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Chapter 1 Measurements

Chapter 1 Measurements. Significant Figures in Calculations. Objectives. To learn how uncertainty in a measurement arises To learn to indicate a measurement ’ s uncertainty by using significant figures To learn to determine the number of significant figures in a calculated result

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Chapter 1 Measurements

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  1. Chapter 1Measurements Significant Figures in Calculations

  2. Objectives • To learn how uncertainty in a measurement arises • To learn to indicate a measurement’s uncertainty by using significant figures • To learn to determine the number of significant figures in a calculated result • To learn the difference between accuracy and precision of a measurement

  3. Significant Figures • When using our calculators we must determine the correct answer; our calculators are mindless drones and don’t know the correct answer. • There are 2 different types of numbers • Exact • Measured • Exact numbers are infinitely important • Measured number = they are measured with a measuring device (name all 4) so these numbers have ERROR. • When you use your calculator your answer can only be as accurate as your worst measurement… 

  4. Exact Numbers An exact number is obtained when you count objects or use a defined relationship. Counting objects are always exact 2 soccer balls 4 pizzas Exact relationships, predefined values, not measured 1 foot = 12 inches 1 meter = 100 cm For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches.

  5. Measured Numbers • Do you see why Measured Numbers have error…you have to make that Guess! • All but one of the significant figures are known with certainty. The last significant figure is only the best possible estimate. • To indicate the precision of a measurement, the value recorded should use all the digits known with certainty.

  6. Measurement and Significant Figures • Every experimental measurement has a degree of uncertainty. • The volume, V, at right is certain in the 10’s place, 10mL<V<20mL • The 1’s digit is also certain, 17mL<V<18mL • A best guess is needed for the tenths place.

  7. ? 8.00 cm or 3 (2.2/8)

  8. A. Uncertainty in Measurement • A measurement always has some degree of uncertainty. • How long is this nail?

  9. A. Uncertainty in Measurement • Different people estimate differently. • Record all certain numbers and one estimated number. • Whenever we measure in chemistry we always record one extra decimal place (estimated number) beyond the markings on the measurement device.

  10. Significant Digits • Significant digits only apply to measurements; therefore there must be a number and a unit. • Without a unit there are no “significant digits” • Significant digits include all measured values and one estimated value • The last number is considered to be estimated

  11. Significant Figures in Measurement • The numbers reported in a measurement are limited by the measuring tool • Significant figures in a measurement include the known digits plus one estimated digit

  12. Learning Check A. Exact numbers are obtained by 1. using a measuring tool 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool 2. counting 3. definition

  13. Solution A. Exact numbers are obtained by 2. counting 3. definition B. Measured numbers are obtained by 1. using a measuring tool

  14. Learning Check Classify each of the following as an exact or a measured number. 1 yard = 3 feet The diameter of a red blood cell is 6 x 10-4 cm. There are 6 hats on the shelf. Gold melts at 1064°C.

  15. Solution Classify each of the following as an exact (1) or a measured(2) number. This is a defined relationship. A measuring tool is used to determine length. The number of hats is obtained by counting. A measuring tool is required.

  16. Significant Figures • Rules for Counting Significant Figures • 100 m __ sig figs • 410100 L __ sig figs • 100. mm __ sig figs • 100.8 s __ sig figs • 100.80 μm __ sig figs • 0.00287 kg __ sig figs • 0.002870 cg __ sig figs • 1.0805 x 104 mm __ sig figs • 1.0805 x 10-12 km __ sig figs • 1.205670 x 10-6 mg __ sig figs

  17. Significant Figures Rules for Counting Significant Figures • Exact numbers -unlimited significant figures • Exact counts have unlimited (infinite) sig figs; they are obtained by simple counting (no uncertainty): • 3 apples, 12 eggs, 1 mole • Conversion Factors have unlimited sig figs also: • 1 in. = 2.54 cm • 1 mole = 6.022 x 1023 atoms

  18. Significant Figures Rules for Counting Significant Figures • Nonzero integers always count as significant figures.1457 mm 4 significant figures

  19. Significant Figures • Rules for Counting Significant Figures • Zeros • Leading zeros -never count0.0025 m 2 significant figures • Captive zero (zeros between 2 non-zero numbers) -always count 1.008 cm 4 significant figures • Trailing zeros - count only if the number is written with a decimal point100 L 1 significant figure • Zeros between a non-zero and a decimal point • 100. kg 3 significant figures • Zeros after a non-zero and a decimal point • 120.0 mg 4 significant figures

  20. Rounding Off • 5 or more to the right of the number being rounded • Raise the score • 4.678 cm rounded to the nearest hundredth • 4.68 cm • 4 or less to the right of the number being rounded • Let it rest!!! • 3.421 cm rounded to the nearest hundredth • 3.42 cm

  21. Rounding Off Numbers • Often when doing arithmetic on a pocket calculator, the answer is displayed with more significant figures than are really justified. • How do you decide how many digits to keep? • Simple rules exist to tell you how.

  22. Note the 4 rules When reading a measured value, all nonzero digits (1, 2, 3, 4, 5, 6, 7, 8, and 9) should be counted as significant. There is a set of rules for determining if a zero in a measurement is significant or not. • RULE 1. Zeros in the middle of a number are like any other digit; they are always significant. Thus, 94.072 g has five significant figures. • RULE 2. Zeros at the beginning of a number are not significant; they act only to locate the decimal point. Thus, 0.0834 cm has three significant figures, and 0.029 07 mL has four.

  23. RULE 3. Zeros at the end of a number and after the decimal point are significant. It is assumed that these zeros would not be shown unless they were significant. 138.200 m has six significant figures. If the value were known to only four significant figures, we would write 138.2 m. • RULE 4. Zeros at the end of a number and before an implied decimal point may or may not be significant. We cannot tell whether they are part of the measurement or whether they act only to locate the unwritten but implied decimal point.

  24. Significant Figures Rules for Addition and Subtraction • The number of decimal places in the result is the same as in the measurement with the smallest number of decimal places(not sig figs!). m m m m m m m m m

  25. Significant Figures Rules for Multiplication and Division • The number of significant figures in the answer is the same as in the measurement with the smallest number of significant figures.

  26. Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

  27. Practice Rule #1 Zeros 6 3 5 5 2 4 6 • All digits count • Leading 0’s don’t • Trailing 0’s do • 0’s count in decimal form • 0’s don’t count w/o decimal • All digits count • 0’s between digits count as well as trailing in decimal form 45.8736 cm .000239 L .00023900 kg 48000. mm 48000 m 3.982106 mm 1.00040 cg

  28. Counting Significant Figures Number of Significant Figures 38.15 cm 4 5.6 ft 2 65.6 lb ___ 122.55 m ___ Complete: All non-zero digits in a measured number are (significant or not significant).

  29. Leading Zeros Number of Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb ____ 0.000262 mL ____ Complete: Leading zeros in decimal numbers are (significant or not significant)

  30. Sandwiched Zeros Number of Significant Figures 50.8 mm 3 2001 min 4 0.702 lb ____ 0.00405 m ____ Complete: Zeros between nonzero numbers are (significant or not significant).

  31. Trailing Zeros Number of Significant Figures25,000 in. 2 200 yr 1 48,600 gal 3 25,005,000 g ____ Complete: Trailing zeros in numbers without decimals are (significant or not significant) if they are serving as place holders.

  32. Significant Figures • Rules for Counting Significant Figures VITALLY IMPORTANT: • For the rest of the year, all calculations must include the correct number of significant figures in order to be fully correct! • on all homework, labs, quizzes, and tests, etc. • even if the directions don’t specifically tell you so

  33. Significant Figures • Numbers recorded in a measurement. • All the certain numbers plus the first estimated number • Accuracy of a measurement – how close your number comes to the actual value • similar to hitting the bull's-eye on a dart board • Precision of a measurement – how close your repeated measurements come to each other (not necessarily the actual value) • how closely grouped are your 3 darts on the board (even if they’re not close to the bull's-eye) • It is possible for measurements to be precise but not accurate, just as it is possible to be accurate but not precise

  34. Once you decide how many digits to retain, the rules for rounding off numbers are straightforward: • RULE 1. If the first digit you remove is 4 or less, drop it and all following digits. 2.4271 g becomes 2.4 gwhen rounded off to two significant figures because the first dropped digit (a 2) is 4 or less. • RULE 2. If the first digit removed is 5 or greater, round up by adding 1 to the last digit kept. 4.5832 m is 4.6 m when rounded off to 2 significant figures since the first dropped digit (an 8) is 5 or greater. • If a calculation has several steps, it is best to round off at the end.

  35. Examples of Rounding For example you want a 4 Sig Fig number 0 is dropped, it is <5 8 is dropped, it is >5; Note you must include the 0’s 5 is dropped it is = 5; note you need a 4 Sig Fig 4965.03 kg 780,582 mm 1999.5 L 4965 kg 780,600 mm 2000. L

  36. Practice Rule #2 Rounding Make the following into a 3 Sig Fig number 1.5587 m 0.0037421 m 1367 m 128,522 m 1.6683 106 m Your Final number must be of the same value as the number you started with, 129,000 m and not 129 m 1.56 m 0.00374 m 1370 m 129,000 m 1.67 106 m

  37. RULE 1. In carrying out a multiplication or division, the answer cannot have more significant figures than either of the original numbers.

  38. RULE 2. In carrying out an addition or subtraction, the answer cannot have more digits after the decimal point than either of the original numbers.

  39. Significant Numbers in Calculations • A calculated answer cannot be more precise than the measuring tool. • A calculated answer must match the least precise measurement. • Significant figures are needed for final answers from 1) adding or subtracting 2) multiplying or dividing

  40. Multiplication and division 49.7 m2 46.4 m/s 0.05985 m2 1.586 107 m2 1.000 m2 32.27 m  1.54 m = 49.6958 m2 3.68 m  .07925 s = 46.4353312 m/s 1.750 m  .0342000 m = 0.05985 m2 3.2650106 m  4.858 m= 1.586137  107 m2 6.0221023 m  1.66110-24 m = 1.000000 m2

  41. Addition/Subtraction 25.5 L 32.72 cm 320 m +34.270 L‑ 0.0049 cm + 12.5 m 59.770 L 32.7151 cm 332.5 m 59.8 L 32.72 cm 330 m

  42. Addition and Subtraction 0.71 g 82000 mL 0.1 g 0 g Look for the last important digit 0.56 g + 0.153 g = 0.713 g 82000 mL + 5.32 mL = 82005.32 mL 10.0 m - 9.8742 m = 0.12580 m 10 g – 9.8742 g = 0.12580 g __ ___ __

  43. Learning Check A. Which answers contain 3 significant figures? 1) 0.4760 m 2) 0.00476 m 3) 4760 m B. All the zeros are significant in 1) 0.00307 m 2) 25.300 m 3) 2.050 x 103 m C. 534,675 rounded to 3 significant figures is 1) 535 m 2) 535,000 m 3) 5.35 x 105 m

  44. Solution A. Which answers contain 3 significant figures? 2) 0.00476 m 3) 4760 m B. All the zeros are significant in 2) 25.300 m 3) 2.050 x 103 m C. 534,675 rounded to 3 significant figures is 2) 535,000 m 3) 5.35 x 105 m

  45. Learning Check In which set(s) do both numbers contain the same number of significant figures? 1) 22.0 m and 22.00 m 2) 400.0 m and 40 m 3) 0.000015 m and 150,000 m

  46. Solution In which set(s) do both numbers contain the samenumber of significant figures? 3) 0.000015 m and 150,000 m

  47. Learning Check SF3 State the number of significant figures in each of the following: A. 0.030 m 1 2 3 B. 4.050 L 2 3 4 C. 0.0008 g 1 2 4 D. 3.00 m 1 2 3 E. 2,080,000 bees 3 5 7

  48. Solution A. 0.030 m 2 B. 4.050 L 4 C. 0.00008 g 1 D. 3.00 m 3 E. 2,080,000 bees 3

  49. Adding and Subtracting The answer has the same number of decimal places as the measurement with the fewest decimal places. 25.2 mmone decimal place + 1.34 mmtwo decimal places 26.54 mm answer 26.5 mm one decimal place

  50. Learning Check In each calculation, round the answer to the correct number of significant figures. A. 235.05 m + 19.6 m + 2.1 m = 1) 256.75 m 2) 256.8 m 3) 257 m B. 58.925 m - 18.2 m = 1) 40.725 m 2) 40.73 m 3) 40.7 m

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