precision = “reproducibility” Do replicate measurements: multiple trials (try to make everything the same) Replicate measurements generally do not yield exact same result, even if all possible systematic errors are avoided. More scatter in values: less precise Better agreement: more precise

10. Random errors Reason for different results from replicate measurements Unavoidable natural fluctuations in value of the property being measured, response of measuring instrument (or person) to the property being measured Equal tendency to be positive or negative Averaging multiple trials minimizes effect of random error on accuracy

11. Describing precision Calculate the range of multiple trials Range = (highest value) – (lowest value) Example: Times for a rock to fall from the top of a building measured four times. Results of 4 trials are: 3.12 s, 3.15 s, 3.20 s, 3.22 s Calculate the range.

12. Reporting measurement Report the average of multiple trials as the measured value Quote the uncertainty (round off to first nonzero digit; may keep extra digit if first nonzero digit is less than 3) Express the measured value to the same number of decimal places as the uncertainty.

13. Example The length of an object was measured; the average of multiple trials is 24.8328 mm and the uncertainty was determined to be +0.0286 mm. How should the length be reported?

14. Estimating uncertainty Use ½ of the range How should the measured value be reported based on the following individual trial results: 27.530, 20.894, 29.227, 21.187

15. Estimating uncertainty More sophisticated method for formal reports: use standard deviation. For each trial value, calculate: deviation = trial value – average Square the deviation Add up all the squared deviations Divide by N-1, where N = #trals This gives us the variance Standard deviation = square root of variance

16. Implying Uncertainty If uncertainty is not quoted, the last digit should reflect magnitude of uncertainty Determine at decimal place of first nonzero digit of uncertainty Express measured value up to that decimal place Digit in this decimal place is the “last significant digit”; digits past this decimal place are “not significant” Example: How should “25.03+0.27 mg” be reported if the uncertainty is not quoted?

17. Reporting individual measurements Digital Scale Examine fluctuation in digital readout; estimate range If readout does not fluctuate; report all digits [instrument is not sensitive enough to detect the random fluctuations] and add an extra digit (0) Analog scale Uncertainty is at least 1/10 of the interval between adjacent marks; at most it is ½ of the interval Example: Adjacent marks on a meter stick are 1 mm apart. How should measurements with this instrument be reported?

18. Significant Figures Significant digits = meaningful digits in a reported measurement; the last significant digit indicates level of uncertainty. Rule 1. If the number is written in scientific notation, all the digits in the coefficient are significant. Examples: How many significant digits are there in 1.50x10 -3 kg?

19. Significant Figures Rule 2. If the number is not written in scientific notation: the first nonzero digit is the first significant digit all digits to the right of the first nonzero digit are also significant Exception: if number has no decimal point, trailing zeros may or may not be significant; unless otherwise specified, assume they are not significant. Example: how many significant digits are in 0.00250 m 73.00 km 3800 mg

20. Significant Figures Avoiding ambiguity: do not report number with trailing zeros if it has no decimal point. Use scientific notation. How should 3800 mg be reported to 2, 3, and 4 significant figures?

21. Significant Figures Watch this video: http://www.youtube.com/watch?v=ZuVPkBb-z2I

22. Test Yourself Which of the following has the fewest number of significant figures? A. 10.50 mg B. 20700 mL C. 0.003100 g D. 1.500x10 -3 m

23. Test Yourself The amount of acid present in a sample was measured several times and the average value was determined to be 3601.2 mg, with an uncertainty of +30 mg. To how many significant figures should the average be reported? A. 2, B. 3, C. 4, D. 5

24. Test Yourself Five replicate measurements of the calcium content of a sample yielded the following results: 5.042%, 5.131%, 5.434%, 4.852%, 4.728%. To how many significant digits should the average be reported? A. 1, B. 2, C. 3, D. 4

25. Test Yourself What is the proper way of reporting the temperature shown on the thermometer scale (Celsius) below? A. 30 o C, B. 30.0 o C, C. 30.00 o C, D. 3.0x10 1 o C

26. Calculating with units When doing calculations, treat units just like any algebraic quantity. Addition or subtraction: 5 kg + 2 kg = (5 + 2) kg = 7 kg Multiplication, division, exponentiation: (2 m) (3 m) = 6 m 2 (6 m) / (2 s) = 3 m/s, or 3 m s -1

27. Adding and subtracting Rule: round off to same number of decimal places as least precise term. Term with largest uncertainty is least precise. Example: 25.0 g + 1.003 g Uncertainty in 25.0 is at least… Uncertainty in 1.003 is at least… Which term is less precise? Therefore, round off answer to…

28. Example Calculate: 4.2 x 10 -3 m + 4.4x10 -4 m - 3.50x10 -5 m Hint: first make all powers of 10 the same

29. Multiplication and Division Rule: round off answer to same number of sig.figs. as the least precise term; least precise term has fewest #sig.figs. Example: 16.00 g / 2.00 mL = ? 16.00 g has how many sig.figs.? 2.00 mL has how many sig.figs.? Which term is less precise? Answer should have how many sig.figs?

30. Multiple Operations Keep one extra digit in intermediate steps to avoid buildup of round off errors. Example: (25.1)(1.0) + 2.5

31. Exact Numbers The concept of significant figures applies only to numbers with uncertainties (measurements). Last significant digit indicates level of uncertainty. Exact numbers have no uncertainties, can be written to as many digits as we want, For calculations, consider exact numbers as having an INFINITE number of significant digits; cannot be the “least precise” term

32. Exact Numbers Example. In the context of the definition: “1 inch = 2.54 cm, exactly” 1 inch can be written as 1.0 inch, 1.00 inch, 1.000 inch, 1.0000 inch, etc. 2.54 cm can be written as 2.54 cm, 2.540 cm, 2.5400 cm, 2.54000 cm, etc. The number of significant figures in “1 inch” and “2.54 cm” are both INFINITE.

33. Example Consider the following formula where 1.8 and 32 are exact: F = 1.8C + 32. Calculate F if C=20.0.

34. Very precise numbers When a number is very precisely known, it can be rounded off for as long as it is not made less precise than the other numbers in a calculation. Example: E = mc 2 c=2.99792458 x10 8 m/s If m=10.0 kg, to what extent can we round off c?

35. Logarithms The number of decimal places in the result should be the same as the number of significant digits in the coefficient if the original number were written in scientific notation Example: log(2.7x10 4 ) = ? Example: 10 -2.699 = ?

36. Test Yourself To how many significant figures should the result of the following calculation be reported: 2500.0 + 25 g? A. 2, B. 3, C. 4, D. 5

37. Test Yourself To how many significant figures should the result of the following calculation be reported: 2500 mL + 25 mL? A. 2, B. 3, C. 4, D. 5

38. Test Yourself How should the result of the following calculation be reported: 80.00 g / 10.0 mL ? A. 8 g/mL, B. 8.0 g/mL, C. 8.00 g/mL, D. 8.000 g/mL

39. Test Yourself How should the result of the following calculation be reported: 0.800 m / 0.010 s? A. 80 m/s, B. 80.0 m/s, C. 8.0x10 1 m/s, D. 8x10 1 m/s

40. Test Yourself How should the result of the following calculation be reported, assuming that 1.8 and 32 are exact numbers: 32 + (1.8)(2.37) A. 36, B. 36.3, C. 36.27, D. 36.266

41. Test Yourself The volume of a sphere of radius r is given by the formula: V = (4/3)(  )r 3 The radius of a sphere is measured to be 5.00 cm. What is the proper way to report the volume? Note:  = 3.141592653589793238462643383279... A. 520 cm 3, B. 523 cm 3, C. 524 cm 3

42. Test Yourself How should the result of the following calculation be reported: 0.0072 + (0.0050)(39.0)? A. 0.20, B. 0.21, C. 0.202, D. 0.2022

43. SI Units Preferred system of units in the sciences and global trade: Systeme Internationale d’Unites Evolved from the metric system Worldwide authority on SI: International Bureau of Weights and Measures (BIPM) http://www.bipm.fr/enus/welcome.html http://www.bipm.fr/enus/welcome.html

44. Base SI Units Base units for seven fundamental quantities were officially defined in 1960 http://www.bipm.fr/enus/3_SI/base_units.html Fundamental quantities mass, length, time, temperature, amount of substance, electric current, luminous intensity, Base Units: kg, m, s, K, mol, A, cd

45. Derived SI Units All other quantities can be thought of as derived from measurements of fundamental quantities. Example: speed = length divided by time Derived unit: m/s or m s -1 Some derived units are given special names (ex. Joule, J = kg m / s 2 ) http://www.bipm.fr/enus/3_SI/si-derived.html http://www.bipm.fr/enus/3_SI/si-derived.html

46. SI Prefixes Why numerous units have been invented: “convenience”! Make number easy to read or write Example: height of a person 6 ft or 0.0011 miles? Prefixes make the use of SI units convenient Prefixes scale units up or down by “powers of 10” to make the numbers easy to read or write (http://www.bipm.fr/enus/3_SI/si-prefixes.html)http://www.bipm.fr/enus/3_SI/si-prefixes.html

47. SI Prefixes Example: Symbol: n Read as: nano Meaning: 10 -9 or 0.000 000 001 Typical distance between two atoms in a molecule: 0.15 nm (or 0.00000000015 m)

48. Metric Prefixes to Memorize Giga (G) = 10 9 Mega (M) = 10 6 Kilo (k) = 10 3 Deci (d) = 10 -1 Centi (c) = 10 -2 Milli (m) = 10 -3 Micro (µ) = 10 -6 (greek letter “mu”) Nano (n) = 10 -9 Pico (p) = 10 -12

49. Commonly used non-SI units Liter: 1 L = 0.001 m 3 or: 1000 L = 1 m 3 Milliliter: 1 mL = 1 cm 3 = 0.001 L [Also: 1 cc] 1 tsp = 5 mL [according to U.S. federal regulation; 21CFR101.9(b)(5)(viii)] 1 tbsp = 3 tsp = 15 mL

50. English system of units Still commonly used in USA and UK By agreement of USA and Commonwealth of Nations in 1959: International avoirdupois pound: 1 lb = 0.45359237 kg, exactly Inch: 1 in = 2.54 cm = 25.4 mm, exactly

51. Unit Conversions Method of algebraic substitution: treat units like variables in algebra Factor label method or dimensional analysis: multiply by a conversion factor so that the original unit cancels out. Conversion factor: ratio of equivalent values NOTE: numbers relating units are generally exact. If rounded off, they must be more precise than the measurement to be converted.

52. Example Convert 3.5 mm to m; Definition: 1 mm = 10 -3 m Algebraic substitution: 3.5 mm = 3.5 (10 -3 m) = 3.5x10 -3 m Factor label method: 3.5 mm x 10 -3 m = 3.5x10 -3 m 1 mm

53. Example Convert 9.8x10 4 m to mm:

54. Example Convert 658 nm to  m.

55. Example Given: 1 L = 10 -3 m 3, 1 atm = 1.01325x10 5 Pa If R=0.08206 L atm mol -1 K -1, express R in m 3 Pa mol -1 K -1.

56. Example Convert: 1.500 x 10 3 ft 2 to in 2 Given: 1 ft = 12 in

57. Example Convert: 2.5 g/mm to g/m

58. Test Yourself Given: 1 inch is exactly equivalent to 2.54 cm, and 12 inches is exactly 1 foot. A measurement of 7.0 cm is equivalent to how many feet?

59. Direct Proportionality Algebraic substitution and factor label methods work only if the quantities expressed in the units involved are directly proportional. When are X and Y directly proportional? If X=0, then Y=0 X/Y = constant If X changes, then Y changes by the same factor Example: length in m and length in cm are directly proportional

60. Temperature Conversions Temperature readings in Celsius, Fahrenheit, and Kelvin scales are not directly proportional. 0 o C = 32 o F = 273.15K Defining equations:

61. Example What readings in the Fahrenheit and Kelvin scales are equivalent to 25.000 o C?

62. Temperature changes Temperature differences or temperature changes are directly proportional.

63. Example If the temperature goes up from 25 o C to 34 o C, by how much did the temperature change in Kelvin?

64. Example If the temperature changes from 50.0 o F to 62.0 o F, by how much did the temperature change in degrees Celsius?

65. Test Yourself During a chemical reaction, the temperature of a mixture dropped by 10.0°C. By how much did the temperature drop in Kelvin? A. 10.0K, B. 283.15K, C. 263.15K

66. Unitless quantities Basis for comparison for unitless quantities are implied in the definition of the quantity. Typically defined as ratios or logarithms of ratios. Example: specific gravity = density of sample divided by density of reference material. Transmittance = intensity of light passing through sample divided by intensity passing through a reference sample. Absorbance = - log (Transmittance)

67. Test Yourself Which of the following is not equivalent to 10 -3 mg? A. 10 -6 g, B. 1  g, C. 10 -4 cg, D. 10 6 ng

68. Test Yourself Which of the following is equivalent to 101.9 MHz? A. 1.019x10 6 Hz, B. 1.019x10 8 Hz, C. 1.019x10 4 Hz, D. 1.019x10 -5 Hz

69. Test Yourself The area of a circle is found to be 2.4 cm 2. What is the area in m 2 ? A. 2.4x10 2 m 2, B. 2.4x10 -2 m 2, C. 2.4x10 -4 m 2, D. 5.8x10 4 m 2

70. Test Yourself The density of air is 1.29x10 -3 g/L. What is the density of air in mg/mL?

71. Test Yourself What temperature reading on the Celsius scale corresponds to 20.0 o F? A. 20.0 o C, B. 68.0 o C, C. 36.0 o C, D. -6.67 o C

72. Test Yourself If the temperature increases by 20.0 degrees Celsius, by how much did it increase in degrees Fahrenheit? A. 36.0°, B. 68.0°, C. 6.67°, D. 20.0°