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Chapter 2

Chapter 2. Measurements. CHAPTER OUTLINE. What is a Measurement?. quantitative observation comparison to an agreed upon standard every measurement has a number and a unit. A Measurement. the unit tells you what standard you are comparing your object to the number tells you

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Chapter 2

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  1. Chapter 2 Measurements

  2. CHAPTER OUTLINE

  3. What is a Measurement? • quantitative observation • comparison to an agreed upon standard • every measurement has a number and a unit

  4. A Measurement • the unit tells you what standard you are comparing your object to • the number tells you • what multiple of the standard the object measures • the uncertainty in the measurement

  5. Scientists have measured the average global temperature rise over the past century to be 0.6°C • °C tells you that the temperature is being compared to the Celsius temperature scale • 0.6 tells you that • the average temperature rise is 0.6 times the standard unit • the uncertainty in the measurement is such that we know the measurement is between 0.5 and 0.7°C

  6. SCIENTIFICNOTATION • Scientific Notation is a convenient way to express very large or very small quantities. • Its general form is A x 10n n = exponent coefficient 1  A < 10

  7. SCIENTIFICNOTATION To convert from decimal to scientific notation: • Move the decimal point in the original number so that it is located after the first nonzero digit. • Follow the new number by a multiplication sign and 10 with an exponent (power). • The exponent is equal to the number of places that the decimal point was shifted. 7 5 0 0 0 0 0 0 7.5 x 10 7

  8. Scientific Notation: Writing Large and Small Numbers • A positive exponent means 1 multiplied by 10 n times. • A negative exponent (–n) means 1 divided by 10 n times.

  9. SCIENTIFICNOTATION • For numbers smaller than 1, the decimal moves to the left and the power becomes negative. 0 0 0 6 4 2 3 6.42 x 10

  10. 1. Write 6419 in scientific notation. Examples: decimal after first nonzero digit power of 10 64.19x102 6.419 x 103 641.9x101 6419. 6419

  11. 2. Write 0.000654 in scientific notation. Examples: power of 10 decimal after first nonzero digit 0.000654 0.00654 x 10-1 0.0654 x 10-2 0.654 x 10-3 6.54 x 10-4

  12. CALCULATIONS WITHSCIENTIFIC NOTATION • To perform multiplication or division with scientific notation: Change numbers to exponential form. Multiply or divide coefficients. Add exponents if multiplying, or subtract exponents if dividing. If needed, reconstruct answer in standard exponential form.

  13. Multiply 30,000 by 600,000 Example 1: Convert to exponential form Multiply coefficients Add exponents Reconstruct answer 9 (3 x 104) (6 x 105) = 18 x 10 1.8 x 1010

  14. Divided 30,000 by 0.006 Example 2: Subtract exponents Reconstruct answer Convert to exponential form Divide coefficients 4 – (-3) (3 x 104) 7 = 0.5 x 10 (6 x 10-3) 5 x 106

  15. Follow-up Problems: (5.5x103)(3.1x105) = 17.05x108 = 1.7x109 (9.7x1014)(4.3x1020) = 41.71x106 = 4.2x105 0.4483x104 = 4.5x103 0.2073x103 = 2.1x102 (3.7x106)(4.0x108) = 14.8x102 = 1.5x103

  16. Follow-up Problems: (8.75x1014)(3.6x108) = 31.5x1022 = 3.2x1023 0.2041x1041 = 2.04x1042

  17. ACCURACY & PRECISION • Precision is the reproducibility of a measurement compared to other similar measurements. • Precision describes how close measurements are to one another. • Precision is affected by random errors. Is this measurement precise? Avg mass = 3.12± 0.01 g This measurement has high precision because the deviation of multiple trials is small.

  18. ACCURACY & PRECISION Is this measurement accurate? • Accuracy is the closeness of a measurement to an accepted value (external standard). • Accuracy describes how true a measurement is. • Accuracy is affected by systematic errors. Avg mass = 3.12± 0.01 g True mass = 3.03 g Accuracy cannot be determined without knowledge of the accepted value. This measurement has low accuracy because the deviation from true value is large.

  19. ACCURACY & PRECISION Poor precision Good precision Good accuracy Poor accuracy Poor precision Good precision Poor accuracy Good accuracy

  20. ACCURACY & PRECISION • Two types of error can affect measurements: • Systematic errors: those errors that are controllable, and cause measurements to be either higher or lower than the actual value. • Random errors: those errors that are uncontrollable, and cause measurements to be both higher and lower than the average value.

  21. ERROR INMEASUREMENTS • Two kinds of numbers are used in science: Counted or defined: exact numbers; have no uncertainty Measured: are subject to error; have uncertainty • Every measurement has uncertainty because of instrument limitations and human error.

  22. ERROR IN MEASUREMENTS certain uncertain 8.65 8.6 uncertain certain • The last digit in any measurement is the estimated one. What is this measurement? What is this measurement?

  23. RECORDING MEASUREMENTSTO THE PROPER NO OF DIGITS What is the correct value for each measurement? a) 28ml (1 certain, 1 uncertain) b) 28.2ml (2 certain, 1 uncertain) c) 28.31ml (3 certain, 1 uncertain)

  24. SIGNIFICANTFIGURES RULES • Significant figures rules are used to determine which digits are significant and which are not. • Significant figures are the certain and uncertain digits in a measurement. All non-zero digits are significant. All sandwiched zeros are significant. Leading zeros (before or after a decimal) are NOT significant. Trailing zeros (after a decimal) are significant. 0 . 0 0 4 0 0 4 5 0 0

  25. Examples: Determine the number of significant figures in each of the following measurements. 93.500 g 461 cm 3 sig figs 5 sig figs 0.006 m 1025 g 1 sig fig 4 sig figs 0.705 mL 5500 km 2 sig figs 3 sig figs

  26. ROUNDING OFFNUMBERS • If rounded digit is less than 5, the digit is dropped. 51.234 Round to 3 sig figs 1.875377 Round to 4 sig figs Less than 5 Less than 5

  27. ROUNDING OFFNUMBERS • If rounded digit is equal to or more than 5, the digit is increased by 1. 51.369 Round to 3 sig figs 4 5.4505 1 Round to 4 sig figs More than 5 Equal to 5

  28. SIGNIFICANTFIGURES & CALCULATIONS • The results of a calculation cannot be more precise than the least precise measurement. • In multiplication or division, the answer must contain the same number of significant figures as in the measurement that has the least number of significant figures. • For addition and subtraction, the answer must have the same number of decimal places as there are in the measurement with the fewest decimal places.

  29. MULTIPLICATION& DIVISION Calculator answer 4 sig figs 3 sig figs (9.2)(6.80)(0.3744) = 23.4225 2 sig figs The answer should have two significant figures because 9.2 is the number with the fewest significant figures. The correct answer is 23

  30. 83.5 23.28 ADDITION &SUBTRACTION Add 83.5 and 23.28 Least precise number Calculator answer 106.78 106.8 Correct answer

  31. Example 1: 5.008 + 16.2 + 13.48 = 34.688 34.7 Least precise number Round to

  32. Example 2: 3 sig figs 6.1788 6.2 2 sig figs Round to

  33. SI UNITS • Measurements are made by scientists to determine size, length and other properties of matter. • For measurements to be useful, a measurement standard must be used. • A standard is an exact quantity that people agree to use for comparison. • SI is the standard system of measurement used worldwide by scientists.

  34. SI (METRIC)BASE UNITS

  35. Basic Units of Measurement • The kilogram is a measure of mass, which is different from weight. • The mass of an object is a measure of the quantity of matter within it. • The weight of an object is a measure of the gravitational pull on that matter. • Consequently, weight depends on gravity while mass does not.

  36. Derived Units • A derived unit is formed from other units. • Many units of volume, a measure of space, are derived units. • Any unit of length, when cubed (raised to the third power), becomes a unit of volume. • Cubic meters (m3), cubic centimeters (cm3), and cubic millimeters (mm3) are all units of volume.

  37. DERIVED UNITS • In addition to the base units, several derived units are commonly used in SI system.

  38. SI PREFIXES • The SI system of units is easy to use because it is based on multiples of ten. • Common prefixes are used with the base units to indicate the multiple of ten that the unit represents. SI Prefixes 106 103 10-2 10-3 10-6

  39. Base Unit 106 103 103 106 10 10 10 10 10 10 10 10 10 micro milli kilo mega SI UNITS & PREFIXES • SI system used a common set of prefixes for use with the base units. 10 10 10 deci centi Smaller units Larger units

  40. Base Unit 106 103 103 106 10 10 10 10 10 10 10 10 10 milli micro kilo mega SI CONVERSION FACTORS 10 10 10 deci centi 1 m = 103 mm or 1 mm = 103 m 1 mm = 103m or 1 m = 103 mm

  41. SI PREFIXES 100000 or 105 How many mm are in a cm? How many cm are in a km? 10x10x10x10x10 10

  42. Prefix Multipliers • Choose the prefix multiplier that is most convenient for a particular measurement. • Pick a unit similar in size to (or smaller than) the quantity you are measuring. • A short chemical bond is about 1.2 × 10–10 m. Which prefix multiplier should you use? • The most convenient one is probably the picometer. Chemical bonds measure about 120 pm.

  43. CONVERSIONFACTORS • Many problems in chemistry and related fields require a change of units. • Any unit can be converted into another by use of the appropriate conversion factor. • Any equality in units can be written in the form of a fraction called a conversion factor. For example: Metric-Metric Factor Equality 1 m = 100 cm 1 m 100 cm Conversion Factors or 100 cm 1 m

  44. CONVERSIONFACTORS Metric-English Factor • Sometimes a conversion factor is given as a percentage. For example: Equality 1 kg = 2.20 lb Conversion Factors or Percentage Factor Percent quantity: 18% body fat by mass Conversion Factors 18 kg body fat 100 kg body mass 2.20 lb 1 kg or 100 kg body mass 18 kg body fat 2.20 lb 1 kg

  45. CONVERSIONOF UNITS • Problems involving conversion of units and other chemistry problems can be solved using the following step-wise method: Plan a sequence of steps to convert the initial unit to the final unit. • 3. Write the conversion factor for each units change in your plan. 4. Set up the problem by arranging cancelling units in the numerator and denominator of the steps involved. • 1. Determine the intial unit given and the final unit needed. final unit beginning unit Conversion factor

  46. Example 1: Convert 164 lb to kg (1 kg = 2.20 lb) Step 1: Given: 164 lb Need: kg Metric-English factor lb kg Step 2: Step 3: or 1 kg 2.20 lb Step 4: 2.20 lb 1 kg

  47. Example 2: The thickness of a book is 2.5 cm. What is this measurement in mm? Step 1: Given: 2.5 cm Need: mm Metric-Metric factor cm mm Step 2: Step 3: or 1 cm 10 mm Step 4: 10 mm 1 cm

  48. Example 3: How many centimeters are in 2.0 ft? (1 in=2.54 cm) Step 1: Given: 2.0 ft Need: cm English-English factor Metric-English factor ft in cm Step 2: 1 in Step 3: and 2.54 cm 1 ft 60.96 cm 61 cm Step 4: 12 in

  49. Example 4: Bronze is 80.0% by mass copper and 20.0% by mass tin. A sculptor is preparing to case a figure that requires 1.75 lb of bronze. How many grams of copper are needed for the brass figure (1lb = 454g)? Step 1: Given: 1.75 lb bronze Need: g of copper lbbrz English-Metric factor gbrz Percentage factor gCu Step 2:

  50. Example 4: 80.0 g Cu 80.0 g Cu Step 3: and 100 g brz 100 g brz Step 4: 1.75 lb brz x x = 635.6 g = 636 g 1 lb 454 g 454 g 1 lb

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