Numbers in Science Chapter 2

1. Numbers in Science Chapter 2

2. Measurement • What is measurement? • Quantitative Observation • Based on a comparison to an accepted scale. • A measurement has 2 Parts – the Number and the Unit • Number Tells Comparison • Unit Tells Scale • There are two common unit scales • English • Metric

3. The Unit

4. The measurement System units • English (US) • Length – inches/feet • Distance – mile • Volume – gallon/quart • Mass- pound • Metric (rest of the world) • Length – meter • Distance – kilometer • Volume – liter • Mass - gram

5. Related Units in the Metric System • All units in the metric system are related to the fundamental unit by a power of 10 • The power of 10 is indicated by a prefix • The prefixes are always the same, regardless of the fundamental unit

6. Fundamental Unit 100

7. Fundamental SI Units • Established in 1960 by an international agreement to standardize science units • These units are in the metric system

8. Length….. • SI unit = meter (m) • About 3½ inches longer than a yard • 1 meter = distance between marks on standard metal rod in a Paris vault or distance covered by a certain number of wavelengths of a special color of light • Commonly use centimeters (cm) • 1 inch (English Units) = 2.54 cm (exactly)

9. Figure 2.1: Comparison of English and metric units for length on a ruler.

10. Volume • Measure of the amount of three-dimensional space occupied by a substance • SI unit = cubic meter (m3) • Commonly measure solid volume in cubic centimeters (cm3) • Commonly measure liquid or gas volume in milliliters (mL) • 1 L is slightly larger than 1 quart • 1 mL = 1 cm3

11. Mass • Measure of the amount of matter present in an object • SI unit = kilogram (kg) • Commonly measure mass in grams (g) or milligrams (mg) • 1 kg = 2.2046 pounds, 1 lbs.. = 453.59 g

12. Temperature Scales Any idea what the three most common temperature scales are? • Fahrenheit Scale, °F • Water’s freezing point = 32°F, boiling point = 212°F • Celsius Scale, °C • Temperature unit larger than the Fahrenheit • Water’s freezing point = 0°C, boiling point = 100°C • Kelvin Scale, K (SI unit) • Temperature unit same size as Celsius • Water’s freezing point = 273 K, boiling point = 373 K

13. Thermometers based on the three temperature scales in (a) ice water and (b) boiling water.

14. The number

15. Scientific Notation • Technique Used to Express Very Large or Very Small Numbers • 135,000,000,000,000,000,000 meters • 0.00000000000465 liters • Based on Powers of 10 • What is power of 10 Big? • 0,10, 100, 1000, 10,000 • 100, 101, 102, 103, 104 • What is the power of 10 Small? • 0.1, 0.01, 0.001, 0.0001 • 10-1, 10-2, 10-3, 10-4

16. Writing Numbers in Scientific Notation 1. Locate the Decimal Point : 1,438. 2. Move the decimal point to the right of the non-zero digit in the largest place - The new number is now between 1 and 10 - 1.438 3. Now, multiply this number by a power of 10 (10n), where n is the number of places you moved the decimal point - In our case, we moved 3 spaces, so n = 3 (103)

17. The final step for the number…… 4. Determine the sign on the exponent n If the decimal point was moved left, n is + If the decimal point was moved right, n is – If the decimal point was not moved, n is 0 • We moved left, so 3 is positive • 1.438 x 103

18. Writing Numbers in Standard Form • Determine the sign of n of 10n • If n is + the decimal point will move to the right • If n is – the decimal point will move to the left • Determine the value of the exponent of 10 • Tells the number of places to move the decimal point • Move the decimal point and rewrite the number Try it for these numbers: 2.687 x 106 and 9.8 x 10-2 • We reverse the process and go from a number in scientific notation to standard form…..

19. Let’s Practice….. • Change these numbers to Scientific Notation: • 1,340,000,000,000 • 697, 000 • 0.00000000000912 • Change these numbers to Standard Form: • 3.76 x 10-5 • 8.2 x 108 • 1.0 x 101 1.34 x 1012 6.97 x 105 9.12 x 10-12 0.0000376 820,000,000 10

20. Are you sure about that number?

21. Uncertainty in Measured Numbers • A measurement always has some amount of uncertainty, you always seem to be guessing what the smallest division is… • To indicate the uncertainty of a single measurement scientists use a system called significant figures • The last digit written in a measurement is the number that is considered to be uncertain cm

22. Rules, Rules, Rules…. • We follow guidelines (i.e. rules) to determine what numbers are significant • Nonzero integers are always significant • 2753 • 89.659 • .281 • Zeros • Captive zeros are always significant (zero sandwich) • 1001.4 • 55.0702 • 4780.012

23. Significant Figures – Tricky Zeros • Zeros • Leading zeros never count as significant figures • 0.00048 • 0.0037009 • 0.0000000802 • Trailing zeros are significant if the number has a decimal point • 22,000 • 63,850. • 0.00630100 • 2.70900 • 100,000

24. Significant Figures Scientific Notation • All numbers before the “x” are significant. Don’t worry about any other rules. • 7.0 x 10-4 g has 2 significant figures • 2.010 x 108 m has 4 significant figures • How many significant figures are in these numbers? • 102,340 0.01796 92,017 • 1.0 x 107 1,200.00 0.1192 • 1,908,021.0 0.000002 8.01010 x 1014

25. Have a little fun remembering sig figs • http://www.youtube.com/watch?v=ZuVPkBb-z2I

26. Exact Numbers • Exact Numbers are numbers known with certainty • Unlimited number of significant figures • They are either • counting numbers • number of sides on a square • or defined • 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm • 1 kg = 1000 g, 1 LB = 16 oz • 1000 mL = 1 L; 1 gal = 4 qts. • 1 minute = 60 seconds

27. Calculations with Significant Figures • Exact numbers do not affect the number of significant figures in an answer • Answers to calculations must be rounded to the proper number of significant figures • round at the end of the calculation • For addition and subtraction, the last digit to the right is the uncertain digit. • Use the least number of decimal places • For multiplication, count the number of sig figs in each number in the calculation, then go with the smallest number of sig figs • Use the least number of significant figures

28. Rules for Rounding Off If the digit to be removed • is less than 5, the preceding digit stays the same • Round 87.482 to 4 sig figs. • is equal to or greater than 5, the preceding digit is increased by 1 • Round 0.00649710 to 3 sig figs. In a series of calculations, carry the extra digits to the final result and then round off Don’t forget to add place-holding zeros if necessary to keep value the same!! Round 80,150,000 to 3 sig figs.

29. Examples of Sig Figs in Math • 5.18 x 0.0208 • 21 + 13.8 + 130.36 • 116.8 – 0.33 Answers must be in the proper number of significant digits!!!

30. Solutions: • 0.107744 round to proper # sig fig • 5.18 has 3 sig figs, 0.0208 has 3 sig figs so answer is 0.108 • 165.47 • Limiting number of sig figs in addition is the smallest number of decimal places = 12 (no decimals) answer is 165 • 116.47 • Same rule as above so answer is 116.5

31. Moving unit to unit: Conversion

32. Exact Numbers • Exact Numbers are numbers known with certainty • They are either • counting numbers • number of sides on a square • or defined • 100 cm = 1 m, 12 in = 1 ft, 1 in = 2.54 cm • 1 kg = 1000 g, 1 LB = 16 oz • 1000 mL = 1 L; 1 gal = 4 qts. • 1 minute = 60 seconds

33. The Metric System Fundamental Unit 100

34. Movement in the Metric system • In the metric system, it is easy it is to convert numbers to different units. • Let’s convert 113 cm to meters • Figure out what you have to begin with and where you need to go.. • How many cm in 1 meter? • 100 cm in 1 meter • Set up the math sentence, and check that the units cancel properly. • 113 cm [1 m/100 cm] = 1.13 m

35. Let’s Practice converting metric units • 250 mL to Liters • 0.250 mL • 1.75 kg to grams • 1,750 grams • 88 µL to mL • 0.088 mL • 475 cg to kg • 47,500,000 or • 4.75 x 107 • 328 mm to dm • 3.28 dm • 0.00075 nL to µL • 0.75 µL

36. Converting Between Metric and non-Metric (English) units

37. Converting non-Metric Units • Many problems involve using equivalence statements to convert one unit of measurement to another • Conversion factors are relationships between two units • Conversion factors are generated from equivalence statements • e.g. 1 inch = 2.54 cm can give or

38. Converting non-Metric Units • Arrange conversion factor so starting unit is on the bottom of the conversion factor • Convert kilometers to miles • You may string conversion factors together for problems that involve more than one conversion factor. • Convert kilometers to inches • Find the relationship(s) between the starting and final units. • Write an equivalence statement and a conversion factor for each relationship. • Arrange the conversion factor(s) to cancel starting unit and result in goal unit.

39. Practice • Convert 1.89 km to miles • Find equivalence statement 1mile = 1.609 km • 1.89 km (1 mile/1.609 km) • 1.17 miles • Convert 5.6 lbs to grams • Find equivalence statement 454 grams = 1 lb • 5.6 lbs(454 grams/1 lb) • 2500 grams • Convert 2.3 L to pints • Find equivalence statements: 1L = 1.06 qts, 1 qt = 2 pints • 2.3 L(1.06 qts/1L)(2 pints/1 qt) • 4.9 pints

40. Temperature Conversions • To find Celsius from Fahrenheit • oC = (oF -32)/1.8 • To find Fahrenheit from Celsius • oF = 1.8(oC) +32 • Celsius to Kelvin • K = oC + 273 • Kelvin to Celsius • oC = K – 273

41. Temperature Conversion Examples • 180°C to Kelvin • To convert Celsius to Kelvin add 273 • 180+ 273 = 453 K • 23°C to Fahrenheit • Use the conversion factor: F = (1.80)C + 32 • F = (1.80)23 + 32 • F=73.4 or 73°F • 87°F to Celsius • Use the conversion factor C=5/9(F-32) • C = 5/9(87-32) • C = 30.5555555… or 31°C • 694 K to Celsius • To convert K to C, subtract 273 • 694-273= 421°C

42. Measurements and Calculations

43. Density • Density is a physical property of matter representing the mass per unit volume • For equal volumes, denser object has larger mass • For equal masses, denser object has small volume • Solids = g/cm3 • Liquids = g/mL • Gases = g/L • Volume of a solid can be determined by water displacement • Density : solids > liquids >>> gases • In a heterogeneous mixture, denser object sinks

44. Using Density in Calculations

46. Density Example Problems • What is the density of a metal with a mass of 11.76 g whose volume occupies 6.30 cm3? • What volume of ethanol (density = 0.785 g/mL) has a mass of 2.04 lbs? • What is the mass (in mg) of a gas that has a density of 0.0125 g/L in a 500. mL container?

47. How could you find your density?

48. Volume by displacement • To determine the volume to insert into the density equation, you must find out the difference between the initial volume and the final volume. • A student attempting to find the density of copper records a mass of 75.2 g. When the copper is inserted into a graduated cylinder, the volume of the cylinder increases from 50.0 mL to 58.5 mL. What is the density of the copper in g/mL?

49. A student masses a piece of unusually shaped metal and determines the mass to be 187.7 grams. After placing the metal in a graduated cylinder, the water level rose from 50.0 mL to 60.2 mL. What is the density of the metal? • A piece of lead (density = 11.34 g/cm3) has a mass of 162.4 g. If a student places the piece of lead in a graduated cylinder, what is the final volume of the graduated cylinder if the initial volume is 10.0 mL?

50. Percent Error • Percent error – absolute value of the error divided by the accepted value, multiplied by 100%. % error = measured value – accepted value x 100% accepted value • Accepted value – correct value based on reliable sources. • Experimental (measured) value – value physically measured in the lab.