# Chapter 2 Measurement and Problem Solving

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1 Introductory Chemistry, 3 rd Edition Nivaldo Tro Measurement and Problem Solving Graph of global Temperature rise in 20 th Century. Cover page Opposite page 11. Roy Kennedy Massachusetts Bay Community College Wellesley Hills, MA 2009, Prentice Hall

2 Outline 2.1 Measuring Global Temperature 2.2 Scientific Notation: Writing Large and Small Numbers 2.3 Significant Figures: Writing Numbers to Reflect Precision 2.4 Significant Figures in Calculations 2.5 The Basic Units of Measure 2.6 Converting One Unit to Another 2.7 Solving Multistep Conversions 2.8 Units Raised to a Power 2.9 Density 2.10 Numerical Problem Solving Strategies and the Solution Map 2

3 2.1 Measuring Global Temperature 3

4 Scientists have Measured the Average Global Temperature Rise over the Past Century to be 0.6 C 0.6 a number, quantitative: between 0.5 and 0.7 C. C unit of a standardized scale C - indicates a number between 0.59 and 0.61 C Number + unit = a measurement 4

5 -Measurement and Problem Solving Topics in this Measurements Precision Basic Units Chapter. Converting Measurements Derived Units 5

6 Scientific Notation A way of writing large and small numbers. A small number : Radius of a hydrogen nucleus is about m (13 zeros between decimal and 5) A large number: Distance to nearest star other than the sun is 24,500,000,000,000 miles 24.5 trillion miles ( zeroes) Tro's Introductory Chemistry, Chapter 2 6

7 Scientific Notation Expresses a number as a number between 1 and 10 multiplied by a power of 10. Example: 3.89 x 10 5 = 389,000 More compact way of writing large or small numbers Easier to compare numbers Easier to determine precision 7

8 Exponents or Powers of Ten Positive exponent = larger number Negative exponent = smaller number Write out powers 10 = /10 = =10 2 1/100 = ,000= /1,000 = ,000 = /10,000 = =

9 2.2 Scientific Notation: Writing Large and Small Numbers 9

10 Scientific Notation Parts of a number in scientific notation Figure from the center of page 12 showing the parts Of a number written in scientific notation 1.35 x 10-8 exponent exponential part (10) decimal point of number between 1 and 10 10

11 Writing Numbers in Scientific Notation for Numbers Greater than Move the decimal to give a number between 1 and 10. Only one digit to the left of the decimal. 2. Decimal from right to left, the number is now smaller than the original number so you must multiply by a number greater than 1. This means increase in the exponent. 3. Example 34,780 move decimal 4 places to left ( ) 4. 34,780 = x 10,000 = x Decimal to left, increase exponent, the number of places you moved the decimal point. 11

12 Our Examples (1) A large number: Distance to nearest star other than the sun is 24,500,000,000,000 miles 24.5 trillion miles ( zeroes) To get a number between 1 and 10 the decimal is moved -- (left) by 13 places. 2,45 is smaller than the actual number so we need a positive exponent x miles 12

13 Writing Numbers in Scientific Notation for Numbers Less than Move the decimal to give a number between 1 and 10. Only one digit will be to the left of the decimal. 2. Decimal from left to right, the number is now larger than the original number so you must multiply by a number less than 1. This means decrease in the exponent. 3. Example move decimal 3 places to right ( ) = 6.7 x 1 / 1,000 = 6.7 x Decimal to right, decrease exponent, the number of places you moved the decimal point. 13

14 Our Examples (2) A small number : Radius of a hydrogen nucleus is about m (13 zeros between decimal and 5) To get a number between 1 and 10 the decimal is moved -- (right) by 14 places. 5. is bigger than the actual number so we need a negative exponent. 5. x m 14

15 Writing a Number in Scientific Notation, Other Examples Continued The U.S. population in 2007 was estimated to be 301,786,000 people. Express this number in scientific notation. 301,786,000 people = x 10 8 people 15

16 Example with a Number that Already has a Power of Ten x 10 5 Move decimal This is left ( ) 2 places, so exponent increase by 2 (5 +2 = 7) x

17 Inputting Scientific Notation into a Calculator Learn how your calculator handles scientific notation. Usually through a key marked EXP 17

18 Reporting Measurements Measurements depend upon the equipment being used. All the digits except the last are certain. The last is an estimate cm 24.7? cm? Is a guess 24.72cm 3 certain one digits + guess = four significant digits Range is to cm 18

19 2.3 Significant Figures: Writing Numbers to Reflect Precision 19

20 Significant Figures Writing numbers to reflect precision. Why do we need significant figures? All measures have some degree of uncertainty. 20

21 Precision Increased by Dividing? Consider traveling 237 miles in 3.7 hours. 237 mi 3.7 h = mi/h 21

22 Significant Figures in a Single Measurement Nonzero numbers are significant (23.89 cm) Zeros Interior zeros always significant ( in) Trailing zeros after a decimal always significant ( L) Leading zeros never significant ( m) Trailing zeros before a decimal ambiguous, avoid by using scientific notation. 22

23 Exact Numbers have Unlimited Significant figures Counted numbers (3 trials of an experiment) Defined numbers (1 in = 2.54 cm) Numbers as part of a formula (2pr = C) 23

24 Examples of Determining the Number of Significant Figures in a Number 24

25 2.4 Significant Figures in Calculations 25

26 Multiplication and Division with Significant Figures The overall answer is limited by the measurement with the fewest number of significant figures , = = 45 3 sig. figs. 5 sig. figs. 2 sig. figs. 2 sig. figs = = sig. figs. 3 sig. figs. 3 sig. figs. 26

27 Rounding Find the number that is the last significant figure. Look at the number to the right. 1. For 0 to 4, keep the last significant figure the same to four significant figures For 5 to 9, add 1 the last significant figure the same to four significant figures

28 Examples of Multiplication and Division 28

29 Addition and Subtraction with Significant Figures The overall answer is limited by the measurement with the fewest number of decimal places decimal places decimal places decimal places decimal places rounds to

30 Both Multiplication/Division and Addition/Subtraction with Significant Figures. Follow the standard order of operations. Please Excuse My Dear Aunt Sally. n - Keep track of number of significant figures at each step, but round at the end. Example next page. 30

31 Example 1.6 Perform the Following Calculations to the Correct Number of Significant Figures a) b) For addition and subtraction, it is useful to draw a line To help determine the correct decimal place

32 Addition or Subtraction and Multiplication and Division Together Do the steps in parentheses and apply the rules to determine the Correct number of significant figures or decimal p[laces in the Intermediate answer. Keep track of significant figures and only round at the last step. 32

33 Both Multiplication/Division and Addition/Subtraction with Significant Figures (cont.) ( ) 2 right 3 right decimal dcecimal (1.614) 2 right decimal (by subt. rules) (1.614) = four three three sig. fig sig. fig. sig. fig (by multip. Rules) 33

34 Example 1.6 Perform the Following Calculations to the Correct Number of Significant Figures, Continued c) d) 34

35 2.5 The Basic Units of Measure 35

36 Measurement = Number + Unit Units - agreed upon standardized quantities Scientists use the SI system which is a form of the metric system SI = Système International Base units are the units for fundamental quantities 36

37 Base Units in the SI System Quantity Unit Symbol Length meter m Mass kilogram kg Time second s Temperature kelvin K 37

38 Length Measure of a 1-D distance SI unit = meter = in, 3.37 in longer than a yard Commonly used centimeters (cm) = 1/100 of a meter. 1 inch = 2.54 cm (exactly, by definition) A figure showing a meterstick slightly longer than a yardstick 38

39 Mass Measure of the amount of matter present in an object. SI unit = kilogram (kg) = 2.2 lb A replica of the standard mass is shown Commonly measure in the lab mass in grams (g) or milligrams (mg). 1 kg = 1,000 g = 1,000,000 mg 1 oz = g 1 lb = g A nickel (5 cent piece) weighs about 5 g Standard Mass Figure 2.6 Top loading Balance Page 23 39

40 Time Measure of the duration of an event. SI units = second (s) 1 s is as a certain number of repeats of a Cs atom in a so called atomic clock. 40

41 Temperature Measure of the average energy of motion of molecules in a sample. Is a result of molecular motions Faster molecular motions = higher temperature Heat flows from warmer objects to cooler objects through molecular collisions until they reach the same temperature. Unit is Kelvin (K). Water boils at K, and freezes at K Three figures Student in 22 C Room temp. Ice at 0 C Death Valley at 40 C 41

42 Nonbase Units in the SI System Units in SI: Prefix (a multiplier) + base unit Example centimeter = centi + meter = centimeter 0.01 x meter = 0.01 meter 42

43 Common Prefixes in the SI System (see the stuff to memorize on the web page) Prefix Symbol Decimal Equivalent Power of 10 mega- M 1,000,000 Base x 10 6 kilo- k 1,000 Base x 10 3 deci- d 0.1 Base x 10-1 centi- c 0.01 Base x 10-2 milli- m Base x 10-3 micro- m or mc Base x 10-6 nano- n Base x

44 Prefixes Indicate What Power of 10 the Base Unit has been Multipied by Kilometer = kilo (1000) + meter = 1000 x meter (km) Nanosecond = nano (10-9 ) + second = 10-9 x second (ns) Milligram = milli (10-3 ) + gram = 10-3 x gram (mg) If the multiplier is greater than 1 the new unit will be larger than the base unit. If the multiplier is less than 1 the new unit will be smaller than the base unit. 44

45 Volume, an Example of a Derived Unit Derived unit. A combination of the base units. This may be a multiplication, division, addition, subtraction or other mathimatical combination Examples Area = length 2 Volume = length 3 SI system = Area = square meter (m 2 ) Volume = cubic meter (m 3 ) A cube 10 cm On a side which Is the volume = 1 Liter Graduated Cylinder 45

46 More Convenient Volume Units Liter (L) is defined as a cubic decimeter (dm 3 ). A dm = 10 cm so a Liter is a cube with 10 cm sides. 10 cm x 10 cm x 10 cm = 1000 cm 3 So a Liter also equals 1000 cm 3 Milliliter = 1/1000 L = 1 cm 3 REMEMBER: 1 ml = 1 cm 3 Solids usually use cm 3 and liquids usually use ml 46

47 Table 2.3 (page 24) Common Units and Their Equivalents Length 1 kilometer (km) = mile (mi) 1 meter (m) = inches (in.) 1 meter (m) = yards (yd) 1 foot (ft) = centimeters (cm) 1 inch (in.) = 2.54 centimeters (cm) exactly 47

48 Table 2.3 (cont.) Common Units and Mass Their Equivalents, Continued 1 kilogram (km) = pounds (lb) 1 pound (lb) = grams (g) 1 ounce (oz) = (g) Volume 1 liter (L) = 1000 milliliters (ml) 1 liter (L) = 1000 cubic centimeters (cm 3 ) 1 liter (L) = quarts (qt) 1 U.S. gallon (gal) = liters (L) 48

49 2.6 Converting One Unit to Another Metric metric (easy, a power of 10) Metric English (a little harder, a conversion factor) Arithmetic operations apply to numbers and units Keep track of units by dimensional analysis 49

50 Dimensional Analysis Using units to help solve problems is called dimensional analysis Units may be operated on mathematically like numbers 50

51 SI to SI Conversions This involves a change in the unit that differs by a power of 10. The number will change by moving the decimal. A way to guide you through the problem is a plan that your text calls solution map. A solution map lists the steps one mus go through to solve the problem 51

52 Example SI to SI How many mm are there in 34.9 m? Solution map: m mm How do we make the change? Using a conversion factor 52

53 Conversion Factor A conversion factor is derived from an equation. It is a fraction = 1 Multiplying by a conversion factor does not change a quantities size, but does change the unit used to measure it. Example from the table 1 mm = m 1mm m m 1mm 53

54 Conversion Factors (cont) Notice the equation gives two conversion factors. The one to choose is determined by which unit is supposed top cancel out Back to our problem 34.9 m is how many mm? 54

55 Conversions Convert an Old Unit to a New Unit new unit old unit = new unit old unit In the solution map Solution map: m mm m = old unit and mm = new unit 55

56 Carrying out the Solution 34.9 m 1 mm = 34,900 mm m 3.49 x 10 4 mm Writing in scientific notation to give 3 sig. fig. 56

57 Problem to Work in Class How many kg are there in 4,392 mm? Solution map: mm m km Could be done in one step, but it is easy to convert to the base unit. 57

58 SI to Other Systems of Measure A piece of notebook paper is 8.50 in wide, what is that length in cm? Solution map: in cm Need a conversion factor from the definition 1 in = 2.54 cm 1in or 2.54 cm 2.54 cm 1in 58

59 Solution 8.50 in 2.54 cm = cm 1 in 21.6 cm ( 3 sig. fig. ) 59

60 2.7 Solving Multistep Conversions 60

61 Some Conversions Require Two or More Conversion Factors Given that 2 Tablespoons (tbsp) = 1 oz, 8 oz = 1 cup, 4 cup = 1 qt, and 1 L = qt How many ml are there in 1.00 tbsp? Solution map: tbsp oz cup qt L ml Each step represents a conversion. 5 steps. (Solution next slide) 61

62 1.00 tbsp 1 oz = oz 2 tbsp oz 1 cup = cup 8 oz cup 1 qt = qt 4 in qt 1 L = L in L 1000 ml = ml 1 L 14.2 ml 62

63 2.8 Units Raised to a Power 63

64 Units Raised to a Power An injection of a drug has a volume of 4.52 cm 3. What is the volume in cubic inches (in 3 )? Solution map: cm 3 in cm 3 1 in = 1.78 in 3 (NO!!!!) 2.54 cm What does cm 3 really mean 64

65 Units Raised to a Power 4.52 cm 3 = 4.53 cm cm cm For squared or cubed units, one must also square or cube the entire conversion factor 4.52 cm 3 1 in 1 in 1 in = 2.54 cm 2.54 cm 2.54 cm or cm 3 1 in = 4.52 cm in 3 = 2.54cm (2.54) 3 cm 3 65

66 Units Raised to a Power 4.52 cm 3 1 in 3 = in cm 3 66

67 2.9 Density 67

68 Mass and Volume Two main characteristics of matter. Mass or volume alone cannot be used to identify what type of matter something is. If you are given a large glass containing 100 g of a clear, colorless liquid and a small glass containing 25 g of a clear, colorless liquid, are both liquids the same stuff? Even though mass and volume are individual properties, for a given type of matter they are related to each other! 68

69 Mass vs. Volume of Brass Mass grams Volume cm 3 69

70 This slide was a graph of the mass as y and the volume as x resulting in a straight line this indicates that the ratio of mass to volume is a constant = the density of a substance For these brass samples d = 8.3 g/ml 70

71 Density Ratio of mass:volume. Its value depends on the kind of material, not the amount. Solids = g/cm 3 1 cm 3 = 1 ml Liquids = g/ml Gases = g/l Density Mass Volume Volume of a solid can be determined by water displacement Archimedes Principle. Explain Density : solids > liquids > gases Except ice is less dense than liquid water! 71

72 For a Table of Sample Densities see Table 2.4 on page 33 72

73 Volume by Water Displacement Graduated cylinder water is added = V initial A solid object is added to the cylinder. Volume of water rises = V final V object = V final - V initial 73

74 Using Density in Calculations Three Possible Formulas Density Volume Mass Volume Mass Density Solution Maps: m, V D m, D V Mass Density Volume V, D m 74

75 Density Problems Think of density as a conversion factor between mass and volume of a given substance. Problems from text for class 2.96, 2.100, 75

76 Problem 2.96 An aluminum engine block has a volume of 4.77 L and a mass of kg. What is the density of the aluminum in grams per cubic centimeter? Solution map Volume, L Volume, ml=cm 3 mass Density = Mass, kg mass, g volume 76

77 Problem 2.96 (cont.) Volume to cm 3 : 4.77 L 1000 ml = 4,770 ml = 4,770 cm3 1 L Mass to g: kg 1000 g = 12,880 g 1 kg Density using the formula: 12,880 g = g / cm g / cm3 4,770 cm3 3 sig. fig. 77

78 Problem Acetone (fingernail polish remover) has a density of g / cm 3. (a) What is the mass in grams of ml of acetone? (b) What is the volume in milliliters of 6.54 g of acetone? There are two basic ways to work this problem. 1)Plug the variables into the formula for density and solve for the unknown. Or 2) Treat the density as a conversion factor between mass and volume of a substance. We will use # 2 Solution maps (a) ml g (b) g ml 78

79 Problem (cont.) A density of g / cm 3 means that 1 cm 3 or ml of the substance Has a mass of g. (i.e. 1 ml = g of acetone) From this equation we can derive conversion factors. (a) The conversion is from ml to g so we want ml to cancel ml g = g (4 sig. fig.) 1 ml (a) The conversion is from g to ml so we want g to cancel 6.54 g 1 ml = 8.32 g (3 sig. fig.) g 79

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