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Measurement and Calculation

Measurement and Calculation. Chapter 2. Zumdahl's Introductory Chemistry - Chapter 2. 1. 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.

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Measurement and Calculation

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  1. Measurement and Calculation Chapter 2 Zumdahl's Introductory Chemistry - Chapter 2 1

  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 Zumdahl's Introductory Chemistry - Chapter 2 2

  3. Scientific Notation • Technique Used to Express Very Large or Very Small Numbers • Based on Powers of 10 • To Compare Numbers Written in Scientific Notation • First Compare Exponents of 10 • Then Compare Numbers Zumdahl's Introductory Chemistry - Chapter 2 3

  4. Writing Numbers in Scientific Notation • Locate the Decimal Point • 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 • Multiply the new number by 10n • where n is the number of places you moved the decimal point • 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 Zumdahl's Introductory Chemistry - Chapter 2 4

  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 Zumdahl's Introductory Chemistry - Chapter 2 5

  6. 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 Zumdahl's Introductory Chemistry - Chapter 2 6

  7. Length • SI unit = meter (m) • About 3› 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) Zumdahl's Introductory Chemistry - Chapter 2 7

  8. 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) • 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 = 103 mL • 1 mL = 0.001 L = 10-3 L • 1 mL = 1 cm3 Zumdahl's Introductory Chemistry - Chapter 2 8

  9. 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 Zumdahl's Introductory Chemistry - Chapter 2 9

  10. 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 Zumdahl's Introductory Chemistry - Chapter 2 10

  11. 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 Zumdahl's Introductory Chemistry - Chapter 2 11

  12. Rules for Counting Significant Figures • Nonzero integers are always significant • Zeros • Leading zeros never count as significant figures • Captive zeros are always significant • Trailing zeros are significant if the number has a decimal point • Exact numbers have an unlimited number of significant figures Zumdahl's Introductory Chemistry - Chapter 2 12

  13. Rules for Rounding Off • If the digit to be removed • is less than 5, the preceding digit stays the same • is equal to or greater than 5, the preceding digit is increased by 1 • 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!! Zumdahl's Introductory Chemistry - Chapter 2 13

  14. 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 Zumdahl's Introductory Chemistry - Chapter 2 14

  15. 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 Zumdahl's Introductory Chemistry - Chapter 2 15

  16. Multiplication/Division with Significant Figures • Result has the same number of significant figures as the measurement with the smallest number of 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 4.5 cm x 0.200 cm = 0.90 cm2 2 sig figs 3 sig figs 2 sig figs Zumdahl's Introductory Chemistry - Chapter 2 16

  17. Adding/Subtracting Numbers with Significant Figures • Result is limited by the number with the smallest number of significant decimal places • Find last significant figure in each measurement • Find which one is “left-most” • Round answer to the same decimal place 450 mL + 27.5 mL = 480 mL precise to 10’s place precise to 0.1’s place precise to 10’s place Zumdahl's Introductory Chemistry - Chapter 2 17

  18. 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 • May be exact or measured • Both parts of the conversion factor should have the same number of significant figures • Conversion factors generated from equivalence statements • e.g. 1 inch = 2.54 cm can give or Zumdahl's Introductory Chemistry - Chapter 2 18

  19. Problem Solving and Dimensional Analysis • Arrange conversion factors so starting unit cancels • Arrange conversion factor so starting unit is on the bottom of the conversion factor • May string conversion factors Zumdahl's Introductory Chemistry - Chapter 2 19

  20. Converting One Unit to Another • Find the relationship(s) between the starting and goal units. Write an equivalence statement for each relationship. • Write a conversion factor for each equivalence statement. • Arrange the conversion factor(s) to cancel starting unit and result in goal unit. Zumdahl's Introductory Chemistry - Chapter 2 20

  21. Converting One Unit to Another • Check that the units cancel properly • Multiply and Divide the numbers to give the answer with the proper unit. • Check your significant figures • Check that your answer makes sense! Zumdahl's Introductory Chemistry - Chapter 2 21

  22. 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 Zumdahl's Introductory Chemistry - Chapter 2 22

  23. 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 Zumdahl's Introductory Chemistry - Chapter 2 23

  24. Using Density in Calculations Zumdahl's Introductory Chemistry - Chapter 2 24

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