Chapter 1 Measurements1.1

Units of Measurement

1

Measurement

You make a measurementevery time you

• measure your height.

• read your watch.

• take your temperature.

• weigh a cantaloupe.

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Measurement in Chemistry

In chemistry we

• measure quantities.

• do experiments.

• calculate results.

• use numbers to report measurements.

• compare results to standards.

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Measurement

In a measurement

• a measuring tool is used to compare some dimension of an object to a standard.

• of the thickness of the skin fold at the waist, calipers are used.

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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 kg

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The Metric System (SI)The metric system or SI (international

system) is

• a decimal system based on 10.

• used in most of the world.

• used everywhere by scientists.

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Units in the Metric SystemIn 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)

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Length Measurement

Length

• is measured using a meterstick.

• uses the unit of meter (m) in both the metric and SI systems.

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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

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Volume Measurement

Volume • is the space occupied by

a substance.

• uses the unit liter (L) in the metric system.

• 1 qt = 946 mL

• uses the unit m3 (cubic meter) in the SI system.

• is measured using a graduated cylinder.

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Mass MeasurementThe 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.

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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 18 ºC or 64 ºF.

• in the SI system uses the Kelvin (K) scale.

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Time Measurement

Time measurement

• uses the unit second (s) in both the metric and SI systems.

• is based on an atomic clock that uses the frequency of cesium atoms.

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Summary of Units of Measurement

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Learning Check

For each of the following, indicate whether the unit describes 1) length, 2) mass, or 3) volume.

____ 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.

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Learning Check

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 in1) 19.6 s. 2) 14.2 min. 3) 3.5 h.

C. The mass of a lemon is1) 12 oz. 2) 0.145 kg. 3) 0.6 lb.

D. The temperature is1) 85 C. 2) 255 K. 3) 45 F.

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Measurements

1.2Scientific Notation

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.

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Numbers in Scientific NotationA number written in scientific notation contains

a• coefficient.• power of 10.

Examples:

coefficient power of ten coefficient power of ten

1.5 x 102 7.35 x 10-4

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Writing Numbers in Scientific Notation

To write a number in scientific notation,

• move the decimal point to give a number 1-9.

• show the spaces moved as a power of 10.

Examples: 52 000. = 5.2 x 10

4 0.00178 = 1.78 x 10-3

4 spaces left 3 spaces right

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Some Powers of 10

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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 Earth12 800 000 m 1.28 x 107 m

Mass of a typical human68 kg 6.8 x 101 kg

Length of a pox virus0.000 03 cm 3 x 10-5 cm

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Study Tip: Scientific Notation

In a number 10 or larger, the decimal point • is moved to the left to give a positive power of 10

In a number less than 1, the decimal point • is moved to the right to give a negative power of

10

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Learning Check

Select the correct scientific notation for each.

A. 0.000 008 m1) 8 x 106 m, 2) 8 x 10-6 m, 3) 0.8 x 10-5 m

B. 72 000 g1) 7.2 x 104 g, 2) 72 x 103 g, 3) 7.2 x 10-4 g

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Learning Check

Write each as a standard number.

A. 2.0 x 10-2 L 1) 200 L, 2) 0.0020 L, 3) 0.020 L

B. 1.8 x 105 g 1) 180 000 g, 2) 0.000 018 g, 3) 18 000 g

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Chapter 1 Measurements1.3

Measured Numbers and Significant Figures

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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.

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Reading a Meterstick

. l2. . . . l . . . . l3 . . . . l . . . . l4. . cm

• The markings on the meterstick at the end of the orange line are read as:

the first digit 3

plus the second digit 3.3

• The last digit is obtained by estimating.

• The end of the line may be estimated between 3.2–3.23 as half way (0.5) or a little more (0.6), which gives a reported length of 3.25 cm or 2.26 cm.

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Known & Estimated DigitsIf the length is reported as 3.26 cm,

• the digits 3 and 2 are certain (known).• the final digit, 6, is estimated (uncertain).• all three digits (2, 7, and 6) are significant,

including the estimated digit.

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Learning Check

. l8. . . . l . . . . l9. . . . l . . . . l10. . cm

What is the length of the blue line?

1) 9.2 cm

2) 9.13 cm

3) 9.19 cm

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Zero as a Measured Number

. l3. . . . l . . . . l4. . . . l . . . . l5. . cm

• For this measurement, the first and second known digits are 4 and 5.

• When a measurement ends on a mark, the estimated digit in the hundredths place is 0.

• This measurement is reported as 4.50 cm.

• What is the measurement of the blue line?31

Significant Figures in Measured NumbersSignificant Figures

• obtained from a measurement include all of the known digits plus the estimated digit.

• reported in a measurement depend on the measuring tool.

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Significant Figures

33

Counting Significant FiguresAll nonzero numbers in a measured number are significant.

Number of Measurement Significant Figures

38.15 cm 45.6 ft 265.6 lb 3122.55 m 5

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Sandwiched Zeros

Sandwiched Zeros• occur between nonzero numbers.• are significant.

Number of Measurement Significant Figures50.8 mm 32001 min 40.0702 lb 30.405 05 m 5

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Trailing ZerosTrailing Zeros• follow nonzero numbers in numbers without

decimal points.• are usually placeholders. • are not significant.

Number of Measurement Significant Figures25 000 cm 2

200 kg 1 48 600 mL 3

25 005 000 g 5

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Leading ZerosLeading Zeros • precede nonzero digits in a decimal number. • are not significant.

Number of Measurement Significant Figures0.008 mm 10.0156 oz 30.0042 lb 20.000 262 mL 3

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Learning Check

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

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Significant Figures in Scientific Notation In scientific notation all digits in the coefficient including zeros are significant.

Number of Measurement Significant Figures

8 x 104 m 18.0 x 104 m 28.00 x 104 m 3

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Study Tip: Significant Figures

The significant figures in a measured number are• all the nonzero numbers.

12.56m 4 significant figures• zeros between nonzero numbers.

4.05 g 3 significant figures

• zeros that follow nonzero numbers in a decimal number.

25.800 L 5 significant figures

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Learning Check

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 (1). 2) two (2). 3) three (3).

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Learning Check

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) 0.000 015 and 150 000

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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 inches1 meter = 100 cm

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Exact Numbers

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Learning Check

A. Exact numbers are obtained by 1. using a measuring tool.

2. counting.3. definition.

B. Measured numbers are obtained by 1. using a measuring tool.

2. counting.3. definition.

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Learning Check

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.

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Chapter 1 Measurements 1.4

Significant Figures in Calculations

47

Rounding Off Calculated Answers

In calculations,

• answers must have the same number of significant figures as the measured numbers.

• a calculator answer often must be rounded off.

• rounding rules are used to obtain the correct number of significant figures.

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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 figuresdrops the digits 884 = 2.5 (increase by

0.1)

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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 4.001.5 1.500.2 0.200

12 12.0

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Learning Check

Round off or add zeros to the following calculated

answers to give three significant figures.

A. 824.75 cm

B. 0.112486 g

C. 8.2 L

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Calculations with Measured Numbers

In calculations with measured numbers, significant figures ordecimal places arecounted to determinethe number of figures inthe final answer.

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Multiplication and Division

When multiplying or dividing

• the final answer must have the same number of significant figures as the measurement with the fewest significant figures.

• use 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 SF53

Learning Check

Select the answer 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

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Addition and Subtraction

When adding or subtracting

• the final answer must have the same number of decimal places as the measurement with the fewest decimal places.

• use 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 final answer with one

decimal place

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Learning Check For each calculation, round off the calculated

answer to give a final answer with 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

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Chapter 1 Measurements

1.5Prefixes and Equalities

57

PrefixesA prefix

in front of a unit increases or decreases the size of that unit.

makes units larger or smaller than the initial unit by one or more factors of 10.

indicates a numerical value.

prefix = value1 kilometer = 1000 meters

1 kilogram = 1000 grams

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Metric and SI Prefixes

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Learning Check

Indicate the unit that matches the description.

1. A mass that is 1000 times greater than 1 gram. 1) kilogram 2) milligram 3) megagram

2. A length that is 1/100 of 1 meter. 1) decimeter 2) centimeter 3) millimeter

3. A unit of time that is 1/1000 of a second. 1) nanosecond 2) microsecond 3) millisecond

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Learning Check

Select the unit you would use to measure

A. your height. 1) millimeters 2) meters 3) kilometers

B. your mass. 1) milligrams 2) grams 3) kilograms

C. the distance between two cities. 1) millimeters 2) meters 3) kilometers

D. the width of an artery. 1) millimeters 2) meters 3) kilometers

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Metric EqualitiesAn 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 cm1 m = 1000 mm

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Measuring Length

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Measuring Volume

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Measuring Mass

Several equalities can be written for mass in the metric (SI) system

1 kg = 1000 g1 g = 1000 mg1 mg = 0.001 g1 mg = 1000 µg

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Learning CheckIndicate the unit that completes each of the followingequalities.

A. 1000 m = ___ 1) 1 mm 2) 1 km 3) 1 dm

B. 0.001 g = ___ 1) 1 mg 2) 1 kg3) 1 dg

C. 0.1 s = ___ 1) 1 ms 2) 1 cs3) 1 ds

D. 0.01 m = ___ 1) 1 mm 2) 1 cm 3) 1 dm

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Learning CheckComplete each of the following equalities.

A. 1 kg = ___ 1) 10 g 2) 100 g 3) 1000 g

B. 1 mm = ___ 1) 0.001 m 2) 0.01 m 3) 0.1 m

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