Chapter 1 Measurements 1.1 Units of Measurement Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Measurement in Chemistry In chemistry we: measure quantities. do experiments. calculate results. use numbers to report measurements. compare results to standards.

Stating a Measurement In every measurement, a number is followed by a unit. Observe the following examples of measurements: Number and Unit 35 m 0.25 L 225 lb 3.4 hr

The Metric System (SI) Metric units or the International System of Units (abbreviated SI units, from the French System International d’Unites) a decimal system based on 10. used in most of the world. used everywhere by scientists.

Units in the Metric System In the metric and SI systems, one unit is used for each type of measurement: Measurement Metric SI Length meter (m) meter (m) Volume liter (L) cubic meter (m3) Mass gram (g) kilogram (kg) Time second (s) second (s) Temperature Celsius (C) Kelvin (K)

Length Length is measured using a meter stick. uses the unit of meter (m) in both the metric and SI systems. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Inches and Centimeters The unit of an inch is equal to exactly 2.54 centimeters in the metric (SI) system. 1 in. = 2.54 cm Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Volume Volume is the space occupied by a substance. uses the unit liter (L) or milliliter (mL). 1 L = 1000 mL is m3(cubic meter) in the SI system. is measured using a graduated cylinder. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Mass The mass of an object is the quantity of material it contains. is measured on a balance. uses the unit gram (g) in the metric system. uses the unit kilogram (kg) in the SI system. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Temperature Measurement The temperature of a substance indicates how hot or cold it is. is measured on the Celsius (C) scale in the metric system. on this thermometer is 18ºC or 64ºF. in the SI system uses the Kelvin (K) scale. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Time Measurement Time measurement uses the unit second(s) in both the metric and SI systems. is based on an atomic clock that uses a frequency emitted by cesium atoms. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Scientific Notation Scientific notation is used to write very large or very small numbers. for the width of a human hair of 0.000 008 m is written 8 x 10-6 m. of a large number such as 4 500 000 s is written 4.5 x 106 s. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Writing Numbers in Scientific Notation A number in scientific notation contains a coefficient and a power of 10. coefficient power of ten coefficient power of ten 1.5 x 102 7.35 x 10-4 To write a number in scientific notation, the decimal point is moved after the first digit. The spaces moved are shown as a power of ten. 52 000. = 5.2 x 10 4 0.00378 = 3.78 x 10-3 4 spaces left 3 spaces right

Some Powers of Ten

Comparing Numbers in Standard and Scientific Notation Here are some numbers written in standard format and in scientific notation. Number in Number in Standard Format Scientific Notation Diameter of the Earth 12 800 000 m 1.28 x 107 m Mass of a human 68 kg 6.8 x 101 kg Length of a pox virus 0.000 03 cm 3 x 10-5 cm

Practice Problems

Learning Check 1 For each of the following, indicate whether the unit describes length, mass, or volume and circle the units ____ A. A bag of tomatoes is 4.6 kg. ____ B. A person is 2.0 m tall. ____ C. A medication contains 0.50 g aspirin. ____ D. A bottle contains 1.5 L of water.

Learning Check 2 Identify the measurement that has an SI unit. A. John’s height is 1) 1.5 yd. 2) 6 ft . 3) 2.1 m. B. The race was won in 1) 19.6 s. 2) 14.2 min. 3) 3.5 hr. C. The mass of a lemon is 1) 12 oz. 2) 0.145 kg. 3) 0.6 lb. D. The temperature is 1) 85C. 2) 255 K. 3) 45F.

Solution 2 A. John’s height is 3) 2.1 m. B. The race was won in C. The mass of a lemon is 2) 0.145 kg. D. The temperature is 2) 255 K.

Learning Check 3 Select the correct scientific notation for each. 1) 8 x 106 2) 8 x 10-6 3) 0.8 x 10-5 B. 72 000 1) 7.2 x 104 2) 72 x 103 3) 7.2 x 10-4

Solution 3 Select the correct scientific notation for each. 2) 8 x 10-6 B. 72 000 1) 7.2 x 104

Learning Check 4 Write each as a standard number. A. 2.0 x 10-2 1) 200 2) 0.0020 3) 0.020 B. 1.8 x 105 1) 180 000 2) 0.000 018 3) 18 000

Solution 4 Write each as a standard number. A. 2.0 x 10-2 3) 0.020 3) 0.020 B. 1.8 x 105 1) 180 000

Measured Numbers & Significant Figures

Measured Numbers A measuring tool is used to determine a quantity such as height or the mass of an object. provides numbers for a measurement called measured numbers.

Reading a Meter Stick . l2. . . . l . . . . l3 . . . . l . . . . l4. . cm The markings on the meter stick at the end of the orange line are read as the first digit 2 plus the second digit 2.7 The last digit is obtained by estimating. The end of the line might be estimated between 2.7–2.8 as half-way (0.5) or a little more (0.6), which gives a reported length of 2.75 cm or 2.76 cm.

Known & Estimated Digits In the length reported as 2.76 cm, The digits 2 and 7 are certain (known). The final digit 6 was estimated (uncertain). All three digits (2.76) are significant including the estimated digit.

Zero as a Measured Number . l3. . . . l . . . . l4. . . . l . . . . l5. . cm For this measurement, the first and second known digits are 4.5. Because the line ends on a mark, the estimated digit in the hundredths place is 0. This measurement is reported as 4.50 cm.

Significant Figures in Measured Numbers obtained from a measurement include all of the known digits plus the estimated digit. reported in a measurement depend on the measuring tool.

Significant Figures

Counting Significant Figures All non-zero numbers in a measured number are significant. Number of Measurement Significant Figures 38.15 cm 4 5.6 ft 2 65.6 lb 3 122.55 m 5

Sandwiched Zeros Sandwiched zeros occur between nonzero numbers. are significant. Number of Measurement Significant Figures 50.8 mm 3 2001 min 4 0.0702 lb 3 0.40505 m 5

Trailing Zeros Trailing zeros follow non-zero numbers in numbers without decimal points. are usually place holders. are not significant. Number of Measurement Significant Figures 25 000 cm 2 200 kg 1 48 600 mL 3 25 005 000 g 5

Leading Zeros Leading zeros precede non-zero digits in a decimal number. are not significant. Number of Measurement Significant Figures 0.008 mm 1 0.0156 oz 3 0.0042 lb 2 0.000262 mL 3

Significant Figures in Scientific Notation In scientific notation all digits including zeros in the coefficient are significant. Number of Measurement Significant Figures 8 x 104 m 1 8.0 x 104 m 2 8.00 x 104 m 3

Examples of Exact Numbers An exact number is obtained when objects are counted. Counted objects 2 soccer balls 4 pizzas from numbers in a defined relationship. Defined relationships 1 foot = 12 inches 1 meter = 100 cm

Exact Numbers

Scientific Notation The number of atoms in 12 g of carbon: 602,200,000,000,000,000,000,000 6.022 x 1023 The mass of a single carbon atom in grams: 0.0000000000000000000000199 1.99 x 10-23 N x 10n N is a number between 1 and 10 n is a positive or negative integer 1.8

Addition & Subtraction w/ Scientific Notation 568.762 0.00000772 move decimal right move decimal left n > 0 n < 0 568.762 = 5.68762 x 102 0.00000772 = 7.72 x 10-6 Addition or Subtraction Write each quantity with the same exponent n Combine N1 and N2 The exponent, n, remains the same 4.31 x 104 + 3.9 x 103 = 4.31 x 104 + 0.39 x 104 = 4.70 x 104 1.8

Multiplication & Division w/ Scientific Notation (4.0 x 10-5) x (7.0 x 103) = (4.0 x 7.0) x (10-5+3) = 28 x 10-2 = 2.8 x 10-1 Multiply N1 and N2 Add exponents n1 and n2 Division 8.5 x 104 ÷ 5.0 x 109 = (8.5 ÷ 5.0) x 104-9 = 1.7 x 10-5 Divide N1 and N2 Subtract exponents n1 and n2 1.8

Practice Problems 1 . l8. . . . l . . . . l9. . . . l . . . . l10. . cm What is the length of the orange line? 1) 9.0 cm 2) 9.03 cm 3) 9.04 cm

Practice Problems 2 State the number of significant figures in each of the following measurements. A. 0.030 m B. 4.050 L C. 0.0008 g D. 2.80 m

Practice Problems 4 Classify each of the following as (1) exact or (2) measured numbers. A.__Gold melts at 1064°C. B.__1 yard = 3 feet C.__The diameter of a red blood cell is 6 x 10-4 cm. D.__There are 6 hats on the shelf. E.__A can of soda contains 355 mL of soda.

Practice Problems 5 A. Which answer(s) contain 3 significant figures? 1) 0.4760 2) 0.00476 3) 4.76 x 103 B. All the zeros are significant in 1) 0.00307 2) 25.300 3) 2.050 x 103 C. The number of significant figures in 5.80 x 102 is 1) one 3) two 3) three

Practice Problems 6 In which set(s) do both numbers contain the same number of significant figures? 1) 22.0 and 22.00 2) 400.0 and 40 3) 0.000015 and 150 000

Significant Figures in Calculations

Rounding Off Calculated Answers In calculations, answers must have the same number of significant figures as the measured numbers. often, a calculator answer must be rounded off. rounding rules are used to obtain the correct number of significant figures. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Rounding Off Calculated Answers When the first digit dropped is 4 or less, the retained numbers remain the same. 45.832 rounded to 3 significant figures drops the digits 32 = 45.8 When the first digit dropped is 5 or greater, the last retained digit is increased by 1. 2.4884 rounded to 2 significant figures drops the digits 884 = 2.5 (increase by 0.1)

Adding Significant Zeros Sometimes a calculated answer requires more significant digits. Then, one or more zeros are added. Calculated Zeros Added to Answer Give 3 Significant Figures 4 4.00 1.5 1.50 0.2 0.200 12 12.0

Calculations with Measured Numbers In calculations with measured numbers, significant figures or decimal places are counted to determine the number of figures in the final answer. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Multiplication and Division When multiplying or dividing use the same number of significant figures as the measurement with the fewest significant figures. rounding rules to obtain the correct number of significant figures. Example: 110.5 x 0.048 = 5.304 = 5.3 (rounded) 4 SF 2 SF calculator 2 SF

Addition and Subtraction When adding or subtracting use the same number of decimal places as the measurement with the fewest decimal places. rounding rules to adjust the number of digits in the answer. 25.2 one decimal place + 1.34 two decimal places 26.54 calculated answer 26.5 answer with one decimal place

Practice Problems 1 For each calculation, round the answer to give the correct number of significant figures. A. 235.05 + 19.6 + 2 = 1) 257 2) 256.7 3) 256.65 B. 58.925 - 18.2 = 1) 40.725 2) 40.73 3) 40.7

Practice Problems 2 Adjust the following calculated answers to give answers with three significant figures. A. 824.75 cm B. 0.112486 g C. 8.2 L

Practice Problems 3 Give an answer for the following with the correct number of significant figures. A. 2.19 x 4.2 = 1) 9 2) 9.2 3) 9.198 B. 4.311 ÷ 0.07 = 1) 61.59 2) 62 3) 60 C. 2.54 x 0.0028 = 0.0105 x 0.060 1) 11.3 2) 11 3) 0.041

Solution 3 A. 2.19 x 4.2 = 2) 9.2 B. 4.311 ÷ 0.07 = 3) 60 C. 2.54 x 0.0028 = 2) 11 0.0105 x 0.060 On a calculator, enter each number followed by the operation key. 2.54 x 0.0028  0.0105  0.060 = 11.28888889 = 11 (rounded)

Prefixes & Equalities

Prefixes A prefix in front of a unit increases or decreases the size of that unit. prefix = value 1 kilometer = 1000 meters 1 kilogram = 1000 grams

Metric and SI Prefixes

Metric Equalities An equality states the same measurement in two different units. can be written using the relationships between two metric units. Example: 1 meter is the same as 100 cm and 1000 mm. 1 m = 100 cm 1 m = 1000 mm

Measuring Length Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Measuring Volume Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Measuring Mass Several equalities can be written for mass in the metric (SI) system 1 kg = 1000 g 1 g = 1000 mg 1 mg = 0.001 g 1 mg = 1000 µg Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Writing Conversion Factors

Equalities Equalities use two different units to describe the same measured amount. are written for relationships between units of the metric system, U.S. units, or between metric and U.S. units. For example, 1 m = 1000 mm 1 lb = 16 oz 2.205 lb = 1 kg

Exact and Measured Numbers in Equalities Equalities between units of the same system are definitions and use exact numbers. different systems (metric and U.S.) use measured numbers and count as significant figures.

Some Common Equalities

Conversion Factors A conversion factor is a fraction obtained from an equality. Equality: 1 in. = 2.54 cm is written as a ratio with a numerator and denominator. can be inverted to give two conversion factors for every equality. 1 in. and 2.54 cm 2.54 cm 1 in.

Conversion Factors in a Problem A conversion factor may be obtained from information in a word problem. is written for that problem only. Example 1: The cost of one gallon (1 gal) of gas is $2.34. 1 gallon of gas and $2.34 $2.34 1 gallon of gas

Percent as a Conversion Factor A percent factor gives the ratio of the parts to the whole. % = Parts x 100 Whole uses the same unit to express the percent. uses the value 100 and a unit for the whole. Example: A food contains 30% (by mass) fat. 30 g fat and 100 g food 100 g food 30 g fat

Percent Factor in a Problem The thickness of the skin fold at the waist indicates 11% body fat. What percent factors can be written for body fat in kg? Percent factors using kg: 11 kg fat and 100 kg mass 100 kg mass 11 kg fat Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Practice Problems 1 Write conversion factors for each pair of units. A. liters and mL B. hours and minutes C. meters and kilometers

Practice Problems 2 Write the equality and conversion factors for each of the following. A. square meters and square centimeters B. jewelry that contains 18% gold C. One gallon of gas is $2.27

Problem Solving

Practice Problem 1 If your pace on a treadmill is 65 meters per minute, how many minutes will it take for you to walk a distance of 7500 feet?

Practice Problem 2 How many lb of sugar are in 120 g of candy if the candy is 25% (by mass) sugar?

Density Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Density Density Compares the mass of an object to its volume. Is the mass of a substance divided by its volume. Density expression Density = mass = g or g = g/cm3 volume mL cm3 Note: 1 mL = 1 cm3

Densities of Common Substances

Volume by Displacement A solid completely submerged in water displaces its own volume of water. The volume of the solid is calculated from the volume difference. 45.0 mL - 35.5 mL = 9.5 mL = 9.5 cm3 Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Density Using Volume Displacement The density of the zinc object is then calculated from its mass and volume. mass = 68.60 g = 7.2 g/cm3 volume 9.5 cm3 Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Sink or Float Ice floats in water because the density of ice is less than the density of water. Aluminum sinks because its density is greater than the density of water. Copyright © 2005 by Pearson Education, Inc. Publishing as Benjamin Cummings

Density as a Conversion Factor Density can be written as an equality. For a substance with a density of 3.8 g/mL, the equality is 3.8 g = 1 mL From this equality, two conversion factors can be written for density. Conversion 3.8 g and 1 mL factors 1 mL 3.8 g

Practice Problems 1 Osmium is a very dense metal. What is its density in g/cm3 if 50.0 g of osmium has a volume of 2.22 cm3?

Practice Problems 2 What is the density (g/cm3) of 48.0 g of a metal if the level of water in a graduated cylinder rises from 25.0 mL to 33.0 mL after the metal is added? 33.0 mL 25.0 mL object

Practice Problems 3 Which diagram correctly represents the liquid layers in the cylinder? Karo (K) syrup (1.4 g/mL), vegetable (V) oil (0.91 g/mL,) water (W) (1.0 g/mL) 1 2 3 K W V V W K V W K

Practice Problems 4 The density of octane, a component of gasoline, is 0.702 g/mL. What is the mass, in kg, of 875 mL of octane?

Practice Problems 5 If olive oil has a density of 0.92 g/mL, how many liters of olive oil are in 285 g of olive oil?

Practice Problems 6 A group of students collected 125 empty aluminum cans to take to the recycling center. If 21 cans make 1.0 lb aluminum, how many liters of aluminum (D=2.70 g/cm3) are obtained from the cans?

Practice Problems 7 1 2 3 25 g of aluminum 2.70 g/mL 45 g of gold Which of the following samples of metals will displace the greatest volume of water? 1 2 3 25 g of aluminum 2.70 g/mL 45 g of gold 19.3 g/mL 75 g of lead 11.3 g/mL