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Measurements and Calculations Chapter 2

Measurements and Calculations Chapter 2. Measurement. Quantitative Observation Comparison Based on an Accepted Scale e.g. Meter Stick Has 2 Parts – the Number and the Unit Number Tells Comparison Unit Tells Scale. Scientific Notation.

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Measurements and Calculations Chapter 2

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  1. Measurements and Calculations Chapter 2

  2. Measurement • Quantitative Observation • Comparison Based on an Accepted Scale • e.g. Meter Stick • Has 2 Parts – the Number and the Unit • Number Tells Comparison • Unit Tells Scale

  3. Scientific Notation • Technique Used to Express Very Large or Very Small Numbers • Based on Powers of 10

  4. Writing Numbers in Scientific Notation 1. Move the decimal point so there is only one non-zero number to the left of it. The new number is now between 1 and 9 2. Multiply the new number by 10n • where n is the number of places you moved the decimal point 3. 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

  5. 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

  6. Standard to Scientific Notation • 0.0000000011 • 8,031,000,000 • 75,000,000 • 0.0005710

  7. Scientific to Standard Notation • 7.10 x 10-5 • 9.38 x 1012 • 2.75 x 10-7 • 5.22 x 104

  8. More practice • Change to standard notation • 0.00065 x 106 • 391 x 10-2 • Change to scientific notation • 41080.642 • 1.8732

  9. 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 or basic unit

  10. Length • SI unit = meter (m) • About 3½ inches longer than a yard • 1 meter = one ten-millionth the distance from the North Pole to the Equator = distance between marks on standard metal rod in a Paris vault = distance covered by a certain number of wavelengths of a special color of light • Commonly use centimeters (cm) • 1 m = 100 cm • 1 cm = 0.01 m = 10 mm • 1 inch = 2.54 cm (exactly)

  11. Figure 2.1: Comparison of English and metricunits for length on a ruler.

  12. 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 (cm x cm x cm)) • 1 m3 = 106 cm3 • 1 cm3 = 10-6 m3 = 0.000001 m3 • Commonly measure liquid or gas volume in milliliters (mL) • 1 L is slightly larger than 1 quart • 1 L = 1 dL3 = 1000 mL = 103mL • 1 mL = 0.001 L = 10-3 L • 1 mL = 1 cm3

  13. Figure 2.3:A 100-mL graduated cylinder.

  14. 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 • 1 kg = 1000 g = 103 g, 1 g = 1000 mg = 103 mg • 1 g = 0.001 kg = 10-3 kg, 1 mg = 0.001 g = 10-3 g

  15. Figure 2.4:An electronic analytical balance used in chemistry labs.

  16. Metric conversions • 250 mL to Liters • 1.75 kg to grams • 88 daL to mL

  17. Metric conversions • 475 cg to mg • 328 hm to Mm • 0.00075 nL to cL

  18. Uncertainty in Measured Numbers • A measurement always has some amount of uncertainty • Uncertainty comes from limitations of the techniques used for comparison • To understand how reliable a measurement is, we need to understand the limitations of the measurement

  19. Reporting Measurements • 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 • Unless stated otherwise, the uncertainty in the last digit is ±1

  20. Rules for Counting Significant Figures • Nonzero integers are always significant • How many significant figures are in the following examples: • 2753 • 89.659 • 0.281

  21. Significant Figures • Zeros • Captive zeros are always significant • How many significant figures are in the following examples: • 1001.4 • 55.0702 • 0.4900008

  22. Significant Figures • Zeros • Leading zeros never count as significant figures • How many significant figures are in the following examples: • 0.00048 • 0.0037009 • 0.0000000802

  23. Significant Figures • Zeros • Trailing zeros are significant if the number has a decimal point • How many significant figures are in the following examples: • 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

  25. 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.

  26. Rules for Rounding Off • In a series of calculations, carry the extra digits to the final result and then round off • Ex: Convert 80,150,000 seconds to years • Don’t forget to add place-holding zeros if necessary to keep value the same!! • Round 80,150,000 to 3 sig figs.

  27. Multiplication/Division with Significant Figures Count the number of significant figures in each measurement • Round the result so it has the same number of significant figures as the measurement with the smallest number of significant figures 14.593 cm x 0.200 cm = 3.7 x 103 x 0.00340 =

  28. Calculations with Significant Figures • Calculators/computers do not know about 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

  29. 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

  30. Problem Solving and Dimensional Analysis • Many problems in chemistry involve using equivalence statements to convert one unit of measurement to another • Conversion factors are relationships between two units • Conversion factors generated from equivalence statements • e.g. 1 inch = 2.54 cm can give or

  31. Problem Solving and Dimensional Analysis • Arrange conversion factor so starting unit is on the bottom of the conversion factor • You may string conversion factors together for problems that involve more than one conversion factor.

  32. Converting One Unit to Another • 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.

  33. Converting One Unit to Another • Check that the units cancel properly • Multiply all the numbers across the top and divide by each number on the bottom to give the answer with the proper unit. • Round your answer to the correct number of significant figures. • Check that your answer makes sense!

  34. English Units Conversions • 155.0 pounds to grams • 2.00 x 108 seconds to hours • 28.5 inches to feet • 4.0 gallons to quarts • 48.39 minutes to hours

  35. More Difficult Conversions • 0.091 ft2 to inches2 • 47.1 mm3 to kL • 682 mg to pounds • 3.5 x 10-4 L to cm3

  36. Complex Conversion Problems • 25 miles per hour to feet per second • 4.70 gallons per minute to mL per year • 5.6 x 10-6 centiliters per square meter (cL/m2) to cubic meters per square foot (m3/ft2)

  37. Temperature Scales • 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 • Temperature unit same size as Celsius • Water’s freezing point = 273 K, boiling point = 373 K

  38. Temperature Conversions • Fahrenheit to Celsius oC = 5/9(oF -32) • Celsius to Fahrenheit oF = 1.8(oC) +32 • Celsius to Kelvin K = oC + 273 • Kelvin to Celsius oC = K – 273

  39. Figure 2.6: Thermometers based on the three temperature scales in (a) ice water and (b) boiling water.

  40. Figure 2.7: The three major temperature scales.

  41. Figure 2.8: Converting 70. 8C to units measured on the Kelvin scale.

  42. Figure 2.9: Comparison of the Celsius and Fahrenheit scales.

  43. Temperature Conversion Examples • 12oC to K • 248 K to oF • 98.6oF to K • 86oF to oC • -5.0oC to oF • 352 K to oC

  44. Density • Density is a 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 • 1 cm3 = 1 mL • 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

  45. Using Density in Calculations

  46. Spherical droplets of mercury, a very dense liquid.

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