Mastering Measurements: Units, Scientific Notation, and Accuracy

1. Lesson 4 Measuring & Significant Digits Topic 1 – Units of Measurement & Metric Prefixes Topic 2 – Scientific Notation Topic 3 – Accuracy, Precision and Error Topic 4 – Significant Digits Anything in black letters = write it in your notes (‘knowts’)

2. Topic 1 – Units of Measurement & Metric Prefixes

3. Measurements without units are useless! “I walked 5 today.” “The speed of light is 186,000 “I weigh 890” “20 of water” All measurements need units!

4. SI – International System of Units We will use all of these in this class

6. Volume - Amount of space occupied by an object (remember?) Normal units used for volume: Solids – m3or cm3 Liquids & Gases – liters (L) or milliliters (ml) 1 L = 1000 mL 1 mL = 1 cm3 The volume of a material changes with temperature, especially for gases.

7. Mass - Measure of inertia (remember?) Weight - Force of gravity on a mass; measured in pounds (lbs) or Newtons. Weight can change with location, mass does not

8. Energy – Ability to do work or produce heat. Normal units used for energy: SI – joule (J) non-SI – calorie (cal) 1 cal = 4.184 J How many joules are in a kilojoule? How many calories are in a kilocalorie?

9. Temperature – measure of how cold or hot an object is.

10. Temperature – measure of the average kinetic energy of molecules. Normal units used for temp: SI – kelvin (K) non-SI – celsius (°C) or Fahrenheit (°F) yucky!

11. Celsius 100 divisions 100°C Boiling point of water 373.15 K 0°C Freezing point of water 273.15 K 100 divisions Kelvin

12. mass Density = volume Normal units for density: g/cm3, g/mL, g/L

14. Topic 2 – Scientific Notation

15. We will often work with really large or really small numbers in this class. Standard Notation Scientific Notation 872,000,000 grams 0.0000056 moles = 8.72 x 108 grams = 5.6 x 10-6 moles

16. exponent coefficient 6.02 x 1023 The coefficient must be a single, nonzero digit, exponent must be an integer.

17. Multiplication and Division To multiply numbers written in scientific notation, multiply the coefficients and add the exponents. (3 x 104) x (2 x 102) = (3 x 2) x 104+2 = 6 x 106 (2.1 x 103) x (4.0 x 10–7) = (2.1 x 4.0) x 103+(–7) = 8.4 x 10–4

18. To divide numbers written in scientific notation, divide the coefficients and subtract the exponents (top – bottom) Coefficient needs to be between 1 and 10

19. Addition and Subtraction When adding or subtracting in Sci. Not., the exponents must be the same. Not the same, need to adjust one of the exponents (5.4 x 103) + (8.0 x 102) = (5.4 x 103) + (0.80 x 103) = (5.4 + 0.80) x 103 = 6.2 x 103

20. Example • Solve each problem and express the answer in scientific notation. • a. (8.0 x 10–2) x (7.0 x 10–5) • b. (7.1 x 10–2) + (5 x 10–3)

21. a. Multiply the coefficients and add the exponents. (8.0 x 10–2) x (7.0 x 10–5) = (8.0 x 7.0) x 10–2 + (–5) = 56 x 10–7 = 5.6 x 10–6

22. b. Rewrite one of the numbers so that the exponents match. Then add the coefficients (7.1 x 10–2) + (5 x 10–3) = (7.1 x 10–2) + (0.5 x 10–2) = (7.1 + 0.5) x 10–2 = 7.6 x 10–2

23. Topic 3 – Accuracy, Precision and Error

24. Accuracy - closeness of a measurement to the actual or accepted value. Precision - closeness of repeated measurements to each other

25. Accuracy and Precision Darts on a dartboard illustrate the difference between accuracy and precision. Poor Accuracy, Poor Precision Good Accuracy, Good Precision Poor Accuracy, Good Precision The closeness of a dart to the bull’s-eye corresponds to the degree of accuracy. The closeness of several darts to one another corresponds to the degree of precision.

26. Error Suppose you measured the melting point of a compound to be 78°C Suppose also, that the actual melting point value (from reference books) is 76°C. The error in your measurement would be 2°C. Error is always a positive value

27. How far off you are in a measurement doesn’t tell you much. For example, lets say you have $1,000,000 in your checking account. When you balance your checkbook at the end of the month, you find that you are off by $175; error = $175 Now, lets be more realistic, you have $225 in your checking account and after balancing you are off by $175!

28. In both cases, there is an error of $175. But in the first, the error is such a small portion of the total that it doesn’t matter as much as the second. So, instead of error, percent error is more valuable. Percent error compares the error to the size of the measurements.

29. Topic 4 – Significant Digits

30. In any measurement, the last digit is estimated 30.2°C The 2 is estimated (uncertain) by the experimenter, another person may say 30.1 or 30.3

31. 9.3 mL 0.72 cm

32. Increasing Precision

33. The significant figures in a measurement are the numbers that are part of the measurement. Zeros that are NOT significant are called placeholders.

34. Rules for determining Significant Figures 1. Every nonzero digit in a reported measurement is assumed to be significant. 2. Zeros appearing between nonzero digits are significant. 3. Leftmost zeros appearing in front of nonzero digits are not significant. They act as placeholders. By writing the measurements in scientific notation, you can eliminate such placeholding zeros. 4. Zeros at the end of a number and to the right of a decimal point are always significant. 5. Zeros at the rightmost end of a measurement that lie to the left of an understood decimal point are not significant if they serve as placeholders to show the magnitude of the number. 5 (continued). If such zeros were known measured values, then they would be significant. Writing the value in scientific notation makes it clear that these zeros are significant. 6. There are two situations in which numbers have an unlimited number of significant figures. The first involves counting. A number that is counted is exact. 6 (continued). The second situation involves exactly defined quantities such as those found within a system of measurement. HOLY SMOKES!!

35. “Line Through” Method for Counting Sig Figs 1. If there is a decimal, start from the left and draw a line through any zeros, the numbers remaining are significant. 2. If there is no decimal, start the line from the right. A Shorter Method SOURCE: Skylar Morben, 2014 MRHS Graduate

36. How many significant digits? • 100 1.00 0.23 • 0.0034 1.01 1005.4 • 0.10 100.0 54.0

37. How many significant digits are in the following measurements? a) 150.31 grams b) 10.03 mL c) 0.045 cm d) 4.00 lbs e) 0.01040 m f) 100.10 cm g) 100 grams h) 1.00 x 102 grams i) 11 cars j) 2 molecules

38. An answer can’t be more accurate than the measurements it was calculated from

39. Rules for Add/Subtracting Sig Figs The answer to an +/- calculation should be rounded to the same number of decimal places as the measurement with the least number of decimal places. 3.2 cm 37.10 g

40. Rules for Mult/Division Sig Figs The answer to a x/÷ calculation should be rounded to the same number of sig figs as the measurement with the least number of sig figs. 3.8 cm2

41. Always round your finalanswer off to the correct number of significant digits.

42. Draw a box around the significant digits in the following measurements. • 2.2000 b) 0.0350 c) 0.0006 • 0.0089 e) 24,000 f) 4.360 x 104 • 0.0708 h) 1200 i) 0.6070 • k) 21.0400 l) 0.007 m) 5.80 x 10-3

43. Round off each of the following numbers to two significant figures. • a) 86.048 b) 29.974 c) 6.1275 • 0.008230 e) 800.7 f) 0.07864 • g) 0.06995 h) 7.096 i) 8000.10

44. Express each of the following numbers in standard scientific notation with the correct number of significant digits. • 0.00000070 • 25.3 • 825,000 • 826.7 • 43,500 • 65.0 • 0.000320 • 0.0432

45. Perform the following arithmetic. Round the answers to the proper number of sig. figs. Box in your final answer and don’t forget units! • 2.41 cm x 3.2 cm b) 4.025 m x 18.2 m • c) 81.4 g  104.2 cm3

46. Perform the following arithmetic. Round the answers to the proper number of sig. figs. Box in your final answer and don’t forget units! • d) 822 mi  0.028 hr • e) 10.89 g / 1 mL